[7a968b] | 1 | //////////////////////////////////////////////////////// |
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| 2 | version="version fpadim.lib 4.1.1.4 Oct_2018 "; // $Id$ |
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| 3 | category="Noncommutative"; |
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| 4 | info=" |
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| 5 | LIBRARY: fpadim.lib Vector space dimension, basis and Hilbert series for finitely presented algebras (Letterplace) |
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| 6 | AUTHORS: Grischa Studzinski, grischa.studzinski at rwth-aachen.de |
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| 7 | @* Viktor Levandovskyy, viktor.levandovskyy at math.rwth-aachen.de |
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| 8 | @* Karim Abou Zeid, karim.abou.zeid at rwth-aachen.de |
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| 9 | |
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| 10 | Support: Joint projects LE 2697/2-1 and KR 1907/3-1 of the Priority Programme SPP 1489: |
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| 11 | 'Algorithmische und Experimentelle Methoden in Algebra, Geometrie und Zahlentheorie' |
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| 12 | of the German DFG (2010-2013) |
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| 13 | and Project II.6 of the transregional collaborative research centre |
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| 14 | SFB-TRR 195 'Symbolic Tools in Mathematics and their Application' of the German DFG (from 2017 on) |
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| 15 | |
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[4cacf6] | 16 | KEYWORDS: finitely presented algebra; Letterplace Groebner basis; K-basis; K-dimension; Hilbert series |
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[7a968b] | 17 | |
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| 18 | PROCEDURES: |
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| 19 | lpKDimCheck(G); checks whether the K-dimension of A/<G> is finite |
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| 20 | lpKDim(G[,d,n]); computes the K-dimension of A/<G> |
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| 21 | lpMonomialBasis(d, donly, J); computes a list of monomials not contained in J |
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| 22 | lpHilbert(G[,d,n]); computes the truncated Hilbert series of A/<G> |
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| 23 | lpSickleDim(G[,d,n]); computes the mistletoes and the K-dimension of A/<G> |
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| 24 | |
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| 25 | NOTE: |
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| 26 | - basering is a Letterplace ring |
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| 27 | - all intvecs correspond to Letterplace monomials |
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| 28 | - if a degree bound d is specified, d <= attrib(basering,uptodeg) holds |
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| 29 | |
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| 30 | In the procedures below, 'iv' stands for intvec representation |
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| 31 | and 'lp' for the letterplace representation of monomials |
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| 32 | |
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| 33 | OVERVIEW: Given the free associative algebra A = K<x_1,...,x_n> and a (finite) Groebner basis |
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| 34 | GB = {g_1,..,g_w}, one is interested in the K-dimension and in the |
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| 35 | explicit monomial K-basis of A/<GB>. |
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| 36 | Therefore one is interested in the following data: |
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| 37 | - the Ufnarovskij graph induced by GB |
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| 38 | - the mistletoes of A/<GB> (special monomials in a basis) |
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| 39 | - the K-dimension of A/<GB> |
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| 40 | - the Hilbert series of A/<GB> |
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| 41 | |
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| 42 | @* The Ufnarovskij graph is used to determine whether A/<GB> has finite |
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| 43 | @* K-dimension. One has to check if the graph contains cycles. |
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| 44 | @* For the whole theory we refer to [Ufn]. Given a |
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| 45 | @* reduced set of monomials GB one can define the basis tree, whose vertex |
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| 46 | @* set V consists of all normal monomials w.r.t. GB. For every two |
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| 47 | @* monomials m_1, m_2 in V there is a direct edge from m_1 to m_2, if and |
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| 48 | @* only if there exists x_k in {x_1,..,x_n}, such that m_1*x_k = m_2. The |
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| 49 | @* set M = {m in V | there is no edge from m to another monomial in V} is |
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| 50 | @* called the set of mistletoes. As one can easily see it consists of |
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| 51 | @* the endpoints of the graph. Since there is a unique path to every |
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| 52 | @* monomial in V the whole graph can be described only from the knowledge |
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| 53 | @* of the mistletoes. Note that V corresponds to a basis of A/<GB>, so |
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| 54 | @* knowing the mistletoes we know a K-basis. The name mistletoes was given |
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| 55 | @* to those points because of these miraculous value and the algorithm is |
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| 56 | @* named sickle, because a sickle is the tool to harvest mistletoes. |
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| 57 | @* For more details see [Stu]. This package uses the Letterplace |
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| 58 | @* format introduced by [LL]. The algebra can either be represented as a |
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| 59 | @* Letterplace ring or via integer vectors: Every variable will only be |
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| 60 | @* represented by its number, so variable one is represented as 1, |
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| 61 | @* variable two as 2 and so on. The monomial x_1*x_3*x_2 for example will |
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| 62 | @* be stored as (1,3,2). Multiplication is concatenation. Note that the |
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| 63 | @* approach in this library does not need an algorithm for computing the normal |
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| 64 | @* form yet. Note that fpa is an acronym for Finitely Presented Algebra. |
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| 65 | @* |
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| 66 | |
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| 67 | REFERENCES: |
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| 68 | @* [Ufn] V. Ufnarovskij: Combinatorical and asymptotic methods in algebra, 1990. |
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| 69 | @* [LL] R. La Scala, V. Levandovskyy: Letterplace ideals and non-commutative |
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| 70 | Groebner bases, Journal of Symbolic Computation, 2009. |
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| 71 | @* [Stu] G. Studzinski: Dimension computations in non-commutative, |
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| 72 | associative algebras, Diploma thesis, RWTH Aachen, 2010. |
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| 73 | |
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| 74 | SEE ALSO: freegb_lib, fpaprops_lib, ncHilb_lib |
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| 75 | "; |
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| 76 | |
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| 77 | // iv2lp, lp2iv etc not in HEADER because they should not be used anymore and will be removed in soon |
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| 78 | |
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| 79 | LIB "freegb.lib"; //for letterplace rings |
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| 80 | LIB "general.lib";//for sorting mistletoes |
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| 81 | |
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| 82 | ///////////////////////////////////////////////////////// |
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| 83 | |
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| 84 | /* very fast and cheap test of consistency and functionality |
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| 85 | DO NOT make it static ! |
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| 86 | after adding the new proc, add it here */ |
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| 87 | proc tstfpadim() |
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| 88 | { |
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| 89 | example ivDHilbert; |
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| 90 | example ivDHilbertSickle; |
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| 91 | example ivKDimCheck; |
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| 92 | example ivHilbert; |
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| 93 | example ivKDim; |
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| 94 | example ivMis2Base; |
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| 95 | example ivMis2Dim; |
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| 96 | example ivOrdMisLex; |
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| 97 | example ivSickle; |
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| 98 | example ivSickleHil; |
