[66c962] | 1 | ////////////////////////////////////////////////////////////////////////////// |
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[341696] | 2 | version="$Id$"; |
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[66c962] | 3 | category="Noncommutative"; |
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| 4 | info=" |
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| 5 | LIBRARY: dmod.lib Algorithms for algebraic D-modules |
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| 6 | AUTHORS: Viktor Levandovskyy, levandov@math.rwth-aachen.de |
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| 7 | @* Jorge Martin Morales, jorge@unizar.es |
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| 8 | |
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[3f4e52] | 9 | THEORY: Let K be a field of characteristic 0. Given a polynomial ring |
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[66c962] | 10 | @* R = K[x_1,...,x_n] and a polynomial F in R, |
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[3f4e52] | 11 | @* one is interested in the R[1/F]-module of rank one, generated by |
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[66c962] | 12 | @* the symbol F^s for a symbolic discrete variable s. |
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[0610f0e] | 13 | @* In fact, the module R[1/F]*F^s has a structure of a D(R)[s]-module, where D(R) |
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[66c962] | 14 | @* is an n-th Weyl algebra K<x_1,...,x_n,d_1,...,d_n | d_j x_j = x_j d_j +1> and |
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| 15 | @* D(R)[s] = D(R) tensored with K[s] over K. |
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| 16 | @* Constructively, one needs to find a left ideal I = I(F^s) in D(R), such |
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| 17 | @* that K[x_1,...,x_n,1/F]*F^s is isomorphic to D(R)/I as a D(R)-module. |
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| 18 | @* We often write just D for D(R) and D[s] for D(R)[s]. |
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| 19 | @* One is interested in the following data: |
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| 20 | @* - Ann F^s = I = I(F^s) in D(R)[s], denoted by LD in the output |
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[3f4e52] | 21 | @* - global Bernstein polynomial in K[s], denoted by bs, |
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[66c962] | 22 | @* - its minimal integer root s0, the list of all roots of bs, which are known |
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| 23 | @* to be rational, with their multiplicities, which is denoted by BS |
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[3f4e52] | 24 | @* - Ann F^s0 = I(F^s0) in D(R), denoted by LD0 in the output |
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[66c962] | 25 | @* (LD0 is a holonomic ideal in D(R)) |
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| 26 | @* - Ann^(1) F^s in D(R)[s], denoted by LD1 (logarithmic derivations) |
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| 27 | @* - an operator in D(R)[s], denoted by PS, such that the functional equality |
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| 28 | @* PS*F^(s+1) = bs*F^s holds in K[x_1,...,x_n,1/F]*F^s. |
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| 29 | |
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[3f4e52] | 30 | REFERENCES: |
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[66c962] | 31 | @* We provide the following implementations of algorithms: |
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[3f4e52] | 32 | @* (OT) the classical Ann F^s algorithm from Oaku and Takayama (Journal of |
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[66c962] | 33 | @* Pure and Applied Math., 1999), |
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| 34 | @* (LOT) Levandovskyy's modification of the Oaku-Takayama algorithm (ISSAC 2007) |
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| 35 | @* (BM) the Ann F^s algorithm by Briancon and Maisonobe (Remarques sur |
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| 36 | @* l'ideal de Bernstein associe a des polynomes, preprint, 2002) |
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| 37 | @* (LM08) V. Levandovskyy and J. Martin-Morales, ISSAC 2008 |
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[3f4e52] | 38 | @* (C) Countinho, A Primer of Algebraic D-Modules, |
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| 39 | @* (SST) Saito, Sturmfels, Takayama 'Groebner Deformations of Hypergeometric |
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[66c962] | 40 | @* Differential Equations', Springer, 2000 |
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| 41 | |
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| 42 | |
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| 43 | GUIDE: |
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| 44 | @* - Ann F^s = I(F^s) = LD in D(R)[s] can be computed by Sannfs [BM, OT, LOT] |
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| 45 | @* - Ann^(1) F^s in D(R)[s] can be computed by Sannfslog |
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| 46 | @* - global Bernstein polynomial bs in K[s] can be computed by bernsteinBM |
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| 47 | @* - Ann F^s0 = I(F^s0) = LD0 in D(R) can be computed by annfs0, annfs, annfsBM, |
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| 48 | @* annfsOT, annfsLOT, annfs2, annfsRB etc. |
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| 49 | @* - all the relevant data to F^s (LD, LD0, bs, PS) are computed by operatorBM |
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| 50 | @* - operator PS can be computed via operatorModulo or operatorBM |
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| 51 | @* |
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| 52 | @* - annihilator of F^{s1} for a number s1 is computed with annfspecial |
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| 53 | @* - annihilator of F_1^s_1 * ... * F_p^s_p is computed with annfsBMI |
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| 54 | @* - computing the multiplicity of a rational number r in the Bernstein poly |
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| 55 | @* of a given ideal goes with checkRoot |
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| 56 | @* - check, whether a given univariate polynomial divides the Bernstein poly |
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| 57 | @* goes with checkFactor |
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| 58 | |
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| 59 | |
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| 60 | MAIN PROCEDURES: |
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| 61 | |
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| 62 | annfs(F[,S,eng]); compute Ann F^s0 in D and Bernstein polynomial for a poly F |
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| 63 | annfspecial(I, F, m, n); compute Ann F^n from Ann F^s for a polynomial F and a number n |
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| 64 | Sannfs(F[,S,eng]); compute Ann F^s in D[s] for a polynomial F |
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| 65 | Sannfslog(F[,eng]); compute Ann^(1) F^s in D[s] for a polynomial F |
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| 66 | bernsteinBM(F[,eng]); compute global Bernstein polynomial for a polynomial F (algorithm of Briancon-Maisonobe) |
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[e64e417] | 67 | bernsteinLift(I,F [,eng]); compute a possible multiple of Bernstein polynomial via lift-like procedure |
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[66c962] | 68 | operatorBM(F[,eng]); compute Ann F^s, Ann F^s0, BS and PS for a polynomial F (algorithm of Briancon-Maisonobe) |
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| 69 | operatorModulo(F, I, b); compute PS via the modulo approach |
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| 70 | annfsParamBM(F[,eng]); compute the generic Ann F^s (algorithm by Briancon and Maisonobe) and exceptional parametric constellations for a polynomial F with parametric coefficients |
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| 71 | annfsBMI(F[,eng]); compute Ann F^s and Bernstein ideal for a polynomial F=f1*..*fP (multivariate algorithm of Briancon-Maisonobe) |
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| 72 | checkRoot(F,a[,S,eng]); check if a given rational is a root of the global Bernstein polynomial of F and compute its multiplicity |
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| 73 | SannfsBFCT(F[,eng]); compute Ann F^s in D[s] for a polynomial F (algorithm of Briancon-Maisonobe, other output ordering) |
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| 74 | annfs0(I,F [,eng]); compute Ann F^s0 in D and Bernstein polynomial from the known Ann F^s in D[s] |
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| 75 | annfs2(I,F [,eng]); compute Ann F^s0 in D and Bernstein polynomial from the known Ann F^s in D[s] by using a trick of Noro |
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[0610f0e] | 76 | annfsRB(I,F [,eng]); compute Ann F^s0 in D and Bernstein polynomial from the known Ann F^s in D[s] by using Jacobian ideal |
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[66c962] | 77 | checkFactor(I,F,q[,eng]); check whether a polynomial q in K[s] is a factor of the global Bernstein polynomial of F from the known Ann F^s in D[s] |
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| 78 | |
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| 79 | AUXILIARY PROCEDURES: |
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| 80 | |
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| 81 | arrange(p); create a poly, describing a full hyperplane arrangement |
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| 82 | reiffen(p,q); create a poly, describing a Reiffen curve |
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| 83 | isHolonomic(M); check whether a module is holonomic |
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| 84 | convloc(L); replace global orderings with local in the ringlist L |
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| 85 | minIntRoot(P,fact); minimal integer root among the roots of a maximal ideal P |
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| 86 | varNum(s); the number of the variable with the name s |
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| 87 | isRational(n); check whether n is a rational number |
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| 88 | |
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[6a07eb] | 89 | SEE ALSO: bfun_lib, dmodapp_lib, dmodvar_lib, gmssing_lib |
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| 90 | |
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| 91 | KEYWORDS: D-module; D-module structure; left annihilator ideal; Bernstein-Sato polynomial; global Bernstein-Sato polynomial; |
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| 92 | Weyl algebra; Bernstein operator; logarithmic annihilator ideal; parametric annihilator; root of Bernstein-Sato polynomial; |
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| 93 | hyperplane arrangement; Oaku-Takayama algorithm; Briancon-Maisonobe algorithm; LOT algorithm; |
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[66c962] | 94 | "; |
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| 95 | |
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[800a62] | 96 | // reworked by JM+VL on 9.3.2010: Sannfslog |
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[e64e417] | 97 | // added by VL on 2.3.2010: bernsteinLift |
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| 98 | // ****** commented out for better readability by VL on 2.3.2010 |
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| 99 | // annfsBM(F[,eng]); compute Ann F^s0 in D and Bernstein polynomial for a polynomial F (algorithm of Briancon-Maisonobe) |
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| 100 | // annfsLOT(F[,eng]); compute Ann F^s0 in D and Bernstein polynomial for a polynomial F (Levandovskyy modification of the Oaku-Takayama algorithm) |
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| 101 | // annfsOT(F[,eng]); compute Ann F^s0 in D and Bernstein polynomial for a polynomial F (algorithm of Oaku-Takayama) |
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| 102 | // SannfsBM(F[,eng]); compute Ann F^s in D[s] for a polynomial F (algorithm of Briancon-Maisonobe) |
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| 103 | // SannfsLOT(F[,eng]); compute Ann F^s in D[s] for a polynomial F (Levandovskyy modification of the Oaku-Takayama algorithm) |
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| 104 | // SannfsOT(F[,eng]); compute Ann F^s in D[s] for a polynomial F (algorithm of Oaku-Takayama) |
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| 105 | // checkRoot1(I,F,a[,eng]); check whether a rational is a root of the global Bernstein polynomial of F from the known Ann F^s in D[s] |
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| 106 | // checkRoot2(I,F,a[,eng]); check whether a rational is a root of the global Bernstein polynomial of F and compute its multiplicity from the known Ann F^s in D[s] |
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| 107 | |
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[66c962] | 108 | LIB "matrix.lib"; // for submat |
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| 109 | LIB "nctools.lib"; // makeModElimRing etc. |
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| 110 | LIB "elim.lib"; // for nselect |
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| 111 | LIB "qhmoduli.lib"; // for Max |
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| 112 | LIB "gkdim.lib"; |
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| 113 | LIB "gmssing.lib"; |
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| 114 | LIB "control.lib"; // for genericity |
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| 115 | LIB "dmodapp.lib"; // for e.g. engine |
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| 116 | LIB "poly.lib"; |
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| 117 | |
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| 118 | proc testdmodlib() |
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| 119 | { |
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| 120 | /* tests all procs for consistency */ |
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| 121 | /* adding the new proc, add it here */ |
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| 122 | |
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| 123 | "MAIN PROCEDURES:"; |
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| 124 | example annfs; |
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| 125 | example Sannfs; |
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| 126 | example Sannfslog; |
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| 127 | example bernsteinBM; |
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| 128 | example operatorBM; |
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| 129 | example annfspecial; |
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| 130 | example annfsParamBM; |
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| 131 | example annfsBMI; |
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| 132 | example checkRoot; |
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| 133 | example annfs0; |
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| 134 | example annfs2; |
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| 135 | example annfsRB; |
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| 136 | example annfs2; |
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| 137 | example operatorModulo; |
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| 138 | "SECONDARY D-MOD PROCEDURES:"; |
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| 139 | example annfsBM; |
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| 140 | example annfsLOT; |
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| 141 | example annfsOT; |
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| 142 | example SannfsBM; |
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| 143 | example SannfsLOT; |
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| 144 | example SannfsOT; |
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| 145 | example SannfsBFCT; |
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| 146 | example checkRoot1; |
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| 147 | example checkRoot2; |
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| 148 | example checkFactor; |
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| 149 | } |
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| 150 | |
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| 151 | |
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| 152 | |
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| 153 | // new top-level procedures |
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| 154 | proc annfs(poly F, list #) |
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| 155 | "USAGE: annfs(f [,S,eng]); f a poly, S a string, eng an optional int |
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| 156 | RETURN: ring |
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[3f4e52] | 157 | PURPOSE: compute the D-module structure of basering[1/f]*f^s with the algorithm |
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[66c962] | 158 | @* given in S and with the Groebner basis engine given in ''eng'' |
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| 159 | NOTE: activate the output ring with the @code{setring} command. |
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| 160 | @* String S; S can be one of the following: |
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| 161 | @* 'bm' (default) - for the algorithm of Briancon and Maisonobe, |
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| 162 | @* 'ot' - for the algorithm of Oaku and Takayama, |
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| 163 | @* 'lot' - for the Levandovskyy's modification of the algorithm of OT. |
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| 164 | @* If eng <>0, @code{std} is used for Groebner basis computations, |
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| 165 | @* otherwise and by default @code{slimgb} is used. |
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| 166 | @* In the output ring: |
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| 167 | @* - the ideal LD (which is a Groebner basis) is the needed D-module structure, |
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| 168 | @* - the list BS contains roots and multiplicities of a BS-polynomial of f. |
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| 169 | DISPLAY: If @code{printlevel}=1, progress debug messages will be printed, |
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| 170 | @* if @code{printlevel}>=2, all the debug messages will be printed. |
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| 171 | EXAMPLE: example annfs; shows examples |
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| 172 | " |
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| 173 | { |
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| 174 | int eng = 0; |
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| 175 | int chs = 0; // choice |
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| 176 | string algo = "bm"; |
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| 177 | string st; |
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| 178 | if ( size(#)>0 ) |
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| 179 | { |
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| 180 | if ( typeof(#[1]) == "string" ) |
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| 181 | { |
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| 182 | st = string(#[1]); |
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| 183 | if ( (st == "BM") || (st == "Bm") || (st == "bM") ||(st == "bm")) |
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| 184 | { |
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| 185 | algo = "bm"; |
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| 186 | chs = 1; |
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| 187 | } |
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| 188 | if ( (st == "OT") || (st == "Ot") || (st == "oT") || (st == "ot")) |
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| 189 | { |
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| 190 | algo = "ot"; |
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| 191 | chs = 1; |
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| 192 | } |
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| 193 | if ( (st == "LOT") || (st == "lOT") || (st == "Lot") || (st == "lot")) |
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| 194 | { |
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| 195 | algo = "lot"; |
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| 196 | chs = 1; |
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| 197 | } |
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| 198 | if (chs != 1) |
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| 199 | { |
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| 200 | // incorrect value of S |
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| 201 | print("Incorrect algorithm given, proceed with the default BM"); |
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| 202 | algo = "bm"; |
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| 203 | } |
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| 204 | // second arg |
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| 205 | if (size(#)>1) |
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| 206 | { |
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| 207 | // exists 2nd arg |
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| 208 | if ( typeof(#[2]) == "int" ) |
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| 209 | { |
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| 210 | // the case: given alg, given engine |
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| 211 | eng = int(#[2]); |
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| 212 | } |
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| 213 | else |
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| 214 | { |
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| 215 | eng = 0; |
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| 216 | } |
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| 217 | } |
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| 218 | else |
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| 219 | { |
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| 220 | // no second arg |
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| 221 | eng = 0; |
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| 222 | } |
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| 223 | } |
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| 224 | else |
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| 225 | { |
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| 226 | if ( typeof(#[1]) == "int" ) |
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| 227 | { |
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| 228 | // the case: default alg, engine |
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| 229 | eng = int(#[1]); |
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| 230 | // algo = "bm"; //is already set |
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| 231 | } |
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| 232 | else |
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| 233 | { |
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| 234 | // incorr. 1st arg |
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| 235 | algo = "bm"; |
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| 236 | } |
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| 237 | } |
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| 238 | } |
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| 239 | |
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| 240 | // size(#)=0, i.e. there is no algorithm and no engine given |
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| 241 | // eng = 0; algo = "bm"; //are already set |
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| 242 | // int ppl = printlevel-voice+2; |
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| 243 | |
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| 244 | int old_printlevel = printlevel; |
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| 245 | printlevel=printlevel+1; |
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| 246 | def save = basering; |
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| 247 | if ( algo=="ot") |
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| 248 | { |
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| 249 | def @A = annfsOT(F,eng); |
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| 250 | } |
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| 251 | else |
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| 252 | { |
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| 253 | if ( algo=="lot") |
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| 254 | { |
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| 255 | def @A = annfsLOT(F,eng); |
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| 256 | } |
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| 257 | else |
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| 258 | { |
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| 259 | // bm = default |
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| 260 | def @A = annfsBM(F,eng); |
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| 261 | } |
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| 262 | } |
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| 263 | printlevel = old_printlevel; |
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| 264 | return(@A); |
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| 265 | } |
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| 266 | example |
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| 267 | { |
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| 268 | "EXAMPLE:"; echo = 2; |
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| 269 | ring r = 0,(x,y,z),Dp; |
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| 270 | poly F = z*x^2+y^3; |
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| 271 | def A = annfs(F); // here, the default BM algorithm will be used |
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| 272 | setring A; // the Weyl algebra in (x,y,z,Dx,Dy,Dz) |
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| 273 | LD; //the annihilator of F^{-1} over A |
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| 274 | BS; // roots with multiplicities of BS polynomial |
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| 275 | } |
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| 276 | |
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| 277 | |
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| 278 | proc Sannfs(poly F, list #) |
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| 279 | "USAGE: Sannfs(f [,S,eng]); f a poly, S a string, eng an optional int |
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| 280 | RETURN: ring |
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[3f4e52] | 281 | PURPOSE: compute the D-module structure of basering[f^s] with the algorithm |
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[66c962] | 282 | @* given in S and with the Groebner basis engine given in eng |
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| 283 | NOTE: activate the output ring with the @code{setring} command. |
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| 284 | @* The value of a string S can be |
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| 285 | @* 'bm' (default) - for the algorithm of Briancon and Maisonobe, |
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| 286 | @* 'lot' - for the Levandovskyy's modification of the algorithm of OT, |
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| 287 | @* 'ot' - for the algorithm of Oaku and Takayama. |
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| 288 | @* If eng <>0, @code{std} is used for Groebner basis computations, |
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| 289 | @* otherwise, and by default @code{slimgb} is used. |
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| 290 | @* In the output ring: |
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| 291 | @* - the ideal LD is the needed D-module structure. |
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| 292 | DISPLAY: If @code{printlevel}=1, progress debug messages will be printed, |
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| 293 | @* if @code{printlevel}>=2, all the debug messages will be printed. |
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| 294 | EXAMPLE: example Sannfs; shows examples |
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| 295 | " |
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| 296 | { |
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| 297 | int eng = 0; |
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| 298 | int chs = 0; // choice |
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| 299 | string algo = "bm"; |
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| 300 | string st; |
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| 301 | if ( size(#)>0 ) |
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| 302 | { |
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| 303 | if ( typeof(#[1]) == "string" ) |
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| 304 | { |
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| 305 | st = string(#[1]); |
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| 306 | if ( (st == "BM") || (st == "Bm") || (st == "bM") ||(st == "bm")) |
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| 307 | { |
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| 308 | algo = "bm"; |
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| 309 | chs = 1; |
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| 310 | } |
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| 311 | if ( (st == "OT") || (st == "Ot") || (st == "oT") || (st == "ot")) |
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| 312 | { |
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| 313 | algo = "ot"; |
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| 314 | chs = 1; |
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| 315 | } |
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| 316 | if ( (st == "LOT") || (st == "lOT") || (st == "Lot") || (st == "lot")) |
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| 317 | { |
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| 318 | algo = "lot"; |
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| 319 | chs = 1; |
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| 320 | } |
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| 321 | if (chs != 1) |
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| 322 | { |
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| 323 | // incorrect value of S |
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| 324 | print("Incorrect algorithm given, proceed with the default BM"); |
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| 325 | algo = "bm"; |
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| 326 | } |
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| 327 | // second arg |
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| 328 | if (size(#)>1) |
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| 329 | { |
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| 330 | // exists 2nd arg |
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| 331 | if ( typeof(#[2]) == "int" ) |
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| 332 | { |
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| 333 | // the case: given alg, given engine |
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| 334 | eng = int(#[2]); |
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| 335 | } |
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| 336 | else |
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| 337 | { |
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| 338 | eng = 0; |
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| 339 | } |
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| 340 | } |
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| 341 | else |
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| 342 | { |
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| 343 | // no second arg |
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| 344 | eng = 0; |
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| 345 | } |
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| 346 | } |
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| 347 | else |
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| 348 | { |
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| 349 | if ( typeof(#[1]) == "int" ) |
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| 350 | { |
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| 351 | // the case: default alg, engine |
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| 352 | eng = int(#[1]); |
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| 353 | // algo = "bm"; //is already set |
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| 354 | } |
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| 355 | else |
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| 356 | { |
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| 357 | // incorr. 1st arg |
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| 358 | algo = "bm"; |
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| 359 | } |
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| 360 | } |
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| 361 | } |
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| 362 | // size(#)=0, i.e. there is no algorithm and no engine given |
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| 363 | // eng = 0; algo = "bm"; //are already set |
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| 364 | // int ppl = printlevel-voice+2; |
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| 365 | printlevel=printlevel+1; |
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| 366 | def save = basering; |
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| 367 | if ( algo=="ot") |
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| 368 | { |
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| 369 | def @A = SannfsOT(F,eng); |
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| 370 | } |
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| 371 | else |
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| 372 | { |
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| 373 | if ( algo=="lot") |
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| 374 | { |
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| 375 | def @A = SannfsLOT(F,eng); |
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| 376 | } |
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| 377 | else |
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| 378 | { |
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| 379 | // bm = default |
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| 380 | def @A = SannfsBM(F,eng); |
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| 381 | } |
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| 382 | } |
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| 383 | printlevel=printlevel-1; |
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| 384 | return(@A); |
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| 385 | } |
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| 386 | example |
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| 387 | { |
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| 388 | "EXAMPLE:"; echo = 2; |
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| 389 | ring r = 0,(x,y,z),Dp; |
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| 390 | poly F = x^3+y^3+z^3; |
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| 391 | printlevel = 0; |
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| 392 | def A = Sannfs(F); // here, the default BM algorithm will be used |
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| 393 | setring A; |
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| 394 | LD; |
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| 395 | } |
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| 396 | |
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| 397 | proc Sannfslog (poly F, list #) |
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| 398 | "USAGE: Sannfslog(f [,eng]); f a poly, eng an optional int |
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| 399 | RETURN: ring |
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| 400 | PURPOSE: compute the D-module structure of basering[1/f]*f^s |
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| 401 | NOTE: activate the output ring with the @code{setring} command. |
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[3f4e52] | 402 | @* In the output ring D[s], the ideal LD1 is generated by the elements |
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[66c962] | 403 | @* in Ann F^s in D[s], coming from logarithmic derivations. |
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| 404 | @* If eng <>0, @code{std} is used for Groebner basis computations, |
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| 405 | @* otherwise, and by default @code{slimgb} is used. |
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| 406 | DISPLAY: If @code{printlevel}=1, progress debug messages will be printed, |
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| 407 | @* if @code{printlevel}>=2, all the debug messages will be printed. |
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| 408 | EXAMPLE: example Sannfslog; shows examples |
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| 409 | " |
---|
| 410 | { |
---|
| 411 | int eng = 0; |
---|
| 412 | if ( size(#)>0 ) |
---|
| 413 | { |
---|
| 414 | if ( typeof(#[1]) == "int" ) |
---|
| 415 | { |
---|
| 416 | eng = int(#[1]); |
---|
| 417 | } |
---|
| 418 | } |
---|
| 419 | int ppl = printlevel-voice+2; |
---|
| 420 | def save = basering; |
---|
| 421 | int N = nvars(basering); |
---|
| 422 | int Nnew = 2*N+1; |
---|
| 423 | int i; |
---|
| 424 | string s; |
---|
| 425 | list RL = ringlist(basering); |
---|
| 426 | list L, Lord; |
---|
| 427 | list tmp; |
---|
| 428 | intvec iv; |
---|
| 429 | L[1] = RL[1]; // char |
---|
| 430 | L[4] = RL[4]; // char, minpoly |
---|
| 431 | // check whether vars have admissible names |
---|
| 432 | list Name = RL[2]; |
---|
| 433 | for (i=1; i<=N; i++) |
---|
| 434 | { |
---|
| 435 | if (Name[i] == "s") |
---|
| 436 | { |
---|
| 437 | ERROR("Variable names should not include s"); |
---|
| 438 | } |
---|
| 439 | } |
---|
| 440 | // the ideal I |
---|
| 441 | ideal I = -F, jacob(F); |
---|
| 442 | dbprint(ppl,"// -1-1- starting the computation of syz(-F,_Dx(F))"); |
---|
| 443 | dbprint(ppl-1, I); |
---|
| 444 | matrix M = syz(I); |
---|
| 445 | M = transpose(M); // it is more usefull working with columns |
---|
| 446 | dbprint(ppl,"// -1-2- the module syz(-F,_Dx(F)) has been computed"); |
---|
| 447 | dbprint(ppl-1, M); |
---|
| 448 | // ------------ the ring @R ------------ |
---|
| 449 | // _x, _Dx, s; elim.ord for _x,_Dx. |
---|
| 450 | // now, create the names for new vars |
---|
| 451 | list DName; |
---|
| 452 | for (i=1; i<=N; i++) |
---|
| 453 | { |
---|
| 454 | DName[i] = "D"+Name[i]; // concat |
---|
| 455 | } |
---|
| 456 | tmp[1] = "s"; |
---|
| 457 | list NName; |
---|
[800a62] | 458 | NName = Name + DName + tmp; |
---|
[66c962] | 459 | L[2] = NName; |
---|
| 460 | tmp = 0; |
---|
[800a62] | 461 | // block ord (dp(N),dp); |
---|
[66c962] | 462 | s = "iv="; |
---|
[800a62] | 463 | for (i=1; i<=Nnew-1; i++) |
---|
[66c962] | 464 | { |
---|
| 465 | s = s+"1,"; |
---|
| 466 | } |
---|
| 467 | s[size(s)]=";"; |
---|
| 468 | execute(s); |
---|
| 469 | tmp[1] = "dp"; // string |
---|
| 470 | tmp[2] = iv; // intvec |
---|
[800a62] | 471 | Lord[1] = tmp; |
---|
| 472 | // continue with dp 1,1,1,1... |
---|
| 473 | tmp[1] = "dp"; // string |
---|
| 474 | s[size(s)] = ","; |
---|
| 475 | s = s+"1;"; |
---|
| 476 | execute(s); |
---|
| 477 | kill s; |
---|
| 478 | kill NName; |
---|
| 479 | tmp[2] = iv; |
---|
| 480 | Lord[2] = tmp; |
---|
| 481 | tmp[1] = "C"; iv = 0; tmp[2]=iv; |
---|
| 482 | Lord[3] = tmp; tmp = 0; |
---|
| 483 | L[3] = Lord; |
---|
[66c962] | 484 | // we are done with the list. Now add a Plural part |
---|
| 485 | def @R@ = ring(L); |
---|
| 486 | setring @R@; |
---|
| 487 | matrix @D[Nnew][Nnew]; |
---|
| 488 | for (i=1; i<=N; i++) |
---|
| 489 | { |
---|
[800a62] | 490 | @D[i,N+i]=1; |
---|
[66c962] | 491 | } |
---|
| 492 | def @R = nc_algebra(1,@D); |
---|
| 493 | setring @R; |
---|
| 494 | kill @R@; |
---|
| 495 | dbprint(ppl,"// -2-1- the ring @R(_x,_Dx,s) is ready"); |
---|
| 496 | dbprint(ppl-1, @R); |
---|
| 497 | matrix M = imap(save,M); |
---|
| 498 | // now, create the vector [-s,_Dx] |
---|
| 499 | vector v = [-s]; // now s is a variable |
---|
| 500 | for (i=1; i<=N; i++) |
---|
| 501 | { |
---|
[800a62] | 502 | v = v + var(i+N)*gen(i+1); |
---|
[66c962] | 503 | } |
---|
| 504 | ideal J = ideal(M*v); |
---|
| 505 | // make leadcoeffs positive |
---|
| 506 | for (i=1; i<= ncols(J); i++) |
---|
| 507 | { |
---|
| 508 | if ( leadcoef(J[i])<0 ) |
---|
| 509 | { |
---|
| 510 | J[i] = -J[i]; |
---|
| 511 | } |
---|
| 512 | } |
---|
| 513 | ideal LD1 = J; |
---|
| 514 | kill J; |
---|
| 515 | export LD1; |
---|
| 516 | return(@R); |
---|
| 517 | } |
---|
| 518 | example |
---|
| 519 | { |
---|
| 520 | "EXAMPLE:"; echo = 2; |
---|
| 521 | ring r = 0,(x,y),Dp; |
---|
[800a62] | 522 | poly F = x4+y5+x*y4; |
---|
[66c962] | 523 | printlevel = 0; |
---|
| 524 | def A = Sannfslog(F); |
---|
| 525 | setring A; |
---|
| 526 | LD1; |
---|
| 527 | } |
---|
| 528 | |
---|
[800a62] | 529 | // JM+VL: output ring restructured into "normal" |
---|
| 530 | |
---|
| 531 | // proc Sannfslog (poly F, list #) |
---|
| 532 | // "USAGE: Sannfslog(f [,eng]); f a poly, eng an optional int |
---|
| 533 | // RETURN: ring |
---|
| 534 | // PURPOSE: compute the D-module structure of basering[1/f]*f^s |
---|
| 535 | // NOTE: activate the output ring with the @code{setring} command. |
---|
| 536 | // @* In the output ring D[s], the ideal LD1 is generated by the elements |
---|
| 537 | // @* in Ann F^s in D[s], coming from logarithmic derivations. |
---|
| 538 | // @* If eng <>0, @code{std} is used for Groebner basis computations, |
---|
| 539 | // @* otherwise, and by default @code{slimgb} is used. |
---|
| 540 | // DISPLAY: If @code{printlevel}=1, progress debug messages will be printed, |
---|
| 541 | // @* if @code{printlevel}>=2, all the debug messages will be printed. |
---|
| 542 | // EXAMPLE: example Sannfslog; shows examples |
---|
| 543 | // " |
---|
| 544 | // { |
---|
| 545 | // int eng = 0; |
---|
| 546 | // if ( size(#)>0 ) |
---|
| 547 | // { |
---|
| 548 | // if ( typeof(#[1]) == "int" ) |
---|
| 549 | // { |
---|
| 550 | // eng = int(#[1]); |
---|
| 551 | // } |
---|
| 552 | // } |
---|
| 553 | // int ppl = printlevel-voice+2; |
---|
| 554 | // def save = basering; |
---|
| 555 | // int N = nvars(basering); |
---|
| 556 | // int Nnew = 2*N+1; |
---|
| 557 | // int i; |
---|
| 558 | // string s; |
---|
| 559 | // list RL = ringlist(basering); |
---|
| 560 | // list L, Lord; |
---|
| 561 | // list tmp; |
---|
| 562 | // intvec iv; |
---|
| 563 | // L[1] = RL[1]; // char |
---|
| 564 | // L[4] = RL[4]; // char, minpoly |
---|
| 565 | // // check whether vars have admissible names |
---|
| 566 | // list Name = RL[2]; |
---|
| 567 | // for (i=1; i<=N; i++) |
---|
| 568 | // { |
---|
| 569 | // if (Name[i] == "s") |
---|
| 570 | // { |
---|
| 571 | // ERROR("Variable names should not include s"); |
---|
| 572 | // } |
---|
| 573 | // } |
---|
| 574 | // // the ideal I |
---|
| 575 | // ideal I = -F, jacob(F); |
---|
| 576 | // dbprint(ppl,"// -1-1- starting the computation of syz(-F,_Dx(F))"); |
---|
| 577 | // dbprint(ppl-1, I); |
---|
| 578 | // matrix M = syz(I); |
---|
| 579 | // M = transpose(M); // it is more usefull working with columns |
---|
| 580 | // dbprint(ppl,"// -1-2- the module syz(-F,_Dx(F)) has been computed"); |
---|
| 581 | // dbprint(ppl-1, M); |
---|
| 582 | // // ------------ the ring @R ------------ |
---|
| 583 | // // _x, _Dx, s; elim.ord for _x,_Dx. |
---|
| 584 | // // now, create the names for new vars |
---|
| 585 | // list DName; |
---|
| 586 | // for (i=1; i<=N; i++) |
---|
| 587 | // { |
---|
| 588 | // DName[i] = "D"+Name[i]; // concat |
---|
| 589 | // } |
---|
| 590 | // tmp[1] = "s"; |
---|
| 591 | // list NName; |
---|
| 592 | // for (i=1; i<=N; i++) |
---|
| 593 | // { |
---|
| 594 | // NName[2*i-1] = Name[i]; |
---|
| 595 | // NName[2*i] = DName[i]; |
---|
| 596 | // //NName[2*i-1] = DName[i]; |
---|
| 597 | // //NName[2*i] = Name[i]; |
---|
| 598 | // } |
---|
| 599 | // NName[Nnew] = tmp[1]; |
---|
| 600 | // L[2] = NName; |
---|
| 601 | // tmp = 0; |
---|
| 602 | // // block ord (a(1,1),a(0,0,1,1),...,dp); |
---|
| 603 | // //list("a",intvec(1,1)), list("a",intvec(0,0,1,1)), ... |
---|
| 604 | // tmp[1] = "a"; // string |
---|
| 605 | // for (i=1; i<=N; i++) |
---|
| 606 | // { |
---|
| 607 | // iv[2*i-1] = 1; |
---|
| 608 | // iv[2*i] = 1; |
---|
| 609 | // tmp[2] = iv; iv = 0; // intvec |
---|
| 610 | // Lord[i] = tmp; |
---|
| 611 | // } |
---|
| 612 | // //list("dp",intvec(1,1,1,1,1,...)) |
---|
| 613 | // s = "iv="; |
---|
| 614 | // for (i=1; i<=Nnew; i++) |
---|
| 615 | // { |
---|
| 616 | // s = s+"1,"; |
---|
| 617 | // } |
---|
| 618 | // s[size(s)]=";"; |
---|
| 619 | // execute(s); |
---|
| 620 | // kill s; |
---|
| 621 | // tmp[1] = "dp"; // string |
---|
| 622 | // tmp[2] = iv; // intvec |
---|
| 623 | // Lord[N+1] = tmp; |
---|
| 624 | // //list("C",intvec(0)) |
---|
| 625 | // tmp[1] = "C"; // string |
---|
| 626 | // iv = 0; |
---|
| 627 | // tmp[2] = iv; // intvec |
---|
| 628 | // Lord[N+2] = tmp; |
---|
| 629 | // tmp = 0; |
---|
| 630 | // L[3] = Lord; |
---|
| 631 | // // we are done with the list. Now add a Plural part |
---|
| 632 | // def @R@ = ring(L); |
---|
| 633 | // setring @R@; |
---|
| 634 | // matrix @D[Nnew][Nnew]; |
---|
| 635 | // for (i=1; i<=N; i++) |
---|
| 636 | // { |
---|
| 637 | // @D[2*i-1,2*i]=1; |
---|
| 638 | // //@D[2*i-1,2*i]=-1; |
---|
| 639 | // } |
---|
| 640 | // def @R = nc_algebra(1,@D); |
---|
| 641 | // setring @R; |
---|
| 642 | // kill @R@; |
---|
| 643 | // dbprint(ppl,"// -2-1- the ring @R(_x,_Dx,s) is ready"); |
---|
| 644 | // dbprint(ppl-1, @R); |
---|
| 645 | // matrix M = imap(save,M); |
---|
| 646 | // // now, create the vector [-s,_Dx] |
---|
| 647 | // vector v = [-s]; // now s is a variable |
---|
| 648 | // for (i=1; i<=N; i++) |
---|
| 649 | // { |
---|
| 650 | // v = v + var(2*i)*gen(i+1); |
---|
| 651 | // //v = v + var(2*i-1)*gen(i+1); |
---|
| 652 | // } |
---|
| 653 | // ideal J = ideal(M*v); |
---|
| 654 | // // make leadcoeffs positive |
---|
| 655 | // for (i=1; i<= ncols(J); i++) |
---|
| 656 | // { |
---|
| 657 | // if ( leadcoef(J[i])<0 ) |
---|
| 658 | // { |
---|
| 659 | // J[i] = -J[i]; |
---|
| 660 | // } |
---|
| 661 | // } |
---|
| 662 | // ideal LD1 = J; |
---|
| 663 | // kill J; |
---|
| 664 | // export LD1; |
---|
| 665 | // return(@R); |
---|
| 666 | // } |
---|
| 667 | // example |
---|
| 668 | // { |
---|
| 669 | // "EXAMPLE:"; echo = 2; |
---|
| 670 | // ring r = 0,(x,y),Dp; |
---|
| 671 | // poly F = x^4+y^5+x*y^4; |
---|
| 672 | // printlevel = 0; |
---|
| 673 | // def A = Sannfslog(F); |
---|
| 674 | // setring A; |
---|
| 675 | // LD1; |
---|
| 676 | // } |
---|
| 677 | |
---|
[66c962] | 678 | |
---|
| 679 | // alternative code for SannfsBM, renamed from annfsBM to ALTannfsBM |
---|
| 680 | // is superfluos - will not be included in the official documentation |
---|
| 681 | static proc ALTannfsBM (poly F, list #) |
---|
| 682 | "USAGE: ALTannfsBM(f [,eng]); f a poly, eng an optional int |
---|
| 683 | RETURN: ring |
---|
[3f4e52] | 684 | PURPOSE: compute the annihilator ideal of f^s in D[s], where D is the Weyl |
---|
[66c962] | 685 | @* algebra, according to the algorithm by Briancon and Maisonobe |
---|
| 686 | NOTE: activate the output ring with the @code{setring} command. In this ring, |
---|
| 687 | @* - the ideal LD is the annihilator of f^s. |
---|
| 688 | @* If eng <>0, @code{std} is used for Groebner basis computations, |
---|
| 689 | @* otherwise, and by default @code{slimgb} is used. |
---|
| 690 | DISPLAY: If @code{printlevel}=1, progress debug messages will be printed, |
---|
| 691 | @* if @code{printlevel}>=2, all the debug messages will be printed. |
---|
| 692 | EXAMPLE: example ALTannfsBM; shows examples |
---|
| 693 | " |
---|
| 694 | { |
---|
| 695 | int eng = 0; |
---|
| 696 | if ( size(#)>0 ) |
---|
| 697 | { |
---|
| 698 | if ( typeof(#[1]) == "int" ) |
---|
| 699 | { |
---|
| 700 | eng = int(#[1]); |
---|
| 701 | } |
---|
| 702 | } |
---|
| 703 | // returns a list with a ring and an ideal LD in it |
---|
| 704 | int ppl = printlevel-voice+2; |
---|
| 705 | // printf("plevel :%s, voice: %s",printlevel,voice); |
---|
| 706 | def save = basering; |
---|
| 707 | int N = nvars(basering); |
---|
| 708 | int Nnew = 2*N+2; |
---|
| 709 | int i,j; |
---|
| 710 | string s; |
---|
| 711 | list RL = ringlist(basering); |
---|
| 712 | list L, Lord; |
---|
| 713 | list tmp; |
---|
| 714 | intvec iv; |
---|
| 715 | L[1] = RL[1]; //char |
---|
| 716 | L[4] = RL[4]; //char, minpoly |
---|
| 717 | // check whether vars have admissible names |
---|
| 718 | list Name = RL[2]; |
---|
| 719 | list RName; |
---|
| 720 | RName[1] = "t"; |
---|
| 721 | RName[2] = "s"; |
---|
| 722 | for (i=1; i<=N; i++) |
---|
| 723 | { |
---|
| 724 | for(j=1; j<=size(RName); j++) |
---|
| 725 | { |
---|
| 726 | if (Name[i] == RName[j]) |
---|
| 727 | { |
---|
| 728 | ERROR("Variable names should not include t,s"); |
---|
| 729 | } |
---|
| 730 | } |
---|
| 731 | } |
---|
| 732 | // now, create the names for new vars |
---|
| 733 | list DName; |
---|
| 734 | for (i=1; i<=N; i++) |
---|
| 735 | { |
---|
| 736 | DName[i] = "D"+Name[i]; //concat |
---|
| 737 | } |
---|
| 738 | tmp[1] = "t"; |
---|
| 739 | tmp[2] = "s"; |
---|
| 740 | list NName = tmp + Name + DName; |
---|
| 741 | L[2] = NName; |
---|
| 742 | // Name, Dname will be used further |
---|
| 743 | kill NName; |
---|
| 744 | // block ord (lp(2),dp); |
---|
| 745 | tmp[1] = "lp"; // string |
---|
| 746 | iv = 1,1; |
---|
| 747 | tmp[2] = iv; //intvec |
---|
| 748 | Lord[1] = tmp; |
---|
| 749 | // continue with dp 1,1,1,1... |
---|
| 750 | tmp[1] = "dp"; // string |
---|
| 751 | s = "iv="; |
---|
| 752 | for (i=1; i<=Nnew; i++) |
---|
| 753 | { |
---|
| 754 | s = s+"1,"; |
---|
| 755 | } |
---|
| 756 | s[size(s)]= ";"; |
---|
| 757 | execute(s); |
---|
| 758 | kill s; |
---|
| 759 | tmp[2] = iv; |
---|
| 760 | Lord[2] = tmp; |
---|
| 761 | tmp[1] = "C"; |
---|
| 762 | iv = 0; |
---|
| 763 | tmp[2] = iv; |
---|
| 764 | Lord[3] = tmp; |
---|
| 765 | tmp = 0; |
---|
| 766 | L[3] = Lord; |
---|
| 767 | // we are done with the list |
---|
| 768 | def @R@ = ring(L); |
---|
| 769 | setring @R@; |
---|
| 770 | matrix @D[Nnew][Nnew]; |
---|
| 771 | @D[1,2]=t; |
---|
| 772 | for(i=1; i<=N; i++) |
---|
| 773 | { |
---|
| 774 | @D[2+i,N+2+i]=1; |
---|
| 775 | } |
---|
| 776 | // L[5] = matrix(UpOneMatrix(Nnew)); |
---|
| 777 | // L[6] = @D; |
---|
| 778 | def @R = nc_algebra(1,@D); |
---|
| 779 | setring @R; |
---|
| 780 | kill @R@; |
---|
| 781 | dbprint(ppl,"// -1-1- the ring @R(t,s,_x,_Dx) is ready"); |
---|
| 782 | dbprint(ppl-1, @R); |
---|
| 783 | // create the ideal I |
---|
| 784 | poly F = imap(save,F); |
---|
| 785 | ideal I = t*F+s; |
---|
| 786 | poly p; |
---|
| 787 | for(i=1; i<=N; i++) |
---|
| 788 | { |
---|
| 789 | p = t; //t |
---|
| 790 | p = diff(F,var(2+i))*p; |
---|
| 791 | I = I, var(N+2+i) + p; |
---|
| 792 | } |
---|
| 793 | // -------- the ideal I is ready ---------- |
---|
| 794 | dbprint(ppl,"// -1-2- starting the elimination of t in @R"); |
---|
| 795 | dbprint(ppl-1, I); |
---|
| 796 | ideal J = engine(I,eng); |
---|
| 797 | ideal K = nselect(J,1); |
---|
| 798 | kill I,J; |
---|
| 799 | dbprint(ppl,"// -1-3- t is eliminated"); |
---|
| 800 | dbprint(ppl-1, K); //K is without t |
---|
| 801 | // create Dn[s], where Dn is the ordinary Weyl Algebra, and put the result into it, |
---|
| 802 | // thus creating the ring @R2 |
---|
| 803 | // keep: N, i,j,s, tmp, RL |
---|
| 804 | setring save; |
---|
| 805 | Nnew = 2*N+1; |
---|
| 806 | // list RL = ringlist(save); //is defined earlier |
---|
| 807 | kill Lord, tmp, iv; |
---|
| 808 | L = 0; |
---|
| 809 | list Lord, tmp; |
---|
| 810 | intvec iv; |
---|
| 811 | L[1] = RL[1]; |
---|
| 812 | L[4] = RL[4]; //char, minpoly |
---|
| 813 | // check whether vars have admissible names -> done earlier |
---|
| 814 | // list Name = RL[2] |
---|
| 815 | // DName is defined earlier |
---|
| 816 | tmp[1] = "s"; |
---|
| 817 | list NName = Name + DName + tmp; |
---|
| 818 | L[2] = NName; |
---|
| 819 | // dp ordering; |
---|
| 820 | string s = "iv="; |
---|
| 821 | for (i=1; i<=Nnew; i++) |
---|
| 822 | { |
---|
| 823 | s = s+"1,"; |
---|
| 824 | } |
---|
| 825 | s[size(s)] = ";"; |
---|
| 826 | execute(s); |
---|
| 827 | kill s; |
---|
| 828 | tmp = 0; |
---|
| 829 | tmp[1] = "dp"; //string |
---|
| 830 | tmp[2] = iv; //intvec |
---|
| 831 | Lord[1] = tmp; |
---|
| 832 | tmp[1] = "C"; |
---|
| 833 | iv = 0; |
---|
| 834 | tmp[2] = iv; |
---|
| 835 | Lord[2] = tmp; |
---|
| 836 | tmp = 0; |
---|
| 837 | L[3] = Lord; |
---|
| 838 | // we are done with the list |
---|
| 839 | // Add: Plural part |
---|
| 840 | def @R2@ = ring(L); |
---|
| 841 | setring @R2@; |
---|
| 842 | matrix @D[Nnew][Nnew]; |
---|
| 843 | for (i=1; i<=N; i++) |
---|
| 844 | { |
---|
| 845 | @D[i,N+i]=1; |
---|
| 846 | } |
---|
| 847 | def @R2 = nc_algebra(1,@D); |
---|
| 848 | setring @R2; |
---|
| 849 | kill @R2@; |
---|
| 850 | dbprint(ppl,"// -2-1- the ring @R2(_x,_Dx,s) is ready"); |
---|
| 851 | dbprint(ppl-1, @R2); |
---|
| 852 | ideal K = imap(@R,K); |
---|
| 853 | option(redSB); |
---|
| 854 | //dbprint(ppl,"// -2-2- the final cosmetic std"); |
---|
| 855 | //K = engine(K,eng); //std does the job too |
---|
| 856 | // total cleanup |
---|
| 857 | kill @R; |
---|
| 858 | ideal LD = K; |
---|
| 859 | export LD; |
---|
| 860 | return(@R2); |
---|
| 861 | } |
---|
| 862 | example |
---|
| 863 | { |
---|
| 864 | "EXAMPLE:"; echo = 2; |
---|
| 865 | ring r = 0,(x,y,z,w),Dp; |
---|
| 866 | poly F = x^3+y^3+z^2*w; |
---|
| 867 | printlevel = 0; |
---|
| 868 | def A = ALTannfsBM(F); |
---|
| 869 | setring A; |
---|
| 870 | LD; |
---|
| 871 | } |
---|
| 872 | |
---|
| 873 | proc bernsteinBM(poly F, list #) |
---|
| 874 | "USAGE: bernsteinBM(f [,eng]); f a poly, eng an optional int |
---|
| 875 | RETURN: list (of roots of the Bernstein polynomial b and their multiplicies) |
---|
[3f4e52] | 876 | PURPOSE: compute the global Bernstein-Sato polynomial for a hypersurface, |
---|
[66c962] | 877 | @* defined by f, according to the algorithm by Briancon and Maisonobe |
---|
| 878 | NOTE: If eng <>0, @code{std} is used for Groebner basis computations, |
---|
| 879 | @* otherwise, and by default @code{slimgb} is used. |
---|
| 880 | DISPLAY: If @code{printlevel}=1, progress debug messages will be printed, |
---|
| 881 | @* if @code{printlevel}>=2, all the debug messages will be printed. |
---|
| 882 | EXAMPLE: example bernsteinBM; shows examples |
---|
| 883 | " |
---|
| 884 | { |
---|
| 885 | int eng = 0; |
---|
| 886 | if ( size(#)>0 ) |
---|
| 887 | { |
---|
| 888 | if ( typeof(#[1]) == "int" ) |
---|
| 889 | { |
---|
| 890 | eng = int(#[1]); |
---|
| 891 | } |
---|
| 892 | } |
---|
| 893 | // returns a list with a ring and an ideal LD in it |
---|
| 894 | int ppl = printlevel-voice+2; |
---|
| 895 | // printf("plevel :%s, voice: %s",printlevel,voice); |
---|
| 896 | def save = basering; |
---|
| 897 | int N = nvars(basering); |
---|
| 898 | int Nnew = 2*N+2; |
---|
| 899 | int i,j; |
---|
| 900 | string s; |
---|
| 901 | list RL = ringlist(basering); |
---|
| 902 | list L, Lord; |
---|
| 903 | list tmp; |
---|
| 904 | intvec iv; |
---|
| 905 | L[1] = RL[1]; //char |
---|
| 906 | L[4] = RL[4]; //char, minpoly |
---|
| 907 | // check whether vars have admissible names |
---|
| 908 | list Name = RL[2]; |
---|
| 909 | list RName; |
---|
| 910 | RName[1] = "t"; |
---|
| 911 | RName[2] = "s"; |
---|
| 912 | for (i=1; i<=N; i++) |
---|
| 913 | { |
---|
| 914 | for(j=1; j<=size(RName); j++) |
---|
| 915 | { |
---|
| 916 | if (Name[i] == RName[j]) |
---|
| 917 | { |
---|
| 918 | ERROR("Variable names should not include t,s"); |
---|
| 919 | } |
---|
| 920 | } |
---|
| 921 | } |
---|
| 922 | // now, create the names for new vars |
---|
| 923 | list DName; |
---|
| 924 | for (i=1; i<=N; i++) |
---|
| 925 | { |
---|
| 926 | DName[i] = "D"+Name[i]; //concat |
---|
| 927 | } |
---|
| 928 | tmp[1] = "t"; |
---|
| 929 | tmp[2] = "s"; |
---|
| 930 | list NName = tmp + Name + DName; |
---|
| 931 | L[2] = NName; |
---|
| 932 | // Name, Dname will be used further |
---|
| 933 | kill NName; |
---|
| 934 | // block ord (lp(2),dp); |
---|
| 935 | tmp[1] = "lp"; // string |
---|
| 936 | iv = 1,1; |
---|
| 937 | tmp[2] = iv; //intvec |
---|
| 938 | Lord[1] = tmp; |
---|
| 939 | // continue with dp 1,1,1,1... |
---|
| 940 | tmp[1] = "dp"; // string |
---|
| 941 | s = "iv="; |
---|
| 942 | for (i=1; i<=Nnew; i++) |
---|
| 943 | { |
---|
| 944 | s = s+"1,"; |
---|
| 945 | } |
---|
| 946 | s[size(s)]= ";"; |
---|
| 947 | execute(s); |
---|
| 948 | kill s; |
---|
| 949 | tmp[2] = iv; |
---|
| 950 | Lord[2] = tmp; |
---|
| 951 | tmp[1] = "C"; |
---|
| 952 | iv = 0; |
---|
| 953 | tmp[2] = iv; |
---|
| 954 | Lord[3] = tmp; |
---|
| 955 | tmp = 0; |
---|
| 956 | L[3] = Lord; |
---|
| 957 | // we are done with the list |
---|
| 958 | def @R@ = ring(L); |
---|
| 959 | setring @R@; |
---|
| 960 | matrix @D[Nnew][Nnew]; |
---|
| 961 | @D[1,2]=t; |
---|
| 962 | for(i=1; i<=N; i++) |
---|
| 963 | { |
---|
| 964 | @D[2+i,N+2+i]=1; |
---|
| 965 | } |
---|
| 966 | // L[5] = matrix(UpOneMatrix(Nnew)); |
---|
| 967 | // L[6] = @D; |
---|
| 968 | def @R = nc_algebra(1,@D); |
---|
| 969 | setring @R; |
---|
| 970 | kill @R@; |
---|
| 971 | dbprint(ppl,"// -1-1- the ring @R(t,s,_x,_Dx) is ready"); |
---|
| 972 | dbprint(ppl-1, @R); |
---|
| 973 | // create the ideal I |
---|
| 974 | poly F = imap(save,F); |
---|
| 975 | ideal I = t*F+s; |
---|
| 976 | poly p; |
---|
| 977 | for(i=1; i<=N; i++) |
---|
| 978 | { |
---|
| 979 | p = t; //t |
---|
| 980 | p = diff(F,var(2+i))*p; |
---|
| 981 | I = I, var(N+2+i) + p; |
---|
| 982 | } |
---|
| 983 | // -------- the ideal I is ready ---------- |
---|
| 984 | dbprint(ppl,"// -1-2- starting the elimination of t in @R"); |
---|
| 985 | dbprint(ppl-1, I); |
---|
| 986 | ideal J = engine(I,eng); |
---|
| 987 | ideal K = nselect(J,1); |
---|
| 988 | kill I,J; |
---|
| 989 | dbprint(ppl,"// -1-3- t is eliminated"); |
---|
| 990 | dbprint(ppl-1, K); //K is without t |
---|
| 991 | // ----------- the ring @R2 ------------ |
---|
| 992 | // _x, _Dx,s; elim.ord for _x,_Dx. |
---|
| 993 | // keep: N, i,j,s, tmp, RL |
---|
| 994 | setring save; |
---|
| 995 | Nnew = 2*N+1; |
---|
| 996 | kill Lord, tmp, iv, RName; |
---|
| 997 | list Lord, tmp; |
---|
| 998 | intvec iv; |
---|
| 999 | L[1] = RL[1]; |
---|
| 1000 | L[4] = RL[4]; //char, minpoly |
---|
| 1001 | // check whether vars hava admissible names -> done earlier |
---|
| 1002 | // now, create the names for new var |
---|
| 1003 | tmp[1] = "s"; |
---|
| 1004 | // DName is defined earlier |
---|
| 1005 | list NName = Name + DName + tmp; |
---|
| 1006 | L[2] = NName; |
---|
| 1007 | tmp = 0; |
---|
| 1008 | // block ord (dp(N),dp); |
---|
| 1009 | string s = "iv="; |
---|
| 1010 | for (i=1; i<=Nnew-1; i++) |
---|
| 1011 | { |
---|
| 1012 | s = s+"1,"; |
---|
| 1013 | } |
---|
| 1014 | s[size(s)]=";"; |
---|
| 1015 | execute(s); |
---|
| 1016 | tmp[1] = "dp"; //string |
---|
| 1017 | tmp[2] = iv; //intvec |
---|
| 1018 | Lord[1] = tmp; |
---|
| 1019 | // continue with dp 1,1,1,1... |
---|
| 1020 | tmp[1] = "dp"; //string |
---|
| 1021 | s[size(s)] = ","; |
---|
| 1022 | s = s+"1;"; |
---|
| 1023 | execute(s); |
---|
| 1024 | kill s; |
---|
| 1025 | kill NName; |
---|
| 1026 | tmp[2] = iv; |
---|
| 1027 | Lord[2] = tmp; |
---|
| 1028 | tmp[1] = "C"; |
---|
| 1029 | iv = 0; |
---|
| 1030 | tmp[2] = iv; |
---|
| 1031 | Lord[3] = tmp; |
---|
| 1032 | tmp = 0; |
---|
| 1033 | L[3] = Lord; |
---|
| 1034 | // we are done with the list. Now add a Plural part |
---|
| 1035 | def @R2@ = ring(L); |
---|
| 1036 | setring @R2@; |
---|
| 1037 | matrix @D[Nnew][Nnew]; |
---|
| 1038 | for (i=1; i<=N; i++) |
---|
| 1039 | { |
---|
| 1040 | @D[i,N+i]=1; |
---|
| 1041 | } |
---|
| 1042 | def @R2 = nc_algebra(1,@D); |
---|
| 1043 | setring @R2; |
---|
| 1044 | kill @R2@; |
---|
| 1045 | dbprint(ppl,"// -2-1- the ring @R2(_x,_Dx,s) is ready"); |
---|
| 1046 | dbprint(ppl-1, @R2); |
---|
| 1047 | ideal MM = maxideal(1); |
---|
| 1048 | MM = 0,s,MM; |
---|
| 1049 | map R01 = @R, MM; |
---|
| 1050 | ideal K = R01(K); |
---|
| 1051 | kill @R, R01; |
---|
| 1052 | poly F = imap(save,F); |
---|
| 1053 | K = K,F; |
---|
| 1054 | dbprint(ppl,"// -2-2- starting the elimination of _x,_Dx in @R2"); |
---|
| 1055 | dbprint(ppl-1, K); |
---|
| 1056 | ideal M = engine(K,eng); |
---|
| 1057 | ideal K2 = nselect(M,1..Nnew-1); |
---|
| 1058 | kill K,M; |
---|
| 1059 | dbprint(ppl,"// -2-3- _x,_Dx are eliminated in @R2"); |
---|
| 1060 | dbprint(ppl-1, K2); |
---|
| 1061 | // the ring @R3 and the search for minimal negative int s |
---|
| 1062 | ring @R3 = 0,s,dp; |
---|
| 1063 | dbprint(ppl,"// -3-1- the ring @R3(s) is ready"); |
---|
| 1064 | ideal K3 = imap(@R2,K2); |
---|
| 1065 | kill @R2; |
---|
| 1066 | poly p = K3[1]; |
---|
| 1067 | dbprint(ppl,"// -3-2- factorization"); |
---|
| 1068 | list P = factorize(p); //with constants and multiplicities |
---|
| 1069 | ideal bs; intvec m; //the Bernstein polynomial is monic, so we are not interested in constants |
---|
| 1070 | for (i=2; i<= size(P[1]); i++) //we delete P[1][1] and P[2][1] |
---|
| 1071 | { |
---|
| 1072 | bs[i-1] = P[1][i]; |
---|
| 1073 | m[i-1] = P[2][i]; |
---|
| 1074 | } |
---|
| 1075 | // "--------- b-function factorizes into ---------"; P; |
---|
| 1076 | //int sP = minIntRoot(bs,1); |
---|
| 1077 | //dbprint(ppl,"// -3-3- minimal integer root found"); |
---|
| 1078 | //dbprint(ppl-1, sP); |
---|
| 1079 | // convert factors to a list of their roots and multiplicities |
---|
| 1080 | bs = normalize(bs); |
---|
| 1081 | bs = -subst(bs,s,0); |
---|
| 1082 | setring save; |
---|
| 1083 | ideal bs = imap(@R3,bs); |
---|
| 1084 | kill @R3; |
---|
| 1085 | list BS = bs,m; |
---|
| 1086 | return(BS); |
---|
| 1087 | } |
---|
| 1088 | example |
---|
| 1089 | { |
---|
| 1090 | "EXAMPLE:"; echo = 2; |
---|
| 1091 | ring r = 0,(x,y,z,w),Dp; |
---|
| 1092 | poly F = x^3+y^3+z^2*w; |
---|
| 1093 | printlevel = 0; |
---|
| 1094 | bernsteinBM(F); |
---|
| 1095 | } |
---|
| 1096 | |
---|
| 1097 | // some changes |
---|
| 1098 | proc annfsBM (poly F, list #) |
---|
| 1099 | "USAGE: annfsBM(f [,eng]); f a poly, eng an optional int |
---|
| 1100 | RETURN: ring |
---|
| 1101 | PURPOSE: compute the D-module structure of basering[1/f]*f^s, according |
---|
| 1102 | @* to the algorithm by Briancon and Maisonobe |
---|
| 1103 | NOTE: activate the output ring with the @code{setring} command. In this ring, |
---|
| 1104 | @* - the ideal LD (which is a Groebner basis) is the needed D-module structure, |
---|
| 1105 | @* which is obtained by substituting the minimal integer root of a Bernstein |
---|
| 1106 | @* polynomial into the s-parametric ideal; |
---|
| 1107 | @* - the list BS is the list of roots and multiplicities of a Bernstein polynomial of f. |
---|
| 1108 | @* If eng <>0, @code{std} is used for Groebner basis computations, |
---|
| 1109 | @* otherwise, and by default @code{slimgb} is used. |
---|
| 1110 | DISPLAY: If @code{printlevel}=1, progress debug messages will be printed, |
---|
| 1111 | @* if @code{printlevel}>=2, all the debug messages will be printed. |
---|
| 1112 | EXAMPLE: example annfsBM; shows examples |
---|
| 1113 | " |
---|
| 1114 | { |
---|
| 1115 | int eng = 0; |
---|
| 1116 | if ( size(#)>0 ) |
---|
| 1117 | { |
---|
| 1118 | if ( typeof(#[1]) == "int" ) |
---|
| 1119 | { |
---|
| 1120 | eng = int(#[1]); |
---|
| 1121 | } |
---|
| 1122 | } |
---|
| 1123 | // returns a list with a ring and an ideal LD in it |
---|
| 1124 | int ppl = printlevel-voice+2; |
---|
| 1125 | // printf("plevel :%s, voice: %s",printlevel,voice); |
---|
| 1126 | def save = basering; |
---|
| 1127 | int N = nvars(basering); |
---|
| 1128 | int Nnew = 2*N+2; |
---|
| 1129 | int i,j; |
---|
| 1130 | string s; |
---|
| 1131 | list RL = ringlist(basering); |
---|
| 1132 | list L, Lord; |
---|
| 1133 | list tmp; |
---|
| 1134 | intvec iv; |
---|
| 1135 | L[1] = RL[1]; //char |
---|
| 1136 | L[4] = RL[4]; //char, minpoly |
---|
| 1137 | // check whether vars have admissible names |
---|
| 1138 | list Name = RL[2]; |
---|
| 1139 | list RName; |
---|
| 1140 | RName[1] = "t"; |
---|
| 1141 | RName[2] = "s"; |
---|
| 1142 | for (i=1; i<=N; i++) |
---|
| 1143 | { |
---|
| 1144 | for(j=1; j<=size(RName); j++) |
---|
| 1145 | { |
---|
| 1146 | if (Name[i] == RName[j]) |
---|
| 1147 | { |
---|
| 1148 | ERROR("Variable names should not include t,s"); |
---|
| 1149 | } |
---|
| 1150 | } |
---|
| 1151 | } |
---|
| 1152 | // now, create the names for new vars |
---|
| 1153 | list DName; |
---|
| 1154 | for (i=1; i<=N; i++) |
---|
| 1155 | { |
---|
| 1156 | DName[i] = "D"+Name[i]; //concat |
---|
| 1157 | } |
---|
| 1158 | tmp[1] = "t"; |
---|
| 1159 | tmp[2] = "s"; |
---|
| 1160 | list NName = tmp + Name + DName; |
---|
| 1161 | L[2] = NName; |
---|
| 1162 | // Name, Dname will be used further |
---|
| 1163 | kill NName; |
---|
| 1164 | // block ord (lp(2),dp); |
---|
| 1165 | tmp[1] = "lp"; // string |
---|
| 1166 | iv = 1,1; |
---|
| 1167 | tmp[2] = iv; //intvec |
---|
| 1168 | Lord[1] = tmp; |
---|
| 1169 | // continue with dp 1,1,1,1... |
---|
| 1170 | tmp[1] = "dp"; // string |
---|
| 1171 | s = "iv="; |
---|
| 1172 | for (i=1; i<=Nnew; i++) |
---|
| 1173 | { |
---|
| 1174 | s = s+"1,"; |
---|
| 1175 | } |
---|
| 1176 | s[size(s)]= ";"; |
---|
| 1177 | execute(s); |
---|
| 1178 | kill s; |
---|
| 1179 | tmp[2] = iv; |
---|
| 1180 | Lord[2] = tmp; |
---|