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| 99 | example ivSickleDim; |
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| 100 | example lpDHilbert; |
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| 101 | example lpDHilbertSickle; |
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| 102 | example lpHilbert; |
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| 103 | example lpKDimCheck; |
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| 104 | example lpKDim; |
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| 105 | example lpMis2Base; |
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| 106 | example lpMis2Dim; |
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| 107 | example lpOrdMisLex; |
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| 108 | example lpSickle; |
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| 109 | example lpSickleHil; |
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| 110 | example lpSickleDim; |
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| 111 | example sickle; |
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| 112 | example ivL2lpI; |
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| 113 | example iv2lp; |
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| 114 | example iv2lpList; |
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| 115 | example iv2lpMat; |
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| 116 | example lp2iv; |
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| 117 | example lp2ivId; |
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| 118 | example lpId2ivLi; |
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| 119 | example lpMonomialBasis; |
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| 120 | } |
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| 121 | |
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| 122 | |
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| 123 | //--------------- auxiliary procedures ------------------ |
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| 124 | |
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| 125 | static proc allVars(list L, intvec P, int n) |
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| 126 | "USAGE: allVars(L,P,n); L a list of intmats, P an intvec, n an integer |
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| 127 | RETURN: int, 0 if all variables are contained in the quotient algebra, 1 otherwise |
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| 128 | " |
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| 129 | {int i,j,r; |
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| 130 | intvec V; |
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| 131 | for (i = 1; i <= size(P); i++) {if (P[i] == 1){ j = i; break;}} |
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| 132 | V = L[j][1..nrows(L[j]),1]; |
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| 133 | for (i = 1; i <= n; i++) {if (isInVec(i,V) == 0) {r = 1; break;}} |
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| 134 | if (r == 0) {return(1);} |
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| 135 | else {return(0);} |
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| 136 | } |
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| 137 | |
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| 138 | static proc checkAssumptions(int d, list L) |
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| 139 | "PURPOSE: Checks, if all the Assumptions are holding |
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| 140 | " |
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| 141 | {if (attrib(basering,"isLetterplaceRing")==0) {ERROR("Basering is not a Letterplace ring!");} |
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| 142 | if (d > attrib(basering,"uptodeg")) {ERROR("Specified degree bound exceeds ring parameter!");} |
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| 143 | int i; |
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| 144 | for (i = 1; i <= size(L); i++) |
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| 145 | {if (entryViolation(L[i], attrib(basering,"isLetterplaceRing"))) |
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| 146 | {ERROR("Not allowed monomial/intvec found!");} |
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| 147 | } |
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| 148 | return(); |
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| 149 | } |
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| 150 | |
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| 151 | static proc createStartMat(int d, int n) |
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| 152 | "USAGE: createStartMat(d,n); d, n integers |
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| 153 | RETURN: intmat |
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| 154 | PURPOSE:Creating the intmat with all normal monomials in n variables and of degree d to start with |
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| 155 | NOTE: d has to be > 0 |
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| 156 | " |
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| 157 | {intmat M[(n^d)][d]; |
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| 158 | int i1,i2,i3,i4; |
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| 159 | for (i1 = 1; i1 <= d; i1++) //Spalten |
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| 160 | {i2 = 1; //durchlaeuft Zeilen |
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| 161 | while (i2 <= (n^d)) |
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| 162 | {for (i3 = 1; i3 <= n; i3++) |
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| 163 | {for (i4 = 1; i4 <= (n^(i1-1)); i4++) |
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| 164 | {M[i2,i1] = i3; |
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| 165 | i2 = i2 + 1; |
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| 166 | } |
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| 167 | } |
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| 168 | } |
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| 169 | } |
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| 170 | return(M); |
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| 171 | } |
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| 172 | |
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| 173 | static proc createStartMat1(int n, intmat M) |
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| 174 | "USAGE: createStartMat1(n,M); n an integer, M an intmat |
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| 175 | RETURN: intmat, with all variables except those in M |
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| 176 | " |
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| 177 | {int i; |
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| 178 | intvec V,Vt; |
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| 179 | V = M[(1..nrows(M)),1]; |
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| 180 | for (i = 1; i <= size(V); i++) {if (isInVec(i,V) == 0) {Vt = Vt,i;}} |
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| 181 | if (Vt == 0) {intmat S; return(S);} |
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| 182 | else {Vt = Vt[2..size(Vt)]; intmat S [size(Vt)][1]; S[1..size(Vt),1] = Vt; return(S);} |
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| 183 | } |
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| 184 | |
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| 185 | static proc entryViolation(intmat M, int n) |
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| 186 | "PURPOSE:checks, if all entries in M are variable-related |
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| 187 | " |
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| 188 | {int i,j; |
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| 189 | for (i = 1; i <= nrows(M); i++) |
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| 190 | {for (j = 1; j <= ncols(M); j++) |
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| 191 | {if(!((1<=M[i,j])&&(M[i,j]<=n))) {return(1);}} |
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| 192 | } |
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| 193 | return(0); |
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| 194 | } |
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| 195 | |
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| 196 | static proc findDimen(intvec V, int n, list L, intvec P, list #) |
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| 197 | "USAGE: findDimen(V,n,L,P,degbound); V,P intvecs, n, an integer, L a list, |
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| 198 | @* degbound an optional integer |
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| 199 | RETURN: int |
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| 200 | PURPOSE:Compute the K-dimension of the quotient algebra |
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| 201 | " |
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| 202 | {int degbound = 0; |
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| 203 | if (size(#) > 0) {if (#[1] > 0) {degbound = #[1];}} |
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| 204 | int dimen,i,j,w,it; |
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| 205 | intvec Vt,Vt2; |
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| 206 | module M; |
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| 207 | if (degbound == 0) |
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| 208 | {for (i = 1; i <= n; i++) |
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| 209 | {Vt = V, i; w = 0; |
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| 210 | for (j = 1; j<= size(P); j++) |
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| 211 | {if (P[j] <= size(Vt)) |
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| 212 | {Vt2 = Vt[(size(Vt)-P[j]+1)..size(Vt)]; |
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| 213 | if (isInMat(Vt2,L[j]) > 0) {w = 1; break;} |
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| 214 | } |
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| 215 | } |
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| 216 | if (w == 0) |
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| 217 | {vector Vtt; |
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| 218 | for (it = 1; it <= size(Vt); it++){Vtt = Vtt + Vt[it]*gen(it);} |