| 1181 | tmp[1] = "C"; |
---|
| 1182 | iv = 0; |
---|
| 1183 | tmp[2] = iv; |
---|
| 1184 | Lord[3] = tmp; |
---|
| 1185 | tmp = 0; |
---|
| 1186 | L[3] = Lord; |
---|
| 1187 | // we are done with the list |
---|
| 1188 | def @R@ = ring(L); |
---|
| 1189 | setring @R@; |
---|
| 1190 | matrix @D[Nnew][Nnew]; |
---|
| 1191 | @D[1,2]=t; |
---|
| 1192 | for(i=1; i<=N; i++) |
---|
| 1193 | { |
---|
| 1194 | @D[2+i,N+2+i]=1; |
---|
| 1195 | } |
---|
| 1196 | // L[5] = matrix(UpOneMatrix(Nnew)); |
---|
| 1197 | // L[6] = @D; |
---|
| 1198 | def @R = nc_algebra(1,@D); |
---|
| 1199 | setring @R; |
---|
| 1200 | kill @R@; |
---|
| 1201 | dbprint(ppl,"// -1-1- the ring @R(t,s,_x,_Dx) is ready"); |
---|
| 1202 | dbprint(ppl-1, @R); |
---|
| 1203 | // create the ideal I |
---|
| 1204 | poly F = imap(save,F); |
---|
| 1205 | ideal I = t*F+s; |
---|
| 1206 | poly p; |
---|
| 1207 | for(i=1; i<=N; i++) |
---|
| 1208 | { |
---|
| 1209 | p = t; //t |
---|
| 1210 | p = diff(F,var(2+i))*p; |
---|
| 1211 | I = I, var(N+2+i) + p; |
---|
| 1212 | } |
---|
| 1213 | // -------- the ideal I is ready ---------- |
---|
| 1214 | dbprint(ppl,"// -1-2- starting the elimination of t in @R"); |
---|
| 1215 | dbprint(ppl-1, I); |
---|
| 1216 | ideal J = engine(I,eng); |
---|
| 1217 | ideal K = nselect(J,1); |
---|
| 1218 | kill I,J; |
---|
| 1219 | dbprint(ppl,"// -1-3- t is eliminated"); |
---|
| 1220 | dbprint(ppl-1, K); //K is without t |
---|
| 1221 | setring save; |
---|
| 1222 | // ----------- the ring @R2 ------------ |
---|
| 1223 | // _x, _Dx,s; elim.ord for _x,_Dx. |
---|
| 1224 | // keep: N, i,j,s, tmp, RL |
---|
| 1225 | Nnew = 2*N+1; |
---|
| 1226 | kill Lord, tmp, iv, RName; |
---|
| 1227 | list Lord, tmp; |
---|
| 1228 | intvec iv; |
---|
| 1229 | L[1] = RL[1]; |
---|
| 1230 | L[4] = RL[4]; //char, minpoly |
---|
| 1231 | // check whether vars hava admissible names -> done earlier |
---|
| 1232 | // now, create the names for new var |
---|
| 1233 | tmp[1] = "s"; |
---|
| 1234 | // DName is defined earlier |
---|
| 1235 | list NName = Name + DName + tmp; |
---|
| 1236 | L[2] = NName; |
---|
| 1237 | tmp = 0; |
---|
| 1238 | // block ord (dp(N),dp); |
---|
| 1239 | string s = "iv="; |
---|
| 1240 | for (i=1; i<=Nnew-1; i++) |
---|
| 1241 | { |
---|
| 1242 | s = s+"1,"; |
---|
| 1243 | } |
---|
| 1244 | s[size(s)]=";"; |
---|
| 1245 | execute(s); |
---|
| 1246 | tmp[1] = "dp"; //string |
---|
| 1247 | tmp[2] = iv; //intvec |
---|
| 1248 | Lord[1] = tmp; |
---|
| 1249 | // continue with dp 1,1,1,1... |
---|
| 1250 | tmp[1] = "dp"; //string |
---|
| 1251 | s[size(s)] = ","; |
---|
| 1252 | s = s+"1;"; |
---|
| 1253 | execute(s); |
---|
| 1254 | kill s; |
---|
| 1255 | kill NName; |
---|
| 1256 | tmp[2] = iv; |
---|
| 1257 | Lord[2] = tmp; |
---|
| 1258 | tmp[1] = "C"; |
---|
| 1259 | iv = 0; |
---|
| 1260 | tmp[2] = iv; |
---|
| 1261 | Lord[3] = tmp; |
---|
| 1262 | tmp = 0; |
---|
| 1263 | L[3] = Lord; |
---|
| 1264 | // we are done with the list. Now add a Plural part |
---|
| 1265 | def @R2@ = ring(L); |
---|
| 1266 | setring @R2@; |
---|
| 1267 | matrix @D[Nnew][Nnew]; |
---|
| 1268 | for (i=1; i<=N; i++) |
---|
| 1269 | { |
---|
| 1270 | @D[i,N+i]=1; |
---|
| 1271 | } |
---|
| 1272 | def @R2 = nc_algebra(1,@D); |
---|
| 1273 | setring @R2; |
---|
| 1274 | kill @R2@; |
---|
| 1275 | dbprint(ppl,"// -2-1- the ring @R2(_x,_Dx,s) is ready"); |
---|
| 1276 | dbprint(ppl-1, @R2); |
---|
| 1277 | ideal MM = maxideal(1); |
---|
| 1278 | MM = 0,s,MM; |
---|
| 1279 | map R01 = @R, MM; |
---|
| 1280 | ideal K = R01(K); |
---|
| 1281 | poly F = imap(save,F); |
---|
| 1282 | K = K,F; |
---|
| 1283 | dbprint(ppl,"// -2-2- starting the elimination of _x,_Dx in @R2"); |
---|
| 1284 | dbprint(ppl-1, K); |
---|
| 1285 | ideal M = engine(K,eng); |
---|
| 1286 | ideal K2 = nselect(M,1..Nnew-1); |
---|
| 1287 | kill K,M; |
---|
| 1288 | dbprint(ppl,"// -2-3- _x,_Dx are eliminated in @R2"); |
---|
| 1289 | dbprint(ppl-1, K2); |
---|
| 1290 | // the ring @R3 and the search for minimal negative int s |
---|
| 1291 | ring @R3 = 0,s,dp; |
---|
| 1292 | dbprint(ppl,"// -3-1- the ring @R3(s) is ready"); |
---|
| 1293 | ideal K3 = imap(@R2,K2); |
---|
| 1294 | poly p = K3[1]; |
---|
| 1295 | dbprint(ppl,"// -3-2- factorization"); |
---|
| 1296 | list P = factorize(p); //with constants and multiplicities |
---|
| 1297 | ideal bs; intvec m; //the Bernstein polynomial is monic, so we are not interested in constants |
---|
| 1298 | for (i=2; i<= size(P[1]); i++) //we ignore P[1][1] (constant) and P[2][1] (its mult.) |
---|
| 1299 | { |
---|
| 1300 | bs[i-1] = P[1][i]; |
---|
| 1301 | m[i-1] = P[2][i]; |
---|
| 1302 | } |
---|
| 1303 | // "--------- b-function factorizes into ---------"; P; |
---|
| 1304 | int sP = minIntRoot(bs,1); |
---|
| 1305 | dbprint(ppl,"// -3-3- minimal integer root found"); |
---|
| 1306 | dbprint(ppl-1, sP); |
---|
| 1307 | // convert factors to a list of their roots |
---|
| 1308 | bs = normalize(bs); |
---|
| 1309 | bs = -subst(bs,s,0); |
---|
| 1310 | list BS = bs,m; |
---|
| 1311 | //TODO: sort BS! |
---|
| 1312 | // --------- substitute s found in the ideal --------- |
---|
| 1313 | // --------- going back to @R and substitute --------- |
---|
| 1314 | setring @R; |
---|
| 1315 | ideal K2 = subst(K,s,sP); |
---|
| 1316 | kill K; |
---|
| 1317 | // create the ordinary Weyl algebra and put the result into it, |
---|
| 1318 | // thus creating the ring @R4 |
---|
| 1319 | // keep: N, i,j,s, tmp, RL |
---|
| 1320 | setring save; |
---|
| 1321 | Nnew = 2*N; |
---|
| 1322 | // list RL = ringlist(save); //is defined earlier |
---|
| 1323 | kill Lord, tmp, iv; |
---|
| 1324 | L = 0; |
---|
| 1325 | list Lord, tmp; |
---|
| 1326 | intvec iv; |
---|
| 1327 | L[1] = RL[1]; |
---|
| 1328 | L[4] = RL[4]; //char, minpoly |
---|
| 1329 | // check whether vars have admissible names -> done earlier |
---|
| 1330 | // list Name = RL[2]M |
---|
| 1331 | // DName is defined earlier |
---|
| 1332 | list NName = Name + DName; |
---|
| 1333 | L[2] = NName; |
---|
| 1334 | // dp ordering; |
---|
| 1335 | string s = "iv="; |
---|
| 1336 | for (i=1; i<=Nnew; i++) |
---|
| 1337 | { |
---|
| 1338 | s = s+"1,"; |
---|
| 1339 | } |
---|
| 1340 | s[size(s)] = ";"; |
---|
| 1341 | execute(s); |
---|
| 1342 | kill s; |
---|
| 1343 | tmp = 0; |
---|
| 1344 | tmp[1] = "dp"; //string |
---|
| 1345 | tmp[2] = iv; //intvec |
---|
| 1346 | Lord[1] = tmp; |
---|
| 1347 | tmp[1] = "C"; |
---|
| 1348 | iv = 0; |
---|
| 1349 | tmp[2] = iv; |
---|
| 1350 | Lord[2] = tmp; |
---|
| 1351 | tmp = 0; |
---|
| 1352 | L[3] = Lord; |
---|
| 1353 | // we are done with the list |
---|
| 1354 | // Add: Plural part |
---|
| 1355 | def @R4@ = ring(L); |
---|
| 1356 | setring @R4@; |
---|
| 1357 | matrix @D[Nnew][Nnew]; |
---|
| 1358 | for (i=1; i<=N; i++) |
---|
| 1359 | { |
---|
| 1360 | @D[i,N+i]=1; |
---|
| 1361 | } |
---|
| 1362 | def @R4 = nc_algebra(1,@D); |
---|
| 1363 | setring @R4; |
---|
| 1364 | kill @R4@; |
---|
| 1365 | dbprint(ppl,"// -4-1- the ring @R4(_x,_Dx) is ready"); |
---|
| 1366 | dbprint(ppl-1, @R4); |
---|
| 1367 | ideal K4 = imap(@R,K2); |
---|
| 1368 | option(redSB); |
---|
| 1369 | dbprint(ppl,"// -4-2- the final cosmetic std"); |
---|
| 1370 | K4 = engine(K4,eng); //std does the job too |
---|
| 1371 | // total cleanup |
---|
| 1372 | kill @R; |
---|
| 1373 | kill @R2; |
---|
| 1374 | list BS = imap(@R3,BS); |
---|
| 1375 | export BS; |
---|
| 1376 | kill @R3; |
---|
| 1377 | ideal LD = K4; |
---|
| 1378 | export LD; |
---|
| 1379 | return(@R4); |
---|
| 1380 | } |
---|
| 1381 | example |
---|
| 1382 | { |
---|
| 1383 | "EXAMPLE:"; echo = 2; |
---|
| 1384 | ring r = 0,(x,y,z),Dp; |
---|
| 1385 | poly F = z*x^2+y^3; |
---|
| 1386 | printlevel = 0; |
---|
| 1387 | def A = annfsBM(F); |
---|
| 1388 | setring A; |
---|
| 1389 | LD; |
---|
| 1390 | BS; |
---|
| 1391 | } |
---|
| 1392 | |
---|
| 1393 | |
---|
| 1394 | // replacing s with -s-1 => data is shorter |
---|
| 1395 | // analogue of annfs0 |
---|
| 1396 | proc annfs2(ideal I, poly F, list #) |
---|
| 1397 | "USAGE: annfs2(I, F [,eng]); I an ideal, F a poly, eng an optional int |
---|
| 1398 | RETURN: ring |
---|
[3f4e52] | 1399 | PURPOSE: compute the annihilator ideal of f^s in the Weyl Algebra, |
---|
[66c962] | 1400 | @* based on the output of Sannfs-like procedure |
---|
| 1401 | @* annfs2 uses shorter expressions in the variable s (the idea of Noro). |
---|
| 1402 | NOTE: activate the output ring with the @code{setring} command. In this ring, |
---|
| 1403 | @* - the ideal LD (which is a Groebner basis) is the annihilator of f^s, |
---|
| 1404 | @* - the list BS contains the roots with multiplicities of the BS polynomial. |
---|
| 1405 | @* If eng <>0, @code{std} is used for Groebner basis computations, |
---|
| 1406 | @* otherwise and by default @code{slimgb} is used. |
---|
| 1407 | DISPLAY: If @code{printlevel}=1, progress debug messages will be printed, |
---|
| 1408 | @* if @code{printlevel}>=2, all the debug messages will be printed. |
---|
| 1409 | EXAMPLE: example annfs2; shows examples |
---|
| 1410 | " |
---|
| 1411 | { |
---|
| 1412 | int eng = 0; |
---|
| 1413 | if ( size(#)>0 ) |
---|
| 1414 | { |
---|
| 1415 | if ( typeof(#[1]) == "int" ) |
---|
| 1416 | { |
---|
| 1417 | eng = int(#[1]); |
---|
| 1418 | } |
---|
| 1419 | } |
---|
| 1420 | def @R2 = basering; |
---|
| 1421 | // we're in D_n[s], where the elim ord for s is set |
---|
| 1422 | ideal J = NF(I,std(F)); |
---|
| 1423 | // make leadcoeffs positive |
---|
| 1424 | int i; |
---|
| 1425 | J = subst(J,s,-s-1); |
---|
| 1426 | for (i=1; i<= ncols(J); i++) |
---|
| 1427 | { |
---|
| 1428 | if (leadcoef(J[i]) <0 ) |
---|
| 1429 | { |
---|
| 1430 | J[i] = -J[i]; |
---|
| 1431 | } |
---|
| 1432 | } |
---|
| 1433 | J = J,F; |
---|
| 1434 | ideal M = engine(J,eng); |
---|
| 1435 | int Nnew = nvars(@R2); |
---|
| 1436 | ideal K2 = nselect(M,1..Nnew-1); |
---|
| 1437 | int ppl = printlevel-voice+2; |
---|
| 1438 | dbprint(ppl,"// -1-1- _x,_Dx are eliminated in basering"); |
---|
| 1439 | dbprint(ppl-1, K2); |
---|
| 1440 | // the ring @R3 and the search for minimal negative int s |
---|
| 1441 | ring @R3 = 0,s,dp; |
---|
| 1442 | dbprint(ppl,"// -2-1- the ring @R3 i.e. K[s] is ready"); |
---|
| 1443 | ideal K3 = imap(@R2,K2); |
---|
| 1444 | poly p = K3[1]; |
---|
| 1445 | dbprint(ppl,"// -2-2- factorization"); |
---|
| 1446 | // ideal P = factorize(p,1); //without constants and multiplicities |
---|
| 1447 | // "--------- b-function factorizes into ---------"; P; |
---|
| 1448 | // convert factors to the list of their roots with mults |
---|
| 1449 | // assume all factors are linear |
---|
| 1450 | // ideal BS = normalize(P); |
---|
| 1451 | // BS = subst(BS,s,0); |
---|
| 1452 | // BS = -BS; |
---|
| 1453 | list P = factorize(p); //with constants and multiplicities |
---|
| 1454 | ideal bs; intvec m; //the Bernstein polynomial is monic, so we are not interested in constants |
---|
| 1455 | for (i=2; i<= size(P[1]); i++) //we delete P[1][1] and P[2][1] |
---|
| 1456 | { |
---|
| 1457 | bs[i-1] = P[1][i]; bs[i-1] = subst(bs[i-1],s,-s-1); |
---|
| 1458 | m[i-1] = P[2][i]; |
---|
| 1459 | } |
---|
| 1460 | int sP = minIntRoot(bs,1); |
---|
| 1461 | bs = normalize(bs); |
---|
| 1462 | bs = -subst(bs,s,0); |
---|
| 1463 | dbprint(ppl,"// -2-3- minimal integer root found"); |
---|
| 1464 | dbprint(ppl-1, sP); |
---|
| 1465 | //TODO: sort BS! |
---|
| 1466 | // --------- substitute s found in the ideal --------- |
---|
| 1467 | // --------- going back to @R and substitute --------- |
---|
| 1468 | setring @R2; |
---|
| 1469 | K2 = subst(I,s,sP); |
---|
| 1470 | // create the ordinary Weyl algebra and put the result into it, |
---|
| 1471 | // thus creating the ring @R5 |
---|
| 1472 | // keep: N, i,j,s, tmp, RL |
---|
| 1473 | Nnew = Nnew - 1; // former 2*N; |
---|
| 1474 | // list RL = ringlist(save); // is defined earlier |
---|
| 1475 | // kill Lord, tmp, iv; |
---|
| 1476 | list L = 0; |
---|
| 1477 | list Lord, tmp; |
---|
| 1478 | intvec iv; |
---|
| 1479 | list RL = ringlist(basering); |
---|
| 1480 | L[1] = RL[1]; |
---|
| 1481 | L[4] = RL[4]; //char, minpoly |
---|
| 1482 | // check whether vars have admissible names -> done earlier |
---|
| 1483 | // list Name = RL[2]M |
---|
| 1484 | // DName is defined earlier |
---|
| 1485 | list NName; // = RL[2]; // skip the last var 's' |
---|
| 1486 | for (i=1; i<=Nnew; i++) |
---|
| 1487 | { |
---|
| 1488 | NName[i] = RL[2][i]; |
---|
| 1489 | } |
---|
| 1490 | L[2] = NName; |
---|
| 1491 | // dp ordering; |
---|
| 1492 | string s = "iv="; |
---|
| 1493 | for (i=1; i<=Nnew; i++) |
---|
| 1494 | { |
---|
| 1495 | s = s+"1,"; |
---|
| 1496 | } |
---|
| 1497 | s[size(s)] = ";"; |
---|
| 1498 | execute(s); |
---|
| 1499 | tmp = 0; |
---|
| 1500 | tmp[1] = "dp"; // string |
---|
| 1501 | tmp[2] = iv; // intvec |
---|
| 1502 | Lord[1] = tmp; |
---|
| 1503 | kill s; |
---|
| 1504 | tmp[1] = "C"; |
---|
| 1505 | iv = 0; |
---|
| 1506 | tmp[2] = iv; |
---|
| 1507 | Lord[2] = tmp; |
---|
| 1508 | tmp = 0; |
---|
| 1509 | L[3] = Lord; |
---|
| 1510 | // we are done with the list |
---|
| 1511 | // Add: Plural part |
---|
| 1512 | def @R4@ = ring(L); |
---|
| 1513 | setring @R4@; |
---|
| 1514 | int N = Nnew/2; |
---|
| 1515 | matrix @D[Nnew][Nnew]; |
---|
| 1516 | for (i=1; i<=N; i++) |
---|
| 1517 | { |
---|
| 1518 | @D[i,N+i]=1; |
---|
| 1519 | } |
---|
| 1520 | def @R4 = nc_algebra(1,@D); |
---|
| 1521 | setring @R4; |
---|
| 1522 | kill @R4@; |
---|
| 1523 | dbprint(ppl,"// -3-1- the ring @R4 is ready"); |
---|
| 1524 | dbprint(ppl-1, @R4); |
---|
| 1525 | ideal K4 = imap(@R2,K2); |
---|
| 1526 | option(redSB); |
---|
| 1527 | dbprint(ppl,"// -3-2- the final cosmetic std"); |
---|
| 1528 | K4 = engine(K4,eng); // std does the job too |
---|
| 1529 | // total cleanup |
---|
| 1530 | ideal bs = imap(@R3,bs); |
---|
| 1531 | kill @R3; |
---|
| 1532 | list BS = bs,m; |
---|
| 1533 | export BS; |
---|
| 1534 | ideal LD = K4; |
---|
| 1535 | export LD; |
---|
| 1536 | return(@R4); |
---|
| 1537 | } |
---|
| 1538 | example |
---|
| 1539 | { "EXAMPLE:"; echo = 2; |
---|
| 1540 | ring r = 0,(x,y,z),Dp; |
---|
| 1541 | poly F = x^3+y^3+z^3; |
---|
| 1542 | printlevel = 0; |
---|
| 1543 | def A = SannfsBM(F); |
---|
| 1544 | setring A; |
---|
| 1545 | LD; |
---|
| 1546 | poly F = imap(r,F); |
---|
| 1547 | def B = annfs2(LD,F); |
---|
| 1548 | setring B; |
---|
| 1549 | LD; |
---|
| 1550 | BS; |
---|
| 1551 | } |
---|
| 1552 | |
---|
| 1553 | // try to replace s with -s-1 => data is shorter as in annfs2 |
---|
[e64e417] | 1554 | // and use what Macaulay2 people call reduceB strategy, that is add |
---|
| 1555 | // not F but Tjurina ideal <F,dF/dx1,...,dF/dxN>; the resulting B-function |
---|
[66c962] | 1556 | // has to be multiplied with (s+1) at the very end |
---|
| 1557 | proc annfsRB(ideal I, poly F, list #) |
---|
| 1558 | "USAGE: annfsRB(I, F [,eng]); I an ideal, F a poly, eng an optional int |
---|
| 1559 | RETURN: ring |
---|
[3f4e52] | 1560 | PURPOSE: compute the annihilator ideal of f^s in the Weyl Algebra, |
---|
[66c962] | 1561 | @* based on the output of Sannfs like procedure |
---|
| 1562 | NOTE: activate the output ring with the @code{setring} command. In this ring, |
---|
| 1563 | @* - the ideal LD (which is a Groebner basis) is the annihilator of f^s, |
---|
| 1564 | @* - the list BS contains the roots with multiplicities of a Bernstein polynomial of f. |
---|
| 1565 | @* If eng <>0, @code{std} is used for Groebner basis computations, |
---|
| 1566 | @* otherwise and by default @code{slimgb} is used. |
---|
[e64e417] | 1567 | @* This procedure uses in addition to F its Jacobian ideal. |
---|
[66c962] | 1568 | DISPLAY: If @code{printlevel}=1, progress debug messages will be printed, |
---|
| 1569 | @* if @code{printlevel}>=2, all the debug messages will be printed. |
---|
| 1570 | EXAMPLE: example annfsRB; shows examples |
---|
| 1571 | " |
---|
| 1572 | { |
---|
| 1573 | int eng = 0; |
---|
| 1574 | if ( size(#)>0 ) |
---|
| 1575 | { |
---|
| 1576 | if ( typeof(#[1]) == "int" ) |
---|
| 1577 | { |
---|
| 1578 | eng = int(#[1]); |
---|
| 1579 | } |
---|
| 1580 | } |
---|
| 1581 | def @R2 = basering; |
---|
| 1582 | int ppl = printlevel-voice+2; |
---|
| 1583 | // we're in D_n[s], where the elim ord for s is set |
---|
| 1584 | // switch to comm. ring in X's and compute the GB of Tjurina ideal |
---|
| 1585 | dbprint(ppl,"// -1-0- creating K[x] and Tjurina ideal"); |
---|
| 1586 | list nL = ringlist(@R2); |
---|
| 1587 | list temp,t2; |
---|
| 1588 | temp[1] = nL[1]; |
---|
| 1589 | temp[4] = nL[4]; |
---|
| 1590 | int @n = int((nvars(@R2)-1)/2); // # of x's |
---|
| 1591 | int i; |
---|
| 1592 | for (i=1; i<=@n; i++) |
---|
| 1593 | { |
---|
| 1594 | t2[i] = nL[2][i]; |
---|
| 1595 | } |
---|
| 1596 | temp[2] = t2; |
---|
| 1597 | t2 = 0; |
---|
| 1598 | t2[1] = nL[3][1]; // more weights than vars? |
---|
| 1599 | t2[2] = nL[3][3]; |
---|
| 1600 | temp[3] = t2; |
---|
| 1601 | def @R22 = ring(temp); |
---|
| 1602 | setring @R22; |
---|
| 1603 | poly F = imap(@R2,F); |
---|
| 1604 | ideal J = F; |
---|
| 1605 | for (i=1; i<=@n; i++) |
---|
| 1606 | { |
---|
| 1607 | J = J, diff(F,var(i)); |
---|
| 1608 | } |
---|
| 1609 | J = engine(J,eng); |
---|
| 1610 | dbprint(ppl,"// -1-1- finished computing the GB of Tjurina ideal"); |
---|
| 1611 | dbprint(ppl-1, J); |
---|
| 1612 | setring @R2; |
---|
| 1613 | ideal JF = imap(@R22,J); |
---|
| 1614 | kill @R22; |
---|
| 1615 | attrib(JF,"isSB",1); // embedded comm ring is used |
---|
| 1616 | ideal J = NF(I,JF); |
---|
| 1617 | dbprint(ppl,"// -1-2- finished computing the NF of I w.r.t. Tjurina ideal"); |
---|
| 1618 | dbprint(ppl-1, J2); |
---|
| 1619 | // make leadcoeffs positive |
---|
| 1620 | J = subst(J,s,-s-1); |
---|
| 1621 | for (i=1; i<= ncols(J); i++) |
---|
| 1622 | { |
---|
| 1623 | if (leadcoef(J[i]) <0 ) |
---|
| 1624 | { |
---|
| 1625 | J[i] = -J[i]; |
---|
| 1626 | } |
---|
| 1627 | } |
---|
| 1628 | J = J,JF; |
---|
| 1629 | ideal M = engine(J,eng); |
---|
| 1630 | int Nnew = nvars(@R2); |
---|
| 1631 | ideal K2 = nselect(M,1..Nnew-1); |
---|
| 1632 | dbprint(ppl,"// -2-0- _x,_Dx are eliminated in basering"); |
---|
| 1633 | dbprint(ppl-1, K2); |
---|
| 1634 | // the ring @R3 and the search for minimal negative int s |
---|
| 1635 | ring @R3 = 0,s,dp; |
---|
| 1636 | dbprint(ppl,"// -2-1- the ring @R3 i.e. K[s] is ready"); |
---|
| 1637 | ideal K3 = imap(@R2,K2); |
---|
| 1638 | poly p = K3[1]; |
---|
| 1639 | p = s*p; // mult with the lost (s+1) factor |
---|
| 1640 | dbprint(ppl,"// -2-2- factorization"); |
---|
| 1641 | // ideal P = factorize(p,1); //without constants and multiplicities |
---|
| 1642 | // "--------- b-function factorizes into ---------"; P; |
---|
| 1643 | // convert factors to the list of their roots with mults |
---|
| 1644 | // assume all factors are linear |
---|
| 1645 | // ideal BS = normalize(P); |
---|
| 1646 | // BS = subst(BS,s,0); |
---|
| 1647 | // BS = -BS; |
---|
| 1648 | list P = factorize(p); //with constants and multiplicities |
---|
| 1649 | ideal bs; intvec m; //the Bernstein polynomial is monic, so we are not interested in constants |
---|
| 1650 | for (i=2; i<= size(P[1]); i++) //we delete P[1][1] and P[2][1] |
---|
| 1651 | { |
---|
| 1652 | bs[i-1] = P[1][i]; bs[i-1] = subst(bs[i-1],s,-s-1); |
---|
| 1653 | m[i-1] = P[2][i]; |
---|
| 1654 | } |
---|
| 1655 | int sP = minIntRoot(bs,1); |
---|
| 1656 | bs = normalize(bs); |
---|
| 1657 | bs = -subst(bs,s,0); |
---|
| 1658 | dbprint(ppl,"// -2-3- minimal integer root found"); |
---|
| 1659 | dbprint(ppl-1, sP); |
---|
| 1660 | //TODO: sort BS! |
---|
| 1661 | // --------- substitute s found in the ideal --------- |
---|
| 1662 | // --------- going back to @R and substitute --------- |
---|
| 1663 | setring @R2; |
---|
| 1664 | K2 = subst(I,s,sP); |
---|
| 1665 | // create the ordinary Weyl algebra and put the result into it, |
---|
| 1666 | // thus creating the ring @R5 |
---|
| 1667 | // keep: N, i,j,s, tmp, RL |
---|
| 1668 | Nnew = Nnew - 1; // former 2*N; |
---|
| 1669 | // list RL = ringlist(save); // is defined earlier |
---|
| 1670 | // kill Lord, tmp, iv; |
---|
| 1671 | list L = 0; |
---|
| 1672 | list Lord, tmp; |
---|
| 1673 | intvec iv; |
---|
| 1674 | list RL = ringlist(basering); |
---|
| 1675 | L[1] = RL[1]; |
---|
| 1676 | L[4] = RL[4]; //char, minpoly |
---|
| 1677 | // check whether vars have admissible names -> done earlier |
---|
| 1678 | // list Name = RL[2]M |
---|
| 1679 | // DName is defined earlier |
---|
| 1680 | list NName; // = RL[2]; // skip the last var 's' |
---|
| 1681 | for (i=1; i<=Nnew; i++) |
---|
| 1682 | { |
---|
| 1683 | NName[i] = RL[2][i]; |
---|
| 1684 | } |
---|
| 1685 | L[2] = NName; |
---|
| 1686 | // dp ordering; |
---|
| 1687 | string s = "iv="; |
---|
| 1688 | for (i=1; i<=Nnew; i++) |
---|
| 1689 | { |
---|
| 1690 | s = s+"1,"; |
---|
| 1691 | } |
---|
| 1692 | s[size(s)] = ";"; |
---|
| 1693 | execute(s); |
---|
| 1694 | tmp = 0; |
---|
| 1695 | tmp[1] = "dp"; // string |
---|
| 1696 | tmp[2] = iv; // intvec |
---|
| 1697 | Lord[1] = tmp; |
---|
| 1698 | kill s; |
---|
| 1699 | tmp[1] = "C"; |
---|
| 1700 | iv = 0; |
---|
| 1701 | tmp[2] = iv; |
---|
| 1702 | Lord[2] = tmp; |
---|
| 1703 | tmp = 0; |
---|
| 1704 | L[3] = Lord; |
---|
| 1705 | // we are done with the list |
---|
| 1706 | // Add: Plural part |
---|
| 1707 | def @R4@ = ring(L); |
---|
| 1708 | setring @R4@; |
---|
| 1709 | int N = Nnew/2; |
---|
| 1710 | matrix @D[Nnew][Nnew]; |
---|
| 1711 | for (i=1; i<=N; i++) |
---|
| 1712 | { |
---|
| 1713 | @D[i,N+i]=1; |
---|
| 1714 | } |
---|
| 1715 | def @R4 = nc_algebra(1,@D); |
---|
| 1716 | setring @R4; |
---|
| 1717 | kill @R4@; |
---|
| 1718 | dbprint(ppl,"// -3-1- the ring @R4 is ready"); |
---|
| 1719 | dbprint(ppl-1, @R4); |
---|
| 1720 | ideal K4 = imap(@R2,K2); |
---|
| 1721 | option(redSB); |
---|
| 1722 | dbprint(ppl,"// -3-2- the final cosmetic std"); |
---|
| 1723 | K4 = engine(K4,eng); // std does the job too |
---|
| 1724 | // total cleanup |
---|
| 1725 | ideal bs = imap(@R3,bs); |
---|
| 1726 | kill @R3; |
---|
| 1727 | list BS = bs,m; |
---|
| 1728 | export BS; |
---|
| 1729 | ideal LD = K4; |
---|
| 1730 | export LD; |
---|
| 1731 | return(@R4); |
---|
| 1732 | } |
---|
| 1733 | example |
---|
| 1734 | { "EXAMPLE:"; echo = 2; |
---|
| 1735 | ring r = 0,(x,y,z),Dp; |
---|
| 1736 | poly F = x^3+y^3+z^3; |
---|
| 1737 | printlevel = 0; |
---|
| 1738 | def A = SannfsBM(F); setring A; |
---|
| 1739 | LD; // s-parametric ahhinilator |
---|
| 1740 | poly F = imap(r,F); |
---|
| 1741 | def B = annfsRB(LD,F); setring B; |
---|
| 1742 | LD; |
---|
| 1743 | BS; |
---|
| 1744 | } |
---|
| 1745 | |
---|
| 1746 | proc operatorBM(poly F, list #) |
---|
| 1747 | "USAGE: operatorBM(f [,eng]); f a poly, eng an optional int |
---|
| 1748 | RETURN: ring |
---|
[3f4e52] | 1749 | PURPOSE: compute the B-operator and other relevant data for Ann F^s, |
---|
[66c962] | 1750 | @* using e.g. algorithm by Briancon and Maisonobe for Ann F^s and BS. |
---|
| 1751 | NOTE: activate the output ring with the @code{setring} command. In the output ring D[s] |
---|
| 1752 | @* - the polynomial F is the same as the input, |
---|
| 1753 | @* - the ideal LD is the annihilator of f^s in Dn[s], |
---|
| 1754 | @* - the ideal LD0 is the needed D-mod structure, where LD0 = LD|s=s0, |
---|
| 1755 | @* - the polynomial bs is the global Bernstein polynomial of f in the variable s, |
---|
| 1756 | @* - the list BS contains all the roots with multiplicities of the global Bernstein polynomial of f, |
---|
| 1757 | @* - the polynomial PS is an operator in Dn[s] such that PS*f^(s+1) = bs*f^s. |
---|
| 1758 | @* If eng <>0, @code{std} is used for Groebner basis computations, |
---|
| 1759 | @* otherwise and by default @code{slimgb} is used. |
---|
| 1760 | DISPLAY: If @code{printlevel}=1, progress debug messages will be printed, |
---|
| 1761 | @* if @code{printlevel}>=2, all the debug messages will be printed. |
---|
| 1762 | EXAMPLE: example operatorBM; shows examples |
---|
| 1763 | " |
---|
| 1764 | { |
---|
| 1765 | int eng = 0; |
---|
| 1766 | if ( size(#)>0 ) |
---|
| 1767 | { |
---|
| 1768 | if ( typeof(#[1]) == "int" ) |
---|
| 1769 | { |
---|
| 1770 | eng = int(#[1]); |
---|
| 1771 | } |
---|
| 1772 | } |
---|
| 1773 | // returns a list with a ring and an ideal LD in it |
---|
| 1774 | int ppl = printlevel-voice+2; |
---|
| 1775 | // printf("plevel :%s, voice: %s",printlevel,voice); |
---|
| 1776 | def save = basering; |
---|
| 1777 | int N = nvars(basering); |
---|
| 1778 | int Nnew = 2*N+2; |
---|
| 1779 | int i,j; |
---|
| 1780 | string s; |
---|
| 1781 | list RL = ringlist(basering); |
---|
| 1782 | list L, Lord; |
---|
| 1783 | list tmp; |
---|
| 1784 | intvec iv; |
---|
| 1785 | L[1] = RL[1]; //char |
---|
| 1786 | L[4] = RL[4]; //char, minpoly |
---|
| 1787 | // check whether vars have admissible names |
---|
| 1788 | list Name = RL[2]; |
---|
| 1789 | list RName; |
---|
| 1790 | RName[1] = "t"; |
---|
| 1791 | RName[2] = "s"; |
---|
| 1792 | for (i=1; i<=N; i++) |
---|
| 1793 | { |
---|
| 1794 | for(j=1; j<=size(RName); j++) |
---|
| 1795 | { |
---|
| 1796 | if (Name[i] == RName[j]) |
---|
| 1797 | { |
---|
| 1798 | ERROR("Variable names should not include t,s"); |
---|
| 1799 | } |
---|
| 1800 | } |
---|
| 1801 | } |
---|
| 1802 | // now, create the names for new vars |
---|
| 1803 | list DName; |
---|
| 1804 | for (i=1; i<=N; i++) |
---|
| 1805 | { |
---|
| 1806 | DName[i] = "D"+Name[i]; //concat |
---|
| 1807 | } |
---|
| 1808 | tmp[1] = "t"; |
---|
| 1809 | tmp[2] = "s"; |
---|
| 1810 | list NName = tmp + Name + DName; |
---|
| 1811 | L[2] = NName; |
---|
| 1812 | // Name, Dname will be used further |
---|
| 1813 | kill NName; |
---|
| 1814 | // block ord (lp(2),dp); |
---|
| 1815 | tmp[1] = "lp"; // string |
---|
| 1816 | iv = 1,1; |
---|
| 1817 | tmp[2] = iv; //intvec |
---|
| 1818 | Lord[1] = tmp; |
---|
| 1819 | // continue with dp 1,1,1,1... |
---|
| 1820 | tmp[1] = "dp"; // string |
---|
| 1821 | s = "iv="; |
---|
| 1822 | for (i=1; i<=Nnew; i++) |
---|
| 1823 | { |
---|
| 1824 | s = s+"1,"; |
---|
| 1825 | } |
---|
| 1826 | s[size(s)]= ";"; |
---|
| 1827 | execute(s); |
---|
| 1828 | kill s; |
---|
| 1829 | tmp[2] = iv; |
---|
| 1830 | Lord[2] = tmp; |
---|
| 1831 | tmp[1] = "C"; |
---|
| 1832 | iv = 0; |
---|
| 1833 | tmp[2] = iv; |
---|
| 1834 | Lord[3] = tmp; |
---|
| 1835 | tmp = 0; |
---|
| 1836 | L[3] = Lord; |
---|
| 1837 | // we are done with the list |
---|
| 1838 | def @R@ = ring(L); |
---|
| 1839 | setring @R@; |
---|
| 1840 | matrix @D[Nnew][Nnew]; |
---|
| 1841 | @D[1,2]=t; |
---|
| 1842 | for(i=1; i<=N; i++) |
---|
| 1843 | { |
---|
| 1844 | @D[2+i,N+2+i]=1; |
---|
| 1845 | } |
---|
| 1846 | // L[5] = matrix(UpOneMatrix(Nnew)); |
---|
| 1847 | // L[6] = @D; |
---|
| 1848 | def @R = nc_algebra(1,@D); |
---|
| 1849 | setring @R; |
---|
| 1850 | kill @R@; |
---|
| 1851 | dbprint(ppl,"// -1-1- the ring @R(t,s,_x,_Dx) is ready"); |
---|
| 1852 | dbprint(ppl-1, @R); |
---|
| 1853 | // create the ideal I |
---|
| 1854 | poly F = imap(save,F); |
---|
| 1855 | ideal I = t*F+s; |
---|
| 1856 | poly p; |
---|
| 1857 | for(i=1; i<=N; i++) |
---|
| 1858 | { |
---|
| 1859 | p = t; //t |
---|
| 1860 | p = diff(F,var(2+i))*p; |
---|
| 1861 | I = I, var(N+2+i) + p; |
---|
| 1862 | } |
---|
| 1863 | // -------- the ideal I is ready ---------- |
---|
| 1864 | dbprint(ppl,"// -1-2- starting the elimination of t in @R"); |
---|
| 1865 | dbprint(ppl-1, I); |
---|
| 1866 | ideal J = engine(I,eng); |
---|
| 1867 | ideal K = nselect(J,1); |
---|
| 1868 | kill I,J; |
---|
| 1869 | dbprint(ppl,"// -1-3- t is eliminated"); |
---|
| 1870 | dbprint(ppl-1, K); //K is without t |
---|
| 1871 | setring save; |
---|
| 1872 | // ----------- the ring @R2 ------------ |
---|
| 1873 | // _x, _Dx,s; elim.ord for _x,_Dx. |
---|
| 1874 | // keep: N, i,j,s, tmp, RL |
---|
| 1875 | Nnew = 2*N+1; |
---|
| 1876 | kill Lord, tmp, iv, RName; |
---|
| 1877 | list Lord, tmp; |
---|
| 1878 | intvec iv; |
---|
| 1879 | L[1] = RL[1]; |
---|
| 1880 | L[4] = RL[4]; //char, minpoly |
---|
| 1881 | // check whether vars hava admissible names -> done earlier |
---|
| 1882 | // now, create the names for new var |
---|
| 1883 | tmp[1] = "s"; |
---|
| 1884 | // DName is defined earlier |
---|
| 1885 | list NName = Name + DName + tmp; |
---|
| 1886 | L[2] = NName; |
---|
| 1887 | tmp = 0; |
---|
| 1888 | // block ord (dp(N),dp); |
---|
| 1889 | string s = "iv="; |
---|
| 1890 | for (i=1; i<=Nnew-1; i++) |
---|
| 1891 | { |
---|
| 1892 | s = s+"1,"; |
---|
| 1893 | } |
---|
| 1894 | s[size(s)]=";"; |
---|
| 1895 | execute(s); |
---|
| 1896 | tmp[1] = "dp"; //string |
---|
| 1897 | tmp[2] = iv; //intvec |
---|
| 1898 | Lord[1] = tmp; |
---|
| 1899 | // continue with dp 1,1,1,1... |
---|
| 1900 | tmp[1] = "dp"; //string |
---|
| 1901 | s[size(s)] = ","; |
---|
| 1902 | s = s+"1;"; |
---|
| 1903 | execute(s); |
---|
| 1904 | kill s; |
---|
| 1905 | kill NName; |
---|
| 1906 | tmp[2] = iv; |
---|
| 1907 | Lord[2] = tmp; |
---|
| 1908 | tmp[1] = "C"; |
---|
| 1909 | iv = 0; |
---|
| 1910 | tmp[2] = iv; |
---|
| 1911 | Lord[3] = tmp; |
---|
| 1912 | tmp = 0; |
---|
| 1913 | L[3] = Lord; |
---|
| 1914 | // we are done with the list. Now add a Plural part |
---|
| 1915 | def @R2@ = ring(L); |
---|
| 1916 | setring @R2@; |
---|
| 1917 | matrix @D[Nnew][Nnew]; |
---|
| 1918 | for (i=1; i<=N; i++) |
---|
| 1919 | { |
---|
| 1920 | @D[i,N+i]=1; |
---|
| 1921 | } |
---|
| 1922 | def @R2 = nc_algebra(1,@D); |
---|
| 1923 | setring @R2; |
---|
| 1924 | kill @R2@; |
---|
| 1925 | dbprint(ppl,"// -2-1- the ring @R2(_x,_Dx,s) is ready"); |
---|
| 1926 | dbprint(ppl-1, @R2); |
---|
| 1927 | ideal MM = maxideal(1); |
---|
| 1928 | MM = 0,s,MM; |
---|
| 1929 | map R01 = @R, MM; |
---|
| 1930 | ideal K = R01(K); |
---|
| 1931 | poly F = imap(save,F); |
---|
| 1932 | K = K,F; |
---|
| 1933 | dbprint(ppl,"// -2-2- starting the elimination of _x,_Dx in @R2"); |
---|
| 1934 | dbprint(ppl-1, K); |
---|
| 1935 | ideal M = engine(K,eng); |
---|
| 1936 | ideal K2 = nselect(M,1..Nnew-1); |
---|
| 1937 | kill K,M; |
---|
| 1938 | dbprint(ppl,"// -2-3- _x,_Dx are eliminated in @R2"); |
---|
| 1939 | dbprint(ppl-1, K2); |
---|
| 1940 | // the ring @R3 and the search for minimal negative int s |
---|
| 1941 | ring @R3 = 0,s,dp; |
---|
| 1942 | dbprint(ppl,"// -3-1- the ring @R3(s) is ready"); |
---|
| 1943 | ideal K3 = imap(@R2,K2); |
---|
| 1944 | kill @R2; |
---|
| 1945 | poly p = K3[1]; |
---|
| 1946 | dbprint(ppl,"// -3-2- factorization"); |
---|
| 1947 | list P = factorize(p); //with constants and multiplicities |
---|
| 1948 | ideal bs; intvec m; //the Bernstein polynomial is monic, so we are not interested in constants |
---|
| 1949 | for (i=2; i<= size(P[1]); i++) //we delete P[1][1] and P[2][1] |
---|
| 1950 | { |
---|
| 1951 | bs[i-1] = P[1][i]; |
---|
| 1952 | m[i-1] = P[2][i]; |
---|
| 1953 | } |
---|
| 1954 | // "--------- b-function factorizes into ---------"; P; |
---|
| 1955 | int sP = minIntRoot(bs,1); |
---|
| 1956 | dbprint(ppl,"// -3-3- minimal integer root found"); |
---|
| 1957 | dbprint(ppl-1, sP); |
---|
| 1958 | // convert factors to a list of their roots with multiplicities |
---|
| 1959 | bs = normalize(bs); |
---|
| 1960 | bs = -subst(bs,s,0); |
---|
| 1961 | list BS = bs,m; |
---|
| 1962 | //TODO: sort BS! |
---|
| 1963 | // --------- substitute s found in the ideal --------- |
---|
| 1964 | // --------- going back to @R and substitute --------- |
---|
| 1965 | setring @R; |
---|
| 1966 | ideal K2 = subst(K,s,sP); |
---|
| 1967 | // create Dn[s], where Dn is the ordinary Weyl algebra, and put the result into it, |
---|
| 1968 | // thus creating the ring @R4 |
---|
| 1969 | // keep: N, i,j,s, tmp, RL |
---|
| 1970 | setring save; |
---|
| 1971 | Nnew = 2*N+1; |
---|
| 1972 | // list RL = ringlist(save); //is defined earlier |
---|
| 1973 | kill Lord, tmp, iv; |
---|
| 1974 | L = 0; |
---|
| 1975 | list Lord, tmp; |
---|
| 1976 | intvec iv; |
---|
| 1977 | L[1] = RL[1]; |
---|
| 1978 | L[4] = RL[4]; //char, minpoly |
---|
| 1979 | // check whether vars have admissible names -> done earlier |
---|
| 1980 | // list Name = RL[2] |
---|
| 1981 | // DName is defined earlier |
---|
| 1982 | tmp[1] = "s"; |
---|
| 1983 | list NName = Name + DName + tmp; |
---|
| 1984 | L[2] = NName; |
---|
| 1985 | // dp ordering; |
---|
| 1986 | string s = "iv="; |
---|
| 1987 | for (i=1; i<=Nnew; i++) |
---|
| 1988 | { |
---|
| 1989 | s = s+"1,"; |
---|
| 1990 | } |
---|
| 1991 | s[size(s)] = ";"; |
---|
| 1992 | execute(s); |
---|
| 1993 | kill s; |
---|
| 1994 | tmp = 0; |
---|
| 1995 | tmp[1] = "dp"; //string |
---|
| 1996 | tmp[2] = iv; //intvec |
---|
| 1997 | Lord[1] = tmp; |
---|
| 1998 | tmp[1] = "C"; |
---|
| 1999 | iv = 0; |
---|
| 2000 | tmp[2] = iv; |
---|
| 2001 | Lord[2] = tmp; |
---|
| 2002 | tmp = 0; |
---|
| 2003 | L[3] = Lord; |
---|
| 2004 | // we are done with the list |
---|
| 2005 | // Add: Plural part |
---|
| 2006 | def @R4@ = ring(L); |
---|
| 2007 | setring @R4@; |
---|
| 2008 | matrix @D[Nnew][Nnew]; |
---|
| 2009 | for (i=1; i<=N; i++) |
---|
| 2010 | { |
---|
| 2011 | @D[i,N+i]=1; |
---|
| 2012 | } |
---|
| 2013 | def @R4 = nc_algebra(1,@D); |
---|
| 2014 | setring @R4; |
---|
| 2015 | kill @R4@; |
---|
| 2016 | dbprint(ppl,"// -4-1- the ring @R4(_x,_Dx,s) is ready"); |
---|
| 2017 | dbprint(ppl-1, @R4); |
---|
| 2018 | ideal LD0 = imap(@R,K2); |
---|
| 2019 | ideal LD = imap(@R,K); |
---|
| 2020 | kill @R; |
---|
| 2021 | poly bs = imap(@R3,p); |
---|
| 2022 | list BS = imap(@R3,BS); |
---|
| 2023 | kill @R3; |
---|
| 2024 | bs = normalize(bs); |
---|
| 2025 | poly F = imap(save,F); |
---|
| 2026 | dbprint(ppl,"// -4-2- starting the computation of PS via lift"); |
---|
| 2027 | //better liftstd, I didn't knot it works also for Plural, liftslimgb? |
---|
| 2028 | // liftstd may give extra coeffs in the resulting ideal |
---|
| 2029 | matrix T = lift(F+LD,bs); |
---|
| 2030 | poly PS = T[1,1]; |
---|
| 2031 | dbprint(ppl,"// -4-3- an operator PS found, PS*f^(s+1) = b(s)*f^s"); |
---|
| 2032 | dbprint(ppl-1,PS); |
---|
| 2033 | option(redSB); |
---|
| 2034 | dbprint(ppl,"// -4-4- the final cosmetic std"); |
---|
| 2035 | LD0 = engine(LD0,eng); //std does the job too |
---|
| 2036 | LD = engine(LD,eng); |
---|
| 2037 | export F,LD,LD0,bs,BS,PS; |
---|
| 2038 | return(@R4); |
---|
| 2039 | } |
---|
| 2040 | example |
---|
| 2041 | { |
---|
| 2042 | "EXAMPLE:"; echo = 2; |
---|
| 2043 | ring r = 0,(x,y,z),Dp; |
---|
| 2044 | poly F = x^3+y^3+z^3; |
---|
| 2045 | printlevel = 0; |
---|
| 2046 | def A = operatorBM(F); |
---|
| 2047 | setring A; |
---|
| 2048 | F; // the original polynomial itself |
---|
| 2049 | LD; // generic annihilator |
---|
| 2050 | LD0; // annihilator |
---|
| 2051 | bs; // normalized Bernstein poly |
---|
| 2052 | BS; // roots and multiplicities of the Bernstein poly |
---|
| 2053 | PS; // the operator, s.t. PS*F^{s+1} = bs*F^s mod LD |
---|
| 2054 | reduce(PS*F-bs,LD); // check the property of PS |
---|
| 2055 | } |
---|
| 2056 | |
---|
| 2057 | // more interesting: |
---|
| 2058 | // ring r = 0,(x,y,z,w),Dp; |
---|
| 2059 | // poly F = x^3+y^3+z^2*w; |
---|
| 2060 | |
---|
| 2061 | // need: (c,<) ordering for such comp's |
---|
| 2062 | |
---|
| 2063 | proc operatorModulo(poly F, ideal I, poly b) |
---|
| 2064 | "USAGE: operatorModulo(f,I,b); f a poly, I an ideal, b a poly |
---|
| 2065 | RETURN: poly |
---|
[3f4e52] | 2066 | PURPOSE: compute the B-operator from the polynomial f, |
---|
| 2067 | @* ideal I = Ann f^s and Bernstein-Sato polynomial b |
---|
[66c962] | 2068 | @* using modulo i.e. kernel of module homomorphism |
---|
[3f4e52] | 2069 | NOTE: The computations take place in the ring, similar to the one |
---|
[66c962] | 2070 | @* returned by Sannfs procedure. |
---|
[3f4e52] | 2071 | @* Note, that operator is not completely reduced wrt Ann f^{s+1}. |
---|
[66c962] | 2072 | @* If printlevel=1, progress debug messages will be printed, |
---|
| 2073 | @* if printlevel>=2, all the debug messages will be printed. |
---|
| 2074 | EXAMPLE: example operatorModulo; shows examples |
---|
| 2075 | " |
---|
| 2076 | { |
---|
| 2077 | int ppl = printlevel-voice+2; |
---|
| 2078 | def save = basering; |
---|
| 2079 | // change the ordering on the currRing |
---|
| 2080 | def mering = makeModElimRing(save); |
---|
| 2081 | setring mering; |
---|
| 2082 | poly b = imap(save, b); |
---|
| 2083 | poly F = imap(save, F); |
---|
| 2084 | ideal I = imap(save, I); |
---|
| 2085 | matrix N = matrix(I); // ann f^s |
---|
| 2086 | // matrix K = hom_kernel(AA,M,N); |
---|
| 2087 | // option(noreturnSB)? |
---|
| 2088 | /// matrix K = modulo(AA,N); // too slow: do it with slim! |
---|
| 2089 | module M = b,-F; |
---|
| 2090 | dbprint(ppl,"starting modulo computation"); |
---|
| 2091 | module K = moduloSlim(M,N); |
---|
| 2092 | dbprint(ppl,"finished modulo computation"); |
---|
| 2093 | // K = transpose(K); |
---|
| 2094 | // matrix M[3][s+2] = F,-b,I[1..s], 1,0:(s+1),0,1,0:(s); |
---|
| 2095 | // module GM = slimgb(M); |
---|
| 2096 | // module GMT = transpose(G); |
---|
| 2097 | // GMT = GMT[2],GMT[3]; // modulo matrix |
---|
[3f4e52] | 2098 | // module K = GMT[2]; |
---|
[66c962] | 2099 | // GMT = transpose(GMT); |
---|
| 2100 | // K = transpose(K); |
---|
| 2101 | // matrix K = GMT; |
---|
| 2102 | ////////////////////////////////////////////////// |
---|
| 2103 | // now select those elts whose 0's entry is nonzero |
---|
| 2104 | // if there is constant => done |
---|
| 2105 | // if not => compute GB and get it |
---|
| 2106 | module L; |
---|
| 2107 | ideal J; |
---|
| 2108 | int i; |
---|
| 2109 | poly t; number n; |
---|
| 2110 | for(i=1; i<=ncols(K); i++) |
---|
| 2111 | { |
---|
| 2112 | if (K[1,i]!=0) |
---|
| 2113 | { |
---|
| 2114 | L = L,K[i]; |
---|
| 2115 | if ( leadmonom(K[1,i]) == 1) |
---|
| 2116 | { |
---|
| 2117 | t = K[2,i]; |
---|
| 2118 | n = leadcoef(K[1,i]); |
---|
| 2119 | t = t/n; |
---|
| 2120 | break; |
---|
[0610f0e] | 2121 | // return(t); |
---|
[66c962] | 2122 | } |
---|
| 2123 | } |
---|
| 2124 | } |
---|
| 2125 | if (n!=0) |
---|
| 2126 | { |
---|
| 2127 | // constant found |
---|
| 2128 | setring save; poly t = imap(mering,t); kill mering; |
---|
| 2129 | return(t); |
---|
| 2130 | } |
---|
[3f4e52] | 2131 | dbprint(ppl,"no explicit constant. Start one more GB computation"); |
---|
[66c962] | 2132 | // else: compute GB and do the same |
---|
| 2133 | L = L[2..ncols(L)]; |
---|
| 2134 | K = slimgb(L); |
---|
| 2135 | dbprint(ppl,"finished GB computation"); |
---|
| 2136 | for(i=1; i<=ncols(K); i++) |
---|
| 2137 | { |
---|
| 2138 | if (K[1,i]!=0) |
---|
| 2139 | { |
---|
| 2140 | if ( leadmonom(K[1,i]) == 1) |
---|
| 2141 | { |
---|
| 2142 | t = K[2,i]; |
---|
| 2143 | n = leadcoef(K[1,i]); |
---|
| 2144 | t = t/n; |
---|
| 2145 | // break; |
---|
| 2146 | setring save; poly t = imap(mering,t); kill mering; |
---|
| 2147 | return(t); |
---|
| 2148 | } |
---|
| 2149 | } |
---|
| 2150 | } |
---|
| 2151 | |
---|
| 2152 | // we are here if no constant found |
---|
| 2153 | "ERROR: must never get here!"; |
---|
| 2154 | return(0); |
---|
| 2155 | // for(i=1; i<=nrows(K); i++) |
---|
| 2156 | // { |
---|
| 2157 | // if (K[i,2]!=0) |
---|
| 2158 | // { |
---|
| 2159 | // if ( leadmonom(K[i,2]) == 1) |
---|
| 2160 | // { |
---|
| 2161 | // t = K[i,1]; |
---|
| 2162 | // n = leadcoef(K[i,2]); |
---|
| 2163 | // t = t/n; |
---|
| 2164 | // // J = J, K[i][2]; |
---|
| 2165 | // break; |
---|
| 2166 | // } |
---|
| 2167 | // } |
---|
| 2168 | // } |
---|
| 2169 | // ideal J = groebner(subst(I,s,s+1)); // for NF |
---|
| 2170 | // t = NF(t,J); |
---|
| 2171 | // "candidate:"; t; |
---|
| 2172 | // J = subst(J,s,s-1); |
---|
| 2173 | // // test: |
---|
| 2174 | // if ( NF(t*F-b,J) !=0) |
---|
| 2175 | // { |
---|
| 2176 | // "Problem: PS does not work on F"; |
---|
| 2177 | // } |
---|
| 2178 | // return(t); |
---|
| 2179 | } |
---|
| 2180 | example |
---|
| 2181 | { |
---|
| 2182 | "EXAMPLE:"; echo = 2; |
---|
| 2183 | // LIB "dmod.lib"; option(prot); option(mem); |
---|
| 2184 | ring r = 0,(x,y),Dp; |
---|
| 2185 | poly F = x^3+y^3+x*y^3; |
---|
| 2186 | def A = Sannfs(F); // here we get LD = ann f^s |
---|
| 2187 | setring A; |
---|
| 2188 | poly F = imap(r,F); |
---|
| 2189 | def B = annfs0(LD,F); // to obtain BS polynomial |
---|
| 2190 | list BS = imap(B,BS); poly bs = fl2poly(BS,"s"); |
---|
| 2191 | poly PS = operatorModulo(F,LD,bs); |
---|
| 2192 | LD = groebner(LD); |
---|
| 2193 | PS = NF(PS,subst(LD,s,s+1));; // reduction modulo Ann s^{s+1} |
---|
[3f4e52] | 2194 | size(PS); |
---|
[66c962] | 2195 | lead(PS); |
---|
| 2196 | reduce(PS*F-bs,LD); // check the defining property of PS |
---|
| 2197 | } |
---|
| 2198 | |
---|
| 2199 | proc annfsParamBM (poly F, list #) |
---|
| 2200 | "USAGE: annfsParamBM(f [,eng]); f a poly, eng an optional int |
---|
| 2201 | RETURN: ring |
---|
| 2202 | PURPOSE: compute the generic Ann F^s and exceptional parametric constellations |
---|
| 2203 | @* of a polynomial with parametric coefficients with the BM algorithm |
---|
| 2204 | NOTE: activate the output ring with the @code{setring} command. In this ring, |
---|
| 2205 | @* - the ideal LD is the D-module structure oa Ann F^s |
---|
| 2206 | @* - the ideal Param contains special parameters as entries |
---|
| 2207 | @* If eng <>0, @code{std} is used for Groebner basis computations, |
---|
| 2208 | @* otherwise, and by default @code{slimgb} is used. |
---|
| 2209 | DISPLAY: If @code{printlevel}=1, progress debug messages will be printed, |
---|
| 2210 | @* if @code{printlevel}>=2, all the debug messages will be printed. |
---|
| 2211 | EXAMPLE: example annfsParamBM; shows examples |
---|
| 2212 | " |
---|
| 2213 | { |
---|
| 2214 | //PURPOSE: compute the list of all possible Bernstein-Sato polynomials for a polynomial with parametric coefficients, according to the algorithm by Briancon and Maisonobe |
---|
| 2215 | // @* - the list BS is the list of roots and multiplicities of a Bernstein polynomial of f. |
---|
| 2216 | // ***** not implented yet **** |
---|
| 2217 | int eng = 0; |
---|
| 2218 | if ( size(#)>0 ) |
---|
| 2219 | { |
---|
| 2220 | if ( typeof(#[1]) == "int" ) |
---|
| 2221 | { |
---|
| 2222 | eng = int(#[1]); |
---|
| 2223 | } |
---|
| 2224 | } |
---|
| 2225 | // returns a list with a ring and an ideal LD in it |
---|
| 2226 | int ppl = printlevel-voice+2; |
---|
| 2227 | // printf("plevel :%s, voice: %s",printlevel,voice); |
---|
| 2228 | def save = basering; |
---|
| 2229 | int N = nvars(basering); |
---|
| 2230 | int Nnew = 2*N+2; |
---|
| 2231 | int i,j; |
---|
| 2232 | string s; |
---|
| 2233 | list RL = ringlist(basering); |
---|
| 2234 | list L, Lord; |
---|
| 2235 | list tmp; |
---|
| 2236 | intvec iv; |
---|
| 2237 | L[1] = RL[1]; //char |
---|
| 2238 | L[4] = RL[4]; //char, minpoly |
---|
| 2239 | // check whether vars have admissible names |
---|
| 2240 | list Name = RL[2]; |
---|
| 2241 | list RName; |
---|
| 2242 | RName[1] = "t"; |
---|
| 2243 | RName[2] = "s"; |
---|
| 2244 | for (i=1; i<=N; i++) |
---|
| 2245 | { |
---|
| 2246 | for(j=1; j<=size(RName); j++) |
---|
| 2247 | { |
---|
| 2248 | if (Name[i] == RName[j]) |
---|
| 2249 | { |
---|
| 2250 | ERROR("Variable names should not include t,s"); |
---|
| 2251 | } |
---|
| 2252 | } |
---|
| 2253 | } |
---|
| 2254 | // now, create the names for new vars |
---|
| 2255 | list DName; |
---|
| 2256 | for (i=1; i<=N; i++) |
---|
| 2257 | { |
---|
| 2258 | DName[i] = "D"+Name[i]; //concat |
---|
| 2259 | } |
---|
| 2260 | tmp[1] = "t"; |
---|
| 2261 | tmp[2] = "s"; |
---|
| 2262 | list NName = tmp + Name + DName; |
---|
| 2263 | L[2] = NName; |
---|
| 2264 | // Name, Dname will be used further |
---|
| 2265 | kill NName; |
---|
| 2266 | // block ord (lp(2),dp); |
---|
| 2267 | tmp[1] = "lp"; // string |
---|
| 2268 | iv = 1,1; |
---|
| 2269 | tmp[2] = iv; //intvec |
---|
| 2270 | Lord[1] = tmp; |
---|
| 2271 | // continue with dp 1,1,1,1... |
---|
| 2272 | tmp[1] = "dp"; // string |
---|
| 2273 | s = "iv="; |
---|
| 2274 | for (i=1; i<=Nnew; i++) |
---|
| 2275 | { |
---|
| 2276 | s = s+"1,"; |
---|
| 2277 | } |
---|
| 2278 | s[size(s)]= ";"; |
---|
| 2279 | execute(s); |
---|
| 2280 | kill s; |
---|
| 2281 | tmp[2] = iv; |
---|
| 2282 | Lord[2] = tmp; |
---|
| 2283 | tmp[1] = "C"; |
---|
| 2284 | iv = 0; |
---|
| 2285 | tmp[2] = iv; |
---|
| 2286 | Lord[3] = tmp; |
---|
| 2287 | tmp = 0; |
---|
| 2288 | L[3] = Lord; |
---|
| 2289 | // we are done with the list |
---|
| 2290 | def @R@ = ring(L); |
---|
| 2291 | setring @R@; |
---|
| 2292 | matrix @D[Nnew][Nnew]; |
---|
| 2293 | @D[1,2]=t; |
---|
| 2294 | for(i=1; i<=N; i++) |
---|
| 2295 | { |
---|
| 2296 | @D[2+i,N+2+i]=1; |
---|
| 2297 | } |
---|
| 2298 | // L[5] = matrix(UpOneMatrix(Nnew)); |
---|
| 2299 | // L[6] = @D; |
---|
| 2300 | def @R = nc_algebra(1,@D); |
---|
| 2301 | setring @R; |
---|
| 2302 | kill @R@; |
---|
| 2303 | dbprint(ppl,"// -1-1- the ring @R(t,s,_x,_Dx) is ready"); |
---|
| 2304 | dbprint(ppl-1, @R); |
---|
| 2305 | // create the ideal I |
---|
| 2306 | poly F = imap(save,F); |
---|
| 2307 | ideal I = t*F+s; |
---|
| 2308 | poly p; |
---|
| 2309 | for(i=1; i<=N; i++) |
---|
| 2310 | { |
---|
| 2311 | p = t; //t |
---|
| 2312 | p = diff(F,var(2+i))*p; |
---|
| 2313 | I = I, var(N+2+i) + p; |
---|
| 2314 | } |
---|
| 2315 | // -------- the ideal I is ready ---------- |
---|
| 2316 | dbprint(ppl,"// -1-2- starting the elimination of t in @R"); |
---|
| 2317 | dbprint(ppl-1, I); |
---|
| 2318 | ideal J = engine(I,eng); |
---|
| 2319 | ideal K = nselect(J,1); |
---|
| 2320 | dbprint(ppl,"// -1-3- t is eliminated"); |
---|
| 2321 | dbprint(ppl-1, K); //K is without t |
---|
| 2322 | // ----- looking for special parameters ----- |
---|
| 2323 | dbprint(ppl,"// -2-1- starting the computation of the transformation matrix (via lift)"); |
---|
| 2324 | J = normalize(J); |
---|
| 2325 | matrix T = lift(I,J); //try also with liftstd |
---|
| 2326 | kill I,J; |
---|
| 2327 | dbprint(ppl,"// -2-2- the transformation matrix has been computed"); |
---|
| 2328 | dbprint(ppl-1, T); //T is the transformation matrix |
---|
| 2329 | dbprint(ppl,"// -2-3- genericity does the job"); |
---|
| 2330 | list lParam = genericity(T); |
---|
| 2331 | int ip = size(lParam); |
---|
| 2332 | int cip; |
---|
| 2333 | string sParam; |
---|
| 2334 | if (sParam[1]=="-") { sParam=""; } //genericity returns "-" |
---|
| 2335 | // if no parameters exist in a basering |
---|
| 2336 | for (cip=1; cip <= ip; cip++) |
---|
| 2337 | { |
---|
| 2338 | sParam = sParam + "," +lParam[cip]; |
---|
| 2339 | } |
---|
| 2340 | if (size(sParam) >=2) |
---|
| 2341 | { |
---|
| 2342 | sParam = sParam[2..size(sParam)]; // removes the 1st colon |
---|
| 2343 | } |
---|
| 2344 | export sParam; |
---|
| 2345 | kill T; |
---|
| 2346 | dbprint(ppl,"// -2-4- the special parameters has been computed"); |
---|
| 2347 | dbprint(ppl, sParam); |
---|
| 2348 | // create Dn[s], where Dn is the ordinary Weyl Algebra, and put the result into it, |
---|
| 2349 | // thus creating the ring @R2 |
---|
| 2350 | // keep: N, i,j,s, tmp, RL |
---|
| 2351 | setring save; |
---|
| 2352 | Nnew = 2*N+1; |
---|
| 2353 | // list RL = ringlist(save); //is defined earlier |
---|
| 2354 | kill Lord, tmp, iv; |
---|
| 2355 | L = 0; |
---|
| 2356 | list Lord, tmp; |
---|
| 2357 | intvec iv; |
---|
| 2358 | L[1] = RL[1]; |
---|
| 2359 | L[4] = RL[4]; //char, minpoly |
---|
| 2360 | // check whether vars have admissible names -> done earlier |
---|
| 2361 | // list Name = RL[2]M |
---|
| 2362 | // DName is defined earlier |
---|
| 2363 | tmp[1] = "s"; |
---|
| 2364 | list NName = Name + DName + tmp; |
---|
| 2365 | L[2] = NName; |
---|
| 2366 | // dp ordering; |
---|
| 2367 | string s = "iv="; |
---|
| 2368 | for (i=1; i<=Nnew; i++) |
---|
| 2369 | { |
---|
| 2370 | s = s+"1,"; |
---|
| 2371 | } |
---|
| 2372 | s[size(s)] = ";"; |
---|
| 2373 | execute(s); |
---|
| 2374 | kill s; |
---|
| 2375 | tmp = 0; |
---|
| 2376 | tmp[1] = "dp"; //string |
---|
| 2377 | tmp[2] = iv; //intvec |
---|
| 2378 | Lord[1] = tmp; |
---|
| 2379 | tmp[1] = "C"; |
---|
| 2380 | iv = 0; |
---|
| 2381 | tmp[2] = iv; |
---|
| 2382 | Lord[2] = tmp; |
---|
| 2383 | tmp = 0; |
---|
| 2384 | L[3] = Lord; |
---|
| 2385 | // we are done with the list |
---|
| 2386 | // Add: Plural part |
---|
| 2387 | def @R2@ = ring(L); |
---|
| 2388 | setring @R2@; |
---|
| 2389 | matrix @D[Nnew][Nnew]; |
---|
| 2390 | for (i=1; i<=N; i++) |
---|
| 2391 | { |
---|
| 2392 | @D[i,N+i]=1; |
---|
| 2393 | } |
---|
| 2394 | def @R2 = nc_algebra(1,@D); |
---|
| 2395 | setring @R2; |
---|
| 2396 | kill @R2@; |
---|
| 2397 | dbprint(ppl,"// -3-1- the ring @R2(_x,_Dx,s) is ready"); |
---|
| 2398 | dbprint(ppl-1, @R2); |
---|
| 2399 | ideal K = imap(@R,K); |
---|
| 2400 | kill @R; |
---|
| 2401 | option(redSB); |
---|
| 2402 | dbprint(ppl,"// -3-2- the final cosmetic std"); |
---|
| 2403 | K = engine(K,eng); //std does the job too |
---|
| 2404 | ideal LD = K; |
---|
| 2405 | export LD; |
---|
| 2406 | if (sParam[1] == ",") |
---|
| 2407 | { |
---|
| 2408 | sParam = sParam[2..size(sParam)]; |
---|
| 2409 | } |
---|
| 2410 | // || ((sParam[1] == " ") && (sParam[2] == ","))) |
---|
| 2411 | execute("ideal Param ="+sParam+";"); |
---|
| 2412 | export Param; |
---|
| 2413 | kill sParam; |
---|
| 2414 | return(@R2); |
---|
| 2415 | } |
---|
| 2416 | example |
---|
| 2417 | { |
---|
| 2418 | "EXAMPLE:"; echo = 2; |
---|
| 2419 | ring r = (0,a,b),(x,y),Dp; |
---|
| 2420 | poly F = x^2 - (y-a)*(y-b); |
---|
| 2421 | printlevel = 0; |
---|
| 2422 | def A = annfsParamBM(F); setring A; |
---|
| 2423 | LD; |
---|
| 2424 | Param; |
---|
| 2425 | setring r; |
---|
| 2426 | poly G = x2-(y-a)^2; // try the exceptional value b=a of parameters |
---|
| 2427 | def B = annfsParamBM(G); setring B; |
---|
| 2428 | LD; |
---|
| 2429 | Param; |
---|
| 2430 | } |
---|
| 2431 | |
---|
| 2432 | // *** the following example is nice, but too complicated for the documentation *** |
---|
| 2433 | // ring r = (0,a),(x,y,z),Dp; |
---|
| 2434 | // poly F = x^4+y^4+z^2+a*x*y*z; |
---|
| 2435 | // printlevel = 2; //0 |
---|
| 2436 | // def A = annfsParamBM(F); |
---|
| 2437 | // setring A; |
---|
| 2438 | // LD; |
---|
| 2439 | // Param; |
---|
| 2440 | |
---|
| 2441 | |
---|
| 2442 | proc annfsBMI(ideal F, list #) |
---|
| 2443 | "USAGE: annfsBMI(F [,eng]); F an ideal, eng an optional int |
---|
| 2444 | RETURN: ring |
---|
[3f4e52] | 2445 | PURPOSE: compute the D-module structure of basering[1/f]*f^s where |
---|
[66c962] | 2446 | @* f = F[1]*..*F[P], according to the algorithm by Briancon and Maisonobe. |
---|
| 2447 | NOTE: activate the output ring with the @code{setring} command. In this ring, |
---|
| 2448 | @* - the ideal LD is the needed D-mod structure, |
---|
| 2449 | @* - the list BS is the Bernstein ideal of a polynomial f = F[1]*..*F[P]. |
---|
| 2450 | @* If eng <>0, @code{std} is used for Groebner basis computations, |
---|
| 2451 | @* otherwise, and by default @code{slimgb} is used. |
---|
| 2452 | @* If printlevel=1, progress debug messages will be printed, |
---|
| 2453 | @* if printlevel>=2, all the debug messages will be printed. |
---|
| 2454 | EXAMPLE: example annfsBMI; shows examples |
---|
| 2455 | " |
---|
| 2456 | { |
---|
| 2457 | int eng = 0; |
---|
| 2458 | if ( size(#)>0 ) |
---|
| 2459 | { |
---|
| 2460 | if ( typeof(#[1]) == "int" ) |
---|
| 2461 | { |
---|
| 2462 | eng = int(#[1]); |
---|
| 2463 | } |
---|
| 2464 | } |
---|
| 2465 | // returns a list with a ring and an ideal LD in it |
---|
| 2466 | int ppl = printlevel-voice+2; |
---|
| 2467 | // printf("plevel :%s, voice: %s",printlevel,voice); |
---|
| 2468 | def save = basering; |
---|
| 2469 | int N = nvars(basering); |
---|
| 2470 | int P = size(F); //if F has some generators which are zero, int P = ncols(I); |
---|
| 2471 | int Nnew = 2*N+2*P; |
---|
| 2472 | int i,j; |
---|
| 2473 | string s; |
---|
| 2474 | list RL = ringlist(basering); |
---|
| 2475 | list L, Lord; |
---|
| 2476 | list tmp; |
---|
| 2477 | intvec iv; |
---|
| 2478 | L[1] = RL[1]; //char |
---|
| 2479 | L[4] = RL[4]; //char, minpoly |
---|
| 2480 | // check whether vars have admissible names |
---|
| 2481 | list Name = RL[2]; |
---|
| 2482 | list RName; |
---|
| 2483 | for (j=1; j<=P; j++) |
---|
| 2484 | { |
---|
| 2485 | RName[j] = "t("+string(j)+")"; |
---|
| 2486 | RName[j+P] = "s("+string(j)+")"; |
---|
| 2487 | } |
---|
| 2488 | for(i=1; i<=N; i++) |
---|
| 2489 | { |
---|
| 2490 | for(j=1; j<=size(RName); j++) |
---|
| 2491 | { |
---|
| 2492 | if (Name[i] == RName[j]) |
---|
| 2493 | { ERROR("Variable names should not include t(i),s(i)"); } |
---|
| 2494 | } |
---|
| 2495 | } |
---|
| 2496 | // now, create the names for new vars |
---|
| 2497 | list DName; |
---|
| 2498 | for(i=1; i<=N; i++) |
---|
| 2499 | { |
---|
| 2500 | DName[i] = "D"+Name[i]; //concat |
---|
| 2501 | } |
---|
| 2502 | list NName = RName + Name + DName; |
---|
| 2503 | L[2] = NName; |
---|
| 2504 | // Name, Dname will be used further |
---|
| 2505 | kill NName; |
---|
| 2506 | // block ord (lp(P),dp); |
---|
| 2507 | tmp[1] = "lp"; //string |
---|
| 2508 | s = "iv="; |
---|
| 2509 | for (i=1; i<=2*P; i++) |
---|
| 2510 | { |
---|
| 2511 | s = s+"1,"; |
---|
| 2512 | } |
---|
| 2513 | s[size(s)]= ";"; |
---|
| 2514 | execute(s); |
---|
| 2515 | tmp[2] = iv; //intvec |
---|
| 2516 | Lord[1] = tmp; |
---|
| 2517 | // continue with dp 1,1,1,1... |
---|
| 2518 | tmp[1] = "dp"; //string |
---|
| 2519 | s = "iv="; |
---|
| 2520 | for (i=1; i<=Nnew; i++) //actually i<=2*N |
---|
| 2521 | { |
---|
| 2522 | s = s+"1,"; |
---|
| 2523 | } |
---|
| 2524 | s[size(s)]= ";"; |
---|
| 2525 | execute(s); |
---|
| 2526 | kill s; |
---|
| 2527 | tmp[2] = iv; |
---|
| 2528 | Lord[2] = tmp; |
---|
| 2529 | tmp[1] = "C"; |
---|
| 2530 | iv = 0; |
---|
| 2531 | tmp[2] = iv; |
---|
| 2532 | Lord[3] = tmp; |
---|
| 2533 | tmp = 0; |
---|
| 2534 | L[3] = Lord; |
---|
| 2535 | // we are done with the list |
---|
| 2536 | def @R@ = ring(L); |
---|
| 2537 | setring @R@; |
---|
| 2538 | matrix @D[Nnew][Nnew]; |
---|
| 2539 | for (i=1; i<=P; i++) |
---|
| 2540 | { |
---|
| 2541 | @D[i,i+P] = t(i); |
---|
| 2542 | } |
---|
| 2543 | for(i=1; i<=N; i++) |
---|
| 2544 | { |
---|
| 2545 | @D[2*P+i,2*P+N+i] = 1; |
---|
| 2546 | } |
---|
| 2547 | // L[5] = matrix(UpOneMatrix(Nnew)); |
---|
| 2548 | // L[6] = @D; |
---|
| 2549 | def @R = nc_algebra(1,@D); |
---|
| 2550 | setring @R; |
---|
| 2551 | kill @R@; |
---|
| 2552 | dbprint(ppl,"// -1-1- the ring @R(_t,_s,_x,_Dx) is ready"); |
---|
| 2553 | dbprint(ppl-1, @R); |
---|
| 2554 | // create the ideal I |
---|
| 2555 | ideal F = imap(save,F); |
---|
| 2556 | ideal I = t(1)*F[1]+s(1); |
---|
| 2557 | for (j=2; j<=P; j++) |
---|
| 2558 | { |
---|
| 2559 | I = I, t(j)*F[j]+s(j); |
---|
| 2560 | } |
---|
| 2561 | poly p,q; |
---|
| 2562 | for (i=1; i<=N; i++) |
---|
| 2563 | { |
---|
| 2564 | p=0; |
---|
| 2565 | for (j=1; j<=P; j++) |
---|
| 2566 | { |
---|
| 2567 | q = t(j); |
---|
| 2568 | q = diff(F[j],var(2*P+i))*q; |
---|
| 2569 | p = p + q; |
---|
| 2570 | } |
---|
| 2571 | I = I, var(2*P+N+i) + p; |
---|
| 2572 | } |
---|
| 2573 | // -------- the ideal I is ready ---------- |
---|
| 2574 | dbprint(ppl,"// -1-2- starting the elimination of "+string(t(1..P))+" in @R"); |
---|
| 2575 | dbprint(ppl-1, I); |
---|
| 2576 | ideal J = engine(I,eng); |
---|
| 2577 | ideal K = nselect(J,1..P); |
---|
| 2578 | kill I,J; |
---|
| 2579 | dbprint(ppl,"// -1-3- all t(i) are eliminated"); |
---|
| 2580 | dbprint(ppl-1, K); //K is without t(i) |
---|
| 2581 | // ----------- the ring @R2 ------------ |
---|
| 2582 | // _x, _Dx,s; elim.ord for _x,_Dx. |
---|
| 2583 | // keep: N, i,j,s, tmp, RL |
---|
| 2584 | setring save; |
---|
| 2585 | Nnew = 2*N+P; |
---|
| 2586 | kill Lord, tmp, iv, RName; |
---|
| 2587 | list Lord, tmp; |
---|
| 2588 | intvec iv; |
---|
| 2589 | L[1] = RL[1]; //char |
---|
| 2590 | L[4] = RL[4]; //char, minpoly |
---|
| 2591 | // check whether vars hava admissible names -> done earlier |
---|
| 2592 | // now, create the names for new var |
---|
| 2593 | for (j=1; j<=P; j++) |
---|
| 2594 | { |
---|
| 2595 | tmp[j] = "s("+string(j)+")"; |
---|
| 2596 | } |
---|
| 2597 | // DName is defined earlier |
---|
| 2598 | list NName = Name + DName + tmp; |
---|
| 2599 | L[2] = NName; |
---|
| 2600 | tmp = 0; |
---|
| 2601 | // block ord (dp(N),dp); |
---|
| 2602 | string s = "iv="; |
---|
| 2603 | for (i=1; i<=Nnew-P; i++) |
---|
| 2604 | { |
---|
| 2605 | s = s+"1,"; |
---|
| 2606 | } |
---|
| 2607 | s[size(s)]=";"; |
---|
| 2608 | execute(s); |
---|
| 2609 | tmp[1] = "dp"; //string |
---|
| 2610 | tmp[2] = iv; //intvec |
---|
| 2611 | Lord[1] = tmp; |
---|
| 2612 | // continue with dp 1,1,1,1... |
---|
| 2613 | tmp[1] = "dp"; //string |
---|
| 2614 | s[size(s)] = ","; |
---|
| 2615 | for (j=1; j<=P; j++) |
---|
| 2616 | { |
---|
| 2617 | s = s+"1,"; |
---|
| 2618 | } |
---|
| 2619 | s[size(s)]=";"; |
---|
| 2620 | execute(s); |
---|
| 2621 | kill s; |
---|
| 2622 | kill NName; |
---|
| 2623 | tmp[2] = iv; |
---|
| 2624 | Lord[2] = tmp; |
---|
| 2625 | tmp[1] = "C"; |
---|
| 2626 | iv = 0; |
---|
| 2627 | tmp[2] = iv; |
---|
| 2628 | Lord[3] = tmp; |
---|
| 2629 | tmp = 0; |
---|
| 2630 | L[3] = Lord; |
---|
| 2631 | // we are done with the list. Now add a Plural part |
---|
| 2632 | def @R2@ = ring(L); |
---|
| 2633 | setring @R2@; |
---|
| 2634 | matrix @D[Nnew][Nnew]; |
---|
| 2635 | for (i=1; i<=N; i++) |
---|
| 2636 | { |
---|
| 2637 | @D[i,N+i]=1; |
---|
| 2638 | } |
---|
| 2639 | def @R2 = nc_algebra(1,@D); |
---|
| 2640 | setring @R2; |
---|
| 2641 | kill @R2@; |
---|
| 2642 | dbprint(ppl,"// -2-1- the ring @R2(_x,_Dx,_s) is ready"); |
---|
| 2643 | dbprint(ppl-1, @R2); |
---|
| 2644 | // ideal MM = maxideal(1); |
---|
| 2645 | // MM = 0,s,MM; |
---|
| 2646 | // map R01 = @R, MM; |
---|
| 2647 | // ideal K = R01(K); |
---|
| 2648 | ideal F = imap(save,F); // maybe ideal F = R01(I); ? |
---|
| 2649 | ideal K = imap(@R,K); // maybe ideal K = R01(I); ? |
---|
| 2650 | poly f=1; |
---|
| 2651 | for (j=1; j<=P; j++) |
---|
| 2652 | { |
---|
| 2653 | f = f*F[j]; |
---|
| 2654 | } |
---|
| 2655 | K = K,f; // to compute B (Bernstein-Sato ideal) |
---|
| 2656 | //j=2; // for example |
---|
| 2657 | //K = K,F[j]; // to compute Bj (see "On the computation of Bernstein-Sato ideals"; Castro, Ucha) |
---|
| 2658 | //K = K,F; // to compute Bsigma (see "On the computation of Bernstein-Sato ideals"; Castro, Ucha) |
---|
| 2659 | dbprint(ppl,"// -2-2- starting the elimination of _x,_Dx in @R2"); |
---|
| 2660 | dbprint(ppl-1, K); |
---|
| 2661 | ideal M = engine(K,eng); |
---|
| 2662 | ideal K2 = nselect(M,1..Nnew-P); |
---|
| 2663 | kill K,M; |
---|
| 2664 | dbprint(ppl,"// -2-3- _x,_Dx are eliminated in @R2"); |
---|
| 2665 | dbprint(ppl-1, K2); |
---|
| 2666 | // the ring @R3 and factorize |
---|
| 2667 | ring @R3 = 0,s(1..P),dp; |
---|
| 2668 | dbprint(ppl,"// -3-1- the ring @R3(_s) is ready"); |
---|
| 2669 | ideal K3 = imap(@R2,K2); |
---|
| 2670 | if (size(K3)==1) |
---|
| 2671 | { |
---|
| 2672 | poly p = K3[1]; |
---|
| 2673 | dbprint(ppl,"// -3-2- factorization"); |
---|
| 2674 | // Warning: now P is an integer |
---|
| 2675 | list Q = factorize(p); //with constants and multiplicities |
---|
| 2676 | ideal bs; intvec m; |
---|
| 2677 | for (i=2; i<=size(Q[1]); i++) //we delete Q[1][1] and Q[2][1] |
---|
| 2678 | { |
---|
| 2679 | bs[i-1] = Q[1][i]; |
---|
| 2680 | m[i-1] = Q[2][i]; |
---|
| 2681 | } |
---|
| 2682 | // "--------- Q-ideal factorizes into ---------"; list(bs,m); |
---|
| 2683 | list BS = bs,m; |
---|
| 2684 | } |
---|
| 2685 | else |
---|
| 2686 | { |
---|
| 2687 | // conjecture: the Bernstein ideal is principal |
---|
| 2688 | dbprint(ppl,"// -3-2- the Bernstein ideal is not principal"); |
---|
| 2689 | ideal BS = K3; |
---|
| 2690 | } |
---|
| 2691 | // create the ring @R4(_x,_Dx,_s) and put the result into it, |
---|
| 2692 | // _x, _Dx,s; ord "dp". |
---|
| 2693 | // keep: N, i,j,s, tmp, RL |
---|
| 2694 | setring save; |
---|
| 2695 | Nnew = 2*N+P; |
---|
| 2696 | // list RL = ringlist(save); //is defined earlier |
---|
| 2697 | kill Lord, tmp, iv; |
---|
| 2698 | L = 0; |
---|
| 2699 | list Lord, tmp; |
---|
| 2700 | intvec iv; |
---|
| 2701 | L[1] = RL[1]; //char |
---|
| 2702 | L[4] = RL[4]; //char, minpoly |
---|
| 2703 | // check whether vars hava admissible names -> done earlier |
---|
| 2704 | // now, create the names for new var |
---|
| 2705 | for (j=1; j<=P; j++) |
---|
| 2706 | { |
---|
| 2707 | tmp[j] = "s("+string(j)+")"; |
---|
| 2708 | } |
---|
| 2709 | // DName is defined earlier |
---|
| 2710 | list NName = Name + DName + tmp; |
---|
| 2711 | L[2] = NName; |
---|
| 2712 | tmp = 0; |
---|
| 2713 | // dp ordering; |
---|
| 2714 | string s = "iv="; |
---|
| 2715 | for (i=1; i<=Nnew; i++) |
---|
| 2716 | { |
---|
| 2717 | s = s+"1,"; |
---|
| 2718 | } |
---|
| 2719 | s[size(s)]=";"; |
---|
| 2720 | execute(s); |
---|
| 2721 | kill s; |
---|
| 2722 | kill NName; |
---|
| 2723 | tmp[1] = "dp"; //string |
---|
| 2724 | tmp[2] = iv; //intvec |
---|
| 2725 | Lord[1] = tmp; |
---|
| 2726 | tmp[1] = "C"; |
---|
| 2727 | iv = 0; |
---|
| 2728 | tmp[2] = iv; |
---|
| 2729 | Lord[2] = tmp; |
---|
| 2730 | tmp = 0; |
---|
| 2731 | L[3] = Lord; |
---|
| 2732 | // we are done with the list. Now add a Plural part |
---|
| 2733 | def @R4@ = ring(L); |
---|
| 2734 | setring @R4@; |
---|
| 2735 | matrix @D[Nnew][Nnew]; |
---|
| 2736 | for (i=1; i<=N; i++) |
---|
| 2737 | { |
---|
| 2738 | @D[i,N+i]=1; |
---|
| 2739 | } |
---|
| 2740 | def @R4 = nc_algebra(1,@D); |
---|
| 2741 | setring @R4; |
---|
| 2742 | kill @R4@; |
---|
| 2743 | dbprint(ppl,"// -4-1- the ring @R4i(_x,_Dx,_s) is ready"); |
---|
| 2744 | dbprint(ppl-1, @R4); |
---|
| 2745 | ideal K4 = imap(@R,K); |
---|
| 2746 | option(redSB); |
---|
| 2747 | dbprint(ppl,"// -4-2- the final cosmetic std"); |
---|
| 2748 | K4 = engine(K4,eng); //std does the job too |
---|
| 2749 | // total cleanup |
---|
| 2750 | kill @R; |
---|
| 2751 | kill @R2; |
---|
| 2752 | def BS = imap(@R3,BS); |
---|
| 2753 | export BS; |
---|
| 2754 | kill @R3; |
---|
| 2755 | ideal LD = K4; |
---|
| 2756 | export LD; |
---|
| 2757 | return(@R4); |
---|
| 2758 | } |
---|
| 2759 | example |
---|
| 2760 | { |
---|
| 2761 | "EXAMPLE:"; echo = 2; |
---|
| 2762 | ring r = 0,(x,y),Dp; |
---|
| 2763 | ideal F = x,y,x+y; |
---|
| 2764 | printlevel = 0; |
---|
| 2765 | def A = annfsBMI(F); |
---|
| 2766 | setring A; |
---|
| 2767 | LD; |
---|
| 2768 | BS; |
---|
| 2769 | } |
---|
| 2770 | |
---|
| 2771 | proc annfsOT(poly F, list #) |
---|
| 2772 | "USAGE: annfsOT(f [,eng]); f a poly, eng an optional int |
---|
| 2773 | RETURN: ring |
---|
[3f4e52] | 2774 | PURPOSE: compute the D-module structure of basering[1/f]*f^s, |
---|
[66c962] | 2775 | @* according to the algorithm by Oaku and Takayama |
---|
| 2776 | NOTE: activate the output ring with the @code{setring} command. In this ring, |
---|
| 2777 | @* - the ideal LD (which is a Groebner basis) is the needed D-module structure, |
---|
| 2778 | @* which is obtained by substituting the minimal integer root of a Bernstein |
---|
| 2779 | @* polynomial into the s-parametric ideal; |
---|
| 2780 | @* - the list BS contains roots with multiplicities of a Bernstein polynomial of f. |
---|
| 2781 | @* If eng <>0, @code{std} is used for Groebner basis computations, |
---|
| 2782 | @* otherwise, and by default @code{slimgb} is used. |
---|
| 2783 | @* If printlevel=1, progress debug messages will be printed, |
---|
| 2784 | @* if printlevel>=2, all the debug messages will be printed. |
---|
| 2785 | EXAMPLE: example annfsOT; shows examples |
---|
| 2786 | " |
---|
| 2787 | { |
---|
| 2788 | int eng = 0; |
---|
| 2789 | if ( size(#)>0 ) |
---|
| 2790 | { |
---|
| 2791 | if ( typeof(#[1]) == "int" ) |
---|
| 2792 | { |
---|
| 2793 | eng = int(#[1]); |
---|
| 2794 | } |
---|
| 2795 | } |
---|
| 2796 | // returns a list with a ring and an ideal LD in it |
---|
| 2797 | int ppl = printlevel-voice+2; |
---|
| 2798 | // printf("plevel :%s, voice: %s",printlevel,voice); |
---|
| 2799 | def save = basering; |
---|
| 2800 | int N = nvars(basering); |
---|
| 2801 | int Nnew = 2*(N+2); |
---|
| 2802 | int i,j; |
---|
| 2803 | string s; |
---|
| 2804 | list RL = ringlist(basering); |
---|
| 2805 | list L, Lord; |
---|
| 2806 | list tmp; |
---|
| 2807 | intvec iv; |
---|
| 2808 | L[1] = RL[1]; // char |
---|
| 2809 | L[4] = RL[4]; // char, minpoly |
---|
| 2810 | // check whether vars have admissible names |
---|
| 2811 | list Name = RL[2]; |
---|
| 2812 | list RName; |
---|
| 2813 | RName[1] = "u"; |
---|
| 2814 | RName[2] = "v"; |
---|
| 2815 | RName[3] = "t"; |
---|
| 2816 | RName[4] = "Dt"; |
---|
| 2817 | for(i=1;i<=N;i++) |
---|
| 2818 | { |
---|
| 2819 | for(j=1; j<=size(RName);j++) |
---|
| 2820 | { |
---|
| 2821 | if (Name[i] == RName[j]) |
---|
| 2822 | { |
---|
| 2823 | ERROR("Variable names should not include u,v,t,Dt"); |
---|
| 2824 | } |
---|
| 2825 | } |
---|
| 2826 | } |
---|
| 2827 | // now, create the names for new vars |
---|
| 2828 | tmp[1] = "u"; |
---|
| 2829 | tmp[2] = "v"; |
---|
| 2830 | list UName = tmp; |
---|
| 2831 | list DName; |
---|
| 2832 | for(i=1;i<=N;i++) |
---|
| 2833 | { |
---|
| 2834 | DName[i] = "D"+Name[i]; // concat |
---|
| 2835 | } |
---|
| 2836 | tmp = 0; |
---|
| 2837 | tmp[1] = "t"; |
---|
| 2838 | tmp[2] = "Dt"; |
---|
| 2839 | list NName = UName + tmp + Name + DName; |
---|
| 2840 | L[2] = NName; |
---|
| 2841 | tmp = 0; |
---|
| 2842 | // Name, Dname will be used further |
---|
| 2843 | kill UName; |
---|
| 2844 | kill NName; |
---|
| 2845 | // block ord (a(1,1),dp); |
---|
| 2846 | tmp[1] = "a"; // string |
---|
| 2847 | iv = 1,1; |
---|
| 2848 | tmp[2] = iv; //intvec |
---|
| 2849 | Lord[1] = tmp; |
---|
| 2850 | // continue with dp 1,1,1,1... |
---|
| 2851 | tmp[1] = "dp"; // string |
---|
| 2852 | s = "iv="; |
---|
| 2853 | for(i=1;i<=Nnew;i++) |
---|
| 2854 | { |
---|
| 2855 | s = s+"1,"; |
---|
| 2856 | } |
---|
| 2857 | s[size(s)]= ";"; |
---|
| 2858 | execute(s); |
---|
| 2859 | tmp[2] = iv; |
---|
| 2860 | Lord[2] = tmp; |
---|
| 2861 | tmp[1] = "C"; |
---|
| 2862 | iv = 0; |
---|
| 2863 | tmp[2] = iv; |
---|
| 2864 | Lord[3] = tmp; |
---|
| 2865 | tmp = 0; |
---|
| 2866 | L[3] = Lord; |
---|
| 2867 | // we are done with the list |
---|
| 2868 | def @R@ = ring(L); |
---|
| 2869 | setring @R@; |
---|
| 2870 | matrix @D[Nnew][Nnew]; |
---|
| 2871 | @D[3,4]=1; |
---|
| 2872 | for(i=1; i<=N; i++) |
---|
| 2873 | { |
---|
| 2874 | @D[4+i,N+4+i]=1; |
---|
| 2875 | } |
---|
| 2876 | // @D[N+3,2*(N+2)]=1; old t,Dt stuff |
---|
| 2877 | // L[5] = matrix(UpOneMatrix(Nnew)); |
---|
| 2878 | // L[6] = @D; |
---|
| 2879 | def @R = nc_algebra(1,@D); |
---|
| 2880 | setring @R; |
---|
| 2881 | kill @R@; |
---|
| 2882 | dbprint(ppl,"// -1-1- the ring @R(u,v,t,Dt,_x,_Dx) is ready"); |
---|
| 2883 | dbprint(ppl-1, @R); |
---|
| 2884 | // create the ideal I |
---|
| 2885 | poly F = imap(save,F); |
---|
| 2886 | ideal I = u*F-t,u*v-1; |
---|
| 2887 | poly p; |
---|
| 2888 | for(i=1; i<=N; i++) |
---|
| 2889 | { |
---|
| 2890 | p = u*Dt; // u*Dt |
---|
| 2891 | p = diff(F,var(4+i))*p; |
---|
| 2892 | I = I, var(N+4+i) + p; |
---|
| 2893 | } |
---|
| 2894 | // -------- the ideal I is ready ---------- |
---|
| 2895 | dbprint(ppl,"// -1-2- starting the elimination of u,v in @R"); |
---|
| 2896 | dbprint(ppl-1, I); |
---|
| 2897 | ideal J = engine(I,eng); |
---|
| 2898 | ideal K = nselect(J,1..2); |
---|
| 2899 | dbprint(ppl,"// -1-3- u,v are eliminated"); |
---|
| 2900 | dbprint(ppl-1, K); // K is without u,v |
---|
| 2901 | setring save; |
---|
| 2902 | // ------------ new ring @R2 ------------------ |
---|
| 2903 | // without u,v and with the elim.ord for t,Dt |
---|