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| 219 | M = M,Vtt; |
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| 220 | kill Vtt; |
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| 221 | } |
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| 222 | } |
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| 223 | if (size(M) == 0) {return(0);} |
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| 224 | else |
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| 225 | {M = simplify(M,2); |
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| 226 | for (i = 1; i <= size(M); i++) |
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| 227 | {kill Vt; intvec Vt; |
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| 228 | for (j =1; j <= size(M[i]); j++){Vt[j] = int(leadcoef(M[i][j]));} |
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| 229 | dimen = dimen + 1 + findDimen(Vt,n,L,P); |
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| 230 | } |
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| 231 | return(dimen); |
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| 232 | } |
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| 233 | } |
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| 234 | else |
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| 235 | {if (size(V) > degbound) {ERROR("monomial exceeds degreebound");} |
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| 236 | if (size(V) == degbound) {return(0);} |
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| 237 | for (i = 1; i <= n; i++) |
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| 238 | {Vt = V, i; w = 0; |
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| 239 | for (j = 1; j<= size(P); j++) |
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| 240 | {if (P[j] <= size(Vt)) |
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| 241 | {Vt2 = Vt[(size(Vt)-P[j]+1)..size(Vt)]; |
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| 242 | if (isInMat(Vt2,L[j]) > 0) {w = 1; break;} |
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| 243 | } |
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| 244 | } |
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| 245 | if (w == 0) {vector Vtt; |
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| 246 | for (it = 1; it <= size(Vt); it++){Vtt = Vtt + Vt[it]*gen(it);} |
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| 247 | M = M,Vtt; |
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| 248 | kill Vtt; |
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| 249 | } |
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| 250 | } |
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| 251 | if (size(M) == 0) {return(0);} |
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| 252 | else |
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| 253 | {M = simplify(M,2); |
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| 254 | for (i = 1; i <= size(M); i++) |
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| 255 | {kill Vt; intvec Vt; |
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| 256 | for (j =1; j <= size(M[i]); j++){Vt[j] = int(leadcoef(M[i][j]));} |
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| 257 | dimen = dimen + 1 + findDimen(Vt,n,L,P,degbound); |
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| 258 | } |
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| 259 | return(dimen); |
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| 260 | } |
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| 261 | } |
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| 262 | } |
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| 263 | |
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| 264 | static proc findCycle(intvec V, list L, intvec P, int n, int ld, module M) |
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| 265 | "USAGE: |
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| 266 | RETURN: int, 1 if Ufn-graph contains a cycle, or 0 otherwise |
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| 267 | PURPOSE:Searching the Ufnarovskij graph for cycles |
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| 268 | " |
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| 269 | {int i,j,w,r;intvec Vt,Vt2; |
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| 270 | int it, it2; |
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| 271 | if (size(V) < ld) |
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| 272 | {for (i = 1; i <= n; i++) |
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| 273 | {Vt = V,i; w = 0; |
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| 274 | for (j = 1; j <= size(P); j++) |
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| 275 | {if (P[j] <= size(Vt)) |
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| 276 | {Vt2 = Vt[(size(Vt)-P[j]+1)..size(Vt)]; |
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| 277 | if (isInMat(Vt2,L[j]) > 0) |
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| 278 | {w = 1; break;} |
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| 279 | } |
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| 280 | } |
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| 281 | if (w == 0) {r = findCycle(Vt,L,P,n,ld,M);} |
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| 282 | if (r == 1) {break;} |
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| 283 | } |
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| 284 | return(r); |
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| 285 | } |
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| 286 | else |
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| 287 | {j = size(M); |
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| 288 | if (j > 0) |
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| 289 | { |
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| 290 | intmat Mt[j][nrows(M)]; |
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| 291 | for (it = 1; it <= j; it++) |
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| 292 | { for(it2 = 1; it2 <= nrows(M);it2++) |
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| 293 | {Mt[it,it2] = int(leadcoef(M[it2,it]));} |
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| 294 | } |
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| 295 | Vt = V[(size(V)-ld+1)..size(V)]; |
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| 296 | //Mt; type(Mt);Vt;type(Vt); |
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| 297 | if (isInMat(Vt,Mt) > 0) {return(1);} |
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| 298 | else |
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| 299 | {vector Vtt; |
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| 300 | for (it =1; it <= size(Vt); it++) |
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| 301 | {Vtt = Vtt + Vt[it]*gen(it);} |
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| 302 | M = M,Vtt; |
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| 303 | kill Vtt; |
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| 304 | for (i = 1; i <= n; i++) |
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| 305 | {Vt = V,i; w = 0; |
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| 306 | for (j = 1; j <= size(P); j++) |
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| 307 | {if (P[j] <= size(Vt)) |
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| 308 | {Vt2 = Vt[(size(Vt)-P[j]+1)..size(Vt)]; |
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| 309 | //L[j]; type(L[j]);Vt2;type(Vt2); |
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| 310 | if (isInMat(Vt2,L[j]) > 0) |
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| 311 | {w = 1; break;} |
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| 312 | } |
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| 313 | } |
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| 314 | if (w == 0) {r = findCycle(Vt,L,P,n,ld,M);} |
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| 315 | if (r == 1) {break;} |
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| 316 | } |
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| 317 | return(r); |
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| 318 | } |
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| 319 | } |
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| 320 | else |
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| 321 | { Vt = V[(size(V)-ld+1)..size(V)]; |
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| 322 | vector Vtt; |
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| 323 | for (it = 1; it <= size(Vt); it++) |
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| 324 | {Vtt = Vtt + Vt[it]*gen(it);} |
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| 325 | M = Vtt; |
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| 326 | kill Vtt; |
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| 327 | for (i = 1; i <= n; i++) |
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| 328 | {Vt = V,i; w = 0; |
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| 329 | for (j = 1; j <= size(P); j++) |
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| 330 | {if (P[j] <= size(Vt)) |
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| 331 | {Vt2 = Vt[(size(Vt)-P[j]+1)..size(Vt)]; |
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| 332 | //L[j]; type(L[j]);Vt2;type(Vt2); |
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| 333 | if (isInMat(Vt2,L[j]) > 0) |
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| 334 | {w = 1; break;} |
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| 335 | } |
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| 336 | } |
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| 337 | if (w == 0) {r = findCycle(Vt,L,P,n,ld,M);} |
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| 338 | if (r == 1) {break;} |
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| 339 | } |
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| 340 | return(r); |
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| 341 | } |
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| 342 | } |
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| 343 | } |
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| 344 | |
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| 345 | |