| 2904 | // tensored with the K[s] |
---|
| 2905 | // keep: N, i,j,s, tmp, RL |
---|
| 2906 | Nnew = 2*N+2+1; |
---|
| 2907 | // list RL = ringlist(save); // is defined earlier |
---|
| 2908 | L = 0; // kill L; |
---|
| 2909 | kill Lord, tmp, iv, RName; |
---|
| 2910 | list Lord, tmp; |
---|
| 2911 | intvec iv; |
---|
| 2912 | L[1] = RL[1]; L[4] = RL[4]; // char, minpoly |
---|
| 2913 | // check whether vars have admissible names -> done earlier |
---|
| 2914 | // list Name = RL[2]; |
---|
| 2915 | list RName; |
---|
| 2916 | RName[1] = "t"; |
---|
| 2917 | RName[2] = "Dt"; |
---|
| 2918 | // now, create the names for new var (here, s only) |
---|
| 2919 | tmp[1] = "s"; |
---|
| 2920 | // DName is defined earlier |
---|
| 2921 | list NName = RName + Name + DName + tmp; |
---|
| 2922 | L[2] = NName; |
---|
| 2923 | tmp = 0; |
---|
| 2924 | // block ord (a(1,1),dp); |
---|
| 2925 | tmp[1] = "a"; iv = 1,1; tmp[2] = iv; //intvec |
---|
| 2926 | Lord[1] = tmp; |
---|
| 2927 | // continue with a(1,1,1,1)... |
---|
| 2928 | tmp[1] = "dp"; s = "iv="; |
---|
| 2929 | for(i=1; i<= Nnew; i++) |
---|
| 2930 | { |
---|
| 2931 | s = s+"1,"; |
---|
| 2932 | } |
---|
| 2933 | s[size(s)]= ";"; execute(s); |
---|
| 2934 | kill NName; |
---|
| 2935 | tmp[2] = iv; |
---|
| 2936 | Lord[2] = tmp; |
---|
| 2937 | // extra block for s |
---|
| 2938 | // tmp[1] = "dp"; iv = 1; |
---|
| 2939 | // s[size(s)]= ","; s = s + "1,1,1;"; execute(s); tmp[2] = iv; |
---|
| 2940 | // Lord[3] = tmp; |
---|
| 2941 | kill s; |
---|
| 2942 | tmp[1] = "C"; iv = 0; tmp[2] = iv; |
---|
| 2943 | Lord[3] = tmp; tmp = 0; |
---|
| 2944 | L[3] = Lord; |
---|
| 2945 | // we are done with the list. Now, add a Plural part |
---|
| 2946 | def @R2@ = ring(L); |
---|
| 2947 | setring @R2@; |
---|
| 2948 | matrix @D[Nnew][Nnew]; |
---|
| 2949 | @D[1,2] = 1; |
---|
| 2950 | for(i=1; i<=N; i++) |
---|
| 2951 | { |
---|
| 2952 | @D[2+i,2+N+i] = 1; |
---|
| 2953 | } |
---|
| 2954 | def @R2 = nc_algebra(1,@D); |
---|
| 2955 | setring @R2; |
---|
| 2956 | kill @R2@; |
---|
| 2957 | dbprint(ppl,"// -2-1- the ring @R2(t,Dt,_x,_Dx,s) is ready"); |
---|
| 2958 | dbprint(ppl-1, @R2); |
---|
| 2959 | ideal MM = maxideal(1); |
---|
| 2960 | MM = 0,0,MM; |
---|
| 2961 | map R01 = @R, MM; |
---|
| 2962 | ideal K = R01(K); |
---|
| 2963 | // ideal K = imap(@R,K); // names of vars are important! |
---|
| 2964 | poly G = t*Dt+s+1; // s is a variable here |
---|
| 2965 | K = NF(K,std(G)),G; |
---|
| 2966 | // -------- the ideal K_(@R2) is ready ---------- |
---|
| 2967 | dbprint(ppl,"// -2-2- starting the elimination of t,Dt in @R2"); |
---|
| 2968 | dbprint(ppl-1, K); |
---|
| 2969 | ideal M = engine(K,eng); |
---|
| 2970 | ideal K2 = nselect(M,1..2); |
---|
| 2971 | dbprint(ppl,"// -2-3- t,Dt are eliminated"); |
---|
| 2972 | dbprint(ppl-1, K2); |
---|
| 2973 | // dbprint(ppl-1+1," -2-4- std of K2"); |
---|
| 2974 | // option(redSB); option(redTail); K2 = std(K2); |
---|
| 2975 | // K2; // without t,Dt, and with s |
---|
| 2976 | // -------- the ring @R3 ---------- |
---|
| 2977 | // _x, _Dx, s; elim.ord for _x,_Dx. |
---|
| 2978 | // keep: N, i,j,s, tmp, RL |
---|
| 2979 | setring save; |
---|
| 2980 | Nnew = 2*N+1; |
---|
| 2981 | // list RL = ringlist(save); // is defined earlier |
---|
| 2982 | // kill L; |
---|
| 2983 | kill Lord, tmp, iv, RName; |
---|
| 2984 | list Lord, tmp; |
---|
| 2985 | intvec iv; |
---|
| 2986 | L[1] = RL[1]; L[4] = RL[4]; // char, minpoly |
---|
| 2987 | // check whether vars have admissible names -> done earlier |
---|
| 2988 | // list Name = RL[2]; |
---|
| 2989 | // now, create the names for new var (here, s only) |
---|
| 2990 | tmp[1] = "s"; |
---|
| 2991 | // DName is defined earlier |
---|
| 2992 | list NName = Name + DName + tmp; |
---|
| 2993 | L[2] = NName; |
---|
| 2994 | tmp = 0; |
---|
| 2995 | // block ord (a(1,1...),dp); |
---|
| 2996 | string s = "iv="; |
---|
| 2997 | for(i=1; i<=Nnew-1; i++) |
---|
| 2998 | { |
---|
| 2999 | s = s+"1,"; |
---|
| 3000 | } |
---|
| 3001 | s[size(s)]= ";"; |
---|
| 3002 | execute(s); |
---|
| 3003 | tmp[1] = "a"; // string |
---|
| 3004 | tmp[2] = iv; //intvec |
---|
| 3005 | Lord[1] = tmp; |
---|
| 3006 | // continue with dp 1,1,1,1... |
---|
| 3007 | tmp[1] = "dp"; // string |
---|
| 3008 | s[size(s)]=","; s= s+"1;"; |
---|
| 3009 | execute(s); |
---|
| 3010 | kill s; |
---|
| 3011 | kill NName; |
---|
| 3012 | tmp[2] = iv; |
---|
| 3013 | Lord[2] = tmp; |
---|
| 3014 | tmp[1] = "C"; iv = 0; tmp[2] = iv; |
---|
| 3015 | Lord[3] = tmp; tmp = 0; |
---|
| 3016 | L[3] = Lord; |
---|
| 3017 | // we are done with the list. Now add a Plural part |
---|
| 3018 | def @R3@ = ring(L); |
---|
| 3019 | setring @R3@; |
---|
| 3020 | matrix @D[Nnew][Nnew]; |
---|
| 3021 | for(i=1; i<=N; i++) |
---|
| 3022 | { |
---|
| 3023 | @D[i,N+i]=1; |
---|
| 3024 | } |
---|
| 3025 | def @R3 = nc_algebra(1,@D); |
---|
| 3026 | setring @R3; |
---|
| 3027 | kill @R3@; |
---|
| 3028 | dbprint(ppl,"// -3-1- the ring @R3(_x,_Dx,s) is ready"); |
---|
| 3029 | dbprint(ppl-1, @R3); |
---|
| 3030 | ideal MM = maxideal(1); |
---|
| 3031 | MM = 0,0,MM; |
---|
| 3032 | map R12 = @R2, MM; |
---|
| 3033 | ideal K = R12(K2); |
---|
| 3034 | poly F = imap(save,F); |
---|
| 3035 | K = K,F; |
---|
| 3036 | dbprint(ppl,"// -3-2- starting the elimination of _x,_Dx in @R3"); |
---|
| 3037 | dbprint(ppl-1, K); |
---|
| 3038 | ideal M = engine(K,eng); |
---|
| 3039 | ideal K3 = nselect(M,1..Nnew-1); |
---|
| 3040 | dbprint(ppl,"// -3-3- _x,_Dx are eliminated in @R3"); |
---|
| 3041 | dbprint(ppl-1, K3); |
---|
| 3042 | // the ring @R4 and the search for minimal negative int s |
---|
| 3043 | ring @R4 = 0,(s),dp; |
---|
| 3044 | dbprint(ppl,"// -4-1- the ring @R4 is ready"); |
---|
| 3045 | ideal K4 = imap(@R3,K3); |
---|
| 3046 | poly p = K4[1]; |
---|
| 3047 | dbprint(ppl,"// -4-2- factorization"); |
---|
| 3048 | //// ideal P = factorize(p,1); // without constants and multiplicities |
---|
| 3049 | list P = factorize(p); // with constants and multiplicities |
---|
| 3050 | ideal bs; intvec m; // the Bernstein polynomial is monic, so we are not interested in constants |
---|
| 3051 | for (i=2; i<=size(P[1]); i++) // we delete P[1][1] and P[2][1] |
---|
| 3052 | { |
---|
| 3053 | bs[i-1] = P[1][i]; |
---|
| 3054 | m[i-1] = P[2][i]; |
---|
| 3055 | } |
---|
| 3056 | // "------ b-function factorizes into ----------"; P; |
---|
| 3057 | //// int sP = minIntRoot(P, 1); |
---|
| 3058 | int sP = minIntRoot(bs,1); |
---|
| 3059 | dbprint(ppl,"// -4-3- minimal integer root found"); |
---|
| 3060 | dbprint(ppl-1, sP); |
---|
| 3061 | // convert factors to a list of their roots |
---|
| 3062 | // assume all factors are linear |
---|
| 3063 | //// ideal BS = normalize(P); |
---|
| 3064 | //// BS = subst(BS,s,0); |
---|
| 3065 | //// BS = -BS; |
---|
| 3066 | bs = normalize(bs); |
---|
| 3067 | bs = subst(bs,s,0); |
---|
| 3068 | bs = -bs; |
---|
| 3069 | list BS = bs,m; |
---|
| 3070 | // TODO: sort BS! |
---|
| 3071 | // ------ substitute s found in the ideal ------ |
---|
| 3072 | // ------- going back to @R2 and substitute -------- |
---|
| 3073 | setring @R2; |
---|
| 3074 | ideal K3 = subst(K2,s,sP); |
---|
| 3075 | // create the ordinary Weyl algebra and put the result into it, |
---|
| 3076 | // thus creating the ring @R5 |
---|
| 3077 | // keep: N, i,j,s, tmp, RL |
---|
| 3078 | setring save; |
---|
| 3079 | Nnew = 2*N; |
---|
| 3080 | // list RL = ringlist(save); // is defined earlier |
---|
| 3081 | kill Lord, tmp, iv; |
---|
| 3082 | L = 0; |
---|
| 3083 | list Lord, tmp; |
---|
| 3084 | intvec iv; |
---|
| 3085 | L[1] = RL[1]; L[4] = RL[4]; // char, minpoly |
---|
| 3086 | // check whether vars have admissible names -> done earlier |
---|
| 3087 | // list Name = RL[2]; |
---|
| 3088 | // DName is defined earlier |
---|
| 3089 | list NName = Name + DName; |
---|
| 3090 | L[2] = NName; |
---|
| 3091 | // dp ordering; |
---|
| 3092 | string s = "iv="; |
---|
| 3093 | for(i=1;i<=Nnew;i++) |
---|
| 3094 | { |
---|
| 3095 | s = s+"1,"; |
---|
| 3096 | } |
---|
| 3097 | s[size(s)]= ";"; |
---|
| 3098 | execute(s); |
---|
| 3099 | tmp = 0; |
---|
| 3100 | tmp[1] = "dp"; // string |
---|
| 3101 | tmp[2] = iv; //intvec |
---|
| 3102 | Lord[1] = tmp; |
---|
| 3103 | kill s; |
---|
| 3104 | tmp[1] = "C"; |
---|
| 3105 | iv = 0; |
---|
| 3106 | tmp[2] = iv; |
---|
| 3107 | Lord[2] = tmp; |
---|
| 3108 | tmp = 0; |
---|
| 3109 | L[3] = Lord; |
---|
| 3110 | // we are done with the list |
---|
| 3111 | // Add: Plural part |
---|
| 3112 | def @R5@ = ring(L); |
---|
| 3113 | setring @R5@; |
---|
| 3114 | matrix @D[Nnew][Nnew]; |
---|
| 3115 | for(i=1; i<=N; i++) |
---|
| 3116 | { |
---|
| 3117 | @D[i,N+i]=1; |
---|
| 3118 | } |
---|
| 3119 | def @R5 = nc_algebra(1,@D); |
---|
| 3120 | setring @R5; |
---|
| 3121 | kill @R5@; |
---|
| 3122 | dbprint(ppl,"// -5-1- the ring @R5 is ready"); |
---|
| 3123 | dbprint(ppl-1, @R5); |
---|
| 3124 | ideal K5 = imap(@R2,K3); |
---|
| 3125 | option(redSB); |
---|
| 3126 | dbprint(ppl,"// -5-2- the final cosmetic std"); |
---|
| 3127 | K5 = engine(K5,eng); // std does the job too |
---|
| 3128 | // total cleanup |
---|
| 3129 | kill @R; |
---|
| 3130 | kill @R2; |
---|
| 3131 | kill @R3; |
---|
| 3132 | //// ideal BS = imap(@R4,BS); |
---|
| 3133 | list BS = imap(@R4,BS); |
---|
| 3134 | export BS; |
---|
| 3135 | ideal LD = K5; |
---|
| 3136 | kill @R4; |
---|
| 3137 | export LD; |
---|
| 3138 | return(@R5); |
---|
| 3139 | } |
---|
| 3140 | example |
---|
| 3141 | { |
---|
| 3142 | "EXAMPLE:"; echo = 2; |
---|
| 3143 | ring r = 0,(x,y,z),Dp; |
---|
| 3144 | poly F = x^2+y^3+z^5; |
---|
| 3145 | printlevel = 0; |
---|
| 3146 | def A = annfsOT(F); |
---|
| 3147 | setring A; |
---|
| 3148 | LD; |
---|
| 3149 | BS; |
---|
| 3150 | } |
---|
| 3151 | |
---|
| 3152 | |
---|
| 3153 | proc SannfsOT(poly F, list #) |
---|
| 3154 | "USAGE: SannfsOT(f [,eng]); f a poly, eng an optional int |
---|
| 3155 | RETURN: ring |
---|
[3f4e52] | 3156 | PURPOSE: compute the D-module structure of basering[1/f]*f^s, according to the |
---|
[66c962] | 3157 | @* 1st step of the algorithm by Oaku and Takayama in the ring D[s] |
---|
| 3158 | NOTE: activate the output ring with the @code{setring} command. |
---|
[3f4e52] | 3159 | @* In the output ring D[s], the ideal LD (which is NOT a Groebner basis) |
---|
[66c962] | 3160 | @* is the needed D-module structure. |
---|
| 3161 | @* If eng <>0, @code{std} is used for Groebner basis computations, |
---|
| 3162 | @* otherwise, and by default @code{slimgb} is used. |
---|
| 3163 | @* If printlevel=1, progress debug messages will be printed, |
---|
| 3164 | @* if printlevel>=2, all the debug messages will be printed. |
---|
| 3165 | EXAMPLE: example SannfsOT; shows examples |
---|
| 3166 | " |
---|
| 3167 | { |
---|
| 3168 | int eng = 0; |
---|
| 3169 | if ( size(#)>0 ) |
---|
| 3170 | { |
---|
| 3171 | if ( typeof(#[1]) == "int" ) |
---|
| 3172 | { |
---|
| 3173 | eng = int(#[1]); |
---|
| 3174 | } |
---|
| 3175 | } |
---|
| 3176 | // returns a list with a ring and an ideal LD in it |
---|
| 3177 | int ppl = printlevel-voice+2; |
---|
| 3178 | // printf("plevel :%s, voice: %s",printlevel,voice); |
---|
| 3179 | def save = basering; |
---|
| 3180 | int N = nvars(basering); |
---|
| 3181 | int Nnew = 2*(N+2); |
---|
| 3182 | int i,j; |
---|
| 3183 | string s; |
---|
| 3184 | list RL = ringlist(basering); |
---|
| 3185 | list L, Lord; |
---|
| 3186 | list tmp; |
---|
| 3187 | intvec iv; |
---|
| 3188 | L[1] = RL[1]; // char |
---|
| 3189 | L[4] = RL[4]; // char, minpoly |
---|
| 3190 | // check whether vars have admissible names |
---|
| 3191 | list Name = RL[2]; |
---|
| 3192 | list RName; |
---|
| 3193 | RName[1] = "u"; |
---|
| 3194 | RName[2] = "v"; |
---|
| 3195 | RName[3] = "t"; |
---|
| 3196 | RName[4] = "Dt"; |
---|
| 3197 | for(i=1;i<=N;i++) |
---|
| 3198 | { |
---|
| 3199 | for(j=1; j<=size(RName);j++) |
---|
| 3200 | { |
---|
| 3201 | if (Name[i] == RName[j]) |
---|
| 3202 | { |
---|
| 3203 | ERROR("Variable names should not include u,v,t,Dt"); |
---|
| 3204 | } |
---|
| 3205 | } |
---|
| 3206 | } |
---|
| 3207 | // now, create the names for new vars |
---|
| 3208 | tmp[1] = "u"; |
---|
| 3209 | tmp[2] = "v"; |
---|
| 3210 | list UName = tmp; |
---|
| 3211 | list DName; |
---|
| 3212 | for(i=1;i<=N;i++) |
---|
| 3213 | { |
---|
| 3214 | DName[i] = "D"+Name[i]; // concat |
---|
| 3215 | } |
---|
| 3216 | tmp = 0; |
---|
| 3217 | tmp[1] = "t"; |
---|
| 3218 | tmp[2] = "Dt"; |
---|
| 3219 | list NName = UName + tmp + Name + DName; |
---|
| 3220 | L[2] = NName; |
---|
| 3221 | tmp = 0; |
---|
| 3222 | // Name, Dname will be used further |
---|
| 3223 | kill UName; |
---|
| 3224 | kill NName; |
---|
| 3225 | // block ord (a(1,1),dp); |
---|
| 3226 | tmp[1] = "a"; // string |
---|
| 3227 | iv = 1,1; |
---|
| 3228 | tmp[2] = iv; //intvec |
---|
| 3229 | Lord[1] = tmp; |
---|
| 3230 | // continue with dp 1,1,1,1... |
---|
| 3231 | tmp[1] = "dp"; // string |
---|
| 3232 | s = "iv="; |
---|
| 3233 | for(i=1;i<=Nnew;i++) |
---|
| 3234 | { |
---|
| 3235 | s = s+"1,"; |
---|
| 3236 | } |
---|
| 3237 | s[size(s)]= ";"; |
---|
| 3238 | execute(s); |
---|
| 3239 | tmp[2] = iv; |
---|
| 3240 | Lord[2] = tmp; |
---|
| 3241 | tmp[1] = "C"; |
---|
| 3242 | iv = 0; |
---|
| 3243 | tmp[2] = iv; |
---|
| 3244 | Lord[3] = tmp; |
---|
| 3245 | tmp = 0; |
---|
| 3246 | L[3] = Lord; |
---|
| 3247 | // we are done with the list |
---|
| 3248 | def @R@ = ring(L); |
---|
| 3249 | setring @R@; |
---|
| 3250 | matrix @D[Nnew][Nnew]; |
---|
| 3251 | @D[3,4]=1; |
---|
| 3252 | for(i=1; i<=N; i++) |
---|
| 3253 | { |
---|
| 3254 | @D[4+i,N+4+i]=1; |
---|
| 3255 | } |
---|
| 3256 | // @D[N+3,2*(N+2)]=1; old t,Dt stuff |
---|
| 3257 | // L[5] = matrix(UpOneMatrix(Nnew)); |
---|
| 3258 | // L[6] = @D; |
---|
| 3259 | def @R = nc_algebra(1,@D); |
---|
| 3260 | setring @R; |
---|
| 3261 | kill @R@; |
---|
| 3262 | dbprint(ppl,"// -1-1- the ring @R(u,v,t,Dt,_x,_Dx) is ready"); |
---|
| 3263 | dbprint(ppl-1, @R); |
---|
| 3264 | // create the ideal I |
---|
| 3265 | poly F = imap(save,F); |
---|
| 3266 | ideal I = u*F-t,u*v-1; |
---|
| 3267 | poly p; |
---|
| 3268 | for(i=1; i<=N; i++) |
---|
| 3269 | { |
---|
| 3270 | p = u*Dt; // u*Dt |
---|
| 3271 | p = diff(F,var(4+i))*p; |
---|
| 3272 | I = I, var(N+4+i) + p; |
---|
| 3273 | } |
---|
| 3274 | // -------- the ideal I is ready ---------- |
---|
| 3275 | dbprint(ppl,"// -1-2- starting the elimination of u,v in @R"); |
---|
| 3276 | dbprint(ppl-1, I); |
---|
| 3277 | ideal J = engine(I,eng); |
---|
| 3278 | ideal K = nselect(J,1..2); |
---|
| 3279 | dbprint(ppl,"// -1-3- u,v are eliminated"); |
---|
| 3280 | dbprint(ppl-1, K); // K is without u,v |
---|
| 3281 | setring save; |
---|
| 3282 | // ------------ new ring @R2 ------------------ |
---|
| 3283 | // without u,v and with the elim.ord for t,Dt |
---|
| 3284 | // tensored with the K[s] |
---|
| 3285 | // keep: N, i,j,s, tmp, RL |
---|
| 3286 | Nnew = 2*N+2+1; |
---|
| 3287 | // list RL = ringlist(save); // is defined earlier |
---|
| 3288 | L = 0; // kill L; |
---|
| 3289 | kill Lord, tmp, iv, RName; |
---|
| 3290 | list Lord, tmp; |
---|
| 3291 | intvec iv; |
---|
| 3292 | L[1] = RL[1]; L[4] = RL[4]; // char, minpoly |
---|
| 3293 | // check whether vars have admissible names -> done earlier |
---|
| 3294 | // list Name = RL[2]; |
---|
| 3295 | list RName; |
---|
| 3296 | RName[1] = "t"; |
---|
| 3297 | RName[2] = "Dt"; |
---|
| 3298 | // now, create the names for new var (here, s only) |
---|
| 3299 | tmp[1] = "s"; |
---|
| 3300 | // DName is defined earlier |
---|
| 3301 | list NName = RName + Name + DName + tmp; |
---|
| 3302 | L[2] = NName; |
---|
| 3303 | tmp = 0; |
---|
| 3304 | // block ord (a(1,1),dp); |
---|
| 3305 | tmp[1] = "a"; iv = 1,1; tmp[2] = iv; //intvec |
---|
| 3306 | Lord[1] = tmp; |
---|
| 3307 | // continue with a(1,1,1,1)... |
---|
| 3308 | tmp[1] = "dp"; s = "iv="; |
---|
| 3309 | for(i=1; i<= Nnew; i++) |
---|
| 3310 | { |
---|
| 3311 | s = s+"1,"; |
---|
| 3312 | } |
---|
| 3313 | s[size(s)]= ";"; execute(s); |
---|
| 3314 | kill NName; |
---|
| 3315 | tmp[2] = iv; |
---|
| 3316 | Lord[2] = tmp; |
---|
| 3317 | // extra block for s |
---|
| 3318 | // tmp[1] = "dp"; iv = 1; |
---|
| 3319 | // s[size(s)]= ","; s = s + "1,1,1;"; execute(s); tmp[2] = iv; |
---|
| 3320 | // Lord[3] = tmp; |
---|
| 3321 | kill s; |
---|
| 3322 | tmp[1] = "C"; iv = 0; tmp[2] = iv; |
---|
| 3323 | Lord[3] = tmp; tmp = 0; |
---|
| 3324 | L[3] = Lord; |
---|
| 3325 | // we are done with the list. Now, add a Plural part |
---|
| 3326 | def @R2@ = ring(L); |
---|
| 3327 | setring @R2@; |
---|
| 3328 | matrix @D[Nnew][Nnew]; |
---|
| 3329 | @D[1,2] = 1; |
---|
| 3330 | for(i=1; i<=N; i++) |
---|
| 3331 | { |
---|
| 3332 | @D[2+i,2+N+i] = 1; |
---|
| 3333 | } |
---|
| 3334 | def @R2 = nc_algebra(1,@D); |
---|
| 3335 | setring @R2; |
---|
| 3336 | kill @R2@; |
---|
| 3337 | dbprint(ppl,"// -2-1- the ring @R2(t,Dt,_x,_Dx,s) is ready"); |
---|
| 3338 | dbprint(ppl-1, @R2); |
---|
| 3339 | ideal MM = maxideal(1); |
---|
| 3340 | MM = 0,0,MM; |
---|
| 3341 | map R01 = @R, MM; |
---|
| 3342 | ideal K = R01(K); |
---|
| 3343 | // ideal K = imap(@R,K); // names of vars are important! |
---|
| 3344 | poly G = t*Dt+s+1; // s is a variable here |
---|
| 3345 | K = NF(K,std(G)),G; |
---|
| 3346 | // -------- the ideal K_(@R2) is ready ---------- |
---|
| 3347 | dbprint(ppl,"// -2-2- starting the elimination of t,Dt in @R2"); |
---|
| 3348 | dbprint(ppl-1, K); |
---|
| 3349 | ideal M = engine(K,eng); |
---|
| 3350 | ideal K2 = nselect(M,1..2); |
---|
| 3351 | dbprint(ppl,"// -2-3- t,Dt are eliminated"); |
---|
| 3352 | dbprint(ppl-1, K2); |
---|
| 3353 | K2 = engine(K2,eng); |
---|
| 3354 | setring save; |
---|
| 3355 | // ----------- the ring @R3 ------------ |
---|
| 3356 | // _x, _Dx,s; elim.ord for _x,_Dx. |
---|
| 3357 | // keep: N, i,j,s, tmp, RL |
---|
| 3358 | Nnew = 2*N+1; |
---|
| 3359 | kill Lord, tmp, iv, RName; |
---|
| 3360 | list Lord, tmp; |
---|
| 3361 | intvec iv; |
---|
| 3362 | L[1] = RL[1]; |
---|
| 3363 | L[4] = RL[4]; // char, minpoly |
---|
| 3364 | // check whether vars hava admissible names -> done earlier |
---|
| 3365 | // now, create the names for new var |
---|
| 3366 | tmp[1] = "s"; |
---|
| 3367 | // DName is defined earlier |
---|
| 3368 | list NName = Name + DName + tmp; |
---|
| 3369 | L[2] = NName; |
---|
| 3370 | tmp = 0; |
---|
| 3371 | // block ord (dp(N),dp); |
---|
| 3372 | string s = "iv="; |
---|
| 3373 | for (i=1; i<=Nnew-1; i++) |
---|
| 3374 | { |
---|
| 3375 | s = s+"1,"; |
---|
| 3376 | } |
---|
| 3377 | s[size(s)]=";"; |
---|
| 3378 | execute(s); |
---|
| 3379 | tmp[1] = "dp"; // string |
---|
| 3380 | tmp[2] = iv; // intvec |
---|
| 3381 | Lord[1] = tmp; |
---|
| 3382 | // continue with dp 1,1,1,1... |
---|
| 3383 | tmp[1] = "dp"; // string |
---|
| 3384 | s[size(s)] = ","; |
---|
| 3385 | s = s+"1;"; |
---|
| 3386 | execute(s); |
---|
| 3387 | kill s; |
---|
| 3388 | kill NName; |
---|
| 3389 | tmp[2] = iv; |
---|
| 3390 | Lord[2] = tmp; |
---|
| 3391 | tmp[1] = "C"; iv = 0; tmp[2]=iv; |
---|
| 3392 | Lord[3] = tmp; tmp = 0; |
---|
| 3393 | L[3] = Lord; |
---|
| 3394 | // we are done with the list. Now add a Plural part |
---|
| 3395 | def @R3@ = ring(L); |
---|
| 3396 | setring @R3@; |
---|
| 3397 | matrix @D[Nnew][Nnew]; |
---|
| 3398 | for (i=1; i<=N; i++) |
---|
| 3399 | { |
---|
| 3400 | @D[i,N+i]=1; |
---|
| 3401 | } |
---|
| 3402 | def @R3 = nc_algebra(1,@D); |
---|
| 3403 | setring @R3; |
---|
| 3404 | kill @R3@; |
---|
| 3405 | dbprint(ppl,"// -3-1- the ring @R3(_x,_Dx,s) is ready"); |
---|
| 3406 | dbprint(ppl-1, @R3); |
---|
| 3407 | ideal MM = maxideal(1); |
---|
| 3408 | MM = 0,s,MM; |
---|
| 3409 | map R01 = @R2, MM; |
---|
| 3410 | ideal K2 = R01(K2); |
---|
| 3411 | // total cleanup |
---|
| 3412 | ideal LD = K2; |
---|
| 3413 | // make leadcoeffs positive |
---|
| 3414 | for (i=1; i<= ncols(LD); i++) |
---|
| 3415 | { |
---|
| 3416 | if (leadcoef(LD[i]) <0 ) |
---|
| 3417 | { |
---|
| 3418 | LD[i] = -LD[i]; |
---|
| 3419 | } |
---|
| 3420 | } |
---|
| 3421 | export LD; |
---|
| 3422 | kill @R; |
---|
| 3423 | kill @R2; |
---|
| 3424 | return(@R3); |
---|
| 3425 | } |
---|
| 3426 | example |
---|
| 3427 | { |
---|
| 3428 | "EXAMPLE:"; echo = 2; |
---|
| 3429 | ring r = 0,(x,y,z),Dp; |
---|
| 3430 | poly F = x^3+y^3+z^3; |
---|
| 3431 | printlevel = 0; |
---|
| 3432 | def A = SannfsOT(F); |
---|
| 3433 | setring A; |
---|
| 3434 | LD; |
---|
| 3435 | } |
---|
| 3436 | |
---|
| 3437 | proc SannfsBM(poly F, list #) |
---|
| 3438 | "USAGE: SannfsBM(f [,eng]); f a poly, eng an optional int |
---|
| 3439 | RETURN: ring |
---|
[3f4e52] | 3440 | PURPOSE: compute the D-module structure of basering[1/f]*f^s, according to the |
---|
[66c962] | 3441 | @* 1st step of the algorithm by Briancon and Maisonobe in the ring D[s]. |
---|
| 3442 | NOTE: activate the output ring with the @code{setring} command. |
---|
[3f4e52] | 3443 | @* In the output ring D[s], the ideal LD (which is NOT a Groebner basis) is |
---|
[66c962] | 3444 | @* the needed D-module structure. |
---|
| 3445 | @* If eng <>0, @code{std} is used for Groebner basis computations, |
---|
| 3446 | @* otherwise, and by default @code{slimgb} is used. |
---|
| 3447 | @* If printlevel=1, progress debug messages will be printed, |
---|
| 3448 | @* if printlevel>=2, all the debug messages will be printed. |
---|
| 3449 | EXAMPLE: example SannfsBM; shows examples |
---|
| 3450 | " |
---|
| 3451 | { |
---|
| 3452 | int eng = 0; |
---|
| 3453 | if ( size(#)>0 ) |
---|
| 3454 | { |
---|
| 3455 | if ( typeof(#[1]) == "int" ) |
---|
| 3456 | { |
---|
| 3457 | eng = int(#[1]); |
---|
| 3458 | } |
---|
| 3459 | } |
---|
| 3460 | // returns a list with a ring and an ideal LD in it |
---|
| 3461 | int ppl = printlevel-voice+2; |
---|
| 3462 | // printf("plevel :%s, voice: %s",printlevel,voice); |
---|
| 3463 | def save = basering; |
---|
| 3464 | int N = nvars(basering); |
---|
| 3465 | int Nnew = 2*N+2; |
---|
| 3466 | int i,j; |
---|
| 3467 | string s; |
---|
| 3468 | list RL = ringlist(basering); |
---|
| 3469 | list L, Lord; |
---|
| 3470 | list tmp; |
---|
| 3471 | intvec iv; |
---|
| 3472 | L[1] = RL[1]; // char |
---|
| 3473 | L[4] = RL[4]; // char, minpoly |
---|
| 3474 | // check whether vars have admissible names |
---|
| 3475 | list Name = RL[2]; |
---|
| 3476 | list RName; |
---|
| 3477 | RName[1] = "t"; |
---|
| 3478 | RName[2] = "s"; |
---|
| 3479 | for(i=1;i<=N;i++) |
---|
| 3480 | { |
---|
| 3481 | for(j=1; j<=size(RName);j++) |
---|
| 3482 | { |
---|
| 3483 | if (Name[i] == RName[j]) |
---|
| 3484 | { |
---|
| 3485 | ERROR("Variable names should not include t,s"); |
---|
| 3486 | } |
---|
| 3487 | } |
---|
| 3488 | } |
---|
| 3489 | // now, create the names for new vars |
---|
| 3490 | list DName; |
---|
| 3491 | for(i=1;i<=N;i++) |
---|
| 3492 | { |
---|
| 3493 | DName[i] = "D"+Name[i]; // concat |
---|
| 3494 | } |
---|
| 3495 | tmp[1] = "t"; |
---|
| 3496 | tmp[2] = "s"; |
---|
| 3497 | list NName = tmp + Name + DName; |
---|
| 3498 | L[2] = NName; |
---|
| 3499 | // Name, Dname will be used further |
---|
| 3500 | kill NName; |
---|
| 3501 | // block ord (lp(2),dp); |
---|
| 3502 | tmp[1] = "lp"; // string |
---|
| 3503 | iv = 1,1; |
---|
| 3504 | tmp[2] = iv; //intvec |
---|
| 3505 | Lord[1] = tmp; |
---|
| 3506 | // continue with dp 1,1,1,1... |
---|
| 3507 | tmp[1] = "dp"; // string |
---|
| 3508 | s = "iv="; |
---|
| 3509 | for(i=1;i<=Nnew;i++) |
---|
| 3510 | { |
---|
| 3511 | s = s+"1,"; |
---|
| 3512 | } |
---|
| 3513 | s[size(s)]= ";"; |
---|
| 3514 | execute(s); |
---|
| 3515 | kill s; |
---|
| 3516 | tmp[2] = iv; |
---|
| 3517 | Lord[2] = tmp; |
---|
| 3518 | tmp[1] = "C"; |
---|
| 3519 | iv = 0; |
---|
| 3520 | tmp[2] = iv; |
---|
| 3521 | Lord[3] = tmp; |
---|
| 3522 | tmp = 0; |
---|
| 3523 | L[3] = Lord; |
---|
| 3524 | // we are done with the list |
---|
| 3525 | def @R@ = ring(L); |
---|
| 3526 | setring @R@; |
---|
| 3527 | matrix @D[Nnew][Nnew]; |
---|
| 3528 | @D[1,2]=t; |
---|
| 3529 | for(i=1; i<=N; i++) |
---|
| 3530 | { |
---|
| 3531 | @D[2+i,N+2+i]=1; |
---|
| 3532 | } |
---|
| 3533 | // L[5] = matrix(UpOneMatrix(Nnew)); |
---|
| 3534 | // L[6] = @D; |
---|
| 3535 | def @R = nc_algebra(1,@D); |
---|
| 3536 | setring @R; |
---|
| 3537 | kill @R@; |
---|
| 3538 | dbprint(ppl,"// -1-1- the ring @R(t,s,_x,_Dx) is ready"); |
---|
| 3539 | dbprint(ppl-1, @R); |
---|
| 3540 | // create the ideal I |
---|
| 3541 | poly F = imap(save,F); |
---|
| 3542 | ideal I = t*F+s; |
---|
| 3543 | poly p; |
---|
| 3544 | for(i=1; i<=N; i++) |
---|
| 3545 | { |
---|
| 3546 | p = t; // t |
---|
| 3547 | p = diff(F,var(2+i))*p; |
---|
| 3548 | I = I, var(N+2+i) + p; |
---|
| 3549 | } |
---|
| 3550 | // -------- the ideal I is ready ---------- |
---|
| 3551 | dbprint(ppl,"// -1-2- starting the elimination of t in @R"); |
---|
| 3552 | dbprint(ppl-1, I); |
---|
| 3553 | ideal J = engine(I,eng); |
---|
| 3554 | ideal K = nselect(J,1); |
---|
| 3555 | dbprint(ppl,"// -1-3- t is eliminated"); |
---|
| 3556 | dbprint(ppl-1, K); // K is without t |
---|
| 3557 | K = engine(K,eng); // std does the job too |
---|
| 3558 | // now, we must change the ordering |
---|
| 3559 | // and create a ring without t, Dt |
---|
| 3560 | // setring S; |
---|
| 3561 | // ----------- the ring @R3 ------------ |
---|
| 3562 | // _x, _Dx,s; elim.ord for _x,_Dx. |
---|
| 3563 | // keep: N, i,j,s, tmp, RL |
---|
| 3564 | Nnew = 2*N+1; |
---|
| 3565 | kill Lord, tmp, iv, RName; |
---|
| 3566 | list Lord, tmp; |
---|
| 3567 | intvec iv; |
---|
| 3568 | list L=imap(save,L); |
---|
| 3569 | list RL=imap(save,RL); |
---|
| 3570 | L[1] = RL[1]; |
---|
| 3571 | L[4] = RL[4]; // char, minpoly |
---|
| 3572 | // check whether vars hava admissible names -> done earlier |
---|
| 3573 | // now, create the names for new var |
---|
| 3574 | tmp[1] = "s"; |
---|
| 3575 | // DName is defined earlier |
---|
| 3576 | list NName = Name + DName + tmp; |
---|
| 3577 | L[2] = NName; |
---|
| 3578 | tmp = 0; |
---|
| 3579 | // block ord (dp(N),dp); |
---|
| 3580 | string s = "iv="; |
---|
| 3581 | for (i=1; i<=Nnew-1; i++) |
---|
| 3582 | { |
---|
| 3583 | s = s+"1,"; |
---|
| 3584 | } |
---|
| 3585 | s[size(s)]=";"; |
---|
| 3586 | execute(s); |
---|
| 3587 | tmp[1] = "dp"; // string |
---|
| 3588 | tmp[2] = iv; // intvec |
---|
| 3589 | Lord[1] = tmp; |
---|
| 3590 | // continue with dp 1,1,1,1... |
---|
| 3591 | tmp[1] = "dp"; // string |
---|
| 3592 | s[size(s)] = ","; |
---|
| 3593 | s = s+"1;"; |
---|
| 3594 | execute(s); |
---|
| 3595 | kill s; |
---|
| 3596 | kill NName; |
---|
| 3597 | tmp[2] = iv; |
---|
| 3598 | Lord[2] = tmp; |
---|
| 3599 | tmp[1] = "C"; iv = 0; tmp[2]=iv; |
---|
| 3600 | Lord[3] = tmp; tmp = 0; |
---|
| 3601 | L[3] = Lord; |
---|
| 3602 | // we are done with the list. Now add a Plural part |
---|
| 3603 | def @R2@ = ring(L); |
---|
| 3604 | setring @R2@; |
---|
| 3605 | matrix @D[Nnew][Nnew]; |
---|
| 3606 | for (i=1; i<=N; i++) |
---|
| 3607 | { |
---|
| 3608 | @D[i,N+i]=1; |
---|
| 3609 | } |
---|
| 3610 | def @R2 = nc_algebra(1,@D); |
---|
| 3611 | setring @R2; |
---|
| 3612 | kill @R2@; |
---|
| 3613 | dbprint(ppl,"// -2-1- the ring @R2(_x,_Dx,s) is ready"); |
---|
| 3614 | dbprint(ppl-1, @R2); |
---|
| 3615 | ideal MM = maxideal(1); |
---|
| 3616 | MM = 0,s,MM; |
---|
| 3617 | map R01 = @R, MM; |
---|
| 3618 | ideal K = R01(K); |
---|
| 3619 | // total cleanup |
---|
| 3620 | ideal LD = K; |
---|
| 3621 | // make leadcoeffs positive |
---|
| 3622 | for (i=1; i<= ncols(LD); i++) |
---|
| 3623 | { |
---|
| 3624 | if (leadcoef(LD[i]) <0 ) |
---|
| 3625 | { |
---|
| 3626 | LD[i] = -LD[i]; |
---|
| 3627 | } |
---|
| 3628 | } |
---|
| 3629 | export LD; |
---|
| 3630 | kill @R; |
---|
| 3631 | return(@R2); |
---|
| 3632 | } |
---|
| 3633 | example |
---|
| 3634 | { |
---|
| 3635 | "EXAMPLE:"; echo = 2; |
---|
| 3636 | ring r = 0,(x,y,z),Dp; |
---|
| 3637 | poly F = x^3+y^3+z^3; |
---|
| 3638 | printlevel = 0; |
---|
| 3639 | def A = SannfsBM(F); |
---|
| 3640 | setring A; |
---|
| 3641 | LD; |
---|
| 3642 | } |
---|
| 3643 | |
---|
| 3644 | static proc safeVarName (string s, list #) |
---|
| 3645 | { |
---|
| 3646 | string S; |
---|
| 3647 | int cv = 1; |
---|
| 3648 | if (size(#)>1) |
---|
| 3649 | { |
---|
| 3650 | if (#[1]=="v") { cv = 0; S = varstr(basering); } |
---|
| 3651 | if (#[1]=="c") { cv = 0; S = charstr(basering); } |
---|
| 3652 | } |
---|
| 3653 | if (cv) { S = charstr(basering) + "," + varstr(basering); } |
---|
| 3654 | S = "," + S + ","; |
---|
| 3655 | s = "," + s + ","; |
---|
| 3656 | while (find(S,s) <> 0) |
---|
| 3657 | { |
---|
| 3658 | s[1] = "@"; |
---|
| 3659 | s = "," + s; |
---|
| 3660 | } |
---|
| 3661 | s = s[2..size(s)-1]; |
---|
| 3662 | return(s) |
---|
| 3663 | } |
---|
| 3664 | |
---|
| 3665 | proc SannfsBFCT(poly F, list #) |
---|
| 3666 | "USAGE: SannfsBFCT(f [,a,b,c]); f a poly, a,b,c optional ints |
---|
| 3667 | RETURN: ring |
---|
| 3668 | PURPOSE: compute a Groebner basis either of Ann(f^s)+<f> or of |
---|
| 3669 | @* Ann(f^s)+<f,f_1,...,f_n> in D[s] |
---|
| 3670 | NOTE: Activate the output ring with the @code{setring} command. |
---|
| 3671 | @* This procedure, unlike SannfsBM, returns the ring D[s] with an anti- |
---|
| 3672 | @* elimination ordering for s. |
---|
| 3673 | @* The output ring contains an ideal @code{LD}, being a Groebner basis |
---|
| 3674 | @* either of Ann(f^s)+<f>, if a=0 (and by default), or of |
---|
| 3675 | @* Ann(f^s)+<f,f_1,...,f_n>, otherwise. |
---|
| 3676 | @* Here, f_i stands for the i-th partial derivative of f. |
---|
| 3677 | @* If b<>0, @code{std} is used for Groebner basis computations, |
---|
| 3678 | @* otherwise, and by default @code{slimgb} is used. |
---|
| 3679 | @* If c<>0, @code{std} is used for Groebner basis computations of |
---|
| 3680 | @* ideals <I+J> when I is already a Groebner basis of <I>. |
---|
| 3681 | @* Otherwise, and by default the engine determined by the switch b is |
---|
| 3682 | @* used. Note that in the case c<>0, the choice for b will be |
---|
| 3683 | @* overwritten only for the types of ideals mentioned above. |
---|
| 3684 | @* This means that if b<>0, specifying c has no effect. |
---|
| 3685 | DISPLAY: If printlevel=1, progress debug messages will be printed, |
---|
| 3686 | @* if printlevel>=2, all the debug messages will be printed. |
---|
| 3687 | EXAMPLE: example SannfsBFCT; shows examples |
---|
| 3688 | " |
---|
| 3689 | { |
---|
| 3690 | int addPD,eng,stdsum; |
---|
| 3691 | if (size(#)>0) |
---|
| 3692 | { |
---|
| 3693 | if (typeof(#[1])=="int" || typeof(#[1])=="number") |
---|
| 3694 | { |
---|
| 3695 | addPD = int(#[1]); |
---|
| 3696 | } |
---|
| 3697 | if (size(#)>1) |
---|
| 3698 | { |
---|
| 3699 | if (typeof(#[2])=="int" || typeof(#[2])=="number") |
---|
| 3700 | { |
---|
[0610f0e] | 3701 | eng = int(#[2]); |
---|
[66c962] | 3702 | } |
---|
| 3703 | if (size(#)>2) |
---|
| 3704 | { |
---|
[0610f0e] | 3705 | if (typeof(#[3])=="int" || typeof(#[3])=="number") |
---|
| 3706 | { |
---|
| 3707 | stdsum = int(#[3]); |
---|
| 3708 | } |
---|
[66c962] | 3709 | } |
---|
| 3710 | } |
---|
| 3711 | } |
---|
| 3712 | int ppl = printlevel-voice+2; |
---|
| 3713 | def save = basering; |
---|
| 3714 | int N = nvars(save); |
---|
| 3715 | intvec optSave = option(get); |
---|
| 3716 | int i,j; |
---|
| 3717 | list RL = ringlist(save); |
---|
| 3718 | // ----- step 1: compute syzigies |
---|
| 3719 | intvec iv; |
---|
| 3720 | list L,Lord; |
---|
| 3721 | iv = 1:N; Lord[1] = list("dp",iv); |
---|
| 3722 | iv = 0; Lord[2] = list("C",iv); |
---|
| 3723 | L = RL; |
---|
| 3724 | L[3] = Lord; |
---|
| 3725 | def @RM = ring(L); |
---|
| 3726 | kill L,Lord; |
---|
| 3727 | setring @RM; |
---|
| 3728 | option(redSB); |
---|
| 3729 | option(redTail); |
---|
| 3730 | def RM = makeModElimRing(@RM); |
---|
| 3731 | setring RM; |
---|
| 3732 | poly F = imap(save,F); |
---|
| 3733 | ideal J = jacob(F); |
---|
| 3734 | J = F,J; |
---|
| 3735 | dbprint(ppl,"// -1-1- Starting the computation of syz(F,_Dx(F))"); |
---|
| 3736 | dbprint(ppl-1, J); |
---|
| 3737 | module M = syz(J); |
---|
| 3738 | dbprint(ppl,"// -1-2- The module syz(F,_Dx(F)) has been computed"); |
---|
| 3739 | dbprint(ppl-1, M); |
---|
| 3740 | dbprint(ppl,"// -1-3- Starting GB computation of syz(F,_Dx(F))"); |
---|
| 3741 | M = engine(M,eng); |
---|
| 3742 | dbprint(ppl,"// -1-4- GB computation finished"); |
---|
| 3743 | dbprint(ppl-1, M); |
---|
| 3744 | // ----- step 2: compute part of Ann(F^s) |
---|
| 3745 | setring save; |
---|
| 3746 | option(set,optSave); |
---|
| 3747 | module M = imap(RM,M); |
---|
| 3748 | kill optSave,RM; |
---|
| 3749 | // ----- create D[s] |
---|
| 3750 | int Nnew = 2*N+1; |
---|
| 3751 | list L, Lord; |
---|
| 3752 | // ----- keep char, minpoly |
---|
| 3753 | L[1] = RL[1]; |
---|
| 3754 | L[4] = RL[4]; |
---|
| 3755 | // ----- create names for new vars |
---|
| 3756 | list Name = RL[2]; |
---|
| 3757 | string newVar@s = safeVarName("s"); |
---|
| 3758 | if (newVar@s[1] == "@") |
---|
| 3759 | { |
---|
| 3760 | print("Name s already assigned to parameter/ringvar."); |
---|
| 3761 | print("Using " + newVar@s + " instead.") |
---|
| 3762 | } |
---|
| 3763 | list DName; |
---|
| 3764 | for (i=1; i<=N; i++) |
---|
| 3765 | { |
---|
| 3766 | DName[i] = safeVarName("D" + Name[i]); |
---|
| 3767 | } |
---|
| 3768 | L[2] = list(newVar@s) + Name + DName; |
---|
| 3769 | // ----- create ordering |
---|
| 3770 | // --- anti-elimination ordering for s |
---|
| 3771 | iv = 1; Lord[1] = list("dp",iv); |
---|
| 3772 | iv = 1:(2*N); Lord[2] = list("dp",iv); |
---|
| 3773 | iv = 0; Lord[3] = list("C",iv); |
---|
| 3774 | L[3] = Lord; |
---|
| 3775 | // ----- create commutative ring |
---|
| 3776 | def @Ds = ring(L); |
---|
| 3777 | kill L,Lord; |
---|
| 3778 | setring @Ds; |
---|
| 3779 | // ----- create nc relations |
---|
| 3780 | matrix Drel[Nnew][Nnew]; |
---|
| 3781 | for (i=1; i<=N; i++) |
---|
| 3782 | { |
---|
| 3783 | Drel[i+1,N+1+i] = 1; |
---|
| 3784 | } |
---|
| 3785 | def Ds = nc_algebra(1,Drel); |
---|
| 3786 | setring Ds; |
---|
| 3787 | kill @Ds; |
---|
| 3788 | dbprint(ppl,"// -2-1- The ring D[s] is ready"); |
---|
| 3789 | dbprint(ppl-1, Ds); |
---|
| 3790 | matrix M = imap(save,M); |
---|
| 3791 | vector v = var(1)*gen(1); |
---|
| 3792 | for (i=1; i<=N; i++) |
---|
| 3793 | { |
---|
| 3794 | v = v + var(i+1+N)*gen(i+1); //[s,_Dx] |
---|
| 3795 | } |
---|
| 3796 | ideal J = transpose(M)*v; |
---|
| 3797 | kill M,v; |
---|
| 3798 | dbprint(ppl,"// -2-2- Compute part of Ann(F^s)"); |
---|
| 3799 | dbprint(ppl-1, J); |
---|
| 3800 | J = engine(J,eng); |
---|
| 3801 | dbprint(ppl,"// -2-3- GB computation finished"); |
---|
| 3802 | dbprint(ppl-1, J); |
---|
| 3803 | // ----- step 3: the full annihilator |
---|
| 3804 | // ----- create D<t,s> |
---|
| 3805 | setring save; |
---|
| 3806 | Nnew = 2*N+2; |
---|
| 3807 | list L, Lord; |
---|
| 3808 | // ----- keep char, minpoly |
---|
| 3809 | L[1] = RL[1]; |
---|
[3f4e52] | 3810 | L[4] = RL[4]; |
---|
[66c962] | 3811 | // ----- create vars |
---|
| 3812 | string newVar@t = safeVarName("t"); |
---|
| 3813 | L[2] = list(newVar@t,newVar@s) + DName + Name; |
---|
| 3814 | // ----- create ordering for elimination of t |
---|
| 3815 | // block ord (lp(2),dp); |
---|
| 3816 | iv = 1,1; Lord[1] = list("lp",iv); |
---|
| 3817 | iv = 1:Nnew; Lord[2] = list("dp",iv); |
---|
| 3818 | iv = 0; Lord[3] = list("C",iv); |
---|
| 3819 | L[3] = Lord; |
---|
| 3820 | def @Dts = ring(L); |
---|
| 3821 | kill RL,L,Lord,Name,DName,newVar@s,newVar@t; |
---|
| 3822 | setring @Dts; |
---|
| 3823 | // ----- create nc relations |
---|
| 3824 | matrix Drel[Nnew][Nnew]; |
---|
| 3825 | Drel[1,2] = var(1); |
---|
| 3826 | for(i=1; i<=N; i++) |
---|
| 3827 | { |
---|
| 3828 | Drel[2+i,N+2+i]=-1; |
---|
| 3829 | } |
---|
| 3830 | def Dts = nc_algebra(1,Drel); |
---|
| 3831 | setring Dts; |
---|
| 3832 | kill @Dts; |
---|
| 3833 | dbprint(ppl,"// -3-1- The ring D<t,s> is ready"); |
---|
| 3834 | dbprint(ppl-1, Dts); |
---|
| 3835 | // ----- create the ideal I following BM |
---|
| 3836 | poly F = imap(save,F); |
---|
| 3837 | ideal I = var(1)*F + var(2); // = t*F + s |
---|
| 3838 | poly p; |
---|
| 3839 | for(i=1; i<=N; i++) |
---|
| 3840 | { |
---|
| 3841 | p = var(1)*diff(F,var(N+2+i)) + var(2+i); // = t*F_i + D_i |
---|
| 3842 | I[i+1] = p; |
---|
| 3843 | } |
---|
| 3844 | // ----- add already computed part to it |
---|
| 3845 | ideal MM = var(2); // s |
---|
| 3846 | for (i=1; i<=N; i++) |
---|
| 3847 | { |
---|
| 3848 | MM[1+i] = var(2+N+i); // _x |
---|
| 3849 | MM[1+N+i] = var(2+i); // _Dx |
---|
| 3850 | } |
---|
| 3851 | map Ds2Dts = Ds,MM; |
---|
| 3852 | ideal J = Ds2Dts(J); |
---|
| 3853 | attrib(J,"isSB",1); |
---|
| 3854 | kill MM,Ds2Dts; |
---|
| 3855 | // ----- start the elimination |
---|
| 3856 | dbprint(ppl,"// -3-2- Starting the elimination of t in D<t,s>"); |
---|
| 3857 | dbprint(ppl-1, I); |
---|
| 3858 | if (stdsum || eng <> 0) |
---|
| 3859 | { |
---|
| 3860 | I = std(J,I); |
---|
| 3861 | } |
---|
| 3862 | else |
---|
| 3863 | { |
---|
| 3864 | I = J,I; |
---|
[3f4e52] | 3865 | I = engine(I,eng); |
---|
[66c962] | 3866 | } |
---|
| 3867 | kill J; |
---|
| 3868 | I = nselect(I,1); |
---|
| 3869 | dbprint(ppl,"// -3-3- t is eliminated"); |
---|
| 3870 | dbprint(ppl-1, I); // I is without t |
---|
| 3871 | // ----- step 4: add F |
---|
| 3872 | // ----- back to D[s] |
---|
| 3873 | setring Ds; |
---|
| 3874 | ideal MM = 0,var(1); // 0,s |
---|
| 3875 | for (i=1; i<=N; i++) |
---|
| 3876 | { |
---|
| 3877 | MM[2+i] = var(1+N+i); // _Dx |
---|
| 3878 | MM[2+N+i] = var(1+i); // _x |
---|
| 3879 | } |
---|
| 3880 | map Dts2Ds = Dts, MM; |
---|
| 3881 | ideal LD = Dts2Ds(I); |
---|
| 3882 | kill J,Dts,Dts2Ds,MM; |
---|
| 3883 | dbprint(ppl,"// -4-1- Starting cosmetic Groebner computation"); |
---|
| 3884 | LD = engine(LD,eng); |
---|
| 3885 | dbprint(ppl,"// -4-2- Finished cosmetic Groebner computation"); |
---|
| 3886 | dbprint(ppl-1, LD); |
---|
| 3887 | // ----- use reduction trick as Macaulay2 does: compute b(s)/(s+1) by adding all partial derivations also |
---|
| 3888 | ideal J; |
---|
| 3889 | if (addPD) |
---|
| 3890 | { |
---|
| 3891 | setring @RM; |
---|
| 3892 | poly F = imap(save,F); |
---|
| 3893 | ideal J = jacob(F); |
---|
| 3894 | J = F,J; |
---|
| 3895 | dbprint(ppl,"// -4-2-1- Start GB computation <f, f_i>"); |
---|
| 3896 | J = engine(J,eng); |
---|
| 3897 | dbprint(ppl,"// -4-2-2- Finished GB computation <f, f_i>"); |
---|
| 3898 | dbprint(ppl-1, J); |
---|
| 3899 | setring Ds; |
---|
| 3900 | J = imap(@RM,J); |
---|
| 3901 | attrib(J,"isSB",1); |
---|
| 3902 | dbprint(ppl,"// -4-3- Start GB computations for Ann f^s + <f, f_i>"); |
---|
| 3903 | } |
---|
| 3904 | else |
---|
| 3905 | { |
---|
| 3906 | J = imap(save,F); |
---|
| 3907 | dbprint(ppl,"// -4-3- Start GB computations for Ann f^s + <f>"); |
---|
| 3908 | } |
---|
| 3909 | kill @RM; |
---|
| 3910 | // ----- the really hard part |
---|
| 3911 | if (stdsum || eng <> 0) |
---|
| 3912 | { |
---|
| 3913 | LD = std(LD,J); |
---|
| 3914 | } |
---|
| 3915 | else |
---|
| 3916 | { |
---|
| 3917 | LD = LD,J; |
---|
[3f4e52] | 3918 | LD = engine(LD,eng); |
---|
[66c962] | 3919 | } |
---|
| 3920 | if (addPD) { dbprint(ppl,"// -4-4- Finished GB computations for Ann f^s + <f, f_i>"); } |
---|
| 3921 | else { dbprint(ppl,"// -4-4- Finished GB computations for Ann f^s + <f>"); } |
---|
| 3922 | dbprint(ppl-1, LD); |
---|
| 3923 | export LD; |
---|
| 3924 | return(Ds); |
---|
| 3925 | } |
---|
| 3926 | example |
---|
| 3927 | { |
---|
| 3928 | "EXAMPLE:"; echo = 2; |
---|
| 3929 | ring r = 0,(x,y,z,w),Dp; |
---|
| 3930 | poly F = x^3+y^3+z^3*w; |
---|
| 3931 | // compute Ann(F^s)+<F> using slimgb only |
---|
| 3932 | def A = SannfsBFCT(F); |
---|
| 3933 | setring A; A; |
---|
| 3934 | LD; |
---|
| 3935 | // the Bernstein-Sato poly of F: |
---|
| 3936 | vec2poly(pIntersect(s,LD)); |
---|
| 3937 | // a fancier example: |
---|
| 3938 | def R = reiffen(4,5); setring R; |
---|
| 3939 | RC; // the Reiffen curve in 4,5 |
---|
| 3940 | // compute Ann(RC^s)+<RC,diff(RC,x),diff(RC,y)> |
---|
| 3941 | // using std for GB computations of ideals <I+J> |
---|
| 3942 | // where I is already a GB of <I> |
---|
| 3943 | // and slimgb for other ideals |
---|
| 3944 | def B = SannfsBFCT(RC,1,0,1); |
---|
| 3945 | setring B; |
---|
| 3946 | // the Bernstein-Sato poly of RC: |
---|
| 3947 | (s-1)*vec2poly(pIntersect(s,LD)); |
---|
| 3948 | } |
---|
| 3949 | |
---|
| 3950 | |
---|
| 3951 | proc SannfsBFCTstd(poly F, list #) |
---|
| 3952 | "USAGE: SannfsBFCTstd(f [,eng]); f a poly, eng an optional int |
---|
| 3953 | RETURN: ring |
---|
| 3954 | PURPOSE: compute Ann f^s and Groebner basis of Ann f^s+f in D[s] |
---|
| 3955 | NOTE: activate the output ring with the @code{setring} command. |
---|
[3f4e52] | 3956 | @* This procedure, unlike SannfsBM, returns a ring with the degrevlex |
---|
[66c962] | 3957 | @* ordering in all variables. |
---|
| 3958 | @* In the ring D[s], the ideal LD (which IS a Groebner basis) is the needed ideal. |
---|
| 3959 | @* In this procedure @code{std} is used for Groebner basis computations. |
---|
| 3960 | DISPLAY: If printlevel=1, progress debug messages will be printed, |
---|
| 3961 | @* if printlevel>=2, all the debug messages will be printed. |
---|
| 3962 | EXAMPLE: example SannfsBFCTstd; shows examples |
---|
| 3963 | " |
---|
| 3964 | { |
---|
[3f4e52] | 3965 | // DEBUG INFO: ordering on the output ring = dp, |
---|
[66c962] | 3966 | // use std(K,F); for reusing the std property of K |
---|
| 3967 | |
---|
| 3968 | int eng = 0; |
---|
| 3969 | if ( size(#)>0 ) |
---|
| 3970 | { |
---|
| 3971 | if ( typeof(#[1]) == "int" ) |
---|
| 3972 | { |
---|
| 3973 | eng = int(#[1]); |
---|
| 3974 | } |
---|
| 3975 | } |
---|
| 3976 | // returns a list with a ring and an ideal LD in it |
---|
| 3977 | int ppl = printlevel-voice+2; |
---|
| 3978 | // printf("plevel :%s, voice: %s",printlevel,voice); |
---|
| 3979 | def save = basering; |
---|
| 3980 | int N = nvars(basering); |
---|
| 3981 | int Nnew = 2*N+2; |
---|
| 3982 | int i,j; |
---|
| 3983 | string s; |
---|
| 3984 | list RL = ringlist(basering); |
---|
| 3985 | list L, Lord; |
---|
| 3986 | list tmp; |
---|
| 3987 | intvec iv; |
---|
| 3988 | L[1] = RL[1]; // char |
---|
| 3989 | L[4] = RL[4]; // char, minpoly |
---|
| 3990 | // check whether vars have admissible names |
---|
| 3991 | list Name = RL[2]; |
---|
| 3992 | list RName; |
---|
| 3993 | RName[1] = "@t"; |
---|
| 3994 | RName[2] = "@s"; |
---|
| 3995 | for(i=1;i<=N;i++) |
---|
| 3996 | { |
---|
| 3997 | for(j=1; j<=size(RName);j++) |
---|
| 3998 | { |
---|
| 3999 | if (Name[i] == RName[j]) |
---|
| 4000 | { |
---|
| 4001 | ERROR("Variable names should not include @t,@s"); |
---|
| 4002 | } |
---|
| 4003 | } |
---|
| 4004 | } |
---|
| 4005 | // now, create the names for new vars |
---|
| 4006 | list DName; |
---|
| 4007 | for(i=1;i<=N;i++) |
---|
| 4008 | { |
---|
| 4009 | DName[i] = "D"+Name[i]; // concat |
---|
| 4010 | } |
---|
| 4011 | tmp[1] = "t"; |
---|
| 4012 | tmp[2] = "s"; |
---|
| 4013 | list NName = tmp + DName + Name ; |
---|
| 4014 | L[2] = NName; |
---|
| 4015 | // Name, Dname will be used further |
---|
| 4016 | kill NName; |
---|
| 4017 | // block ord (lp(2),dp); |
---|
| 4018 | tmp[1] = "lp"; // string |
---|
| 4019 | iv = 1,1; |
---|
| 4020 | tmp[2] = iv; //intvec |
---|
| 4021 | Lord[1] = tmp; |
---|
| 4022 | // continue with dp 1,1,1,1... |
---|
| 4023 | tmp[1] = "dp"; // string |
---|
| 4024 | s = "iv="; |
---|
| 4025 | for(i=1;i<=Nnew;i++) |
---|
| 4026 | { |
---|
| 4027 | s = s+"1,"; |
---|
| 4028 | } |
---|
| 4029 | s[size(s)]= ";"; |
---|
| 4030 | execute(s); |
---|
| 4031 | kill s; |
---|
| 4032 | tmp[2] = iv; |
---|
| 4033 | Lord[2] = tmp; |
---|
| 4034 | tmp[1] = "C"; |
---|
| 4035 | iv = 0; |
---|
| 4036 | tmp[2] = iv; |
---|
| 4037 | Lord[3] = tmp; |
---|
| 4038 | tmp = 0; |
---|
| 4039 | L[3] = Lord; |
---|
| 4040 | // we are done with the list |
---|
| 4041 | def @R@ = ring(L); |
---|
| 4042 | setring @R@; |
---|
| 4043 | matrix @D[Nnew][Nnew]; |
---|
| 4044 | @D[1,2]=t; |
---|
| 4045 | for(i=1; i<=N; i++) |
---|
| 4046 | { |
---|
| 4047 | @D[2+i,N+2+i]=-1; |
---|
| 4048 | } |
---|
| 4049 | // L[5] = matrix(UpOneMatrix(Nnew)); |
---|
| 4050 | // L[6] = @D; |
---|
| 4051 | def @R = nc_algebra(1,@D); |
---|
| 4052 | setring @R; |
---|
| 4053 | kill @R@; |
---|
| 4054 | dbprint(ppl,"// -1-1- the ring @R(t,s,_Dx,_x) is ready"); |
---|
| 4055 | dbprint(ppl-1, @R); |
---|
| 4056 | // create the ideal I |
---|
| 4057 | poly F = imap(save,F); |
---|
| 4058 | ideal I = t*F+s; |
---|
| 4059 | poly p; |
---|
| 4060 | for(i=1; i<=N; i++) |
---|
| 4061 | { |
---|
| 4062 | p = t; // t |
---|
| 4063 | p = diff(F,var(N+2+i))*p; |
---|
| 4064 | I = I, var(2+i) + p; |
---|
| 4065 | } |
---|
| 4066 | // -------- the ideal I is ready ---------- |
---|
| 4067 | dbprint(ppl,"// -1-2- starting the elimination of t in @R"); |
---|
| 4068 | dbprint(ppl-1, I); |
---|
| 4069 | ideal J = engine(I,eng); |
---|
| 4070 | ideal K = nselect(J,1); |
---|
| 4071 | dbprint(ppl,"// -1-3- t is eliminated"); |
---|
| 4072 | dbprint(ppl-1, K); // K is without t |
---|
| 4073 | K = engine(K,eng); // std does the job too |
---|
| 4074 | // now, we must change the ordering |
---|
| 4075 | // and create a ring without t |
---|
| 4076 | // setring S; |
---|
| 4077 | // ----------- the ring @R3 ------------ |
---|
| 4078 | // _Dx,_x,s; +fast ord ! |
---|
| 4079 | // keep: N, i,j,s, tmp, RL |
---|
| 4080 | Nnew = 2*N+1; |
---|
| 4081 | kill Lord, tmp, iv, RName; |
---|
| 4082 | list Lord, tmp; |
---|
| 4083 | intvec iv; |
---|
| 4084 | list L=imap(save,L); |
---|
| 4085 | list RL=imap(save,RL); |
---|
| 4086 | L[1] = RL[1]; |
---|
| 4087 | L[4] = RL[4]; // char, minpoly |
---|
| 4088 | // check whether vars hava admissible names -> done earlier |
---|
| 4089 | // now, create the names for new var |
---|
| 4090 | tmp[1] = "s"; |
---|
| 4091 | // DName is defined earlier |
---|
| 4092 | list NName = DName + Name + tmp; |
---|
| 4093 | L[2] = NName; |
---|
| 4094 | tmp = 0; |
---|
| 4095 | // just dp |
---|
| 4096 | // block ord (dp(N),dp); |
---|
| 4097 | string s = "iv="; |
---|
| 4098 | for (i=1; i<=Nnew; i++) |
---|
| 4099 | { |
---|
| 4100 | s = s+"1,"; |
---|
| 4101 | } |
---|
| 4102 | s[size(s)]=";"; |
---|
| 4103 | execute(s); |
---|
| 4104 | tmp[1] = "dp"; // string |
---|
| 4105 | tmp[2] = iv; // intvec |
---|
| 4106 | Lord[1] = tmp; |
---|
| 4107 | kill s; |
---|
| 4108 | kill NName; |
---|
| 4109 | tmp[1] = "C"; |
---|
| 4110 | Lord[2] = tmp; tmp = 0; |
---|
| 4111 | L[3] = Lord; |
---|
| 4112 | // we are done with the list. Now add a Plural part |
---|
| 4113 | def @R2@ = ring(L); |
---|
| 4114 | setring @R2@; |
---|
| 4115 | matrix @D[Nnew][Nnew]; |
---|
| 4116 | for (i=1; i<=N; i++) |
---|
| 4117 | { |
---|
| 4118 | @D[i,N+i]=-1; |
---|
| 4119 | } |
---|
| 4120 | def @R2 = nc_algebra(1,@D); |
---|
| 4121 | setring @R2; |
---|
| 4122 | kill @R2@; |
---|
| 4123 | dbprint(ppl,"// -2-1- the ring @R2(_Dx,_x,s) is ready"); |
---|
| 4124 | dbprint(ppl-1, @R2); |
---|
| 4125 | ideal MM = maxideal(1); |
---|
| 4126 | MM = 0,s,MM; |
---|
| 4127 | map R01 = @R, MM; |
---|
| 4128 | ideal K = R01(K); |
---|
| 4129 | // total cleanup |
---|
| 4130 | poly F = imap(save, F); |
---|
| 4131 | // ideal LD = K,F; |
---|
| 4132 | dbprint(ppl,"// -2-2- start GB computations for Ann f^s + f"); |
---|
| 4133 | // dbprint(ppl-1, LD); |
---|
| 4134 | ideal LD = std(K,F); |
---|
| 4135 | // LD = engine(LD,eng); |
---|
| 4136 | dbprint(ppl,"// -2-3- finished GB computations for Ann f^s + f"); |
---|
| 4137 | dbprint(ppl-1, LD); |
---|
| 4138 | // make leadcoeffs positive |
---|
| 4139 | for (i=1; i<= ncols(LD); i++) |
---|
| 4140 | { |
---|
| 4141 | if (leadcoef(LD[i]) <0 ) |
---|
| 4142 | { |
---|
| 4143 | LD[i] = -LD[i]; |
---|
| 4144 | } |
---|
| 4145 | } |
---|
| 4146 | export LD; |
---|
| 4147 | kill @R; |
---|
| 4148 | return(@R2); |
---|
| 4149 | } |
---|
| 4150 | example |
---|
| 4151 | { |
---|
| 4152 | "EXAMPLE:"; echo = 2; |
---|
| 4153 | ring r = 0,(x,y,z,w),Dp; |
---|
| 4154 | poly F = x^3+y^3+z^3*w; |
---|
| 4155 | printlevel = 0; |
---|
| 4156 | def A = SannfsBFCT(F); setring A; |
---|
| 4157 | intvec v = 1,2,3,4,5,6,7,8; |
---|
| 4158 | // are there polynomials, depending on s only? |
---|
| 4159 | nselect(LD,v); |
---|
| 4160 | // a fancier example: |
---|
| 4161 | def R = reiffen(4,5); setring R; |
---|
| 4162 | v = 1,2,3,4; |
---|
| 4163 | RC; // the Reiffen curve in 4,5 |
---|
| 4164 | def B = SannfsBFCT(RC); |
---|
| 4165 | setring B; |
---|
| 4166 | // Are there polynomials, depending on s only? |
---|
| 4167 | nselect(LD,v); |
---|
| 4168 | // It is not the case. Are there leading monomials in s only? |
---|
| 4169 | nselect(lead(LD),v); |
---|
| 4170 | } |
---|
| 4171 | |
---|
| 4172 | // use a finer ordering |
---|
| 4173 | |
---|
| 4174 | proc SannfsLOT(poly F, list #) |
---|
| 4175 | "USAGE: SannfsLOT(f [,eng]); f a poly, eng an optional int |
---|
| 4176 | RETURN: ring |
---|
[3f4e52] | 4177 | PURPOSE: compute the D-module structure of basering[1/f]*f^s, according to the |
---|
[66c962] | 4178 | @* Levandovskyy's modification of the algorithm by Oaku and Takayama in D[s] |
---|
| 4179 | NOTE: activate the output ring with the @code{setring} command. |
---|
[3f4e52] | 4180 | @* In the ring D[s], the ideal LD (which is NOT a Groebner basis) is |
---|
[66c962] | 4181 | @* the needed D-module structure. |
---|
| 4182 | @* If eng <>0, @code{std} is used for Groebner basis computations, |
---|
| 4183 | @* otherwise, and by default @code{slimgb} is used. |
---|
| 4184 | @* If printlevel=1, progress debug messages will be printed, |
---|
| 4185 | @* if printlevel>=2, all the debug messages will be printed. |
---|
| 4186 | EXAMPLE: example SannfsLOT; shows examples |
---|
| 4187 | " |
---|
| 4188 | { |
---|
| 4189 | int eng = 0; |
---|
| 4190 | if ( size(#)>0 ) |
---|
| 4191 | { |
---|
| 4192 | if ( typeof(#[1]) == "int" ) |
---|
| 4193 | { |
---|
| 4194 | eng = int(#[1]); |
---|
| 4195 | } |
---|
| 4196 | } |
---|
| 4197 | // returns a list with a ring and an ideal LD in it |
---|
| 4198 | int ppl = printlevel-voice+2; |
---|
| 4199 | // printf("plevel :%s, voice: %s",printlevel,voice); |
---|
| 4200 | def save = basering; |
---|
| 4201 | int N = nvars(basering); |
---|
| 4202 | // int Nnew = 2*(N+2); |
---|
| 4203 | int Nnew = 2*(N+1)+1; //removed u,v; added s |
---|
| 4204 | int i,j; |
---|
| 4205 | string s; |
---|
| 4206 | list RL = ringlist(basering); |
---|
| 4207 | list L, Lord; |
---|
| 4208 | list tmp; |
---|
| 4209 | intvec iv; |
---|
| 4210 | L[1] = RL[1]; // char |
---|
| 4211 | L[4] = RL[4]; // char, minpoly |
---|
| 4212 | // check whether vars have admissible names |
---|
| 4213 | list Name = RL[2]; |
---|
| 4214 | list RName; |
---|
| 4215 | // RName[1] = "u"; |
---|
| 4216 | // RName[2] = "v"; |
---|
| 4217 | RName[1] = "t"; |
---|
| 4218 | RName[2] = "Dt"; |
---|
| 4219 | for(i=1;i<=N;i++) |
---|
| 4220 | { |
---|
| 4221 | for(j=1; j<=size(RName);j++) |
---|
| 4222 | { |
---|
| 4223 | if (Name[i] == RName[j]) |
---|
| 4224 | { |
---|
| 4225 | ERROR("Variable names should not include t,Dt"); |
---|
| 4226 | } |
---|
| 4227 | } |
---|
| 4228 | } |
---|
| 4229 | // now, create the names for new vars |
---|
| 4230 | // tmp[1] = "u"; |
---|
| 4231 | // tmp[2] = "v"; |
---|
| 4232 | // list UName = tmp; |
---|
| 4233 | list DName; |
---|
| 4234 | for(i=1;i<=N;i++) |
---|
| 4235 | { |
---|
| 4236 | DName[i] = "D"+Name[i]; // concat |
---|
| 4237 | } |
---|
| 4238 | tmp = 0; |
---|
| 4239 | tmp[1] = "t"; |
---|
| 4240 | tmp[2] = "Dt"; |
---|
| 4241 | list SName ; SName[1] = "s"; |
---|
| 4242 | // list NName = tmp + Name + DName + SName; |
---|
| 4243 | list NName = tmp + SName + Name + DName; |
---|
| 4244 | L[2] = NName; |
---|
| 4245 | tmp = 0; |
---|
| 4246 | // Name, Dname will be used further |
---|
| 4247 | // kill UName; |
---|
| 4248 | kill NName; |
---|
| 4249 | // block ord (a(1,1),dp); |
---|
| 4250 | tmp[1] = "a"; // string |
---|
| 4251 | iv = 1,1; |
---|
| 4252 | tmp[2] = iv; //intvec |
---|
| 4253 | Lord[1] = tmp; |
---|
| 4254 | // continue with a(0,0,1) |
---|
| 4255 | tmp[1] = "a"; // string |
---|
| 4256 | iv = 0,0,1; |
---|
| 4257 | tmp[2] = iv; //intvec |
---|
| 4258 | Lord[2] = tmp; |
---|
| 4259 | // continue with dp 1,1,1,1... |
---|
| 4260 | tmp[1] = "dp"; // string |
---|
| 4261 | s = "iv="; |
---|
| 4262 | for(i=1;i<=Nnew;i++) |
---|
| 4263 | { |
---|
| 4264 | s = s+"1,"; |
---|
| 4265 | } |
---|
| 4266 | s[size(s)]= ";"; |
---|
| 4267 | execute(s); |
---|
| 4268 | tmp[2] = iv; |
---|
| 4269 | Lord[3] = tmp; |
---|
| 4270 | tmp[1] = "C"; |
---|
| 4271 | iv = 0; |
---|
| 4272 | tmp[2] = iv; |
---|
| 4273 | Lord[4] = tmp; |
---|
| 4274 | tmp = 0; |
---|
| 4275 | L[3] = Lord; |
---|
| 4276 | // we are done with the list |
---|
| 4277 | def @R@ = ring(L); |
---|
| 4278 | setring @R@; |
---|
| 4279 | matrix @D[Nnew][Nnew]; |
---|
| 4280 | @D[1,2]=1; |
---|
| 4281 | for(i=1; i<=N; i++) |
---|
| 4282 | { |
---|
| 4283 | @D[3+i,N+3+i]=1; |
---|
| 4284 | } |
---|
| 4285 | // ADD [s,t]=-t, [s,Dt]=Dt |
---|
| 4286 | @D[1,3] = -var(1); |
---|
| 4287 | @D[2,3] = var(2); |
---|
| 4288 | // @D[N+3,2*(N+2)]=1; old t,Dt stuff |
---|
| 4289 | // L[5] = matrix(UpOneMatrix(Nnew)); |
---|
| 4290 | // L[6] = @D; |
---|
| 4291 | def @R = nc_algebra(1,@D); |
---|
| 4292 | setring @R; |
---|
| 4293 | kill @R@; |
---|
| 4294 | dbprint(ppl,"// -1-1- the ring @R(t,Dt,s,_x,_Dx) is ready"); |
---|
| 4295 | dbprint(ppl-1, @R); |
---|
| 4296 | // create the ideal I |
---|
| 4297 | poly F = imap(save,F); |
---|
| 4298 | // ideal I = u*F-t,u*v-1; |
---|
| 4299 | ideal I = F-t; |
---|
| 4300 | poly p; |
---|
| 4301 | for(i=1; i<=N; i++) |
---|
| 4302 | { |
---|
| 4303 | // p = u*Dt; // u*Dt |
---|
| 4304 | p = Dt; |
---|
| 4305 | p = diff(F,var(3+i))*p; |
---|
| 4306 | I = I, var(N+3+i) + p; |
---|
| 4307 | } |
---|
| 4308 | // I = I, var(1)*var(2) + var(Nnew) +1; // reduce it with t-f!!! |
---|
| 4309 | // t*Dt + s +1 reduced with t-f gives f*Dt + s |
---|
| 4310 | I = I, F*var(2) + var(3); |
---|
| 4311 | // -------- the ideal I is ready ---------- |
---|
| 4312 | dbprint(ppl,"// -1-2- starting the elimination of t,Dt in @R"); |
---|
| 4313 | dbprint(ppl-1, I); |
---|
| 4314 | ideal J = engine(I,eng); |
---|
| 4315 | ideal K = nselect(J,1..2); |
---|
| 4316 | dbprint(ppl,"// -1-3- t,Dt are eliminated"); |
---|
| 4317 | dbprint(ppl-1, K); // K is without t, Dt |
---|
| 4318 | K = engine(K,eng); // std does the job too |
---|
| 4319 | // now, we must change the ordering |
---|
| 4320 | // and create a ring without t, Dt |
---|
| 4321 | setring save; |
---|
| 4322 | // ----------- the ring @R3 ------------ |
---|
| 4323 | // _x, _Dx,s; elim.ord for _x,_Dx. |
---|
| 4324 | // keep: N, i,j,s, tmp, RL |
---|
| 4325 | Nnew = 2*N+1; |
---|
| 4326 | kill Lord, tmp, iv, RName; |
---|
| 4327 | list Lord, tmp; |
---|
| 4328 | intvec iv; |
---|
| 4329 | L[1] = RL[1]; |
---|
| 4330 | L[4] = RL[4]; // char, minpoly |
---|
| 4331 | // check whether vars hava admissible names -> done earlier |
---|
| 4332 | // now, create the names for new var |
---|
| 4333 | tmp[1] = "s"; |
---|
| 4334 | // DName is defined earlier |
---|
| 4335 | list NName = Name + DName + tmp; |
---|
| 4336 | L[2] = NName; |
---|
| 4337 | tmp = 0; |
---|
| 4338 | // block ord (dp(N),dp); |
---|
| 4339 | // string s is already defined |
---|
| 4340 | s = "iv="; |
---|
| 4341 | for (i=1; i<=Nnew-1; i++) |
---|
| 4342 | { |
---|
| 4343 | s = s+"1,"; |
---|
| 4344 | } |
---|
| 4345 | s[size(s)]=";"; |
---|
| 4346 | execute(s); |
---|
| 4347 | tmp[1] = "dp"; // string |
---|
| 4348 | tmp[2] = iv; // intvec |
---|
| 4349 | Lord[1] = tmp; |
---|
| 4350 | // continue with dp 1,1,1,1... |
---|
| 4351 | tmp[1] = "dp"; // string |
---|
| 4352 | s[size(s)] = ","; |
---|
| 4353 | s = s+"1;"; |
---|
| 4354 | execute(s); |
---|
| 4355 | kill s; |
---|
| 4356 | kill NName; |
---|
| 4357 | tmp[2] = iv; |
---|
| 4358 | Lord[2] = tmp; |
---|
| 4359 | tmp[1] = "C"; iv = 0; tmp[2]=iv; |
---|
| 4360 | Lord[3] = tmp; tmp = 0; |
---|
| 4361 | L[3] = Lord; |
---|
| 4362 | // we are done with the list. Now add a Plural part |
---|
| 4363 | def @R2@ = ring(L); |
---|
| 4364 | setring @R2@; |
---|
| 4365 | matrix @D[Nnew][Nnew]; |
---|
| 4366 | for (i=1; i<=N; i++) |
---|
| 4367 | { |
---|
| 4368 | @D[i,N+i]=1; |
---|
| 4369 | } |
---|
| 4370 | def @R2 = nc_algebra(1,@D); |
---|
| 4371 | setring @R2; |
---|
| 4372 | kill @R2@; |
---|
| 4373 | dbprint(ppl,"// -2-1- the ring @R2(_x,_Dx,s) is ready"); |
---|
| 4374 | dbprint(ppl-1, @R2); |
---|
| 4375 | ideal MM = maxideal(1); |
---|
| 4376 | // MM = 0,s,MM; |
---|
| 4377 | MM = 0,0,s,MM[1..size(MM)-1]; |
---|
| 4378 | map R01 = @R, MM; |
---|
| 4379 | ideal K = R01(K); |
---|
| 4380 | // total cleanup |
---|
| 4381 | ideal LD = K; |
---|
| 4382 | // make leadcoeffs positive |
---|
| 4383 | for (i=1; i<= ncols(LD); i++) |
---|
| 4384 | { |
---|
| 4385 | if (leadcoef(LD[i]) <0 ) |
---|
| 4386 | { |
---|
| 4387 | LD[i] = -LD[i]; |
---|
| 4388 | } |
---|
| 4389 | } |
---|
| 4390 | export LD; |
---|
| 4391 | kill @R; |
---|
| 4392 | return(@R2); |
---|
| 4393 | } |
---|
| 4394 | example |
---|
| 4395 | { |
---|
| 4396 | "EXAMPLE:"; echo = 2; |
---|
| 4397 | ring r = 0,(x,y,z),Dp; |
---|
| 4398 | poly F = x^3+y^3+z^3; |
---|
| 4399 | printlevel = 0; |
---|
| 4400 | def A = SannfsLOT(F); |
---|
| 4401 | setring A; |
---|
| 4402 | LD; |
---|
| 4403 | } |
---|
| 4404 | |
---|
| 4405 | /* |
---|
| 4406 | proc SannfsLOTold(poly F, list #) |
---|
| 4407 | "USAGE: SannfsLOT(f [,eng]); f a poly, eng an optional int |
---|
| 4408 | RETURN: ring |
---|
| 4409 | PURPOSE: compute the D-module structure of basering[1/f]*f^s, according to the Levandovskyy's modification of the algorithm by Oaku and Takayama in the ring D[s], where D is the Weyl algebra |
---|
| 4410 | NOTE: activate the output ring with the @code{setring} command. |
---|
| 4411 | @* In the ring D[s], the ideal LD (which is NOT a Groebner basis) is the needed D-module structure. |
---|
| 4412 | @* If eng <>0, @code{std} is used for Groebner basis computations, |
---|
| 4413 | @* otherwise, and by default @code{slimgb} is used. |
---|
| 4414 | @* If printlevel=1, progress debug messages will be printed, |
---|
| 4415 | @* if printlevel>=2, all the debug messages will be printed. |
---|
| 4416 | EXAMPLE: example SannfsLOT; shows examples |
---|
| 4417 | " |
---|
| 4418 | { |
---|
| 4419 | int eng = 0; |
---|
| 4420 | if ( size(#)>0 ) |
---|
| 4421 | { |
---|
| 4422 | if ( typeof(#[1]) == "int" ) |
---|
| 4423 | { |
---|
| 4424 | eng = int(#[1]); |
---|
| 4425 | } |
---|
| 4426 | } |
---|
| 4427 | // returns a list with a ring and an ideal LD in it |
---|
| 4428 | int ppl = printlevel-voice+2; |
---|
| 4429 | // printf("plevel :%s, voice: %s",printlevel,voice); |
---|
| 4430 | def save = basering; |
---|
| 4431 | int N = nvars(basering); |
---|
| 4432 | // int Nnew = 2*(N+2); |
---|
| 4433 | int Nnew = 2*(N+1)+1; //removed u,v; added s |
---|
| 4434 | int i,j; |
---|
| 4435 | string s; |
---|
| 4436 | list RL = ringlist(basering); |
---|
| 4437 | list L, Lord; |
---|
| 4438 | list tmp; |
---|
| 4439 | intvec iv; |
---|
| 4440 | L[1] = RL[1]; // char |
---|
| 4441 | L[4] = RL[4]; // char, minpoly |
---|
| 4442 | // check whether vars have admissible names |
---|
| 4443 | list Name = RL[2]; |
---|
| 4444 | list RName; |
---|
| 4445 | // RName[1] = "u"; |
---|
| 4446 | // RName[2] = "v"; |
---|
| 4447 | RName[1] = "t"; |
---|
| 4448 | RName[2] = "Dt"; |
---|
| 4449 | for(i=1;i<=N;i++) |
---|
| 4450 | { |
---|
| 4451 | for(j=1; j<=size(RName);j++) |
---|
| 4452 | { |
---|
| 4453 | if (Name[i] == RName[j]) |
---|
| 4454 | { |
---|
| 4455 | ERROR("Variable names should not include t,Dt"); |
---|
| 4456 | } |
---|
| 4457 | } |
---|
| 4458 | } |
---|
| 4459 | // now, create the names for new vars |
---|
| 4460 | // tmp[1] = "u"; |
---|
| 4461 | // tmp[2] = "v"; |
---|
| 4462 | // list UName = tmp; |
---|
| 4463 | list DName; |
---|
| 4464 | for(i=1;i<=N;i++) |
---|
| 4465 | { |
---|
| 4466 | DName[i] = "D"+Name[i]; // concat |
---|
| 4467 | } |
---|
| 4468 | tmp = 0; |
---|
| 4469 | tmp[1] = "t"; |
---|
| 4470 | tmp[2] = "Dt"; |
---|
| 4471 | list SName ; SName[1] = "s"; |
---|
| 4472 | // list NName = UName + tmp + Name + DName; |
---|
| 4473 | list NName = tmp + Name + DName + SName; |
---|
| 4474 | L[2] = NName; |
---|
| 4475 | tmp = 0; |
---|
| 4476 | // Name, Dname will be used further |
---|
| 4477 | // kill UName; |
---|
| 4478 | kill NName; |
---|
| 4479 | // block ord (a(1,1),dp); |
---|
| 4480 | tmp[1] = "a"; // string |
---|
| 4481 | iv = 1,1; |
---|
| 4482 | tmp[2] = iv; //intvec |
---|
| 4483 | Lord[1] = tmp; |
---|
| 4484 | // continue with dp 1,1,1,1... |
---|
| 4485 | tmp[1] = "dp"; // string |
---|
| 4486 | s = "iv="; |
---|
| 4487 | for(i=1;i<=Nnew;i++) |
---|
| 4488 | { |
---|
| 4489 | s = s+"1,"; |
---|
| 4490 | } |
---|
| 4491 | s[size(s)]= ";"; |
---|
| 4492 | execute(s); |
---|
| 4493 | tmp[2] = iv; |
---|
| 4494 | Lord[2] = tmp; |
---|
| 4495 | tmp[1] = "C"; |
---|
| 4496 | iv = 0; |
---|
| 4497 | tmp[2] = iv; |
---|
| 4498 | Lord[3] = tmp; |
---|
| 4499 | tmp = 0; |
---|
| 4500 | L[3] = Lord; |
---|
| 4501 | // we are done with the list |
---|
| 4502 | def @R@ = ring(L); |
---|
| 4503 | setring @R@; |
---|
| 4504 | matrix @D[Nnew][Nnew]; |
---|
| 4505 | @D[1,2]=1; |
---|
| 4506 | for(i=1; i<=N; i++) |
---|
| 4507 | { |
---|
| 4508 | @D[2+i,N+2+i]=1; |
---|
| 4509 | } |
---|
| 4510 | // ADD [s,t]=-t, [s,Dt]=Dt |
---|
| 4511 | @D[1,Nnew] = -var(1); |
---|
| 4512 | @D[2,Nnew] = var(2); |
---|
| 4513 | // @D[N+3,2*(N+2)]=1; old t,Dt stuff |
---|
| 4514 | // L[5] = matrix(UpOneMatrix(Nnew)); |
---|
| 4515 | // L[6] = @D; |
---|
| 4516 | def @R = nc_algebra(1,@D); |
---|
| 4517 | setring @R; |
---|
| 4518 | kill @R@; |
---|
| 4519 | dbprint(ppl,"// -1-1- the ring @R(t,Dt,_x,_Dx,s) is ready"); |
---|
| 4520 | dbprint(ppl-1, @R); |
---|
| 4521 | // create the ideal I |
---|
| 4522 | poly F = imap(save,F); |
---|
| 4523 | // ideal I = u*F-t,u*v-1; |
---|
| 4524 | ideal I = F-t; |
---|
| 4525 | poly p; |
---|
| 4526 | for(i=1; i<=N; i++) |
---|
| 4527 | { |
---|
| 4528 | // p = u*Dt; // u*Dt |
---|
| 4529 | p = Dt; |
---|
| 4530 | p = diff(F,var(2+i))*p; |
---|
| 4531 | I = I, var(N+2+i) + p; |
---|
| 4532 | } |
---|
| 4533 | // I = I, var(1)*var(2) + var(Nnew) +1; // reduce it with t-f!!! |
---|
| 4534 | // t*Dt + s +1 reduced with t-f gives f*Dt + s |
---|
| 4535 | I = I, F*var(2) + var(Nnew); |
---|
| 4536 | // -------- the ideal I is ready ---------- |
---|
| 4537 | dbprint(ppl,"// -1-2- starting the elimination of t,Dt in @R"); |
---|
| 4538 | dbprint(ppl-1, I); |
---|
| 4539 | ideal J = engine(I,eng); |
---|
| 4540 | ideal K = nselect(J,1..2); |
---|
| 4541 | dbprint(ppl,"// -1-3- t,Dt are eliminated"); |
---|
| 4542 | dbprint(ppl-1, K); // K is without t, Dt |
---|
| 4543 | K = engine(K,eng); // std does the job too |
---|
| 4544 | // now, we must change the ordering |
---|
| 4545 | // and create a ring without t, Dt |
---|
| 4546 | setring save; |
---|
| 4547 | // ----------- the ring @R3 ------------ |
---|
| 4548 | // _x, _Dx,s; elim.ord for _x,_Dx. |
---|
| 4549 | // keep: N, i,j,s, tmp, RL |
---|
| 4550 | Nnew = 2*N+1; |
---|
| 4551 | kill Lord, tmp, iv, RName; |
---|
| 4552 | list Lord, tmp; |
---|
| 4553 | intvec iv; |
---|
| 4554 | L[1] = RL[1]; |
---|
| 4555 | L[4] = RL[4]; // char, minpoly |
---|
| 4556 | // check whether vars hava admissible names -> done earlier |
---|
| 4557 | // now, create the names for new var |
---|
| 4558 | tmp[1] = "s"; |
---|
| 4559 | // DName is defined earlier |
---|
| 4560 | list NName = Name + DName + tmp; |
---|
| 4561 | L[2] = NName; |
---|
| 4562 | tmp = 0; |
---|
| 4563 | // block ord (dp(N),dp); |
---|
| 4564 | // string s is already defined |
---|
| 4565 | s = "iv="; |
---|
| 4566 | for (i=1; i<=Nnew-1; i++) |
---|
| 4567 | { |
---|
| 4568 | s = s+"1,"; |
---|
| 4569 | } |
---|
| 4570 | s[size(s)]=";"; |
---|
| 4571 | execute(s); |
---|
| 4572 | tmp[1] = "dp"; // string |
---|
| 4573 | tmp[2] = iv; // intvec |
---|
| 4574 | Lord[1] = tmp; |
---|
| 4575 | // continue with dp 1,1,1,1... |
---|
| 4576 | tmp[1] = "dp"; // string |
---|
| 4577 | s[size(s)] = ","; |
---|
| 4578 | s = s+"1;"; |
---|
| 4579 | execute(s); |
---|
| 4580 | kill s; |
---|
| 4581 | kill NName; |
---|
| 4582 | tmp[2] = iv; |
---|
| 4583 | Lord[2] = tmp; |
---|
| 4584 | tmp[1] = "C"; iv = 0; tmp[2]=iv; |
---|
| 4585 | Lord[3] = tmp; tmp = 0; |
---|
| 4586 | L[3] = Lord; |
---|
| 4587 | // we are done with the list. Now add a Plural part |
---|
| 4588 | def @R2@ = ring(L); |
---|
| 4589 | setring @R2@; |
---|
| 4590 | matrix @D[Nnew][Nnew]; |
---|
| 4591 | for (i=1; i<=N; i++) |
---|
| 4592 | { |
---|
| 4593 | @D[i,N+i]=1; |
---|
| 4594 | } |
---|
| 4595 | def @R2 = nc_algebra(1,@D); |
---|
| 4596 | setring @R2; |
---|
| 4597 | kill @R2@; |
---|
| 4598 | dbprint(ppl,"// -2-1- the ring @R2(_x,_Dx,s) is ready"); |
---|
| 4599 | dbprint(ppl-1, @R2); |
---|
| 4600 | ideal MM = maxideal(1); |
---|
| 4601 | MM = 0,s,MM; |
---|
| 4602 | map R01 = @R, MM; |
---|
| 4603 | ideal K = R01(K); |
---|
| 4604 | // total cleanup |
---|
| 4605 | ideal LD = K; |
---|
| 4606 | // make leadcoeffs positive |
---|
| 4607 | for (i=1; i<= ncols(LD); i++) |
---|
| 4608 | { |
---|
| 4609 | if (leadcoef(LD[i]) <0 ) |
---|
| 4610 | { |
---|
| 4611 | LD[i] = -LD[i]; |
---|
| 4612 | } |
---|
| 4613 | } |
---|
| 4614 | export LD; |
---|
| 4615 | kill @R; |
---|
| 4616 | return(@R2); |
---|
| 4617 | } |
---|
| 4618 | example |
---|
| 4619 | { |
---|
| 4620 | "EXAMPLE:"; echo = 2; |
---|
| 4621 | ring r = 0,(x,y,z),Dp; |
---|
| 4622 | poly F = x^3+y^3+z^3; |
---|
| 4623 | printlevel = 0; |
---|
| 4624 | def A = SannfsLOTold(F); |
---|
| 4625 | setring A; |
---|
| 4626 | LD; |
---|
| 4627 | } |
---|
| 4628 | |
---|
| 4629 | */ |
---|
| 4630 | |
---|
| 4631 | proc annfsLOT(poly F, list #) |
---|
| 4632 | "USAGE: annfsLOT(F [,eng]); F a poly, eng an optional int |
---|
| 4633 | RETURN: ring |
---|
[3f4e52] | 4634 | PURPOSE: compute the D-module structure of basering[1/f]*f^s, according to |
---|
[66c962] | 4635 | @* the Levandovskyy's modification of the algorithm by Oaku and Takayama |
---|
| 4636 | NOTE: activate the output ring with the @code{setring} command. In this ring, |
---|
| 4637 | @* - the ideal LD (which is a Groebner basis) is the needed D-module structure, |
---|
| 4638 | @* which is obtained by substituting the minimal integer root of a Bernstein |
---|
| 4639 | @* polynomial into the s-parametric ideal; |
---|
| 4640 | @* - the list BS contains the roots with multiplicities of BS polynomial of f. |
---|
| 4641 | @* If eng <>0, @code{std} is used for Groebner basis computations, |
---|
| 4642 | @* otherwise and by default @code{slimgb} is used. |
---|
| 4643 | @* If printlevel=1, progress debug messages will be printed, |
---|
| 4644 | @* if printlevel>=2, all the debug messages will be printed. |
---|
| 4645 | EXAMPLE: example annfsLOT; shows examples |
---|
| 4646 | " |
---|
| 4647 | { |
---|
| 4648 | int eng = 0; |
---|
| 4649 | if ( size(#)>0 ) |
---|
| 4650 | { |
---|
| 4651 | if ( typeof(#[1]) == "int" ) |
---|
| 4652 | { |
---|
| 4653 | eng = int(#[1]); |
---|
| 4654 | } |
---|
| 4655 | } |
---|
| 4656 | printlevel=printlevel+1; |
---|
| 4657 | def save = basering; |
---|
| 4658 | def @A = SannfsLOT(F,eng); |
---|
| 4659 | setring @A; |
---|
| 4660 | poly F = imap(save,F); |
---|
| 4661 | def B = annfs0(LD,F,eng); |
---|
| 4662 | return(B); |
---|
| 4663 | } |
---|
| 4664 | example |
---|
| 4665 | { |
---|
| 4666 | "EXAMPLE:"; echo = 2; |
---|
| 4667 | ring r = 0,(x,y,z),Dp; |
---|