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| 346 | static proc findCycleDFS(int i, intmat T, intvec V) |
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| 347 | " |
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| 348 | PURPOSE: |
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| 349 | this is a classical deep-first search for cycles contained in a graph given by an intmat |
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| 350 | " |
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| 351 | { |
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| 352 | intvec rV; |
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| 353 | int k,k1,t; |
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| 354 | int j = V[size(V)]; |
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| 355 | if (T[j,i] > 0) {return(V);} |
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| 356 | else |
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| 357 | { |
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| 358 | for (k = 1; k <= ncols(T); k++) |
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| 359 | { |
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| 360 | t = 0; |
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| 361 | if (T[j,k] > 0) |
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| 362 | { |
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| 363 | for (k1 = 1; k1 <= size(V); k1++) {if (V[k1] == k) {t = 1; break;}} |
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| 364 | if (t == 0) |
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| 365 | { |
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| 366 | rV = V; |
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| 367 | rV[size(rV)+1] = k; |
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| 368 | rV = findCycleDFS(i,T,rV); |
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| 369 | if (rV[1] > -1) {return(rV);} |
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| 370 | } |
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| 371 | } |
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| 372 | } |
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| 373 | } |
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| 374 | return(intvec(-1)); |
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| 375 | } |
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| 376 | |
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| 377 | |
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| 378 | |
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| 379 | static proc findHCoeff(intvec V,int n,list L,intvec P,intvec H,list #) |
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| 380 | "USAGE: findHCoeff(V,n,L,P,H,degbound); L a list of intmats, degbound an integer |
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| 381 | RETURN: intvec |
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| 382 | PURPOSE:Compute the coefficient of the Hilbert series (upto degree degbound) |
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| 383 | NOTE: Starting with a part of the Hilbert series we change the coefficient |
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| 384 | @* depending on how many basis elements we found on the actual branch |
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| 385 | " |
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| 386 | {int degbound = 0; |
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| 387 | if (size(#) > 0){if (#[1] > 0){degbound = #[1];}} |
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| 388 | int i,w,j,it; |
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| 389 | int h1 = 0; |
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| 390 | intvec Vt,Vt2,H1; |
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| 391 | module M; |
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| 392 | if (degbound == 0) |
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| 393 | {for (i = 1; i <= n; i++) |
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| 394 | {Vt = V, i; w = 0; |
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| 395 | for (j = 1; j<= size(P); j++) |
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| 396 | {if (P[j] <= size(Vt)) |
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| 397 | {Vt2 = Vt[(size(Vt)-P[j]+1)..size(Vt)]; |
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| 398 | if (isInMat(Vt2,L[j]) > 0) {w = 1; break;} |
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| 399 | } |
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| 400 | } |
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| 401 | if (w == 0) |
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| 402 | {vector Vtt; |
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| 403 | for (it = 1; it <= size(Vt); it++){Vtt = Vtt + Vt[it]*gen(it);} |
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| 404 | M = M,Vtt; |
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| 405 | kill Vtt; |
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| 406 | } |
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| 407 | } |
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| 408 | if (size(M) == 0) {return(H);} |
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| 409 | else |
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| 410 | {M = simplify(M,2); |
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| 411 | for (i = 1; i <= size(M); i++) |
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| 412 | {kill Vt; intvec Vt; |
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| 413 | for (j =1; j <= size(M[i]); j++) {Vt[j] = int(leadcoef(M[i][j]));} |
---|
| 414 | h1 = h1 + 1; H1 = findHCoeff(Vt,n,L,P,H1); |
---|
| 415 | } |
---|
| 416 | if (size(H1) < (size(V)+2)) {H1[(size(V)+2)] = h1;} |
---|
| 417 | else {H1[(size(V)+2)] = H1[(size(V)+2)] + h1;} |
---|
| 418 | H1 = H1 + H; |
---|
| 419 | return(H1); |
---|
| 420 | } |
---|
| 421 | } |
---|
| 422 | else |
---|
| 423 | {if (size(V) > degbound) {ERROR("monomial exceeds degreebound");} |
---|
| 424 | if (size(V) == degbound) {return(H);} |
---|
| 425 | for (i = 1; i <= n; i++) |
---|
| 426 | {Vt = V, i; w = 0; |
---|
| 427 | for (j = 1; j<= size(P); j++) |
---|
| 428 | {if (P[j] <= size(Vt)) |
---|
| 429 | {Vt2 = Vt[(size(Vt)-P[j]+1)..size(Vt)]; |
---|
| 430 | if (isInMat(Vt2,L[j]) > 0) {w = 1; break;} |
---|
| 431 | } |
---|
| 432 | } |
---|
| 433 | if (w == 0) |
---|
| 434 | {vector Vtt; |
---|
| 435 | for (it = 1; it <= size(Vt); it++){Vtt = Vtt + Vt[it]*gen(it);} |
---|
| 436 | M = M,Vtt; |
---|
| 437 | kill Vtt; |
---|
| 438 | } |
---|
| 439 | } |
---|
| 440 | if (size(M) == 0) {return(H);} |
---|
| 441 | else |
---|
| 442 | {M = simplify(M,2); |
---|
| 443 | for (i = 1; i <= size(M); i++) |
---|
| 444 | {kill Vt; intvec Vt; |
---|
| 445 | for (j =1; j <= size(M[i]); j++) |
---|
| 446 | {Vt[j] = int(leadcoef(M[i][j]));} |
---|
| 447 | h1 = h1 + 1; H1 = findHCoeff(Vt,n,L,P,H1,degbound); |
---|
| 448 | } |
---|
| 449 | if (size(H1) < (size(V)+2)) { H1[(size(V)+2)] = h1;} |
---|
| 450 | else {H1[(size(V)+2)] = H1[(size(V)+2)] + h1;} |
---|
| 451 | H1 = H1 + H; |
---|
| 452 | return(H1); |
---|
| 453 | } |
---|
| 454 | } |
---|
| 455 | } |
---|
| 456 | |
---|
| 457 | static proc findHCoeffMis(intvec V, int n, list L, intvec P, list R,list #) |
---|
| 458 | "USAGE: findHCoeffMis(V,n,L,P,R,degbound); degbound an optional integer, L a |
---|
| 459 | @* list of Intmats, R |
---|
| 460 | RETURN: list |
---|
| 461 | PURPOSE:Compute the coefficients of the Hilbert series and the Mistletoes all |
---|
| 462 | @* at once |
---|
| 463 | " |
---|
| 464 | {int degbound = 0; |
---|
| 465 | if (size(#) > 0) {if (#[1] > 0) {degbound = #[1];}} |
---|
| 466 | int i,w,j,h1; |
---|
| 467 | intvec Vt,Vt2,H1; int it; |
---|
| 468 | module M; |
---|
| 469 | if (degbound == 0) |
---|
| 470 | {for (i = 1; i <= n; i++) |
---|
| 471 | {Vt = V, i; w = 0; |
---|
| 472 | for (j = 1; j<= size(P); j++) |
---|
| 473 | {if (P[j] <= size(Vt)) |
---|
| 474 | {Vt2 = Vt[(size(Vt)-P[j]+1)..size(Vt)]; |
---|
| 475 | if (isInMat(Vt2,L[j]) > 0) {w = 1; break;} |
---|
| 476 | } |
---|
| 477 | } |
---|
| 478 | if (w == 0) |
---|
| 479 | {vector Vtt; |
---|
| 480 | for (it = 1; it <= size(Vt); it++){Vtt = Vtt + Vt[it]*gen(it);} |
---|
| 481 | M = M,Vtt; |
---|
| 482 | kill Vtt; |
---|
| 483 | } |
---|
| 484 | } |
---|
| 485 | if (size(M) == 0) {if (size(R) < 2){R[2] = list(V);} else {R[2] = R[2] + list(V);} return(R);} |
---|
| 486 | else |
---|
| 487 | {M = simplify(M,2); |
---|
| 488 | for (i = 1; i <= size(M); i++) |
---|
| 489 | {kill Vt; intvec Vt; |
---|
| 490 | for (j =1; j <= size(M[i]); j++) |
---|
| 491 | {Vt[j] = int(leadcoef(M[i][j]));} |
---|
| 492 | if (size(R[1]) < (size(V)+2)) { R[1][(size(V)+2)] = 1;} |
---|
| 493 | else |
---|
| 494 | {R[1][(size(V)+2)] = R[1][(size(V)+2)] + 1;} |
---|
| 495 | R = findHCoeffMis(Vt,n,L,P,R); |
---|
| 496 | } |
---|
| 497 | return(R); |
---|
| 498 | } |
---|
| 499 | } |
---|
| 500 | else |
---|
| 501 | {if (size(V) > degbound) {ERROR("monomial exceeds degreebound");} |
---|
| 502 | if (size(V) == degbound) |
---|
| 503 | {if (size(R) < 2){R[2] = list (V);} |
---|
| 504 | else{R[2] = R[2] + list (V);} |
---|
| 505 | return(R); |
---|
| 506 | } |
---|
| 507 | for (i = 1; i <= n; i++) |
---|
| 508 | {Vt = V, i; w = 0; |
---|
| 509 | for (j = 1; j<= size(P); j++) |
---|
| 510 | {if (P[j] <= size(Vt)) |
---|
| 511 | {Vt2 = Vt[(size(Vt)-P[j]+1)..size(Vt)]; |
---|
| 512 | if (isInMat(Vt2,L[j]) > 0) {w = 1; break;} |
---|
| 513 | } |
---|
| 514 | } |
---|
| 515 | if (w == 0) |
---|
| 516 | {vector Vtt; |
---|
| 517 | for (it = 1; it <= size(Vt); it++){Vtt = Vtt + Vt[it]*gen(it);} |
---|
| 518 | M = M,Vtt; |
---|
| 519 | kill Vtt; |
---|
| 520 | } |
---|
| 521 | } |
---|
| 522 | if (size(M) == 0) {if (size(R) < 2){R[2] = list(V);} else {R[2] = R[2] + list(V);} return(R);} |
---|
| 523 | else |
---|
| 524 | {M = simplify(M,2); |
---|
| 525 | for (i = 1; i <= ncols(M); i++) |
---|
| 526 | {kill Vt; intvec Vt; |
---|
| 527 | for (j =1; j <= size(M[i]); j++) |
---|
| 528 | {Vt[j] = int(leadcoef(M[i][j]));} |
---|
| 529 | if (size(R[1]) < (size(V)+2)) { R[1][(size(V)+2)] = 1;} |
---|
| 530 | else |
---|
| 531 | {R[1][(size(V)+2)] = R[1][(size(V)+2)] + 1;} |
---|
| 532 | R = findHCoeffMis(Vt,n,L,P,R,degbound); |
---|
| 533 | } |
---|
| 534 | return(R); |
---|
| 535 | } |
---|
| 536 | } |
---|
| 537 | } |
---|
| 538 | |
---|
| 539 | |
---|
| 540 | static proc findMisDim(intvec V,int n,list L,intvec P,list R,list #) |
---|
| 541 | "USAGE: |
---|
| 542 | RETURN: list |
---|
| 543 | PURPOSE:Compute the K-dimension and the Mistletoes all at once |
---|
| 544 | " |
---|
| 545 | {int degbound = 0; |
---|
| 546 | if (size(#) > 0) {if (#[1] > 0) {degbound = #[1];}} |
---|
| 547 | int dimen,i,j,w; |
---|
| 548 | intvec Vt,Vt2; int it; |
---|
| 549 | module M; |
---|
| 550 | if (degbound == 0) |
---|
| 551 | {for (i = 1; i <= n; i++) |
---|
| 552 | {Vt = V, i; w = 0; |
---|
| 553 | for (j = 1; j<= size(P); j++) |
---|
| 554 | {if (P[j] <= size(Vt)) |
---|
| 555 | {Vt2 = Vt[(size(Vt)-P[j]+1)..size(Vt)]; |
---|
| 556 | if (isInMat(Vt2,L[j]) > 0) {w = 1; break;} |
---|
| 557 | } |
---|
| 558 | } |
---|
| 559 | if (w == 0) |
---|
| 560 | {vector Vtt; |
---|
| 561 | for (it = 1; it <= size(Vt); it++){Vtt = Vtt + Vt[it]*gen(it);} |
---|
| 562 | M = M,Vtt; |
---|
| 563 | kill Vtt; |
---|
| 564 | } |
---|
| 565 | } |
---|
| 566 | if (size(M) == 0) |
---|
| 567 | {if (size(R) < 2){R[2] = list (V);} |
---|
| 568 | else{R[2] = R[2] + list(V);} |
---|
| 569 | return(R); |
---|
| 570 | } |
---|
| 571 | else |
---|
| 572 | {M = simplify(M,2); |
---|
| 573 | for (i = 1; i <= size(M); i++) |
---|
| 574 | {kill Vt; intvec Vt; |
---|
| 575 | for (j =1; j <= size(M[i]); j++){Vt[j] = int(leadcoef(M[i][j]));} |
---|
| 576 | R[1] = R[1] + 1; R = findMisDim(Vt,n,L,P,R); |
---|
| 577 | } |
---|
| 578 | return(R); |
---|
| 579 | } |
---|
| 580 | } |
---|
| 581 | else |
---|
| 582 | {if (size(V) > degbound) {ERROR("monomial exceeds degreebound");} |
---|
| 583 | if (size(V) == degbound) |
---|
| 584 | {if (size(R) < 2){R[2] = list (V);} |
---|
| 585 | else{R[2] = R[2] + list (V);} |
---|
| 586 | return(R); |
---|
| 587 | } |
---|
| 588 | for (i = 1; i <= n; i++) |
---|
| 589 | {Vt = V, i; w = 0; |
---|
| 590 | for (j = 1; j<= size(P); j++) |
---|
| 591 | {if (P[j] <= size(Vt)) |
---|
| 592 | {Vt2 = Vt[(size(Vt)-P[j]+1)..size(Vt)]; |
---|
| 593 | if (isInMat(Vt2,L[j]) > 0) {w = 1; break;} |
---|
| 594 | } |
---|
| 595 | } |
---|
| 596 | if (w == 0) |
---|
| 597 | {vector Vtt; |
---|
| 598 | for (it = 1; it <= size(Vt); it++){Vtt = Vtt + Vt[it]*gen(it);} |
---|
| 599 | M = M,Vtt; |
---|
| 600 | kill Vtt; |
---|
| 601 | } |
---|
| 602 | } |
---|
| 603 | if (size(M) == 0) |
---|
| 604 | {if (size(R) < 2){R[2] = list (V);} |
---|
| 605 | else{R[2] = R[2] + list(V);} |
---|
| 606 | return(R); |
---|
| 607 | } |
---|
| 608 | else |
---|
| 609 | {M = simplify(M,2); |
---|
| 610 | for (i = 1; i <= size(M); i++) |
---|
| 611 | {kill Vt; intvec Vt; |
---|
| 612 | for (j =1; j <= size(M[i]); j++){Vt[j] = int(leadcoef(M[i][j]));} |
---|
| 613 | R[1] = R[1] + 1; R = findMisDim(Vt,n,L,P,R,degbound); |
---|
| 614 | } |
---|
| 615 | return(R); |
---|
| 616 | } |
---|
| 617 | } |
---|
| 618 | } |
---|
| 619 | |
---|
| 620 | |
---|
| 621 | static proc findmistletoes(intvec V, int n, list L, intvec P, list #) |
---|
| 622 | "USAGE: findmistletoes(V,n,L,P,degbound); V a normal word, n the number of |
---|
| 623 | @* variables, L the GB, P the occuring degrees, |
---|
| 624 | @* and degbound the (optional) degreebound |
---|
| 625 | RETURN: list |
---|
| 626 | PURPOSE:Compute mistletoes starting in V |
---|
| 627 | NOTE: V has to be normal w.r.t. L, it will not be checked for being so |
---|
| 628 | " |
---|
| 629 | {int degbound = 0; |
---|
| 630 | if (size(#) > 0) {if (#[1] > 0) {degbound = #[1];}} |
---|
| 631 | list R; intvec Vt,Vt2; int it; |
---|
| 632 | int i,j; |
---|
| 633 | module M; |
---|
| 634 | if (degbound == 0) |
---|
| 635 | {int w; |
---|
| 636 | for (i = 1; i <= n; i++) |
---|
| 637 | {Vt = V,i; w = 0; |
---|
| 638 | for (j = 1; j <= size(P); j++) |
---|
| 639 | {if (P[j] <= size(Vt)) |
---|
| 640 | {Vt2 = Vt[(size(Vt)-P[j]+1)..size(Vt)]; |
---|
| 641 | if (isInMat(Vt2,L[j]) > 0) |
---|
| 642 | {w = 1; break;} |
---|
| 643 | } |
---|
| 644 | } |
---|
| 645 | if (w == 0) |
---|
| 646 | {vector Vtt; |
---|
| 647 | for (it = 1; it <= size(Vt); it++){Vtt = Vtt + Vt[it]*gen(it);} |
---|
| 648 | M = M,Vtt; |
---|
| 649 | kill Vtt; |
---|
| 650 | } |
---|
| 651 | } |
---|
| 652 | if (size(M)==0) {R = V; return(R);} |
---|
| 653 | else |
---|
| 654 | {M = simplify(M,2); |
---|
| 655 | for (i = 1; i <= size(M); i++) |
---|
| 656 | {kill Vt; intvec Vt; |
---|
| 657 | for (j =1; j <= size(M[i]); j++){Vt[j] = int(leadcoef(M[i][j]));} |
---|
| 658 | R = R + findmistletoes(Vt,n,L,P); |
---|
| 659 | } |
---|
| 660 | return(R); |
---|
| 661 | } |
---|
| 662 | } |
---|
| 663 | else |
---|
| 664 | {if (size(V) > degbound) {ERROR("monomial exceeds degreebound");} |
---|
| 665 | if (size(V) == degbound) {R = V; return(R);} |
---|
| 666 | int w; |
---|
| 667 | for (i = 1; i <= n; i++) |
---|
| 668 | {Vt = V,i; w = 0; |
---|
| 669 | for (j = 1; j <= size(P); j++) |
---|
| 670 | {if (P[j] <= size(Vt)) |
---|
| 671 | {Vt2 = Vt[(size(Vt)-P[j]+1)..size(Vt)]; |
---|
| 672 | if (isInMat(Vt2,L[j]) > 0){w = 1; break;} |
---|
| 673 | } |
---|
| 674 | } |
---|
| 675 | if (w == 0) |
---|
| 676 | {vector Vtt; |
---|
| 677 | for (it = 1; it <= size(Vt); it++){Vtt = Vtt + Vt[it]*gen(it);} |
---|
| 678 | M = M,Vtt; |
---|
| 679 | kill Vtt; |
---|
| 680 | } |
---|
| 681 | } |
---|
| 682 | if (size(M) == 0) {R = V; return(R);} |
---|
| 683 | else |
---|
| 684 | {M = simplify(M,2); |
---|
| 685 | for (i = 1; i <= ncols(M); i++) |
---|
| 686 | {kill Vt; intvec Vt; |
---|
| 687 | for (j =1; j <= size(M[i]); j++) |
---|
| 688 | {Vt[j] = int(leadcoef(M[i][j]));} |
---|
| 689 | //Vt; typeof(Vt); size(Vt); |
---|
| 690 | R = R + findmistletoes(Vt,n,L,P,degbound); |
---|
| 691 | } |
---|
| 692 | return(R); |
---|
| 693 | } |
---|
| 694 | } |
---|
| 695 | } |
---|
| 696 | |
---|
| 697 | static proc growthAlg(intmat T, list #) |
---|
| 698 | " |
---|
| 699 | real algorithm for checking the growth of an algebra |
---|
| 700 | " |
---|
| 701 | { |
---|
| 702 | int s = 1; |
---|
| 703 | if (size(#) > 0) { s = #[1];} |
---|
| 704 | int j; |
---|
| 705 | int n = ncols(T); |
---|
| 706 | intvec NV,C; NV[n] = 0; int m,i; |
---|
| 707 | intmat T2[n][n] = T[1..n,1..n]; intmat N[n][n]; |
---|
| 708 | if (T2 == N) |
---|
| 709 | { |
---|
| 710 | for (i = 1; i <= n; i++) |
---|
| 711 | { |
---|
| 712 | if (m < T[n+1,i]) { m = T[n+1,i];} |
---|
| 713 | } |
---|
| 714 | return(m); |
---|
| 715 | } |
---|
| 716 | |
---|
| 717 | //first part: the diagonals |
---|
| 718 | for (i = s; i <= n; i++) |
---|
| 719 | { |
---|
| 720 | if (T[i,i] > 0) |
---|
| 721 | { |
---|
| 722 | if ((T[i,i] >= 1) && (T[n+1,i] > 0)) {return(-1);} |
---|
| 723 | if ((T[i,i] == 1) && (T[n+1,i] == 0)) |
---|
| 724 | { |
---|
| 725 | T[i,i] = 0; |
---|
| 726 | T[n+1,i] = 1; |
---|
| 727 | return(growthAlg(T)); |
---|
| 728 | } |
---|
| 729 | } |
---|
| 730 | } |
---|
| 731 | |
---|
| 732 | //second part: searching for the last but one vertices |
---|
| 733 | T2 = T2*T2; |
---|
| 734 | for (i = s; i <= n; i++) |
---|
| 735 | { |
---|
| 736 | if ((intvec(T[i,1..n]) <> intvec(0)) && (intvec(T2[i,1..n]) == intvec(0))) |
---|
| 737 | { |
---|
| 738 | for (j = 1; j <= n; j++) |
---|
| 739 | { |
---|
| 740 | if ((T[i,j] > 0) && (m < T[n+1,j])) {m = T[n+1,j];} |
---|
| 741 | } |
---|
| 742 | T[n+1,i] = T[n+1,i] + m; |
---|
| 743 | T[i,1..n] = NV; |
---|
| 744 | return(growthAlg(T)); |
---|
| 745 | } |
---|
| 746 | } |
---|
| 747 | m = 0; |
---|
| 748 | |
---|
| 749 | //third part: searching for circles |
---|
| 750 | for (i = s; i <= n; i++) |
---|
| 751 | { |
---|
| 752 | T2 = T[1..n,1..n]; |
---|
| 753 | C = findCycleDFS(i,T2, intvec(i)); |
---|
| 754 | if (C[1] > 0) |
---|
| 755 | { |
---|
| 756 | for (j = 2; j <= size(C); j++) |
---|
| 757 | { |
---|
| 758 | T[i,1..n] = T[i,1..n] + T[C[j],1..n]; |
---|
| 759 | T[C[j],1..n] = NV; |
---|
| 760 | } |
---|
| 761 | for (j = 2; j <= size(C); j++) |
---|
| 762 | { |
---|
| 763 | T[1..n,i] = T[1..n,i] + T[1..n,C[j]]; |
---|
| 764 | T[1..n,C[j]] = NV; |
---|
| 765 | } |
---|
| 766 | T[i,i] = T[i,i] - size(C) + 1; |
---|
| 767 | m = 0; |
---|
| 768 | for (j = 1; j <= size(C); j++) |
---|
| 769 | { |
---|
| 770 | m = m + T[n+1,C[j]]; |
---|
| 771 | } |
---|
| 772 | for (j = 1; j <= size(C); j++) |
---|
| 773 | { |
---|
| 774 | T[n+1,C[j]] = m; |
---|
| 775 | } |
---|
| 776 | return(growthAlg(T,i)); |
---|
| 777 | } |
---|
| 778 | else {ERROR("No Cycle found, something seems wrong! Please contact the authors.");} |
---|
| 779 | } |
---|
| 780 | |
---|
| 781 | m = 0; |
---|
| 782 | for (i = 1; i <= n; i++) |
---|
| 783 | { |
---|
| 784 | if (m < T[n+1,i]) |
---|
| 785 | { |
---|
| 786 | m = T[n+1,i]; |
---|
| 787 | } |
---|
| 788 | } |
---|
| 789 | return(m); |
---|
| 790 | } |
---|
| 791 | |
---|
| 792 | static proc GlDimSuffix(intvec v, intvec g) |
---|
| 793 | { |
---|
| 794 | //Computes the shortest r such that g is a suffix for vr |
---|
| 795 | //only valid for lex orderings? |
---|
| 796 | intvec r,gt,vt,lt,g2; |
---|
| 797 | int lg,lv,l,i,c,f; |
---|
| 798 | lg = size(g); lv = size(v); |
---|
| 799 | if (lg <= lv) |
---|
| 800 | { |
---|
| 801 | l = lv-lg; |
---|
| 802 | } |
---|
| 803 | else |
---|
| 804 | { |
---|
| 805 | l = 0; g2 = g[(lv+1)..lg]; |
---|
| 806 | g = g[1..lv]; lg = size(g); |
---|
| 807 | c = 1; |
---|
| 808 | } |
---|
| 809 | while (l < lv) |
---|
| 810 | { |
---|
| 811 | vt = v[(l+1)..lv]; |
---|
| 812 | gt = g[1..(lv-l)]; |
---|
| 813 | lt = size(gt); |
---|
| 814 | for (i = 1; i <= lt; i++) |
---|
| 815 | { |
---|
| 816 | if (vt[i]<>gt[i]) {l++; break;} |
---|
| 817 | } |
---|
| 818 | if (lt <=i ) { f = 1; break;} |
---|
| 819 | } |
---|
| 820 | if (f == 0) {return(g);} |
---|
| 821 | r = g[(lv-l+1)..lg]; |
---|
| 822 | if (c == 1) {r = r,g2;} |
---|
| 823 | return(r); |
---|
| 824 | } |
---|
| 825 | |
---|
| 826 | static proc isNormal(intvec V, list G) |
---|
| 827 | { |
---|
| 828 | int i,j,k,l; |
---|
| 829 | k = 0; |
---|
| 830 | for (i = 1; i <= size(G); i++) |
---|
| 831 | { |
---|
| 832 | if ( size(G[i]) <= size(V) ) |
---|
| 833 | { |
---|
| 834 | while ( size(G[i])+k <= size(V) ) |
---|
| 835 | { |
---|
| 836 | if ( G[i] == V[(1+k)..size(V)] ) {return(1);} |
---|
| 837 | } |
---|
| 838 | } |
---|
| 839 | } |
---|
| 840 | return(0); |
---|
| 841 | } |
---|
| 842 | |
---|
| 843 | static proc findDChain(list L) |
---|
| 844 | { |
---|
| 845 | list Li; int i,j; |
---|
| 846 | for (i = 1; i <= size(L); i++) {Li[i] = size(L[i]);} |
---|
| 847 | Li = sort(Li); Li = Li[1]; |
---|
| 848 | return(Li[size(Li)]); |
---|
| 849 | } |
---|
| 850 | |
---|
| 851 | static proc isInList(intvec V, list L) |
---|
| 852 | "USAGE: isInList(V,L); V an intvec, L a list of intvecs |