| 4668 | poly F = z*x^2+y^3; |
---|
| 4669 | printlevel = 0; |
---|
| 4670 | def A = annfsLOT(F); |
---|
| 4671 | setring A; |
---|
| 4672 | LD; |
---|
| 4673 | BS; |
---|
| 4674 | } |
---|
| 4675 | |
---|
| 4676 | proc annfs0(ideal I, poly F, list #) |
---|
| 4677 | "USAGE: annfs0(I, F [,eng]); I an ideal, F a poly, eng an optional int |
---|
| 4678 | RETURN: ring |
---|
[3f4e52] | 4679 | PURPOSE: compute the annihilator ideal of f^s in the Weyl Algebra, based |
---|
[66c962] | 4680 | @* on the output of Sannfs-like procedure |
---|
| 4681 | NOTE: activate the output ring with the @code{setring} command. In this ring, |
---|
| 4682 | @* - the ideal LD (which is a Groebner basis) is the annihilator of f^s, |
---|
| 4683 | @* - the list BS contains the roots with multiplicities of BS polynomial of f. |
---|
| 4684 | @* If eng <>0, @code{std} is used for Groebner basis computations, |
---|
| 4685 | @* otherwise and by default @code{slimgb} is used. |
---|
| 4686 | @* If printlevel=1, progress debug messages will be printed, |
---|
| 4687 | @* if printlevel>=2, all the debug messages will be printed. |
---|
| 4688 | EXAMPLE: example annfs0; shows examples |
---|
| 4689 | " |
---|
| 4690 | { |
---|
| 4691 | int eng = 0; |
---|
| 4692 | if ( size(#)>0 ) |
---|
| 4693 | { |
---|
| 4694 | if ( typeof(#[1]) == "int" ) |
---|
| 4695 | { |
---|
| 4696 | eng = int(#[1]); |
---|
| 4697 | } |
---|
| 4698 | } |
---|
| 4699 | def @R2 = basering; |
---|
| 4700 | // we're in D_n[s], where the elim ord for s is set |
---|
| 4701 | ideal J = NF(I,std(F)); |
---|
| 4702 | // make leadcoeffs positive |
---|
| 4703 | int i; |
---|
| 4704 | for (i=1; i<= ncols(J); i++) |
---|
| 4705 | { |
---|
| 4706 | if (leadcoef(J[i]) <0 ) |
---|
| 4707 | { |
---|
| 4708 | J[i] = -J[i]; |
---|
| 4709 | } |
---|
| 4710 | } |
---|
| 4711 | J = J,F; |
---|
| 4712 | ideal M = engine(J,eng); |
---|
| 4713 | int Nnew = nvars(@R2); |
---|
| 4714 | ideal K2 = nselect(M,1..Nnew-1); |
---|
| 4715 | int ppl = printlevel-voice+2; |
---|
| 4716 | dbprint(ppl,"// -1-1- _x,_Dx are eliminated in basering"); |
---|
| 4717 | dbprint(ppl-1, K2); |
---|
| 4718 | // the ring @R3 and the search for minimal negative int s |
---|
| 4719 | ring @R3 = 0,s,dp; |
---|
| 4720 | dbprint(ppl,"// -2-1- the ring @R3 i.e. K[s] is ready"); |
---|
| 4721 | ideal K3 = imap(@R2,K2); |
---|
| 4722 | poly p = K3[1]; |
---|
| 4723 | dbprint(ppl,"// -2-2- factorization"); |
---|
| 4724 | // ideal P = factorize(p,1); //without constants and multiplicities |
---|
| 4725 | // "--------- b-function factorizes into ---------"; P; |
---|
| 4726 | // convert factors to the list of their roots with mults |
---|
| 4727 | // assume all factors are linear |
---|
| 4728 | // ideal BS = normalize(P); |
---|
| 4729 | // BS = subst(BS,s,0); |
---|
| 4730 | // BS = -BS; |
---|
| 4731 | list P = factorize(p); //with constants and multiplicities |
---|
| 4732 | ideal bs; intvec m; //the Bernstein polynomial is monic, so we are not interested in constants |
---|
| 4733 | for (i=2; i<= size(P[1]); i++) //we delete P[1][1] and P[2][1] |
---|
| 4734 | { |
---|
| 4735 | bs[i-1] = P[1][i]; |
---|
| 4736 | m[i-1] = P[2][i]; |
---|
| 4737 | } |
---|
| 4738 | int sP = minIntRoot(bs,1); |
---|
| 4739 | bs = normalize(bs); |
---|
| 4740 | bs = -subst(bs,s,0); |
---|
| 4741 | dbprint(ppl,"// -2-3- minimal integer root found"); |
---|
| 4742 | dbprint(ppl-1, sP); |
---|
| 4743 | //TODO: sort BS! |
---|
| 4744 | // --------- substitute s found in the ideal --------- |
---|
| 4745 | // --------- going back to @R and substitute --------- |
---|
| 4746 | setring @R2; |
---|
| 4747 | K2 = subst(I,s,sP); |
---|
| 4748 | // create the ordinary Weyl algebra and put the result into it, |
---|
| 4749 | // thus creating the ring @R5 |
---|
| 4750 | // keep: N, i,j,s, tmp, RL |
---|
| 4751 | Nnew = Nnew - 1; // former 2*N; |
---|
| 4752 | // list RL = ringlist(save); // is defined earlier |
---|
| 4753 | // kill Lord, tmp, iv; |
---|
| 4754 | list L = 0; |
---|
| 4755 | list Lord, tmp; |
---|
| 4756 | intvec iv; |
---|
| 4757 | list RL = ringlist(basering); |
---|
| 4758 | L[1] = RL[1]; |
---|
| 4759 | L[4] = RL[4]; //char, minpoly |
---|
| 4760 | // check whether vars have admissible names -> done earlier |
---|
| 4761 | // list Name = RL[2]M |
---|
| 4762 | // DName is defined earlier |
---|
| 4763 | list NName; // = RL[2]; // skip the last var 's' |
---|
| 4764 | for (i=1; i<=Nnew; i++) |
---|
| 4765 | { |
---|
| 4766 | NName[i] = RL[2][i]; |
---|
| 4767 | } |
---|
| 4768 | L[2] = NName; |
---|
| 4769 | // dp ordering; |
---|
| 4770 | string s = "iv="; |
---|
| 4771 | for (i=1; i<=Nnew; i++) |
---|
| 4772 | { |
---|
| 4773 | s = s+"1,"; |
---|
| 4774 | } |
---|
| 4775 | s[size(s)] = ";"; |
---|
| 4776 | execute(s); |
---|
| 4777 | tmp = 0; |
---|
| 4778 | tmp[1] = "dp"; // string |
---|
| 4779 | tmp[2] = iv; // intvec |
---|
| 4780 | Lord[1] = tmp; |
---|
| 4781 | kill s; |
---|
| 4782 | tmp[1] = "C"; |
---|
| 4783 | iv = 0; |
---|
| 4784 | tmp[2] = iv; |
---|
| 4785 | Lord[2] = tmp; |
---|
| 4786 | tmp = 0; |
---|
| 4787 | L[3] = Lord; |
---|
| 4788 | // we are done with the list |
---|
| 4789 | // Add: Plural part |
---|
| 4790 | def @R4@ = ring(L); |
---|
| 4791 | setring @R4@; |
---|
| 4792 | int N = Nnew/2; |
---|
| 4793 | matrix @D[Nnew][Nnew]; |
---|
| 4794 | for (i=1; i<=N; i++) |
---|
| 4795 | { |
---|
| 4796 | @D[i,N+i]=1; |
---|
| 4797 | } |
---|
| 4798 | def @R4 = nc_algebra(1,@D); |
---|
| 4799 | setring @R4; |
---|
| 4800 | kill @R4@; |
---|
| 4801 | dbprint(ppl,"// -3-1- the ring @R4 is ready"); |
---|
| 4802 | dbprint(ppl-1, @R4); |
---|
| 4803 | ideal K4 = imap(@R2,K2); |
---|
| 4804 | option(redSB); |
---|
| 4805 | dbprint(ppl,"// -3-2- the final cosmetic std"); |
---|
| 4806 | K4 = engine(K4,eng); // std does the job too |
---|
| 4807 | // total cleanup |
---|
| 4808 | ideal bs = imap(@R3,bs); |
---|
| 4809 | kill @R3; |
---|
| 4810 | list BS = bs,m; |
---|
| 4811 | export BS; |
---|
| 4812 | ideal LD = K4; |
---|
| 4813 | export LD; |
---|
| 4814 | return(@R4); |
---|
| 4815 | } |
---|
| 4816 | example |
---|
| 4817 | { "EXAMPLE:"; echo = 2; |
---|
| 4818 | ring r = 0,(x,y,z),Dp; |
---|
| 4819 | poly F = x^3+y^3+z^3; |
---|
| 4820 | printlevel = 0; |
---|
| 4821 | def A = SannfsBM(F); setring A; |
---|
| 4822 | // alternatively, one can use SannfsOT or SannfsLOT |
---|
| 4823 | LD; |
---|
| 4824 | poly F = imap(r,F); |
---|
| 4825 | def B = annfs0(LD,F); setring B; |
---|
| 4826 | LD; |
---|
| 4827 | BS; |
---|
| 4828 | } |
---|
| 4829 | |
---|
| 4830 | // proc annfsgms(poly F, list #) |
---|
| 4831 | // "USAGE: annfsgms(f [,eng]); f a poly, eng an optional int |
---|
| 4832 | // ASSUME: f has an isolated critical point at 0 |
---|
| 4833 | // RETURN: ring |
---|
| 4834 | // PURPOSE: compute the D-module structure of basering[1/f]*f^s |
---|
| 4835 | // NOTE: activate the output ring with the @code{setring} command. In this ring, |
---|
| 4836 | // @* - the ideal LD is the needed D-mod structure, |
---|
| 4837 | // @* - the ideal BS is the list of roots of a Bernstein polynomial of f. |
---|
| 4838 | // @* If eng <>0, @code{std} is used for Groebner basis computations, |
---|
| 4839 | // @* otherwise (and by default) @code{slimgb} is used. |
---|
| 4840 | // @* If printlevel=1, progress debug messages will be printed, |
---|
| 4841 | // @* if printlevel>=2, all the debug messages will be printed. |
---|
| 4842 | // EXAMPLE: example annfsgms; shows examples |
---|
| 4843 | // " |
---|
| 4844 | // { |
---|
| 4845 | // LIB "gmssing.lib"; |
---|
| 4846 | // int eng = 0; |
---|
| 4847 | // if ( size(#)>0 ) |
---|
| 4848 | // { |
---|
| 4849 | // if ( typeof(#[1]) == "int" ) |
---|
| 4850 | // { |
---|
| 4851 | // eng = int(#[1]); |
---|
| 4852 | // } |
---|
| 4853 | // } |
---|
| 4854 | // int ppl = printlevel-voice+2; |
---|
| 4855 | // // returns a ring with the ideal LD in it |
---|
| 4856 | // def save = basering; |
---|
| 4857 | // // compute the Bernstein polynomial from gmssing.lib |
---|
| 4858 | // list RL = ringlist(basering); |
---|
| 4859 | // // in the descr. of the ordering, replace "p" by "s" |
---|
| 4860 | // list NL = convloc(RL); |
---|
| 4861 | // // create a ring with the ordering, converted to local |
---|
| 4862 | // def @LR = ring(NL); |
---|
| 4863 | // setring @LR; |
---|
| 4864 | // poly F = imap(save, F); |
---|
| 4865 | // ideal B = bernstein(F)[1]; |
---|
| 4866 | // // since B may not contain (s+1) [following gmssing.lib] |
---|
| 4867 | // // add it! |
---|
| 4868 | // B = B,-1; |
---|
| 4869 | // B = simplify(B,2+4); // erase zero and repeated entries |
---|
| 4870 | // // find the minimal integer value |
---|
| 4871 | // int S = minIntRoot(B,0); |
---|
| 4872 | // dbprint(ppl,"// -0- minimal integer root found"); |
---|
| 4873 | // dbprint(ppl-1,S); |
---|
| 4874 | // setring save; |
---|
| 4875 | // int N = nvars(basering); |
---|
| 4876 | // int Nnew = 2*(N+2); |
---|
| 4877 | // int i,j; |
---|
| 4878 | // string s; |
---|
| 4879 | // // list RL = ringlist(basering); |
---|
| 4880 | // list L, Lord; |
---|
| 4881 | // list tmp; |
---|
| 4882 | // intvec iv; |
---|
| 4883 | // L[1] = RL[1]; // char |
---|
| 4884 | // L[4] = RL[4]; // char, minpoly |
---|
| 4885 | // // check whether vars have admissible names |
---|
| 4886 | // list Name = RL[2]; |
---|
| 4887 | // list RName; |
---|
| 4888 | // RName[1] = "u"; |
---|
| 4889 | // RName[2] = "v"; |
---|
| 4890 | // RName[3] = "t"; |
---|
| 4891 | // RName[4] = "Dt"; |
---|
| 4892 | // for(i=1;i<=N;i++) |
---|
| 4893 | // { |
---|
| 4894 | // for(j=1; j<=size(RName);j++) |
---|
| 4895 | // { |
---|
| 4896 | // if (Name[i] == RName[j]) |
---|
| 4897 | // { |
---|
| 4898 | // ERROR("Variable names should not include u,v,t,Dt"); |
---|
| 4899 | // } |
---|
| 4900 | // } |
---|
| 4901 | // } |
---|
| 4902 | // // now, create the names for new vars |
---|
| 4903 | // // tmp[1] = "u"; tmp[2] = "v"; tmp[3] = "t"; tmp[4] = "Dt"; |
---|
| 4904 | // list UName = RName; |
---|
| 4905 | // list DName; |
---|
| 4906 | // for(i=1;i<=N;i++) |
---|
| 4907 | // { |
---|
| 4908 | // DName[i] = "D"+Name[i]; // concat |
---|
| 4909 | // } |
---|
| 4910 | // list NName = UName + Name + DName; |
---|
| 4911 | // L[2] = NName; |
---|
| 4912 | // tmp = 0; |
---|
| 4913 | // // Name, Dname will be used further |
---|
| 4914 | // kill UName; |
---|
| 4915 | // kill NName; |
---|
| 4916 | // // block ord (a(1,1),dp); |
---|
| 4917 | // tmp[1] = "a"; // string |
---|
| 4918 | // iv = 1,1; |
---|
| 4919 | // tmp[2] = iv; //intvec |
---|
| 4920 | // Lord[1] = tmp; |
---|
| 4921 | // // continue with dp 1,1,1,1... |
---|
| 4922 | // tmp[1] = "dp"; // string |
---|
| 4923 | // s = "iv="; |
---|
| 4924 | // for(i=1; i<=Nnew; i++) // need really all vars! |
---|
| 4925 | // { |
---|
| 4926 | // s = s+"1,"; |
---|
| 4927 | // } |
---|
| 4928 | // s[size(s)]= ";"; |
---|
| 4929 | // execute(s); |
---|
| 4930 | // tmp[2] = iv; |
---|
| 4931 | // Lord[2] = tmp; |
---|
| 4932 | // tmp[1] = "C"; |
---|
| 4933 | // iv = 0; |
---|
| 4934 | // tmp[2] = iv; |
---|
| 4935 | // Lord[3] = tmp; |
---|
| 4936 | // tmp = 0; |
---|
| 4937 | // L[3] = Lord; |
---|
| 4938 | // // we are done with the list |
---|
| 4939 | // def @R = ring(L); |
---|
| 4940 | // setring @R; |
---|
| 4941 | // matrix @D[Nnew][Nnew]; |
---|
| 4942 | // @D[3,4] = 1; // t,Dt |
---|
| 4943 | // for(i=1; i<=N; i++) |
---|
| 4944 | // { |
---|
| 4945 | // @D[4+i,4+N+i]=1; |
---|
| 4946 | // } |
---|
| 4947 | // // L[5] = matrix(UpOneMatrix(Nnew)); |
---|
| 4948 | // // L[6] = @D; |
---|
| 4949 | // nc_algebra(1,@D); |
---|
| 4950 | // dbprint(ppl,"// -1-1- the ring @R is ready"); |
---|
| 4951 | // dbprint(ppl-1,@R); |
---|
| 4952 | // // create the ideal |
---|
| 4953 | // poly F = imap(save,F); |
---|
| 4954 | // ideal I = u*F-t,u*v-1; |
---|
| 4955 | // poly p; |
---|
| 4956 | // for(i=1; i<=N; i++) |
---|
| 4957 | // { |
---|
| 4958 | // p = u*Dt; // u*Dt |
---|
| 4959 | // p = diff(F,var(4+i))*p; |
---|
| 4960 | // I = I, var(N+4+i) + p; // Dx, Dy |
---|
| 4961 | // } |
---|
| 4962 | // // add the relations between t,Dt and s |
---|
| 4963 | // // I = I, t*Dt+1+S; |
---|
| 4964 | // // -------- the ideal I is ready ---------- |
---|
| 4965 | // dbprint(ppl,"// -1-2- starting the elimination of u,v in @R"); |
---|
| 4966 | // ideal J = engine(I,eng); |
---|
| 4967 | // ideal K = nselect(J,1..2); |
---|
| 4968 | // dbprint(ppl,"// -1-3- u,v are eliminated in @R"); |
---|
| 4969 | // dbprint(ppl-1,K); // without u,v: not yet our answer |
---|
| 4970 | // //----- create a ring with elim.ord for t,Dt ------- |
---|
| 4971 | // setring save; |
---|
| 4972 | // // ------------ new ring @R2 ------------------ |
---|
| 4973 | // // without u,v and with the elim.ord for t,Dt |
---|
| 4974 | // // keep: N, i,j,s, tmp, RL |
---|
| 4975 | // Nnew = 2*N+2; |
---|
| 4976 | // // list RL = ringlist(save); // is defined earlier |
---|
| 4977 | // kill Lord,tmp,iv, RName; |
---|
| 4978 | // L = 0; |
---|
| 4979 | // list Lord, tmp; |
---|
| 4980 | // intvec iv; |
---|
| 4981 | // L[1] = RL[1]; // char |
---|
| 4982 | // L[4] = RL[4]; // char, minpoly |
---|
| 4983 | // // check whether vars have admissible names -> done earlier |
---|
| 4984 | // // list Name = RL[2]; |
---|
| 4985 | // list RName; |
---|
| 4986 | // RName[1] = "t"; |
---|
| 4987 | // RName[2] = "Dt"; |
---|
| 4988 | // // DName is defined earlier |
---|
| 4989 | // list NName = RName + Name + DName; |
---|
| 4990 | // L[2] = NName; |
---|
| 4991 | // tmp = 0; |
---|
| 4992 | // // block ord (a(1,1),dp); |
---|
| 4993 | // tmp[1] = "a"; // string |
---|
| 4994 | // iv = 1,1; |
---|
| 4995 | // tmp[2] = iv; //intvec |
---|
| 4996 | // Lord[1] = tmp; |
---|
| 4997 | // // continue with dp 1,1,1,1... |
---|
| 4998 | // tmp[1] = "dp"; // string |
---|
| 4999 | // s = "iv="; |
---|
| 5000 | // for(i=1;i<=Nnew;i++) |
---|
| 5001 | // { |
---|
| 5002 | // s = s+"1,"; |
---|
| 5003 | // } |
---|
| 5004 | // s[size(s)]= ";"; |
---|
| 5005 | // execute(s); |
---|
| 5006 | // kill s; |
---|
| 5007 | // kill NName; |
---|
| 5008 | // tmp[2] = iv; |
---|
| 5009 | // Lord[2] = tmp; |
---|
| 5010 | // tmp[1] = "C"; |
---|
| 5011 | // iv = 0; |
---|
| 5012 | // tmp[2] = iv; |
---|
| 5013 | // Lord[3] = tmp; |
---|
| 5014 | // tmp = 0; |
---|
| 5015 | // L[3] = Lord; |
---|
| 5016 | // // we are done with the list |
---|
| 5017 | // // Add: Plural part |
---|
| 5018 | // def @R2 = ring(L); |
---|
| 5019 | // setring @R2; |
---|
| 5020 | // matrix @D[Nnew][Nnew]; |
---|
| 5021 | // @D[1,2]=1; |
---|
| 5022 | // for(i=1; i<=N; i++) |
---|
| 5023 | // { |
---|
| 5024 | // @D[2+i,2+N+i]=1; |
---|
| 5025 | // } |
---|
| 5026 | // nc_algebra(1,@D); |
---|
| 5027 | // dbprint(ppl,"// -2-1- the ring @R2 is ready"); |
---|
| 5028 | // dbprint(ppl-1,@R2); |
---|
| 5029 | // ideal MM = maxideal(1); |
---|
| 5030 | // MM = 0,0,MM; |
---|
| 5031 | // map R01 = @R, MM; |
---|
| 5032 | // ideal K2 = R01(K); |
---|
| 5033 | // // add the relations between t,Dt and s |
---|
| 5034 | // // K2 = K2, t*Dt+1+S; |
---|
| 5035 | // poly G = t*Dt+S+1; |
---|
| 5036 | // K2 = NF(K2,std(G)),G; |
---|
| 5037 | // dbprint(ppl,"// -2-2- starting elimination for t,Dt in @R2"); |
---|
| 5038 | // ideal J = engine(K2,eng); |
---|
| 5039 | // ideal K = nselect(J,1..2); |
---|
| 5040 | // dbprint(ppl,"// -2-3- t,Dt are eliminated"); |
---|
| 5041 | // dbprint(ppl-1,K); |
---|
| 5042 | // // "------- produce a final result --------"; |
---|
| 5043 | // // ----- create the ordinary Weyl algebra and put |
---|
| 5044 | // // ----- the result into it |
---|
| 5045 | // // ------ create the ring @R5 |
---|
| 5046 | // // keep: N, i,j,s, tmp, RL |
---|
| 5047 | // setring save; |
---|
| 5048 | // Nnew = 2*N; |
---|
| 5049 | // // list RL = ringlist(save); // is defined earlier |
---|
| 5050 | // kill Lord, tmp, iv; |
---|
| 5051 | // // kill L; |
---|
| 5052 | // L = 0; |
---|
| 5053 | // list Lord, tmp; |
---|
| 5054 | // intvec iv; |
---|
| 5055 | // L[1] = RL[1]; // char |
---|
| 5056 | // L[4] = RL[4]; // char, minpoly |
---|
| 5057 | // // check whether vars have admissible names -> done earlier |
---|
| 5058 | // // list Name = RL[2]; |
---|
| 5059 | // // DName is defined earlier |
---|
| 5060 | // list NName = Name + DName; |
---|
| 5061 | // L[2] = NName; |
---|
| 5062 | // // dp ordering; |
---|
| 5063 | // string s = "iv="; |
---|
| 5064 | // for(i=1;i<=2*N;i++) |
---|
| 5065 | // { |
---|
| 5066 | // s = s+"1,"; |
---|
| 5067 | // } |
---|
| 5068 | // s[size(s)]= ";"; |
---|
| 5069 | // execute(s); |
---|
| 5070 | // tmp = 0; |
---|
| 5071 | // tmp[1] = "dp"; // string |
---|
| 5072 | // tmp[2] = iv; //intvec |
---|
| 5073 | // Lord[1] = tmp; |
---|
| 5074 | // kill s; |
---|
| 5075 | // tmp[1] = "C"; |
---|
| 5076 | // iv = 0; |
---|
| 5077 | // tmp[2] = iv; |
---|
| 5078 | // Lord[2] = tmp; |
---|
| 5079 | // tmp = 0; |
---|
| 5080 | // L[3] = Lord; |
---|
| 5081 | // // we are done with the list |
---|
| 5082 | // // Add: Plural part |
---|
| 5083 | // def @R5 = ring(L); |
---|
| 5084 | // setring @R5; |
---|
| 5085 | // matrix @D[Nnew][Nnew]; |
---|
| 5086 | // for(i=1; i<=N; i++) |
---|
| 5087 | // { |
---|
| 5088 | // @D[i,N+i]=1; |
---|
| 5089 | // } |
---|
| 5090 | // nc_algebra(1,@D); |
---|
| 5091 | // dbprint(ppl,"// -3-1- the ring @R5 is ready"); |
---|
| 5092 | // dbprint(ppl-1,@R5); |
---|
| 5093 | // ideal K5 = imap(@R2,K); |
---|
| 5094 | // option(redSB); |
---|
| 5095 | // dbprint(ppl,"// -3-2- the final cosmetic std"); |
---|
| 5096 | // K5 = engine(K5,eng); // std does the job too |
---|
| 5097 | // // total cleanup |
---|
| 5098 | // kill @R; |
---|
| 5099 | // kill @R2; |
---|
| 5100 | // ideal LD = K5; |
---|
| 5101 | // ideal BS = imap(@LR,B); |
---|
| 5102 | // kill @LR; |
---|
| 5103 | // export BS; |
---|
| 5104 | // export LD; |
---|
| 5105 | // return(@R5); |
---|
| 5106 | // } |
---|
| 5107 | // example |
---|
| 5108 | // { |
---|
| 5109 | // "EXAMPLE:"; echo = 2; |
---|
| 5110 | // ring r = 0,(x,y,z),Dp; |
---|
| 5111 | // poly F = x^2+y^3+z^5; |
---|
| 5112 | // def A = annfsgms(F); |
---|
| 5113 | // setring A; |
---|
| 5114 | // LD; |
---|
| 5115 | // print(matrix(BS)); |
---|
| 5116 | // } |
---|
| 5117 | |
---|
| 5118 | |
---|
| 5119 | proc convloc(list @NL) |
---|
| 5120 | "USAGE: convloc(L); L a list |
---|
| 5121 | RETURN: list |
---|
| 5122 | PURPOSE: convert a ringlist L into another ringlist, |
---|
| 5123 | @* where all the 'p' orderings are replaced with the 's' orderings, e.g. @code{dp} by @code{ds}. |
---|
| 5124 | ASSUME: L is a result of a ringlist command |
---|
| 5125 | EXAMPLE: example convloc; shows examples |
---|
| 5126 | " |
---|
| 5127 | { |
---|
| 5128 | list NL = @NL; |
---|
| 5129 | // given a ringlist, returns a new ringlist, |
---|
| 5130 | // where all the p-orderings are replaced with s-ord's |
---|
| 5131 | int i,j,k,found; |
---|
| 5132 | int nblocks = size(NL[3]); |
---|
| 5133 | for(i=1; i<=nblocks; i++) |
---|
| 5134 | { |
---|
| 5135 | for(j=1; j<=size(NL[3][i]); j++) |
---|
| 5136 | { |
---|
| 5137 | if (typeof(NL[3][i][j]) == "string" ) |
---|
| 5138 | { |
---|
| 5139 | found = 0; |
---|
| 5140 | for (k=1; k<=size(NL[3][i][j]); k++) |
---|
| 5141 | { |
---|
| 5142 | if (NL[3][i][j][k]=="p") |
---|
| 5143 | { |
---|
| 5144 | NL[3][i][j][k]="s"; |
---|
| 5145 | found = 1; |
---|
| 5146 | // printf("replaced at %s,%s,%s",i,j,k); |
---|
| 5147 | } |
---|
| 5148 | } |
---|
| 5149 | } |
---|
| 5150 | } |
---|
| 5151 | } |
---|
| 5152 | return(NL); |
---|
| 5153 | } |
---|
| 5154 | example |
---|
| 5155 | { |
---|
| 5156 | "EXAMPLE:"; echo = 2; |
---|
| 5157 | ring r = 0,(x,y,z),(Dp(2),dp(1)); |
---|
| 5158 | list L = ringlist(r); |
---|
| 5159 | list N = convloc(L); |
---|
| 5160 | def rs = ring(N); |
---|
| 5161 | setring rs; |
---|
| 5162 | rs; |
---|
| 5163 | } |
---|
| 5164 | |
---|
| 5165 | proc annfspecial(ideal I, poly F, int mir, number n) |
---|
| 5166 | "USAGE: annfspecial(I,F,mir,n); I an ideal, F a poly, int mir, number n |
---|
| 5167 | RETURN: ideal |
---|
[3f4e52] | 5168 | PURPOSE: compute the annihilator ideal of F^n in the Weyl Algebra |
---|
[66c962] | 5169 | @* for the given rational number n |
---|
| 5170 | ASSUME: The basering is D[s] and contains 's' explicitly as a variable, |
---|
| 5171 | @* the ideal I is the Ann F^s in D[s] (obtained with e.g. SannfsBM), |
---|
| 5172 | @* the integer 'mir' is the minimal integer root of the BS polynomial of F, |
---|
| 5173 | @* and the number n is rational. |
---|
[3f4e52] | 5174 | NOTE: We compute the real annihilator for any rational value of n (both |
---|
| 5175 | @* generic and exceptional). The implementation goes along the lines of |
---|
| 5176 | @* the Algorithm 5.3.15 from Saito-Sturmfels-Takayama. |
---|
[66c962] | 5177 | DISPLAY: If printlevel=1, progress debug messages will be printed, |
---|
| 5178 | @* if printlevel>=2, all the debug messages will be printed. |
---|
| 5179 | EXAMPLE: example annfspecial; shows examples |
---|
| 5180 | " |
---|
| 5181 | { |
---|
| 5182 | |
---|
| 5183 | if (!isRational(n)) |
---|
| 5184 | { |
---|
| 5185 | "ERROR: n must be a rational number!"; |
---|
| 5186 | } |
---|
| 5187 | int ppl = printlevel-voice+2; |
---|
| 5188 | // int mir = minIntRoot(L[1],0); |
---|
| 5189 | int ns = varNum("s"); |
---|
| 5190 | if (!ns) |
---|
| 5191 | { |
---|
| 5192 | ERROR("Variable s expected in the ideal AnnFs"); |
---|
| 5193 | } |
---|
| 5194 | int d; |
---|
| 5195 | ideal P = subst(I,var(ns),n); |
---|
| 5196 | if (denominator(n) == 1) |
---|
| 5197 | { |
---|
| 5198 | // n is fraction-free |
---|
| 5199 | d = int(numerator(n)); |
---|
| 5200 | if ( (!d) && (n!=0)) |
---|
| 5201 | { |
---|
| 5202 | ERROR("no parametric special values are allowed"); |
---|
| 5203 | } |
---|
| 5204 | d = d - mir; |
---|
| 5205 | if (d>0) |
---|
| 5206 | { |
---|
| 5207 | dbprint(ppl,"// -1-1- starting syzygy computations"); |
---|
| 5208 | matrix J[1][1] = F^d; |
---|
| 5209 | dbprint(ppl-1,"// -1-1-1- of the polynomial ideal"); |
---|
| 5210 | dbprint(ppl-1,J); |
---|
| 5211 | matrix K[1][size(I)] = subst(I,var(ns),mir); |
---|
| 5212 | dbprint(ppl-1,"// -1-1-2- modulo ideal:"); |
---|
| 5213 | dbprint(ppl-1, K); |
---|
| 5214 | module M = modulo(J,K); |
---|
| 5215 | dbprint(ppl-1,"// -1-1-3- getting the result:"); |
---|
| 5216 | dbprint(ppl-1, M); |
---|
| 5217 | P = P, ideal(M); |
---|
| 5218 | dbprint(ppl,"// -1-2- finished syzygy computations"); |
---|
| 5219 | } |
---|
| 5220 | else |
---|
| 5221 | { |
---|
| 5222 | dbprint(ppl,"// -1-1- d<=0, no syzygy computations needed"); |
---|
| 5223 | dbprint(ppl-1,"// -1-2- for d ="); |
---|
| 5224 | dbprint(ppl-1, d); |
---|
| 5225 | } |
---|
| 5226 | } |
---|
| 5227 | // also the else case: d<=0 or n is not an integer |
---|
| 5228 | dbprint(ppl,"// -2-1- starting final Groebner basis"); |
---|
| 5229 | P = groebner(P); |
---|
| 5230 | dbprint(ppl,"// -2-2- finished final Groebner basis"); |
---|
| 5231 | return(P); |
---|
| 5232 | } |
---|
| 5233 | example |
---|
| 5234 | { |
---|
| 5235 | "EXAMPLE:"; echo = 2; |
---|
| 5236 | ring r = 0,(x,y),dp; |
---|
| 5237 | poly F = x3-y2; |
---|
| 5238 | def B = annfs(F); setring B; |
---|
| 5239 | minIntRoot(BS[1],0); |
---|
| 5240 | // So, the minimal integer root is -1 |
---|
| 5241 | setring r; |
---|
| 5242 | def A = SannfsBM(F); |
---|
| 5243 | setring A; |
---|
| 5244 | poly F = x3-y2; |
---|
| 5245 | annfspecial(LD,F,-1,3/4); // generic root |
---|
| 5246 | annfspecial(LD,F,-1,-2); // integer but still generic root |
---|
| 5247 | annfspecial(LD,F,-1,1); // exceptional root |
---|
| 5248 | } |
---|
| 5249 | |
---|
| 5250 | /* |
---|
| 5251 | //static proc new_ex_annfspecial() |
---|
| 5252 | { |
---|
| 5253 | //another example for annfspecial: x3+y3+z3 |
---|
| 5254 | ring r = 0,(x,y,z),dp; |
---|
| 5255 | poly F = x3+y3+z3; |
---|
| 5256 | def B = annfs(F); setring B; |
---|
| 5257 | minIntRoot(BS[1],0); |
---|
| 5258 | // So, the minimal integer root is -1 |
---|
| 5259 | setring r; |
---|
| 5260 | def A = SannfsBM(F); |
---|
| 5261 | setring A; |
---|
| 5262 | poly F = x3+y3+z3; |
---|
| 5263 | annfspecial(LD,F,-1,3/4); // generic root |
---|
| 5264 | annfspecial(LD,F,-1,-2); // integer but still generic root |
---|
| 5265 | annfspecial(LD,F,-1,1); // exceptional root |
---|
| 5266 | } |
---|
| 5267 | */ |
---|
| 5268 | |
---|
| 5269 | proc minIntRoot(ideal P, int fact) |
---|
| 5270 | "USAGE: minIntRoot(P, fact); P an ideal, fact an int |
---|
| 5271 | RETURN: int |
---|
| 5272 | PURPOSE: minimal integer root of a maximal ideal P |
---|
| 5273 | NOTE: if fact==1, P is the result of some 'factorize' call, |
---|
| 5274 | @* else P is treated as the result of bernstein::gmssing.lib |
---|
| 5275 | @* in both cases without constants and multiplicities |
---|
| 5276 | EXAMPLE: example minIntRoot; shows examples |
---|
| 5277 | " |
---|
| 5278 | { |
---|
| 5279 | // ideal P = factorize(p,1); |
---|
| 5280 | // or ideal P = bernstein(F)[1]; |
---|
| 5281 | intvec vP; |
---|
| 5282 | number nP; |
---|
| 5283 | int i; |
---|
| 5284 | if ( fact ) |
---|
| 5285 | { |
---|
| 5286 | // the result comes from "factorize" |
---|
| 5287 | P = normalize(P); // now leadcoef = 1 |
---|
| 5288 | // TODO: hunt for units and kill then !!! |
---|
[867e1a3] | 5289 | P = matrix(lead(P))-P; |
---|
[66c962] | 5290 | // nP = leadcoef(P[i]-lead(P[i])); // for 1 var only, extract the coeff |
---|
| 5291 | } |
---|
| 5292 | else |
---|
| 5293 | { |
---|
| 5294 | // bernstein deletes -1 usually |
---|
| 5295 | P = P, -1; |
---|
| 5296 | } |
---|
| 5297 | // for both situations: |
---|
| 5298 | // now we have an ideal of fractions of type "number" |
---|
| 5299 | int sP = size(P); |
---|
| 5300 | for(i=1; i<=sP; i++) |
---|
| 5301 | { |
---|
| 5302 | nP = leadcoef(P[i]); |
---|
| 5303 | if ( (nP - int(nP)) == 0 ) |
---|
| 5304 | { |
---|
| 5305 | vP = vP,int(nP); |
---|
| 5306 | } |
---|
| 5307 | } |
---|
| 5308 | if ( size(vP)>=2 ) |
---|
| 5309 | { |
---|
| 5310 | vP = vP[2..size(vP)]; |
---|
| 5311 | } |
---|
| 5312 | sP = -Max(-vP); |
---|
| 5313 | if (sP == 0) |
---|
| 5314 | { |
---|
| 5315 | "Warning: zero root present!"; |
---|
| 5316 | } |
---|
| 5317 | return(sP); |
---|
| 5318 | } |
---|
| 5319 | example |
---|
| 5320 | { |
---|
| 5321 | "EXAMPLE:"; echo = 2; |
---|
| 5322 | ring r = 0,(x,y),ds; |
---|
| 5323 | poly f1 = x*y*(x+y); |
---|
| 5324 | ideal I1 = bernstein(f1)[1]; // a local Bernstein poly |
---|
| 5325 | I1; |
---|
| 5326 | minIntRoot(I1,0); |
---|
| 5327 | poly f2 = x2-y3; |
---|
| 5328 | ideal I2 = bernstein(f2)[1]; |
---|
| 5329 | I2; |
---|
| 5330 | minIntRoot(I2,0); |
---|
| 5331 | // now we illustrate the behaviour of factorize |
---|
| 5332 | // together with a global ordering |
---|
| 5333 | ring r2 = 0,x,dp; |
---|
| 5334 | poly f3 = 9*(x+2/3)*(x+1)*(x+4/3); //global b-polynomial of f1=x*y*(x+y) |
---|
| 5335 | ideal I3 = factorize(f3,1); |
---|
| 5336 | I3; |
---|
| 5337 | minIntRoot(I3,1); |
---|
| 5338 | // and a more interesting situation |
---|
| 5339 | ring s = 0,(x,y,z),ds; |
---|
| 5340 | poly f = x3 + y3 + z3; |
---|
| 5341 | ideal I = bernstein(f)[1]; |
---|
| 5342 | I; |
---|
| 5343 | minIntRoot(I,0); |
---|
| 5344 | } |
---|
| 5345 | |
---|
| 5346 | proc isHolonomic(def M) |
---|
| 5347 | "USAGE: isHolonomic(M); M an ideal/module/matrix |
---|
| 5348 | RETURN: int, 1 if M is holonomic over the base ring, and 0 otherwise |
---|
| 5349 | ASSUME: basering is a Weyl algebra in characteristic 0 |
---|
| 5350 | PURPOSE: check whether M is holonomic over the base ring |
---|
| 5351 | NOTE: M is holonomic if 2*dim(M) = dim(R), where R is the |
---|
| 5352 | base ring; dim stays for Gelfand-Kirillov dimension |
---|
| 5353 | EXAMPLE: example isHolonomic; shows examples |
---|
| 5354 | " |
---|
| 5355 | { |
---|
| 5356 | if (dmodappassumeViolation()) |
---|
| 5357 | { |
---|
| 5358 | ERROR("Basering is inappropriate: characteristic>0 or qring present"); |
---|
| 5359 | } |
---|
| 5360 | if (!isWeyl(basering)) |
---|
| 5361 | { |
---|
| 5362 | ERROR("Basering is not a Weyl algebra"); |
---|
| 5363 | } |
---|
| 5364 | |
---|
| 5365 | if ( (typeof(M) != "ideal") && (typeof(M) != "module") && (typeof(M) != "matrix") ) |
---|
| 5366 | { |
---|
| 5367 | // print(typeof(M)); |
---|
| 5368 | ERROR("an argument of type ideal/module/matrix expected"); |
---|
| 5369 | } |
---|
| 5370 | if (attrib(M,"isSB")!=1) |
---|
| 5371 | { |
---|
| 5372 | M = std(M); |
---|
| 5373 | } |
---|
| 5374 | int dimR = gkdim(std(0)); |
---|
| 5375 | int dimM = gkdim(M); |
---|
| 5376 | return( (dimR==2*dimM) ); |
---|
| 5377 | } |
---|
| 5378 | example |
---|
| 5379 | { |
---|
| 5380 | "EXAMPLE:"; echo = 2; |
---|
| 5381 | ring R = 0,(x,y),dp; |
---|
| 5382 | poly F = x*y*(x+y); |
---|
| 5383 | def A = annfsBM(F,0); |
---|
| 5384 | setring A; |
---|
| 5385 | LD; |
---|
| 5386 | isHolonomic(LD); |
---|
| 5387 | ideal I = std(LD[1]); |
---|
| 5388 | I; |
---|
| 5389 | isHolonomic(I); |
---|
| 5390 | } |
---|
| 5391 | |
---|
| 5392 | proc reiffen(int p, int q) |
---|
| 5393 | "USAGE: reiffen(p, q); int p, int q |
---|
| 5394 | RETURN: ring |
---|
| 5395 | PURPOSE: set up the polynomial, describing a Reiffen curve |
---|
| 5396 | NOTE: activate the output ring with the @code{setring} command and |
---|
| 5397 | @* find the curve as a polynomial RC. |
---|
| 5398 | @* A Reiffen curve is defined as RC = x^p + y^q + xy^{q-1}, q >= p+1 >= 5 |
---|
| 5399 | |
---|
| 5400 | EXAMPLE: example reiffen; shows examples |
---|
| 5401 | " |
---|
| 5402 | { |
---|
| 5403 | // we allow also other numbers, p \geq 1, q\geq 1 |
---|
| 5404 | // a Reiffen curve is defined as |
---|
| 5405 | // F = x^p + y^q +x*y^{q-1}, q \geq p+1 \geq 5 |
---|
| 5406 | |
---|
| 5407 | // ASSUME: q >= p+1 >= 5. Otherwise an error message is returned |
---|
| 5408 | |
---|
| 5409 | // if ( (p<4) || (q<5) || (q-p<1) ) |
---|
| 5410 | // { |
---|
| 5411 | // ERROR("Some of conditions p>=4, q>=5 or q>=p+1 is not satisfied!"); |
---|
| 5412 | // } |
---|
| 5413 | if ( (p<1) || (q<1) ) |
---|
| 5414 | { |
---|
| 5415 | ERROR("Some of conditions p>=1, q>=1 is not satisfied!"); |
---|
| 5416 | } |
---|
| 5417 | ring @r = 0,(x,y),dp; |
---|
| 5418 | poly RC = y^q +x^p + x*y^(q-1); |
---|
| 5419 | export RC; |
---|
| 5420 | return(@r); |
---|
| 5421 | } |
---|
| 5422 | example |
---|
| 5423 | { |
---|
| 5424 | "EXAMPLE:"; echo = 2; |
---|
| 5425 | def r = reiffen(4,5); |
---|
| 5426 | setring r; |
---|
| 5427 | RC; |
---|
| 5428 | } |
---|
| 5429 | |
---|
| 5430 | proc arrange(int p) |
---|
| 5431 | "USAGE: arrange(p); int p |
---|
| 5432 | RETURN: ring |
---|
| 5433 | PURPOSE: set up the polynomial, describing a hyperplane arrangement |
---|
| 5434 | NOTE: must be executed in a commutative ring |
---|
| 5435 | ASSUME: basering is present and it is commutative |
---|
| 5436 | EXAMPLE: example arrange; shows examples |
---|
| 5437 | " |
---|
| 5438 | { |
---|
| 5439 | int UseBasering = 0 ; |
---|
| 5440 | if (p<2) |
---|
| 5441 | { |
---|
| 5442 | ERROR("p>=2 is required!"); |
---|
| 5443 | } |
---|
| 5444 | if ( nameof(basering)!="basering" ) |
---|
| 5445 | { |
---|
| 5446 | if (p > nvars(basering)) |
---|
| 5447 | { |
---|
| 5448 | ERROR("too big p"); |
---|
| 5449 | } |
---|
| 5450 | else |
---|
| 5451 | { |
---|
| 5452 | def @r = basering; |
---|
| 5453 | UseBasering = 1; |
---|
| 5454 | } |
---|
| 5455 | } |
---|
| 5456 | else |
---|
| 5457 | { |
---|
| 5458 | // no basering found |
---|
| 5459 | ERROR("no basering found!"); |
---|
| 5460 | // ring @r = 0,(x(1..p)),dp; |
---|
| 5461 | } |
---|
| 5462 | int i,j,sI; |
---|
| 5463 | poly q = 1; |
---|
| 5464 | list ar; |
---|
| 5465 | ideal tmp; |
---|
| 5466 | tmp = ideal(var(1)); |
---|
| 5467 | ar[1] = tmp; |
---|
| 5468 | for (i = 2; i<=p; i++) |
---|
| 5469 | { |
---|
| 5470 | // add i-nary sums to the product |
---|
| 5471 | ar = indAR(ar,i); |
---|
| 5472 | } |
---|
| 5473 | for (i = 1; i<=p; i++) |
---|
| 5474 | { |
---|
| 5475 | tmp = ar[i]; sI = size(tmp); |
---|
| 5476 | for (j = 1; j<=sI; j++) |
---|
| 5477 | { |
---|
| 5478 | q = q*tmp[j]; |
---|
| 5479 | } |
---|
| 5480 | } |
---|
| 5481 | if (UseBasering) |
---|
| 5482 | { |
---|
| 5483 | return(q); |
---|
| 5484 | } |
---|
| 5485 | // poly AR = q; export AR; |
---|
| 5486 | // return(@r); |
---|
| 5487 | } |
---|
| 5488 | example |
---|
| 5489 | { |
---|
| 5490 | "EXAMPLE:"; echo = 2; |
---|
| 5491 | ring X = 0,(x,y,z,t),dp; |
---|
| 5492 | poly q = arrange(3); |
---|
| 5493 | factorize(q,1); |
---|
| 5494 | } |
---|
| 5495 | |
---|
| 5496 | proc checkRoot(poly F, number a, list #) |
---|
| 5497 | "USAGE: checkRoot(f,alpha [,S,eng]); poly f, number alpha, string S, int eng |
---|
| 5498 | RETURN: int |
---|
[6439db] | 5499 | ASSUME: Basering is a commutative ring, alpha is a positive rational number. |
---|
| 5500 | PURPOSE: check whether a negative rational number -alpha is a root of the global |
---|
[66c962] | 5501 | @* Bernstein-Sato polynomial of f and compute its multiplicity, |
---|
| 5502 | @* with the algorithm given by S and with the Groebner basis engine given by eng. |
---|
[3f4e52] | 5503 | NOTE: The annihilator of f^s in D[s] is needed and hence it is computed with the |
---|
[66c962] | 5504 | @* algorithm by Briancon and Maisonobe. The value of a string S can be |
---|
| 5505 | @* 'alg1' (default) - for the algorithm 1 of [LM08] |
---|
| 5506 | @* 'alg2' - for the algorithm 2 of [LM08] |
---|
| 5507 | @* Depending on the value of S, the output of type int is: |
---|
| 5508 | @* 'alg1': 1 only if -alpha is a root of the global Bernstein-Sato polynomial |
---|
| 5509 | @* 'alg2': the multiplicity of -alpha as a root of the global Bernstein-Sato |
---|
| 5510 | @* polynomial of f. If -alpha is not a root, the output is 0. |
---|
| 5511 | @* If eng <>0, @code{std} is used for Groebner basis computations, |
---|
| 5512 | @* otherwise (and by default) @code{slimgb} is used. |
---|