---|
| 853 | RETURN: int |
---|
| 854 | PURPOSE:Finding the position of V in L, returns 0, if V is not in M |
---|
| 855 | " |
---|
| 856 | {int i,n; |
---|
| 857 | n = 0; |
---|
| 858 | for (i = 1; i <= size(L); i++) {if (L[i] == V) {n = i; break;}} |
---|
| 859 | return(n); |
---|
| 860 | } |
---|
| 861 | |
---|
| 862 | static proc isInMat(intvec V, intmat M) |
---|
| 863 | "USAGE: isInMat(V,M);V an intvec, M an intmat |
---|
| 864 | RETURN: int |
---|
| 865 | PURPOSE:Finding the position of V in M, returns 0, if V is not in M |
---|
| 866 | " |
---|
| 867 | {if (size(V) <> ncols(M)) {return(0);} |
---|
| 868 | int i; |
---|
| 869 | intvec Vt; |
---|
| 870 | for (i = 1; i <= nrows(M); i++) |
---|
| 871 | {Vt = M[i,1..ncols(M)]; |
---|
| 872 | if ((V-Vt) == 0){return(i);} |
---|
| 873 | } |
---|
| 874 | return(0); |
---|
| 875 | } |
---|
| 876 | |
---|
| 877 | static proc isInVec(int v,intvec V) |
---|
| 878 | "USAGE: isInVec(v,V); v an integer,V an intvec |
---|
| 879 | RETURN: int |
---|
| 880 | PURPOSE:Finding the position of v in V, returns 0, if v is not in V |
---|
| 881 | " |
---|
| 882 | {int i,n; |
---|
| 883 | n = 0; |
---|
| 884 | for (i = 1; i <= size(V); i++) {if (V[i] == v) {n = i; break;}} |
---|
| 885 | return(n); |
---|
| 886 | } |
---|
| 887 | |
---|
| 888 | |
---|
| 889 | static proc isPF(intvec P, intvec I) |
---|
| 890 | " |
---|
| 891 | PURPOSE: |
---|
| 892 | checks, if a word P is a praefix of another word I |
---|
| 893 | " |
---|
| 894 | { |
---|
| 895 | int n = size(P); |
---|
| 896 | if (n <= 0 || P == 0) {return(1);} |
---|
| 897 | if (size(I) < n) {return(0);} |
---|
| 898 | intvec IP = I[1..n]; |
---|
| 899 | if (IP == P) {return(1);} |
---|
| 900 | else {return(0);} |
---|
| 901 | } |
---|
| 902 | |
---|
| 903 | proc ivL2lpI(list L) |
---|
| 904 | "USAGE: ivL2lpI(L); L a list of intvecs (deprecated, will be removed soon) |
---|
| 905 | RETURN: ideal |
---|
| 906 | PURPOSE:Transforming a list of intvecs into an ideal of Letterplace monomials |
---|
| 907 | ASSUME: - Intvec corresponds to a Letterplace monomial |
---|
| 908 | @* - basering has to be a Letterplace ring |
---|
| 909 | NOTE: - Assumptions will not be checked! |
---|
| 910 | EXAMPLE: example ivL2lpI; shows examples |
---|
| 911 | " |
---|
| 912 | { |
---|
| 913 | int i; ideal G; |
---|
| 914 | poly p; |
---|
| 915 | for (i = 1; i <= size(L); i++) |
---|
| 916 | {p = iv2lp(L[i]); |
---|
| 917 | G[(size(G) + 1)] = p; |
---|
| 918 | } |
---|
| 919 | return(G); |
---|
| 920 | } |
---|
| 921 | example |
---|
| 922 | { |
---|
| 923 | "EXAMPLE:"; echo = 2; |
---|
| 924 | ring r = 0,(x,y,z),dp; |
---|
| 925 | def R = makeLetterplaceRing(5);// constructs a Letterplace ring |
---|
| 926 | setring R; //sets basering to Letterplace ring |
---|
| 927 | intvec u = 1,1,2; intvec v = 2,1,3; intvec w = 3,1,1; |
---|
| 928 | // u = x^2y, v = yxz, w = zx^2 in intvec representation |
---|
| 929 | list L = u,v,w; |
---|
| 930 | ivL2lpI(L);// invokes the procedure, returns the ideal containing u,v,w |
---|
| 931 | } |
---|
| 932 | |
---|
| 933 | proc iv2lp(intvec I) |
---|
| 934 | "USAGE: iv2lp(I); I an intvec (deprecated, will be removed soon) |
---|
| 935 | RETURN: poly |
---|
| 936 | PURPOSE:Transforming an intvec into the corresponding Letterplace polynomial |
---|
| 937 | ASSUME: - Intvec corresponds to a Letterplace monomial |
---|
| 938 | @* - basering has to be a Letterplace ring |
---|
| 939 | NOTE: - Assumptions will not be checked! |
---|
| 940 | EXAMPLE: example iv2lp; shows examples |
---|
| 941 | " |
---|
| 942 | {if (I[1] == 0) {return(1);} |
---|
| 943 | int i = size(I); |
---|
| 944 | if (i > attrib(basering,"uptodeg")) {ERROR("polynomial exceeds degreebound");} |
---|
| 945 | int j; poly p = 1; |
---|
| 946 | for (j = 1; j <= i; j++) {if (I[j] > 0) { p = p*var(I[j]);}} //ignore zeroes, because they correspond to 1 |
---|
| 947 | return(p); |
---|
| 948 | } |
---|
| 949 | example |
---|
| 950 | { |
---|
| 951 | "EXAMPLE:"; echo = 2; |
---|
| 952 | ring r = 0,(x,y,z),dp; |
---|
| 953 | def R = makeLetterplaceRing(5); // constructs a Letterplace ring |
---|
| 954 | setring R; //sets basering to Letterplace ring |
---|
| 955 | intvec u = 1,1,2; intvec v = 2,1,3; intvec w = 3,1,1; |
---|
| 956 | // u = x^2y, v = yxz, w = zx^2 in intvec representation |
---|
| 957 | iv2lp(u); // invokes the procedure and returns the corresponding poly |
---|
| 958 | iv2lp(v); |
---|
| 959 | iv2lp(w); |
---|
| 960 | } |
---|
| 961 | |
---|
| 962 | proc iv2lpList(list L) |
---|
| 963 | "USAGE: iv2lpList(L); L a list of intmats (deprecated, will be removed soon) |
---|
| 964 | RETURN: ideal |
---|
| 965 | PURPOSE:Converting a list of intmats into an ideal of corresponding monomials |
---|
| 966 | ASSUME: - The rows of each intmat in L must correspond to a Letterplace monomial |
---|
| 967 | @* - basering has to be a Letterplace ring |
---|
| 968 | EXAMPLE: example iv2lpList; shows examples |
---|
| 969 | " |
---|
| 970 | {checkAssumptions(0,L); |
---|
| 971 | ideal G; |
---|
| 972 | int i; |
---|
| 973 | for (i = 1; i <= size(L); i++){G = G + iv2lpMat(L[i]);} |
---|
| 974 | return(G); |
---|
| 975 | } |
---|
| 976 | example |
---|
| 977 | { |
---|
| 978 | "EXAMPLE:"; echo = 2; |
---|
| 979 | ring r = 0,(x,y,z),dp; |
---|
| 980 | def R = makeLetterplaceRing(5); // constructs a Letterplace ring |
---|
| 981 | setring R; // sets basering to Letterplace ring |
---|
| 982 | intmat u[3][1] = 1,1,2; intmat v[1][3] = 2,1,3; intmat w[2][3] = 3,1,1,2,3,1; |
---|
| 983 | // defines intmats of different size containing intvec representations of |
---|
| 984 | // monomials as rows |
---|
| 985 | list L = u,v,w; |
---|
| 986 | print(u); print(v); print(w); // shows the intmats contained in L |
---|
| 987 | iv2lpList(L); // returns the corresponding monomials as an ideal |
---|
| 988 | } |
---|
| 989 | |
---|
| 990 | |
---|
| 991 | proc iv2lpMat(intmat M) |
---|
| 992 | "USAGE: iv2lpMat(M); M an intmat (deprecated, will be removed soon) |
---|
| 993 | RETURN: ideal |
---|
| 994 | PURPOSE:Converting an intmat into an ideal of the corresponding monomials |
---|
| 995 | ASSUME: - The rows of M must correspond to Letterplace monomials |
---|
| 996 | @* - basering has to be a Letterplace ring |
---|
| 997 | EXAMPLE: example iv2lpMat; shows examples |
---|
| 998 | " |
---|
| 999 | {list L = M; |
---|
| 1000 | checkAssumptions(0,L); |
---|
| 1001 | kill L; |
---|
| 1002 | ideal G; poly p; |
---|
| 1003 | int i; intvec I; |
---|
| 1004 | for (i = 1; i <= nrows(M); i++) |
---|
| 1005 | { I = M[i,1..ncols(M)]; |
---|
| 1006 | p = iv2lp(I); |
---|
| 1007 | G[size(G)+1] = p; |
---|
| 1008 | } |
---|
| 1009 | return(G); |
---|
| 1010 | } |
---|
| 1011 | example |
---|
| 1012 | { |
---|
| 1013 | "EXAMPLE:"; echo = 2; |
---|
| 1014 | ring r = 0,(x,y,z),dp; |
---|
| 1015 | def R = makeLetterplaceRing(5); // constructs a Letterplace ring |
---|
| 1016 | setring R; // sets basering to Letterplace ring |
---|
| 1017 | intmat u[3][1] = 1,1,2; intmat v[1][3] = 2,1,3; intmat w[2][3] = 3,1,1,2,3,1; |
---|
| 1018 | // defines intmats of different size containing intvec representations of |
---|
| 1019 | // monomials as rows |
---|
| 1020 | iv2lpMat(u); // returns the monomials contained in u |
---|
| 1021 | iv2lpMat(v); // returns the monomials contained in v |
---|
| 1022 | iv2lpMat(w); // returns the monomials contained in w |
---|
| 1023 | } |
---|
| 1024 | |
---|
| 1025 | proc lpId2ivLi(ideal G) |
---|
| 1026 | "USAGE: lpId2ivLi(G); G an ideal (deprecated, will be removed soon) |
---|
| 1027 | RETURN: list |
---|
| 1028 | PURPOSE:Transforming an ideal into the corresponding list of intvecs |
---|
| 1029 | ASSUME: - basering has to be a Letterplace ring |
---|
| 1030 | EXAMPLE: example lpId2ivLi; shows examples |
---|
| 1031 | " |
---|
| 1032 | { |
---|
| 1033 | int i,j,k; |
---|
| 1034 | list M; |
---|
| 1035 | checkAssumptions(0,M); |
---|
| 1036 | for (i = 1; i <= size(G); i++) {M[i] = lp2iv(G[i]);} |
---|
| 1037 | return(M); |
---|
| 1038 | } |
---|
| 1039 | example |
---|
| 1040 | { |
---|
| 1041 | "EXAMPLE:"; echo = 2; |
---|
| 1042 | ring r = 0,(x,y),dp; |
---|
| 1043 | def R = makeLetterplaceRing(5); // constructs a Letterplace ring |
---|
| 1044 | setring R; // sets basering to Letterplace ring |
---|
| 1045 | ideal L = x*x,y*y,x*y*x; |
---|
| 1046 | lpId2ivLi(L); // returns the corresponding intvecs as a list |
---|
| 1047 | } |
---|
| 1048 | |
---|
| 1049 | proc lp2iv(poly p) |
---|
| 1050 | "USAGE: lp2iv(p); p a poly (deprecated, will be removed soon) |
---|
| 1051 | RETURN: intvec |
---|
| 1052 | PURPOSE:Transforming a monomial into the corresponding intvec |
---|
| 1053 | ASSUME: - basering has to be a Letterplace ring |
---|
| 1054 | NOTE: - Assumptions will not be checked! |
---|
| 1055 | EXAMPLE: example lp2iv; shows examples |
---|
| 1056 | " |
---|
| 1057 | {p = normalize(lead(p)); |
---|
| 1058 | intvec I; |
---|
| 1059 | int i,j; |
---|
| 1060 | if (deg(p) > attrib(basering,"uptodeg")) {ERROR("Monomial exceeds degreebound");} |
---|
| 1061 | if (p == 1) {return(I);} |
---|
| 1062 | if (p == 0) {ERROR("Monomial is not allowed to equal zero");} |
---|
| 1063 | intvec lep = leadexp(p); |
---|
| 1064 | for ( i = 1; i <= attrib(basering,"isLetterplaceRing"); i++) {if (lep[i] == 1) {I = i; break;}} |
---|
| 1065 | for (i = (attrib(basering,"isLetterplaceRing")+1); i <= size(lep); i++) |
---|
| 1066 | {if (lep[i] == 1) |
---|
| 1067 | { j = (i mod attrib(basering,"isLetterplaceRing")); |
---|
| 1068 | if (j == 0) {I = I,attrib(basering,"isLetterplaceRing");} |
---|
| 1069 | else {I = I,j;} |
---|
| 1070 | } |
---|
| 1071 | else { if (lep[i] > 1) {ERROR("monomial has a not allowed multidegree");}} |
---|
| 1072 | } |
---|
| 1073 | if (I[1] == 0) {ERROR("monomial has a not allowed multidegree");} |
---|
| 1074 | |
---|
| 1075 | return(I); |
---|
| 1076 | } |
---|
| 1077 | example |
---|
| 1078 | { |
---|
| 1079 | "EXAMPLE:"; echo = 2; |
---|
| 1080 | ring r = 0,(x,y,z),dp; |
---|
| 1081 | def R = makeLetterplaceRing(5); // constructs a Letterplace ring |
---|
| 1082 | setring R; // sets basering to Letterplace ring |
---|
| 1083 | poly p = x*x*z; poly q = y*y*x*x; |
---|
| 1084 | poly w= z*y*x*z*z; |
---|
| 1085 | // p,q,w are some polynomials we want to transform into their |
---|
| 1086 | // intvec representation |
---|
| 1087 | lp2iv(p); lp2iv(q); lp2iv(w); |
---|
| 1088 | } |
---|
| 1089 | |
---|
| 1090 | proc lp2ivId(ideal G) |
---|
| 1091 | "USAGE: lp2ivId(G); G an ideal (deprecated, will be removed soon) |
---|
| 1092 | RETURN: list |
---|
| 1093 | PURPOSE:Converting an ideal into an list of intmats, |
---|
| 1094 | @* the corresponding intvecs forming the rows |
---|
| 1095 | ASSUME: - basering has to be a Letterplace ring |
---|
| 1096 | EXAMPLE: example lp2ivId; shows examples |
---|
| 1097 | " |
---|
| 1098 | {G = normalize(lead(G)); |
---|
| 1099 | intvec I; list L; |
---|
| 1100 | checkAssumptions(0,L); |
---|
| 1101 | int i,md; |
---|
| 1102 | for (i = 1; i <= size(G); i++) { if (md <= deg(G[i])) {md = deg(G[i]);}} |
---|
| 1103 | while (size(G) > 0) |
---|
| 1104 | {ideal Gt; |
---|
| 1105 | for (i = 1; i <= ncols(G); i++) {if (md == deg(G[i])) {Gt = Gt + G[i]; G[i] = 0;}} |
---|
| 1106 | if (size(Gt) > 0) |
---|
| 1107 | {G = simplify(G,2); |
---|
| 1108 | intmat M [size(Gt)][md]; |
---|
| 1109 | for (i = 1; i <= size(Gt); i++) {M[i,1..md] = lp2iv(Gt[i]);} |
---|
| 1110 | L = insert(L,M); |
---|
| 1111 | kill M; kill Gt; |
---|
| 1112 | md = md - 1; |
---|
| 1113 | } |
---|
| 1114 | else {kill Gt; md = md - 1;} |
---|
| 1115 | } |
---|
| 1116 | return(L); |
---|
| 1117 | } |
---|
| 1118 | example |
---|
| 1119 | { |
---|
| 1120 | "EXAMPLE:"; echo = 2; |
---|
| 1121 | ring r = 0,(x,y,z),dp; |
---|
| 1122 | def R = makeLetterplaceRing(5); // constructs a Letterplace ring |
---|
| 1123 | setring R; // sets basering to Letterplace ring |
---|
| 1124 | poly p = x*x*z; poly q = y*y*x*x; |
---|
| 1125 | poly w = z*y*x*z; |
---|
| 1126 | // p,q,w are some polynomials we want to transform into their |
---|
| 1127 | // intvec representation |
---|
| 1128 | ideal G = p,q,w; |
---|
| 1129 | // define the ideal containing p,q and w |
---|
| 1130 | lp2ivId(G); // and return the list of intmats for this ideal |
---|
| 1131 | } |
---|
| 1132 | |
---|
| 1133 | // -----------------main procedures---------------------- |
---|
| 1134 | |
---|
| 1135 | static proc lpGraphOfNormalWords(ideal G) |
---|
| 1136 | "USAGE: lpGraphOfNormalWords(G); G a set of monomials in a letterplace ring |
---|
| 1137 | RETURN: intmat |
---|
| 1138 | PURPOSE: Constructs the graph of normal words induced by G |
---|
| 1139 | @*: the adjacency matrix of the graph of normal words induced by G |
---|
| 1140 | ASSUME: - basering is a Letterplace ring |
---|
| 1141 | - G are the leading monomials of a Groebner basis |
---|
| 1142 | " |
---|
| 1143 | { |
---|
| 1144 | // construct the Graph of normal words [Studzinski page 78] |
---|
| 1145 | // construct set of vertices |
---|
| 1146 | int v = attrib(basering,"isLetterplaceRing"); int d = attrib(basering,"uptodeg"); |
---|
| 1147 | ideal V; poly p,q,w; |
---|
| 1148 | ideal LG = lead(G); |
---|
| 1149 | int i,j,k,b; intvec E,Et; |
---|
| 1150 | for (i = 1; i <= v; i++){V = V, var(i);} |
---|
| 1151 | for (i = 1; i <= size(LG); i++) |
---|
| 1152 | { |
---|
| 1153 | E = leadexp(LG[i]); |
---|
| 1154 | if (E == intvec(0)) {V = V,monomial(intvec(0));} |
---|
| 1155 | else |
---|
| 1156 | { |
---|
| 1157 | for (j = 1; j < d; j++) |
---|
| 1158 | { |
---|
| 1159 | Et = E[(j*v+1)..(d*v)]; |
---|
| 1160 | if (Et == intvec(0)) {break;} |
---|
| 1161 | else {V = V, monomial(Et);} |
---|
| 1162 | } |
---|
| 1163 | } |
---|
| 1164 | } |
---|
| 1165 | V = simplify(V,2+4); |
---|
| 1166 | printf("V = %p", V); |
---|
| 1167 | |
---|
| 1168 | |
---|
| 1169 | // construct incidence matrix |
---|
| 1170 | |
---|
| 1171 | list LV = lpId2ivLi(V); |
---|
| 1172 | intvec Ip,Iw; |
---|
| 1173 | int n = size(V); |
---|
| 1174 | intmat T[n+1][n]; |
---|
| 1175 | for (i = 1; i <= n; i++) |
---|
| 1176 | { |
---|
| 1177 | // printf("for1 (i=%p, n=%p)", i, n); |
---|
| 1178 | p = V[i]; Ip = lp2iv(p); |
---|
| 1179 | for (j = 1; j <= n; j++) |
---|
| 1180 | { |
---|
| 1181 | // printf("for2 (j=%p, n=%p)", j, n); |
---|
| 1182 | k = 1; b = 1; |
---|
| 1183 | q = V[j]; |
---|
| 1184 | w = lpNF(p*q,LG); |
---|
| 1185 | if (w <> 0) |
---|
| 1186 | { |
---|
| 1187 | Iw = lp2iv(w); |
---|
| 1188 | while (k <= n) |
---|
| 1189 | { |
---|
| 1190 | // printf("while (k=%p, n=%p)", k, n); |
---|
| 1191 | if (isPF(LV[k],Iw) > 0) |
---|
| 1192 | {if (isPF(LV[k],Ip) == 0) {b = 0; k = n+1;} else {k++;} |
---|
| 1193 | } |
---|
| 1194 | else {k++;} |
---|
| 1195 | } |
---|
| 1196 | T[i,j] = b; |
---|
| 1197 | // print("Incidence Matrix:"); |
---|
| 1198 | // print(T); |
---|
| 1199 | } |
---|
| 1200 | } |
---|
| 1201 | } |
---|
| 1202 | return(T); |
---|
| 1203 | } |
---|
| 1204 | |
---|
| 1205 | // This proc is deprecated, see lpGkDim() in fpaprops.lib |
---|
| 1206 | /* proc lpGkDim(ideal G) */ |
---|
| 1207 | /* "USAGE: lpGkDim(G); G an ideal in a letterplace ring */ |
---|
| 1208 | /* RETURN: int */ |
---|
| 1209 | /* PURPOSE: Determines the Gelfand Kirillov dimension of A/<G> */ |
---|
| 1210 | /* @*: -1 means it is infinite */ |
---|
| 1211 | /* ASSUME: - basering is a Letterplace ring */ |
---|
| 1212 | /* - G is a Groebner basis */ |
---|
| 1213 | /* NOTE: see fpaprops.lib for a faster and more up to date version of this method */ |
---|
| 1214 | /* " */ |
---|
| 1215 | /* { */ |
---|
| 1216 | /* return(growthAlg(lpGraphOfNormalWords(G))); */ |
---|
| 1217 | /* } */ |
---|
| 1218 | |
---|
| 1219 | static proc ivDHilbert(list L, int n, list #) |
---|
| 1220 | "USAGE: ivDHilbert(L,n[,degbound]); L a list of intmats, n an integer, |
---|
| 1221 | @* degbound an optional integer |
---|
| 1222 | RETURN: list |
---|
| 1223 | PURPOSE:Compute the K-dimension and the Hilbert series |
---|
| 1224 | ASSUME: - basering is a Letterplace ring |
---|
| 1225 | @* - all rows of each intmat correspond to a Letterplace monomial |
---|
| 1226 | @* - if you specify a different degree bound degbound, |
---|
| 1227 | @* degbound <= attrib(basering,uptodeg) holds |
---|
| 1228 | NOTE: - If L is the list returned, then L[1] is an integer corresponding to the |
---|
| 1229 | @* dimension, L[2] is an intvec which contains the coefficients of the |
---|
| 1230 | @* Hilbert series |
---|
| 1231 | @* - If degbound is set, there will be a degree bound added. By default there |
---|
| 1232 | @* is no degree bound |
---|
| 1233 | @* - n is the number of variables |
---|
| 1234 | @* - If I = L[2] is the intvec returned, then I[k] is the (k-1)-th coefficient of |
---|
| 1235 | @* the Hilbert series. |
---|
| 1236 | @* - If the K-dimension is known to be infinite, a degree bound is needed |
---|
| 1237 | EXAMPLE: example ivDHilbert; shows examples |
---|
| 1238 | " |
---|
| 1239 | {int degbound = 0; |
---|
| 1240 | if (size(#) > 0){if (typeof(#[1])=="int"){if (#[1] > 0){degbound = #[1];}}} |
---|
| 1241 | checkAssumptions(degbound,L); |
---|
| 1242 | intvec H; int i,dimen; |
---|
| 1243 | H = ivHilbert(L,n,degbound); |
---|
| 1244 | for (i = 1; i <= size(H); i++){dimen = dimen + H[i];} |
---|
| 1245 | L = dimen,H; |
---|
| 1246 | return(L); |
---|
| 1247 | } |
---|
| 1248 | example |
---|
| 1249 | { |
---|
| 1250 | "EXAMPLE:"; echo = 2; |
---|
| 1251 | ring r = 0,(x,y),dp; |
---|
| 1252 | def R = makeLetterplaceRing(5); // constructs a Letterplace ring |
---|
| 1253 | R; |
---|
| 1254 | setring R; // sets basering to Letterplace ring |
---|
| 1255 | //some intmats, which contain monomials in intvec representation as rows |
---|
| 1256 | intmat I1 [2][2] = 1,1,2,2; intmat I2 [1][3] = 1,2,1; |
---|
| 1257 | intmat J1 [1][2] = 1,1; intmat J2 [2][3] = 2,1,2,1,2,1; |
---|
| 1258 | print(I1); |
---|
| 1259 | print(I2); |
---|
| 1260 | print(J1); |
---|
| 1261 | print(J2); |
---|
| 1262 | list G = I1,I2; // ideal, which is already a Groebner basis |
---|
| 1263 | list I = J1,J2; // ideal, which is already a Groebner basis |
---|
| 1264 | //the procedure without a degree bound |
---|
| 1265 | ivDHilbert(G,2); |
---|
| 1266 | // the procedure with degree bound 5 |
---|
| 1267 | ivDHilbert(I,2,5); |
---|
| 1268 | } |
---|
| 1269 | |
---|
| 1270 | static proc ivDHilbertSickle(list L, int n, list #) |
---|
| 1271 | "USAGE: ivDHilbertSickle(L,n[,degbound]); L a list of intmats, n an integer, |
---|
| 1272 | @* degbound an optional integer |
---|
| 1273 | RETURN: list |
---|
| 1274 | PURPOSE:Compute the K-dimension, Hilbert series and mistletoes |
---|
| 1275 | ASSUME: - basering is a Letterplace ring. |
---|
| 1276 | @* - All rows of each intmat correspond to a Letterplace monomial. |
---|
| 1277 | @* - If you specify a different degree bound degbound, |
---|
| 1278 | @* degbound <= attrib(basering,uptodeg) holds. |
---|
| 1279 | NOTE: - If L is the list returned, then L[1] is an integer, L[2] is an intvec |
---|
| 1280 | @* which contains the coefficients of the Hilbert series and L[3] |
---|
| 1281 | @* is a list, containing the mistletoes as intvecs. |
---|
| 1282 | @* - If degbound is set, a degree bound will be added. By default there |
---|
| 1283 | @* is no degree bound. |
---|
| 1284 | @* - n is the number of variables. |
---|
| 1285 | @* - If I = L[2] is the intvec returned, then I[k] is the (k-1)-th |
---|
| 1286 | @* coefficient of the Hilbert series. |
---|
| 1287 | @* - If the K-dimension is known to be infinite, a degree bound is needed |
---|
| 1288 | EXAMPLE: example ivDHilbertSickle; shows examples |
---|
| 1289 | " |
---|
| 1290 | {int degbound = 0; |
---|
| 1291 | if (size(#) > 0){if (typeof(#[1])=="int"){if (#[1] > 0){degbound = #[1];}}} |
---|
| 1292 | checkAssumptions(degbound,L); |
---|
| 1293 | int i,dimen; list R; |
---|
| 1294 | R = ivSickleHil(L,n,degbound); |
---|
| 1295 | for (i = 1; i <= size(R[1]); i++){dimen = dimen + R[1][i];} |
---|
| 1296 | R[3] = R[2]; R[2] = R[1]; R[1] = dimen; |
---|
| 1297 | return(R); |
---|
| 1298 | } |
---|
| 1299 | example |
---|
| 1300 | { |
---|
| 1301 | "EXAMPLE:"; echo = 2; |
---|
| 1302 | ring r = 0,(x,y),dp; |
---|
| 1303 | def R = makeLetterplaceRing(5); // constructs a Letterplace ring |
---|
| 1304 | R; |
---|
| 1305 | setring R; // sets basering to Letterplace ring |
---|
| 1306 | //some intmats, which contain monomials in intvec representation as rows |
---|
| 1307 | intmat I1 [2][2] = 1,1,2,2; intmat I2 [1][3] = 1,2,1; |
---|
| 1308 | intmat J1 [1][2] = 1,1; intmat J2 [2][3] = 2,1,2,1,2,1; |
---|
| 1309 | print(I1); |
---|
| 1310 | print(I2); |
---|
| 1311 | print(J1); |
---|
| 1312 | print(J2); |
---|
| 1313 | list G = I1,I2;// ideal, which is already a Groebner basis |
---|
| 1314 | list I = J1,J2; // ideal, which is already a Groebner basis |
---|
| 1315 | ivDHilbertSickle(G,2); // invokes the procedure without a degree bound |
---|
| 1316 | ivDHilbertSickle(I,2,3); // invokes the procedure with degree bound 3 |
---|
| 1317 | } |
---|
| 1318 | |
---|
| 1319 | static proc ivKDimCheck(list L, int n) |
---|
| 1320 | "USAGE: ivKDimCheck(L,n); L a list of intmats, n an integer |