| 5513 | DISPLAY: If printlevel=1, progress debug messages will be printed, |
---|
| 5514 | @* if printlevel>=2, all the debug messages will be printed. |
---|
| 5515 | EXAMPLE: example checkRoot; shows examples |
---|
| 5516 | " |
---|
| 5517 | { |
---|
| 5518 | int eng = 0; |
---|
| 5519 | int chs = 0; // choice |
---|
| 5520 | string algo = "alg1"; |
---|
| 5521 | string st; |
---|
| 5522 | if ( size(#)>0 ) |
---|
| 5523 | { |
---|
| 5524 | if ( typeof(#[1]) == "string" ) |
---|
| 5525 | { |
---|
| 5526 | st = string(#[1]); |
---|
| 5527 | if ( (st == "alg1") || (st == "ALG1") || (st == "Alg1") ||(st == "aLG1")) |
---|
| 5528 | { |
---|
| 5529 | algo = "alg1"; |
---|
| 5530 | chs = 1; |
---|
| 5531 | } |
---|
| 5532 | if ( (st == "alg2") || (st == "ALG2") || (st == "Alg2") || (st == "aLG2")) |
---|
| 5533 | { |
---|
| 5534 | algo = "alg2"; |
---|
| 5535 | chs = 1; |
---|
| 5536 | } |
---|
| 5537 | if (chs != 1) |
---|
| 5538 | { |
---|
| 5539 | // incorrect value of S |
---|
| 5540 | print("Incorrect algorithm given, proceed with the default alg1"); |
---|
| 5541 | algo = "alg1"; |
---|
| 5542 | } |
---|
| 5543 | // second arg |
---|
| 5544 | if (size(#)>1) |
---|
| 5545 | { |
---|
| 5546 | // exists 2nd arg |
---|
| 5547 | if ( typeof(#[2]) == "int" ) |
---|
| 5548 | { |
---|
| 5549 | // the case: given alg, given engine |
---|
| 5550 | eng = int(#[2]); |
---|
| 5551 | } |
---|
| 5552 | else |
---|
| 5553 | { |
---|
| 5554 | eng = 0; |
---|
| 5555 | } |
---|
| 5556 | } |
---|
| 5557 | else |
---|
| 5558 | { |
---|
| 5559 | // no second arg |
---|
| 5560 | eng = 0; |
---|
| 5561 | } |
---|
| 5562 | } |
---|
| 5563 | else |
---|
| 5564 | { |
---|
| 5565 | if ( typeof(#[1]) == "int" ) |
---|
| 5566 | { |
---|
| 5567 | // the case: default alg, engine |
---|
| 5568 | eng = int(#[1]); |
---|
| 5569 | // algo = "alg1"; //is already set |
---|
| 5570 | } |
---|
| 5571 | else |
---|
| 5572 | { |
---|
| 5573 | // incorr. 1st arg |
---|
| 5574 | algo = "alg1"; |
---|
| 5575 | } |
---|
| 5576 | } |
---|
| 5577 | } |
---|
| 5578 | // size(#)=0, i.e. there is no algorithm and no engine given |
---|
| 5579 | // eng = 0; algo = "alg1"; //are already set |
---|
| 5580 | // int ppl = printlevel-voice+2; |
---|
[6439db] | 5581 | // check assume: a is positive rational number |
---|
| 5582 | if (!isRational(a)) |
---|
| 5583 | { |
---|
| 5584 | ERROR("rational root expected for checking"); |
---|
| 5585 | } |
---|
| 5586 | if (numerator(a) < 0 ) |
---|
| 5587 | { |
---|
| 5588 | ERROR("expected positive -alpha"); |
---|
| 5589 | // the following is more user-friendly but less correct |
---|
| 5590 | // print("proceeding with the negated root"); |
---|
| 5591 | // a = -a; |
---|
| 5592 | } |
---|
[66c962] | 5593 | printlevel=printlevel+1; |
---|
| 5594 | def save = basering; |
---|
| 5595 | def @A = SannfsBM(F); |
---|
| 5596 | setring @A; |
---|
| 5597 | poly F = imap(save,F); |
---|
| 5598 | number a = imap(save,a); |
---|
| 5599 | if ( algo=="alg1") |
---|
| 5600 | { |
---|
| 5601 | int output = checkRoot1(LD,F,a,eng); |
---|
| 5602 | } |
---|
| 5603 | else |
---|
| 5604 | { |
---|
| 5605 | if ( algo=="alg2") |
---|
| 5606 | { |
---|
| 5607 | int output = checkRoot2(LD,F,a,eng); |
---|
| 5608 | } |
---|
| 5609 | } |
---|
| 5610 | printlevel=printlevel-1; |
---|
| 5611 | return(output); |
---|
| 5612 | } |
---|
| 5613 | example |
---|
| 5614 | { |
---|
| 5615 | "EXAMPLE:"; echo = 2; |
---|
| 5616 | printlevel=0; |
---|
| 5617 | ring r = 0,(x,y),Dp; |
---|
| 5618 | poly F = x^4+y^5+x*y^4; |
---|
| 5619 | checkRoot(F,11/20); // -11/20 is a root of bf |
---|
| 5620 | poly G = x*y; |
---|
| 5621 | checkRoot(G,1,"alg2"); // -1 is a root of bg with multiplicity 2 |
---|
| 5622 | } |
---|
| 5623 | |
---|
| 5624 | proc checkRoot1(ideal I, poly F, number a, list #) |
---|
| 5625 | "USAGE: checkRoot1(I,f,alpha [,eng]); ideal I, poly f, number alpha, int eng |
---|
| 5626 | ASSUME: Basering is D[s], I is the annihilator of f^s in D[s], |
---|
| 5627 | @* that is basering and I are the output of Sannfs-like procedure, |
---|
| 5628 | @* f is a polynomial in K[x] and alpha is a rational number. |
---|
| 5629 | RETURN: int, 1 if -alpha is a root of the Bernstein-Sato polynomial of f |
---|
| 5630 | PURPOSE: check, whether alpha is a root of the global Bernstein-Sato polynomial of f |
---|
| 5631 | NOTE: If eng <>0, @code{std} is used for Groebner basis computations, |
---|
| 5632 | @* otherwise (and by default) @code{slimgb} is used. |
---|
| 5633 | DISPLAY: If printlevel=1, progress debug messages will be printed, |
---|
| 5634 | @* if printlevel>=2, all the debug messages will be printed. |
---|
| 5635 | EXAMPLE: example checkRoot1; shows examples |
---|
| 5636 | " |
---|
| 5637 | { |
---|
| 5638 | // to check: alpha is rational (has char 0 check inside) |
---|
| 5639 | if (!isRational(a)) |
---|
| 5640 | { |
---|
| 5641 | "ERROR: alpha must be a rational number!"; |
---|
| 5642 | } |
---|
| 5643 | // no qring |
---|
| 5644 | if ( size(ideal(basering)) >0 ) |
---|
| 5645 | { |
---|
| 5646 | "ERROR: no qring is allowed"; |
---|
| 5647 | } |
---|
| 5648 | int eng = 0; |
---|
| 5649 | if ( size(#)>0 ) |
---|
| 5650 | { |
---|
| 5651 | if ( typeof(#[1]) == "int" ) |
---|
| 5652 | { |
---|
| 5653 | eng = int(#[1]); |
---|
| 5654 | } |
---|
| 5655 | } |
---|
| 5656 | int ppl = printlevel-voice+2; |
---|
| 5657 | dbprint(ppl,"// -0-1- starting the procedure checkRoot1"); |
---|
| 5658 | def save = basering; |
---|
| 5659 | int N = nvars(basering); |
---|
| 5660 | int Nnew = N-1; |
---|
| 5661 | int n = Nnew / 2; |
---|
| 5662 | int i; |
---|
| 5663 | string s; |
---|
| 5664 | list RL = ringlist(basering); |
---|
| 5665 | list L, Lord; |
---|
| 5666 | list tmp; |
---|
| 5667 | intvec iv; |
---|
| 5668 | L[1] = RL[1]; // char |
---|
| 5669 | L[4] = RL[4]; // char, minpoly |
---|
| 5670 | // check whether basering is D[s]=K(_x,_Dx,s) |
---|
| 5671 | list Name = RL[2]; |
---|
| 5672 | // for (i=1; i<=n; i++) |
---|
| 5673 | // { |
---|
| 5674 | // if ( bracket(var(i+n),var(i))!=1 ) |
---|
| 5675 | // { |
---|
| 5676 | // ERROR("basering should be D[s]=K(_x,_Dx,s)"); |
---|
| 5677 | // } |
---|
| 5678 | // } |
---|
| 5679 | if ( Name[N]!="s" ) |
---|
| 5680 | { |
---|
| 5681 | ERROR("the last variable of basering should be s"); |
---|
| 5682 | } |
---|
| 5683 | // now, create the new vars |
---|
| 5684 | list NName; |
---|
| 5685 | for (i=1; i<=Nnew; i++) |
---|
| 5686 | { |
---|
| 5687 | NName[i] = Name[i]; |
---|
| 5688 | } |
---|
| 5689 | L[2] = NName; |
---|
| 5690 | kill Name,NName; |
---|
| 5691 | // block ord (dp); |
---|
| 5692 | tmp[1] = "dp"; // string |
---|
| 5693 | s = "iv="; |
---|
| 5694 | for (i=1; i<=Nnew; i++) |
---|
| 5695 | { |
---|
| 5696 | s = s+"1,"; |
---|
| 5697 | } |
---|
| 5698 | s[size(s)]=";"; |
---|
| 5699 | execute(s); |
---|
| 5700 | kill s; |
---|
| 5701 | tmp[2] = iv; |
---|
| 5702 | Lord[1] = tmp; |
---|
| 5703 | tmp[1] = "C"; |
---|
| 5704 | iv = 0; |
---|
| 5705 | tmp[2] = iv; |
---|
| 5706 | Lord[2] = tmp; |
---|
| 5707 | tmp = 0; |
---|
| 5708 | L[3] = Lord; |
---|
| 5709 | // we are done with the list |
---|
| 5710 | def @R@ = ring(L); |
---|
| 5711 | setring @R@; |
---|
| 5712 | matrix @D[Nnew][Nnew]; |
---|
| 5713 | for (i=1; i<=n; i++) |
---|
| 5714 | { |
---|
| 5715 | @D[i,i+n]=1; |
---|
| 5716 | } |
---|
| 5717 | def @R = nc_algebra(1,@D); |
---|
| 5718 | setring @R; |
---|
| 5719 | kill @R@; |
---|
| 5720 | dbprint(ppl,"// -1-1- the ring @R(_x,_Dx) is ready"); |
---|
| 5721 | dbprint(ppl-1, S); |
---|
| 5722 | // create the ideal K = ann_D[s](f^s)_{s=-alpha} + < f > |
---|
| 5723 | setring save; |
---|
| 5724 | ideal K = subst(I,s,-a); |
---|
| 5725 | dbprint(ppl,"// -1-2- the variable s has been substituted by "+string(-a)); |
---|
| 5726 | dbprint(ppl-1, K); |
---|
| 5727 | K = NF(K,std(F)); |
---|
| 5728 | // make leadcoeffs positive |
---|
| 5729 | for (i=1; i<=ncols(K); i++) |
---|
| 5730 | { |
---|
| 5731 | if ( leadcoef(K[i])<0 ) |
---|
| 5732 | { |
---|
| 5733 | K[i] = -K[i]; |
---|
| 5734 | } |
---|
| 5735 | } |
---|
| 5736 | K = K,F; |
---|
| 5737 | // ------------ the ideal K is ready ------------ |
---|
| 5738 | setring @R; |
---|
| 5739 | ideal K = imap(save,K); |
---|
| 5740 | dbprint(ppl,"// -1-3- starting the computation of a Groebner basis of K in @R"); |
---|
| 5741 | dbprint(ppl-1, K); |
---|
| 5742 | ideal G = engine(K,eng); |
---|
| 5743 | dbprint(ppl,"// -1-4- the Groebner basis has been computed"); |
---|
| 5744 | dbprint(ppl-1, G); |
---|
| 5745 | return(G[1]!=1); |
---|
| 5746 | } |
---|
| 5747 | example |
---|
| 5748 | { |
---|
| 5749 | "EXAMPLE:"; echo = 2; |
---|
| 5750 | ring r = 0,(x,y),Dp; |
---|
| 5751 | poly F = x^4+y^5+x*y^4; |
---|
| 5752 | printlevel = 0; |
---|
| 5753 | def A = Sannfs(F); |
---|
| 5754 | setring A; |
---|
| 5755 | poly F = imap(r,F); |
---|
| 5756 | checkRoot1(LD,F,31/20); // -31/20 is not a root of bs |
---|
| 5757 | checkRoot1(LD,F,11/20); // -11/20 is a root of bs |
---|
| 5758 | } |
---|
| 5759 | |
---|
| 5760 | proc checkRoot2 (ideal I, poly F, number a, list #) |
---|
| 5761 | "USAGE: checkRoot2(I,f,a [,eng]); I an ideal, f a poly, alpha a number, eng an optional int |
---|
[3f4e52] | 5762 | ASSUME: I is the annihilator of f^s in D[s], basering is D[s], |
---|
[66c962] | 5763 | @* that is basering and I are the output os Sannfs-like procedure, |
---|
| 5764 | @* f is a polynomial in K[_x] and alpha is a rational number. |
---|
[3f4e52] | 5765 | RETURN: int, the multiplicity of -alpha as a root of the BS polynomial of f. |
---|
[66c962] | 5766 | PURPOSE: check whether a rational number alpha is a root of the global Bernstein- |
---|
| 5767 | @* Sato polynomial of f and compute its multiplicity from the known Ann F^s in D[s] |
---|
| 5768 | NOTE: If -alpha is not a root, the output is 0. |
---|
| 5769 | @* If eng <>0, @code{std} is used for Groebner basis computations, |
---|
| 5770 | @* otherwise (and by default) @code{slimgb} is used. |
---|
| 5771 | DISPLAY: If printlevel=1, progress debug messages will be printed, |
---|
| 5772 | @* if printlevel>=2, all the debug messages will be printed. |
---|
| 5773 | EXAMPLE: example checkRoot2; shows examples |
---|
| 5774 | " |
---|
| 5775 | { |
---|
| 5776 | |
---|
[3f4e52] | 5777 | |
---|
[66c962] | 5778 | // to check: alpha is rational (has char 0 check inside) |
---|
| 5779 | if (!isRational(a)) |
---|
| 5780 | { |
---|
| 5781 | "ERROR: alpha must be a rational number!"; |
---|
| 5782 | } |
---|
| 5783 | // no qring |
---|
| 5784 | if ( size(ideal(basering)) >0 ) |
---|
| 5785 | { |
---|
| 5786 | "ERROR: no qring is allowed"; |
---|
| 5787 | } |
---|
| 5788 | |
---|
| 5789 | int eng = 0; |
---|
| 5790 | if ( size(#)>0 ) |
---|
| 5791 | { |
---|
| 5792 | if ( typeof(#[1]) == "int" ) |
---|
| 5793 | { |
---|
| 5794 | eng = int(#[1]); |
---|
| 5795 | } |
---|
| 5796 | } |
---|
| 5797 | int ppl = printlevel-voice+2; |
---|
| 5798 | dbprint(ppl,"// -0-1- starting the procedure checkRoot2"); |
---|
| 5799 | def save = basering; |
---|
| 5800 | int N = nvars(basering); |
---|
| 5801 | int n = (N-1) / 2; |
---|
| 5802 | int i; |
---|
| 5803 | string s; |
---|
| 5804 | list RL = ringlist(basering); |
---|
| 5805 | list L, Lord; |
---|
| 5806 | list tmp; |
---|
| 5807 | intvec iv; |
---|
| 5808 | L[1] = RL[1]; // char |
---|
| 5809 | L[4] = RL[4]; // char, minpoly |
---|
| 5810 | // check whether basering is D[s]=K(_x,_Dx,s) |
---|
| 5811 | list Name = RL[2]; |
---|
| 5812 | for (i=1; i<=n; i++) |
---|
| 5813 | { |
---|
| 5814 | if ( bracket(var(i+n),var(i))!=1 ) |
---|
| 5815 | { |
---|
| 5816 | ERROR("basering should be D[s]=K(_x,_Dx,s)"); |
---|
| 5817 | } |
---|
| 5818 | } |
---|
| 5819 | if ( Name[N]!="s" ) |
---|
| 5820 | { |
---|
| 5821 | ERROR("the last variable of basering should be s"); |
---|
| 5822 | } |
---|
| 5823 | // now, create the new vars |
---|
| 5824 | L[2] = Name; |
---|
| 5825 | kill Name; |
---|
| 5826 | // block ord (dp); |
---|
| 5827 | tmp[1] = "dp"; // string |
---|
| 5828 | s = "iv="; |
---|
| 5829 | for (i=1; i<=N; i++) |
---|
| 5830 | { |
---|
| 5831 | s = s+"1,"; |
---|
| 5832 | } |
---|
| 5833 | s[size(s)]=";"; |
---|
| 5834 | execute(s); |
---|
| 5835 | kill s; |
---|
| 5836 | tmp[2] = iv; |
---|
| 5837 | Lord[1] = tmp; |
---|
| 5838 | tmp[1] = "C"; |
---|
| 5839 | iv = 0; |
---|
| 5840 | tmp[2] = iv; |
---|
| 5841 | Lord[2] = tmp; |
---|
| 5842 | tmp = 0; |
---|
| 5843 | L[3] = Lord; |
---|
| 5844 | // we are done with the list |
---|
| 5845 | def @R@ = ring(L); |
---|
| 5846 | setring @R@; |
---|
| 5847 | matrix @D[N][N]; |
---|
| 5848 | for (i=1; i<=n; i++) |
---|
| 5849 | { |
---|
| 5850 | @D[i,i+n]=1; |
---|
| 5851 | } |
---|
| 5852 | def @R = nc_algebra(1,@D); |
---|
| 5853 | setring @R; |
---|
| 5854 | kill @R@; |
---|
| 5855 | dbprint(ppl,"// -1-1- the ring @R(_x,_Dx,s) is ready"); |
---|
| 5856 | dbprint(ppl-1, @R); |
---|
| 5857 | // now, continue with the algorithm |
---|
| 5858 | ideal I = imap(save,I); |
---|
| 5859 | poly F = imap(save,F); |
---|
| 5860 | number a = imap(save,a); |
---|
| 5861 | ideal II = NF(I,std(F)); |
---|
| 5862 | // make leadcoeffs positive |
---|
| 5863 | for (i=1; i<=ncols(II); i++) |
---|
| 5864 | { |
---|
| 5865 | if ( leadcoef(II[i])<0 ) |
---|
| 5866 | { |
---|
| 5867 | II[i] = -II[i]; |
---|
| 5868 | } |
---|
| 5869 | } |
---|
| 5870 | ideal J,G; |
---|
| 5871 | int m; // the output (multiplicity) |
---|
| 5872 | dbprint(ppl,"// -2- starting the bucle"); |
---|
| 5873 | for (i=0; i<=n; i++) // the multiplicity has to be <= n |
---|
| 5874 | { |
---|
| 5875 | // create the ideal Ji = ann_D[s](f^s) + < f, (s+alpha)^{i+1} > |
---|
| 5876 | // (s+alpha)^i in Ji <==> -alpha is a root with multiplicity >= i |
---|
| 5877 | J = II,F,(s+a)^(i+1); |
---|
| 5878 | // ------------ the ideal Ji is ready ----------- |
---|
| 5879 | dbprint(ppl,"// -2-"+string(i+1)+"-1- starting the computation of a Groebner basis of J"+string(i)+" in @R"); |
---|
| 5880 | dbprint(ppl-1, J); |
---|
| 5881 | G = engine(J,eng); |
---|
| 5882 | dbprint(ppl,"// -2-"+string(i+1)+"-2- the Groebner basis has been computed"); |
---|
| 5883 | dbprint(ppl-1, G); |
---|
| 5884 | if ( NF((s+a)^i,G)==0 ) |
---|
| 5885 | { |
---|
| 5886 | dbprint(ppl,"// -2-"+string(i+1)+"-3- the number "+string(-a)+" has not multiplicity "+string(i+1)); |
---|
| 5887 | m = i; |
---|
| 5888 | break; |
---|
| 5889 | } |
---|
| 5890 | dbprint(ppl,"// -2-"+string(i+1)+"-3- the number "+string(-a)+" has multiplicity at least "+string(i+1)); |
---|
| 5891 | } |
---|
| 5892 | dbprint(ppl,"// -3- the bucle has finished"); |
---|
| 5893 | return(m); |
---|
| 5894 | } |
---|
| 5895 | example |
---|
| 5896 | { |
---|
| 5897 | "EXAMPLE:"; echo = 2; |
---|
| 5898 | ring r = 0,(x,y,z),Dp; |
---|
| 5899 | poly F = x*y*z; |
---|
| 5900 | printlevel = 0; |
---|
| 5901 | def A = Sannfs(F); |
---|
| 5902 | setring A; |
---|
| 5903 | poly F = imap(r,F); |
---|
| 5904 | checkRoot2(LD,F,1); // -1 is a root of bs with multiplicity 3 |
---|
| 5905 | checkRoot2(LD,F,1/3); // -1/3 is not a root |
---|
| 5906 | } |
---|
| 5907 | |
---|
| 5908 | proc checkFactor(ideal I, poly F, poly q, list #) |
---|
| 5909 | "USAGE: checkFactor(I,f,qs [,eng]); I an ideal, f a poly, qs a poly, eng an optional int |
---|
| 5910 | ASSUME: checkFactor is called from the basering, created by Sannfs-like proc, |
---|
| 5911 | @* that is, from the Weyl algebra in x1,..,xN,d1,..,dN tensored with K[s]. |
---|
[3f4e52] | 5912 | @* The ideal I is the annihilator of f^s in D[s], that is the ideal, computed |
---|
[66c962] | 5913 | @* by Sannfs-like procedure (usually called LD there). |
---|
| 5914 | @* Moreover, f is a polynomial in K[x1,..,xN] and qs is a polynomial in K[s]. |
---|
| 5915 | RETURN: int, 1 if qs is a factor of the global Bernstein polynomial of f and 0 otherwise |
---|
[3f4e52] | 5916 | PURPOSE: check whether a univariate polynomial qs is a factor of the |
---|
[66c962] | 5917 | @* Bernstein-Sato polynomial of f without explicit knowledge of the latter. |
---|
| 5918 | NOTE: If eng <>0, @code{std} is used for Groebner basis computations, |
---|
| 5919 | @* otherwise (and by default) @code{slimgb} is used. |
---|
| 5920 | DISPLAY: If printlevel=1, progress debug messages will be printed, |
---|
| 5921 | @* if printlevel>=2, all the debug messages will be printed. |
---|
| 5922 | EXAMPLE: example checkFactor; shows examples |
---|
| 5923 | " |
---|
| 5924 | { |
---|
| 5925 | |
---|
| 5926 | // ASSUME too complicated, cannot check it. |
---|
| 5927 | |
---|
| 5928 | int eng = 0; |
---|
| 5929 | if ( size(#)>0 ) |
---|
| 5930 | { |
---|
| 5931 | if ( typeof(#[1]) == "int" ) |
---|
| 5932 | { |
---|
| 5933 | eng = int(#[1]); |
---|
| 5934 | } |
---|
| 5935 | } |
---|
| 5936 | int ppl = printlevel-voice+2; |
---|
| 5937 | def @R2 = basering; |
---|
| 5938 | int N = nvars(@R2); |
---|
| 5939 | int i; |
---|
| 5940 | // we're in D_n[s], where the elim ord for s is set |
---|
| 5941 | dbprint(ppl,"// -0-1- starting the procedure checkFactor"); |
---|
| 5942 | dbprint(ppl,"// -1-1- the ring @R2(_x,_Dx,s) is ready"); |
---|
| 5943 | dbprint(ppl-1, @R2); |
---|
| 5944 | // create the ideal J = ann_D[s](f^s) + < f,q > |
---|
| 5945 | ideal J = NF(I,std(F)); |
---|
| 5946 | // make leadcoeffs positive |
---|
| 5947 | for (i=1; i<=ncols(J); i++) |
---|
| 5948 | { |
---|
| 5949 | if ( leadcoef(J[i])<0 ) |
---|
| 5950 | { |
---|
| 5951 | J[i] = -J[i]; |
---|
| 5952 | } |
---|
| 5953 | } |
---|
| 5954 | J = J,F,q; |
---|
| 5955 | // ------------ the ideal J is ready ----------- |
---|
| 5956 | dbprint(ppl,"// -1-2- starting the elimination of _x,_Dx in @R2"); |
---|
| 5957 | dbprint(ppl-1, J); |
---|
| 5958 | ideal G = engine(J,eng); |
---|
| 5959 | ideal K = nselect(G,1..N-1); |
---|
| 5960 | kill J,G; |
---|
| 5961 | dbprint(ppl,"// -1-3- _x,_Dx are eliminated"); |
---|
| 5962 | dbprint(ppl-1, K); |
---|
| 5963 | //q is a factor of bs if and only if K = < q > |
---|
| 5964 | //K = normalize(K); |
---|
| 5965 | //q = normalize(q); |
---|
| 5966 | //return( (K[1]==q) ); |
---|
| 5967 | return( NF(K[1],std(q))==0 ); |
---|
| 5968 | } |
---|
| 5969 | example |
---|
| 5970 | { |
---|
| 5971 | "EXAMPLE:"; echo = 2; |
---|
| 5972 | ring r = 0,(x,y),Dp; |
---|
| 5973 | poly F = x^4+y^5+x*y^4; |
---|
| 5974 | printlevel = 0; |
---|
| 5975 | def A = Sannfs(F); |
---|
| 5976 | setring A; |
---|
| 5977 | poly F = imap(r,F); |
---|
| 5978 | checkFactor(LD,F,20*s+31); // -31/20 is not a root of bs |
---|
| 5979 | checkFactor(LD,F,20*s+11); // -11/20 is a root of bs |
---|
| 5980 | checkFactor(LD,F,(20*s+11)^2); // the multiplicity of -11/20 is 1 |
---|
| 5981 | } |
---|
| 5982 | |
---|
| 5983 | proc varNum(string s) |
---|
| 5984 | "USAGE: varNum(s); string s |
---|
| 5985 | RETURN: int |
---|
| 5986 | PURPOSE: returns the number of the variable with the name s |
---|
| 5987 | @* among the variables of basering or 0 if there is no such variable |
---|
| 5988 | EXAMPLE: example varNum; shows examples |
---|
| 5989 | " |
---|
| 5990 | { |
---|
| 5991 | int i; |
---|
| 5992 | for (i=1; i<= nvars(basering); i++) |
---|
| 5993 | { |
---|
| 5994 | if ( string(var(i)) == s ) |
---|
| 5995 | { |
---|
| 5996 | return(i); |
---|
| 5997 | } |
---|
| 5998 | } |
---|
| 5999 | return(0); |
---|
| 6000 | } |
---|
| 6001 | example |
---|
| 6002 | { |
---|
| 6003 | "EXAMPLE:"; echo = 2; |
---|
| 6004 | ring X = 0,(x,y1,t,z(0),z,tTa),dp; |
---|
| 6005 | varNum("z"); |
---|
| 6006 | varNum("t"); |
---|
| 6007 | varNum("xyz"); |
---|
| 6008 | } |
---|
| 6009 | |
---|
| 6010 | static proc indAR(list L, int n) |
---|
| 6011 | "USAGE: indAR(L,n); list L, int n |
---|
| 6012 | RETURN: list |
---|
[3f4e52] | 6013 | PURPOSE: computes arrangement inductively, using L and |
---|
[66c962] | 6014 | @* var(n) as the next variable |
---|
| 6015 | ASSUME: L has a structure of an arrangement |
---|
| 6016 | EXAMPLE: example indAR; shows examples |
---|
| 6017 | " |
---|
| 6018 | { |
---|
| 6019 | if ( (n<2) || (n>nvars(basering)) ) |
---|
| 6020 | { |
---|
| 6021 | ERROR("incorrect n"); |
---|
| 6022 | } |
---|
| 6023 | int sl = size(L); |
---|
| 6024 | list K; |
---|
| 6025 | ideal tmp; |
---|
| 6026 | poly @t = L[sl][1] + var(n); //1 elt |
---|
| 6027 | K[sl+1] = ideal(@t); |
---|
| 6028 | tmp = L[1]+var(n); |
---|
| 6029 | K[1] = tmp; tmp = 0; |
---|
| 6030 | int i,j,sI; |
---|
| 6031 | ideal I; |
---|
| 6032 | for(i=sl; i>=2; i--) |
---|
| 6033 | { |
---|
| 6034 | I = L[i-1]; sI = size(I); |
---|
| 6035 | for(j=1; j<=sI; j++) |
---|
| 6036 | { |
---|
| 6037 | I[j] = I[j] + var(n); |
---|
| 6038 | } |
---|
| 6039 | tmp = L[i],I; |
---|
| 6040 | K[i] = tmp; |
---|
| 6041 | I = 0; tmp = 0; |
---|
| 6042 | } |
---|
| 6043 | kill I; kill tmp; |
---|
| 6044 | return(K); |
---|
| 6045 | } |
---|
| 6046 | example |
---|
| 6047 | { |
---|
| 6048 | "EXAMPLE:"; echo = 2; |
---|
| 6049 | ring r = 0,(x,y,z,t,v),dp; |
---|
| 6050 | list L; |
---|
| 6051 | L[1] = ideal(x); |
---|
| 6052 | list K = indAR(L,2); |
---|
| 6053 | K; |
---|
| 6054 | list M = indAR(K,3); |
---|
| 6055 | M; |
---|
| 6056 | M = indAR(M,4); |
---|
| 6057 | M; |
---|
| 6058 | } |
---|
| 6059 | |
---|
| 6060 | proc isRational(number n) |
---|
[3f4e52] | 6061 | "USAGE: isRational(n); n number |
---|
[66c962] | 6062 | RETURN: int |
---|
| 6063 | PURPOSE: determine whether n is a rational number, |
---|
| 6064 | @* that is it does not contain parameters. |
---|
| 6065 | ASSUME: ground field is of characteristic 0 |
---|
| 6066 | EXAMPLE: example indAR; shows examples |
---|
| 6067 | " |
---|
| 6068 | { |
---|
| 6069 | if (char(basering) != 0) |
---|
| 6070 | { |
---|
| 6071 | ERROR("The ground field must be of characteristic 0!"); |
---|
| 6072 | } |
---|
| 6073 | number dn = denominator(n); |
---|
| 6074 | number nn = numerator(n); |
---|
| 6075 | return( ((int(dn)==dn) && (int(nn)==nn)) ); |
---|
| 6076 | } |
---|
| 6077 | example |
---|
| 6078 | { |
---|
| 6079 | "EXAMPLE:"; echo = 2; |
---|
| 6080 | ring r = (0,a),(x,y),dp; |
---|
| 6081 | number n1 = 11/73; |
---|
| 6082 | isRational(n1); |
---|
| 6083 | number n2 = (11*a+3)/72; |
---|
| 6084 | isRational(n2); |
---|
| 6085 | } |
---|
| 6086 | |
---|
[e64e417] | 6087 | proc bernsteinLift(ideal I, poly F, list #) |
---|
| 6088 | "USAGE: bernsteinLift(I, F [,eng]); I an ideal, F a poly, eng an optional int |
---|
| 6089 | RETURN: list |
---|
[0610f0e] | 6090 | PURPOSE: compute the (multiple of) Bernstein-Sato polynomial with lift-like method, |
---|
[e64e417] | 6091 | @* based on the output of Sannfs-like procedure |
---|
[0610f0e] | 6092 | NOTE: the output list contains the roots with multiplicities of the candidate |
---|
[e64e417] | 6093 | @* for being Bernstein-Sato polynomial of f. |
---|
| 6094 | @* If eng <>0, @code{std} is used for Groebner basis computations, |
---|
| 6095 | @* otherwise and by default @code{slimgb} is used. |
---|
| 6096 | @* If printlevel=1, progress debug messages will be printed, |
---|
| 6097 | @* if printlevel>=2, all the debug messages will be printed. |
---|
| 6098 | EXAMPLE: example bernsteinLift; shows examples |
---|
| 6099 | " |
---|
| 6100 | { |
---|
| 6101 | // assume: s is the last variable! check in the code |
---|
| 6102 | int eng = 0; |
---|
| 6103 | if ( size(#)>0 ) |
---|
| 6104 | { |
---|
| 6105 | if ( typeof(#[1]) == "int" ) |
---|
| 6106 | { |
---|
| 6107 | eng = int(#[1]); |
---|
| 6108 | } |
---|
| 6109 | } |
---|
| 6110 | def @R2 = basering; |
---|
| 6111 | int Nnew = nvars(@R2); |
---|
| 6112 | int N = Nnew/2; |
---|
| 6113 | int ppl = printlevel-voice+2; |
---|
| 6114 | // we're in D_n[s], where the elim ord for s is set |
---|
| 6115 | // create D_n(s) |
---|
| 6116 | // create the ordinary Weyl algebra and put the result into it, |
---|
| 6117 | // keep: N, i,j,s, tmp, RL |
---|
| 6118 | Nnew = Nnew - 1; // former 2*N; |
---|
| 6119 | list L = 0; |
---|
| 6120 | list Lord, tmp; |
---|
| 6121 | intvec iv; int i; |
---|
| 6122 | list RL = ringlist(basering); |
---|
| 6123 | // if we work over alg. extension => problem! |
---|
| 6124 | if (size(RL[1]) > 1) |
---|
| 6125 | { |
---|
| 6126 | ERROR("cannot work over algebraic field extension"); |
---|
| 6127 | } |
---|
| 6128 | tmp[1] = RL[1]; // char |
---|
| 6129 | tmp[2] = list("s"); |
---|
| 6130 | tmp[3] = list(list("lp",int(1))); |
---|
| 6131 | tmp[4] = ideal(0); |
---|
| 6132 | L[1] = tmp; // field |
---|
| 6133 | tmp = 0; |
---|
| 6134 | L[4] = RL[4]; // factor ideal |
---|
| 6135 | |
---|
| 6136 | // check whether vars have admissible names -> done earlier |
---|
| 6137 | // list Name = RL[2]M |
---|
| 6138 | // DName is defined earlier |
---|
| 6139 | list NName; // = RL[2]; // skip the last var 's' |
---|
| 6140 | for (i=1; i<=Nnew; i++) |
---|
| 6141 | { |
---|
| 6142 | NName[i] = RL[2][i]; |
---|
| 6143 | } |
---|
| 6144 | L[2] = NName; |
---|
| 6145 | // (c, ) ordering: |
---|
| 6146 | tmp[1] = "c"; |
---|
| 6147 | iv = 0; |
---|
| 6148 | tmp[2] = iv; |
---|
| 6149 | Lord[1] = tmp; |
---|
| 6150 | tmp=0; |
---|
| 6151 | // dp ordering; |
---|
| 6152 | string s = "iv="; |
---|
| 6153 | for (i=1; i<=Nnew; i++) |
---|
| 6154 | { |
---|
| 6155 | s = s+"1,"; |
---|
| 6156 | } |
---|
| 6157 | s[size(s)] = ";"; |
---|
| 6158 | execute(s); |
---|
| 6159 | tmp = 0; |
---|
| 6160 | tmp[1] = "dp"; // string |
---|
| 6161 | tmp[2] = iv; // intvec |
---|
| 6162 | Lord[2] = tmp; |
---|
| 6163 | kill s; |
---|
| 6164 | tmp = 0; |
---|
| 6165 | L[3] = Lord; |
---|
| 6166 | // we are done with the list |
---|
| 6167 | // Add: Plural part |
---|
| 6168 | def @R4@ = ring(L); |
---|
| 6169 | setring @R4@; |
---|
| 6170 | matrix @D[Nnew][Nnew]; |
---|
| 6171 | for (i=1; i<=N; i++) |
---|
| 6172 | { |
---|
| 6173 | @D[i,N+i]=1; |
---|
| 6174 | } |
---|
| 6175 | def @R4 = nc_algebra(1,@D); |
---|
| 6176 | setring @R4; |
---|
| 6177 | kill @R4@; |
---|
| 6178 | dbprint(ppl,"// -3-1- the ring K(s)<x,dx> is ready"); |
---|
| 6179 | dbprint(ppl-1, @R4); |
---|
| 6180 | // map things correctly, using names |
---|
| 6181 | ideal J = imap(@R2, I), imap(@R2,F); |
---|
| 6182 | module M; |
---|
| 6183 | // make leadcoeffs positive |
---|
| 6184 | for (i=1; i<= ncols(J); i++) |
---|
| 6185 | { |
---|
[0610f0e] | 6186 | if (J[i]!=0) |
---|
[e64e417] | 6187 | { |
---|
| 6188 | M[i] = J[i]*gen(1) + gen(1+i); |
---|
| 6189 | } |
---|
| 6190 | } |
---|
| 6191 | dbprint(ppl,"// -3-2- starting GB of the assoc. module M"); |
---|
| 6192 | M = engine(M,eng); |
---|
| 6193 | dbprint(ppl,"// -3-3- finished GB of the assoc. module M"); |
---|
| 6194 | dbprint(ppl-1, M); |
---|
| 6195 | // now look for (1) entry with 1st comp nonzero |
---|
| 6196 | // determine whether there are several 1st comps nonzero |
---|
| 6197 | module M2; |
---|
| 6198 | for (i=1; i<= ncols(M); i++) |
---|
| 6199 | { |
---|
[0610f0e] | 6200 | if (M[1,i]!=0) |
---|
[e64e417] | 6201 | { |
---|
| 6202 | M2 = M2, M[i]; |
---|
| 6203 | } |
---|
| 6204 | } |
---|
| 6205 | M2 = simplify(M2,2); // skip 0s |
---|
| 6206 | if (ncols(M2) > 1) |
---|
| 6207 | { |
---|
| 6208 | dbprint(ppl,"// -*- more than 1 element with nonzero leading component"); |
---|
| 6209 | option(redSB); option(redTail); // set them back? |
---|
[0610f0e] | 6210 | M2 = interred(M2); |
---|
[e64e417] | 6211 | if (ncols(M2) > 1) |
---|
| 6212 | { |
---|
| 6213 | ERROR("more than one leading component after interred: assume violation!"); |
---|
| 6214 | } |
---|
| 6215 | if (leadexp(M2[1]) != 0) |
---|
| 6216 | { |
---|
| 6217 | ERROR("nonconstant entry after interred: assume violation!"); |
---|
| 6218 | } |
---|
| 6219 | } |
---|
| 6220 | // now there's only one el-t with leadcomp<>0 |
---|
| 6221 | vector V = M2[1]; |
---|
| 6222 | number bcand = leadcoef(V[1]); // 1st component |
---|
| 6223 | V[1]=0; |
---|
| 6224 | number ct = content(V); // content of the cofactors |
---|
| 6225 | poly CF = ct*V[ncols(J)]; // polynomial in K[s]<x,dx>, cofactor to F |
---|
| 6226 | dbprint(ppl,"// -3-4- the cofactor candidate found"); |
---|
| 6227 | dbprint(ppl-1,CF); |
---|
| 6228 | dbprint(ppl,"// -3-5- the entry as it is"); |
---|
| 6229 | dbprint(ppl-1,bcand); |
---|
| 6230 | bcand = bcand*ct; // a product of both |
---|
| 6231 | dbprint(ppl,"// -3-6- the content of the rest vector"); |
---|
| 6232 | dbprint(ppl-1,ct); |
---|
| 6233 | ring @R3 = 0,s,dp; |
---|
| 6234 | dbprint(ppl,"// -4-1- the ring @R3 i.e. K[s] is ready"); |
---|
| 6235 | poly bcand = imap(@R4,bcand); |
---|
| 6236 | dbprint(ppl,"// -4-2- factorization"); |
---|
| 6237 | list P = factorize(bcand); //with constants and multiplicities |
---|
| 6238 | ideal bs; intvec m; //the Bernstein polynomial is monic, so we are not interested in constants |
---|
| 6239 | for (i=2; i<= size(P[1]); i++) //we delete P[1][1] and P[2][1] |
---|
| 6240 | { |
---|
| 6241 | bs[i-1] = P[1][i]; |
---|
| 6242 | m[i-1] = P[2][i]; |
---|
| 6243 | } |
---|
| 6244 | bs = normalize(bs); bs = -subst(bs,s,0); // to get roots only |
---|
| 6245 | setring @R2; // the ring the story started with |
---|
| 6246 | ideal bs = imap(@R3,bs); // intvec m is global |
---|
| 6247 | intvec mm = m; m = 0; |
---|
| 6248 | kill @R3; // kills m as well.... |
---|
[0610f0e] | 6249 | list @L = list(bs, mm); |
---|
[e64e417] | 6250 | // look for (2) return the GB of syzygies? |
---|
| 6251 | return(@L); |
---|
| 6252 | } |
---|
| 6253 | example |
---|
| 6254 | { "EXAMPLE:"; echo = 2; |
---|
| 6255 | ring r = 0,(x,y,z),Dp; |
---|
| 6256 | poly F = x^3+y^3+z^3; |
---|
| 6257 | printlevel = 0; |
---|
| 6258 | def A = Sannfs(F); setring A; |
---|
| 6259 | LD; |
---|
| 6260 | poly F = imap(r,F); |
---|
| 6261 | list L = bernsteinLift(LD,F); L; |
---|
| 6262 | poly bs = fl2poly(L,"s"); bs; // the candidate for Bernstein-Sato polynomial |
---|
| 6263 | } |
---|
| 6264 | |
---|
[66c962] | 6265 | /// ****** EXAMPLES ************ |
---|
| 6266 | |
---|
| 6267 | /* |
---|
| 6268 | |
---|
| 6269 | //static proc exCheckGenericity() |
---|
| 6270 | { |
---|
| 6271 | LIB "control.lib"; |
---|
| 6272 | ring r = (0,a,b,c),x,dp; |
---|
| 6273 | poly p = (x-a)*(x-b)*(x-c); |
---|
| 6274 | def A = annfsBM(p); |
---|
| 6275 | setring A; |
---|
| 6276 | ideal J = slimgb(LD); |
---|
| 6277 | matrix T = lift(LD,J); |
---|
| 6278 | T = normalize(T); |
---|
| 6279 | genericity(T); |
---|
| 6280 | // Ann =x^3*Dx+3*x^2*t*Dt+(-a-b-c)*x^2*Dx+(-2*a-2*b-2*c)*x*t*Dt+3*x^2+(a*b+a*c+b*c)*x*Dx+(a*b+a*c+b*c)*t*Dt+(-2*a-2*b-2*c)*x+(-a*b*c)*Dx+(a*b+a*c+b*c) |
---|
| 6281 | // genericity: g = a2-ab-ac+b2-bc+c2 =0 |
---|
| 6282 | // g = (a -(b+c)/2)^2 + (3/4)*(b-c)^2; |
---|
| 6283 | // g ==0 <=> a=b=c |
---|
| 6284 | // indeed, Ann = (x-a)^2*(x*Dx+3*t*Dt+(-a)*Dx+3) |
---|
| 6285 | // -------------------------------------------- |
---|
| 6286 | // BUT a direct computation shows |
---|
| 6287 | // when a=b=c, |
---|
| 6288 | // Ann = x*Dx+3*t*Dt+(-a)*Dx+3 |
---|
| 6289 | } |
---|
| 6290 | |
---|
| 6291 | //static proc exOT_17() |
---|
| 6292 | { |
---|
| 6293 | // Oaku-Takayama, p.208 |
---|
| 6294 | ring R = 0,(x,y),dp; |
---|
| 6295 | poly F = x^3-y^2; // x^2+x*y+y^2; |
---|
| 6296 | option(prot); |
---|
| 6297 | option(mem); |
---|
| 6298 | // option(redSB); |
---|
| 6299 | def A = annfsOT(F,0); |
---|
| 6300 | setring A; |
---|
| 6301 | LD; |
---|
| 6302 | gkdim(LD); // a holonomic check |
---|
| 6303 | // poly F = x^3-y^2; // = x^7 - y^5; // x^3-y^4; // x^5 - y^4; |
---|
| 6304 | } |
---|
| 6305 | |
---|
| 6306 | //static proc exOT_16() |
---|
| 6307 | { |
---|
| 6308 | // Oaku-Takayama, p.208 |
---|
| 6309 | ring R = 0,(x),dp; |
---|
| 6310 | poly F = x*(1-x); |
---|
| 6311 | option(prot); |
---|
| 6312 | option(mem); |
---|
| 6313 | // option(redSB); |
---|
| 6314 | def A = annfsOT(F,0); |
---|
| 6315 | setring A; |
---|
| 6316 | LD; |
---|
| 6317 | gkdim(LD); // a holonomic check |
---|
| 6318 | } |
---|
| 6319 | |
---|
| 6320 | //static proc ex_bcheck() |
---|
| 6321 | { |
---|
| 6322 | ring R = 0,(x,y),dp; |
---|
| 6323 | poly F = x*y*(x+y); |
---|
| 6324 | option(prot); |
---|
| 6325 | option(mem); |
---|
| 6326 | int eng = 0; |
---|
| 6327 | // option(redSB); |
---|
| 6328 | def A = annfsOT(F,eng); |
---|
| 6329 | setring A; |
---|
| 6330 | LD; |
---|
| 6331 | } |
---|
| 6332 | |
---|
| 6333 | //static proc ex_bcheck2() |
---|
| 6334 | { |
---|
| 6335 | ring R = 0,(x,y),dp; |
---|
| 6336 | poly F = x*y*(x+y); |
---|
| 6337 | int eng = 0; |
---|
| 6338 | def A = annfsBM(F,eng); |
---|
| 6339 | setring A; |
---|
| 6340 | LD; |
---|
| 6341 | } |
---|
| 6342 | |
---|
| 6343 | //static proc ex_BMI() |
---|
| 6344 | { |
---|
| 6345 | // a hard example |
---|
| 6346 | ring r = 0,(x,y),Dp; |
---|
| 6347 | poly F1 = (x2-y3)*(x3-y2); |
---|
| 6348 | poly F2 = (x2-y3)*(xy4+y5+x4); |
---|
| 6349 | ideal F = F1,F2; |
---|
| 6350 | def A = annfsBMI(F); |
---|
| 6351 | setring A; |
---|
| 6352 | LD; |
---|
| 6353 | BS; |
---|
| 6354 | } |
---|
| 6355 | |
---|
| 6356 | //static proc ex2_BMI() |
---|
| 6357 | { |
---|
| 6358 | // this example was believed to be intractable in 2005 by Gago-Vargas, Castro and Ucha |
---|
| 6359 | ring r = 0,(x,y),Dp; |
---|
| 6360 | option(prot); |
---|
| 6361 | option(mem); |
---|
| 6362 | ideal F = x2+y3,x3+y2; |
---|
| 6363 | printlevel = 2; |
---|
| 6364 | def A = annfsBMI(F); |
---|
| 6365 | setring A; |
---|
| 6366 | LD; |
---|
| 6367 | BS; |
---|
| 6368 | } |
---|
| 6369 | |
---|
| 6370 | //static proc ex_operatorBM() |
---|
| 6371 | { |
---|
| 6372 | ring r = 0,(x,y,z,w),Dp; |
---|
| 6373 | poly F = x^3+y^3+z^2*w; |
---|
| 6374 | printlevel = 0; |
---|
| 6375 | def A = operatorBM(F); |
---|
| 6376 | setring A; |
---|
| 6377 | F; // the original polynomial itself |
---|
| 6378 | LD; // generic annihilator |
---|
| 6379 | LD0; // annihilator |
---|
| 6380 | bs; // normalized Bernstein poly |
---|
| 6381 | BS; // root and multiplicities of the Bernstein poly |
---|
| 6382 | PS; // the operator, s.t. PS*F^{s+1} = bs*F^s mod LD |
---|
| 6383 | reduce(PS*F-bs,LD); // check the property of PS |
---|
| 6384 | } |
---|
| 6385 | |
---|
| 6386 | */ |
---|