---|
| 1321 | RETURN: int, 0 if the dimension is finite, or 1 otherwise |
---|
| 1322 | PURPOSE:Decides, whether the K-dimension is finite or not |
---|
| 1323 | ASSUME: - basering is a Letterplace ring. |
---|
| 1324 | @* - All rows of each intmat correspond to a Letterplace monomial. |
---|
| 1325 | NOTE: - n is the number of variables. |
---|
| 1326 | EXAMPLE: example ivKDimCheck; shows examples |
---|
| 1327 | " |
---|
| 1328 | {checkAssumptions(0,L); |
---|
| 1329 | int i,r; |
---|
| 1330 | intvec P,H; |
---|
| 1331 | for (i = 1; i <= size(L); i++) |
---|
| 1332 | {P[i] = ncols(L[i]); |
---|
| 1333 | if (P[i] == 1) {if (isInMat(H,L[i]) > 0) {ERROR("Quotient algebra is trivial");}} |
---|
| 1334 | } |
---|
| 1335 | if (size(L) == 0) {ERROR("GB is empty, quotient algebra corresponds to free algebra");} |
---|
| 1336 | kill H; |
---|
| 1337 | intmat S; int sd,ld; intvec V; |
---|
| 1338 | sd = P[1]; ld = P[1]; |
---|
| 1339 | for (i = 2; i <= size(P); i++) |
---|
| 1340 | {if (P[i] < sd) {sd = P[i];} |
---|
| 1341 | if (P[i] > ld) {ld = P[i];} |
---|
| 1342 | } |
---|
| 1343 | sd = (sd - 1); ld = ld - 1; |
---|
| 1344 | if (ld == 0) { return(allVars(L,P,n));} |
---|
| 1345 | if (sd == 0) { for (i = 1; i <= size(L); i++){if (ncols(L[i]) == 1){S = createStartMat1(n,L[i]); break;}}} |
---|
| 1346 | else {S = createStartMat(sd,n);} |
---|
| 1347 | module M; |
---|
| 1348 | for (i = 1; i <= nrows(S); i++) |
---|
| 1349 | {V = S[i,1..ncols(S)]; |
---|
| 1350 | if (findCycle(V,L,P,n,ld,M)) {r = 1; break;} |
---|
| 1351 | } |
---|
| 1352 | return(r); |
---|
| 1353 | } |
---|
| 1354 | example |
---|
| 1355 | { |
---|
| 1356 | "EXAMPLE:"; echo = 2; |
---|
| 1357 | ring r = 0,(x,y),dp; |
---|
| 1358 | def R = makeLetterplaceRing(5); // constructs a Letterplace ring |
---|
| 1359 | R; |
---|
| 1360 | setring R; // sets basering to Letterplace ring |
---|
| 1361 | //some intmats, which contain monomials in intvec representation as rows |
---|
| 1362 | intmat I1 [2][2] = 1,1,2,2; intmat I2 [1][3] = 1,2,1; |
---|
| 1363 | intmat J1 [1][2] = 1,1; intmat J2 [2][3] = 2,1,2,1,2,1; |
---|
| 1364 | print(I1); |
---|
| 1365 | print(I2); |
---|
| 1366 | print(J1); |
---|
| 1367 | print(J2); |
---|
| 1368 | list G = I1,I2;// ideal, which is already a Groebner basis |
---|
| 1369 | list I = J1,J2; // ideal, which is already a Groebner basis and which |
---|
| 1370 | ivKDimCheck(G,2); // invokes the procedure, factor is of finite K-dimension |
---|
| 1371 | ivKDimCheck(I,2); // invokes the procedure, factor is not of finite K-dimension |
---|
| 1372 | } |
---|
| 1373 | |
---|
| 1374 | static proc ivHilbert(list L, int n, list #) |
---|
| 1375 | "USAGE: ivHilbert(L,n[,degbound]); L a list of intmats, n an integer, |
---|
| 1376 | @* degbound an optional integer |
---|
| 1377 | RETURN: intvec, containing the coefficients of the Hilbert series |
---|
| 1378 | PURPOSE:Compute the Hilbert series |
---|
| 1379 | ASSUME: - basering is a Letterplace ring. |
---|
| 1380 | @* - all rows of each intmat correspond to a Letterplace monomial |
---|
| 1381 | @* - if you specify a different degree bound degbound, |
---|
| 1382 | @* degbound <= attrib(basering,uptodeg) holds. |
---|
| 1383 | NOTE: - If degbound is set, a degree bound will be added. By default there |
---|
| 1384 | @* is no degree bound. |
---|
| 1385 | @* - n is the number of variables. |
---|
| 1386 | @* - If I is returned, then I[k] is the (k-1)-th coefficient of the Hilbert |
---|
| 1387 | @* series. |
---|
| 1388 | @* - If the K-dimension is known to be infinite, a degree bound is needed |
---|
| 1389 | EXAMPLE: example ivHilbert; shows examples |
---|
| 1390 | " |
---|
| 1391 | {int degbound = 0; |
---|
| 1392 | if (size(#) > 0) {if (typeof(#[1])=="int"){if (#[1] > 0) {degbound = #[1];}}} |
---|
| 1393 | intvec P,H; int i; |
---|
| 1394 | for (i = 1; i <= size(L); i++) |
---|
| 1395 | {P[i] = ncols(L[i]); |
---|
| 1396 | if (P[i] == 1) {if ( isInMat(H,L[i]) > 0) {ERROR("Quotient algebra is trivial");}} |
---|
| 1397 | } |
---|
| 1398 | if (size(L) == 0) {ERROR("GB is empty, quotient algebra corresponds to free algebra");} |
---|
| 1399 | H[1] = 1; |
---|
| 1400 | checkAssumptions(degbound,L); |
---|
| 1401 | if (degbound == 0) |
---|
| 1402 | {int sd; |
---|
| 1403 | intmat S; |
---|
| 1404 | sd = P[1]; |
---|
| 1405 | for (i = 2; i <= size(P); i++) {if (P[i] < sd) {sd = P[i];}} |
---|
| 1406 | sd = (sd - 1); |
---|
| 1407 | if (sd == 0) { for (i = 1; i <= size(L); i++){if (ncols(L[i]) == 1){S = createStartMat1(n,L[i]); break;}}} |
---|
| 1408 | else {S = createStartMat(sd,n);} |
---|
| 1409 | if (intvec(S) == 0) {return(H);} |
---|
| 1410 | for (i = 1; i <= sd; i++) {H = H,(n^i);} |
---|
| 1411 | for (i = 1; i <= nrows(S); i++) |
---|
| 1412 | {intvec St = S[i,1..ncols(S)]; |
---|
| 1413 | H = findHCoeff(St,n,L,P,H); |
---|
| 1414 | kill St; |
---|
| 1415 | } |
---|
| 1416 | return(H); |
---|
| 1417 | } |
---|
| 1418 | else |
---|
| 1419 | {for (i = 1; i <= size(P); i++) |
---|
| 1420 | {if (P[i] > degbound) {ERROR("degreebound is too small, GB contains elements of higher degree");}} |
---|
| 1421 | int sd; |
---|
| 1422 | intmat S; |
---|
| 1423 | sd = P[1]; |
---|
| 1424 | for (i = 2; i <= size(P); i++) {if (P[i] < sd) {sd = P[i];}} |
---|
| 1425 | sd = (sd - 1); |
---|
| 1426 | if (sd == 0) { for (i = 1; i <= size(L); i++){if (ncols(L[i]) == 1){S = createStartMat1(n,L[i]); break;}}} |
---|
| 1427 | else {S = createStartMat(sd,n);} |
---|
| 1428 | if (intvec(S) == 0) {return(H);} |
---|
| 1429 | for (i = 1; i <= sd; i++) {H = H,(n^i);} |
---|
| 1430 | for (i = 1; i <= nrows(S); i++) |
---|
| 1431 | {intvec St = S[i,1..ncols(S)]; |
---|
| 1432 | H = findHCoeff(St,n,L,P,H,degbound); |
---|
| 1433 | kill St; |
---|
| 1434 | } |
---|
| 1435 | return(H); |
---|
| 1436 | } |
---|
| 1437 | } |
---|
| 1438 | example |
---|
| 1439 | { |
---|
| 1440 | "EXAMPLE:"; echo = 2; |
---|
| 1441 | ring r = 0,(x,y),dp; |
---|
| 1442 | def R = makeLetterplaceRing(5); // constructs a Letterplace ring |
---|
| 1443 | R; |
---|
| 1444 | setring R; // sets basering to Letterplace ring |
---|
| 1445 | //some intmats, which contain monomials in intvec representation as rows |
---|
| 1446 | intmat I1 [2][2] = 1,1,2,2; intmat I2 [1][3] = 1,2,1; |
---|
| 1447 | intmat J1 [1][2] = 1,1; intmat J2 [2][3] = 2,1,2,1,2,1; |
---|
| 1448 | print(I1); |
---|
| 1449 | print(I2); |
---|
| 1450 | print(J1); |
---|
| 1451 | print(J2); |
---|
| 1452 | list G = I1,I2; // ideal, which is already a Groebner basis |
---|
| 1453 | list I = J1,J2; // ideal, which is already a Groebner basis |
---|
| 1454 | ivHilbert(G,2); // invokes the procedure without any degree bound |
---|
| 1455 | ivHilbert(I,2,5); // invokes the procedure with degree bound 5 |
---|
| 1456 | } |
---|
| 1457 | |
---|
| 1458 | |
---|
| 1459 | static proc ivKDim(list L, int n, list #) |
---|
| 1460 | "USAGE: ivKDim(L,n[,degbound]); L a list of intmats, |
---|
| 1461 | @* n an integer, degbound an optional integer |
---|
| 1462 | RETURN: int, the K-dimension of A/<L> |
---|
| 1463 | PURPOSE:Compute the K-dimension of A/<L> |
---|
| 1464 | ASSUME: - basering is a Letterplace ring. |
---|
| 1465 | @* - all rows of each intmat correspond to a Letterplace monomial |
---|
| 1466 | @* - if you specify a different degree bound degbound, |
---|
| 1467 | @* degbound <= attrib(basering,uptodeg) holds. |
---|
| 1468 | NOTE: - If degbound is set, a degree bound will be added. By default there |
---|
| 1469 | @* is no degree bound. |
---|
| 1470 | @* - n is the number of variables. |
---|
| 1471 | @* - If the K-dimension is known to be infinite, a degree bound is needed |
---|
| 1472 | EXAMPLE: example ivKDim; shows examples |
---|
| 1473 | " |
---|
| 1474 | {int degbound = 0; |
---|
| 1475 | if (size(#) > 0) {if (typeof(#[1])=="int"){if (#[1] > 0) {degbound = #[1];}}} |
---|
| 1476 | intvec P,H; int i; |
---|
| 1477 | for (i = 1; i <= size(L); i++) |
---|
| 1478 | {P[i] = ncols(L[i]); |
---|
| 1479 | if (P[i] == 1) {if ( isInMat(H,L[i]) > 0) {ERROR("Quotient algebra is trivial");}} |
---|
| 1480 | } |
---|
| 1481 | if (size(L) == 0) {ERROR("GB is empty, quotient algebra corresponds to free algebra");} |
---|
| 1482 | kill H; |
---|
| 1483 | checkAssumptions(degbound,L); |
---|
| 1484 | if (degbound == 0) |
---|
| 1485 | {int sd; int dimen = 1; |
---|
| 1486 | intmat S; |
---|
| 1487 | sd = P[1]; |
---|
| 1488 | for (i = 2; i <= size(P); i++) {if (P[i] < sd) {sd = P[i];}} |
---|
| 1489 | sd = (sd - 1); |
---|
| 1490 | if (sd == 0) { for (i = 1; i <= size(L); i++){if (ncols(L[i]) == 1){S = createStartMat1(n,L[i]); break;}}} |
---|
| 1491 | else {S = createStartMat(sd,n);} |
---|
| 1492 | if (intvec(S) == 0) {return(dimen);} |
---|
| 1493 | for (i = 1; i <= sd; i++) {dimen = dimen +(n^i);} |
---|
| 1494 | for (i = 1; i <= nrows(S); i++) |
---|
| 1495 | {intvec St = S[i,1..ncols(S)]; |
---|
| 1496 | dimen = dimen + findDimen(St,n,L,P); |
---|
| 1497 | kill St; |
---|
| 1498 | } |
---|
| 1499 | return(dimen); |
---|
| 1500 | } |
---|
| 1501 | else |
---|
| 1502 | {for (i = 1; i <= size(P); i++) |
---|
| 1503 | {if (P[i] > degbound) {ERROR("degreebound is too small, GB contains elements of higher degree");}} |
---|
| 1504 | int sd; int dimen = 1; |
---|
| 1505 | intmat S; |
---|
| 1506 | sd = P[1]; |
---|
| 1507 | for (i = 2; i <= size(P); i++) {if (P[i] < sd) {sd = P[i];}} |
---|
| 1508 | sd = (sd - 1); |
---|
| 1509 | if (sd == 0) { for (i = 1; i <= size(L); i++){if (ncols(L[i]) == 1){S = createStartMat1(n,L[i]); break;}}} |
---|
| 1510 | else {S = createStartMat(sd,n);} |
---|
| 1511 | if (intvec(S) == 0) {return(dimen);} |
---|
| 1512 | for (i = 1; i <= sd; i++) {dimen = dimen +(n^i);} |
---|
| 1513 | for (i = 1; i <= nrows(S); i++) |
---|
| 1514 | {intvec St = S[i,1..ncols(S)]; |
---|
| 1515 | dimen = dimen + findDimen(St,n,L,P, degbound); |
---|
| 1516 | kill St; |
---|
| 1517 | } |
---|
| 1518 | return(dimen); |
---|
| 1519 | } |
---|
| 1520 | } |
---|
| 1521 | example |
---|
| 1522 | { |
---|
| 1523 | "EXAMPLE:"; echo = 2; |
---|
| 1524 | ring r = 0,(x,y),dp; |
---|
| 1525 | def R = makeLetterplaceRing(5); // constructs a Letterplace ring |
---|
| 1526 | R; |
---|
| 1527 | setring R; // sets basering to Letterplace ring |
---|
| 1528 | //some intmats, which contain monomials in intvec representation as rows |
---|
| 1529 | intmat I1 [2][2] = 1,1,2,2; intmat I2 [1][3] = 1,2,1; |
---|
| 1530 | intmat J1 [1][2] = 1,1; intmat J2 [2][3] = 2,1,2,1,2,1; |
---|
| 1531 | print(I1); |
---|
| 1532 | print(I2); |
---|
| 1533 | print(J1); |
---|
| 1534 | print(J2); |
---|
| 1535 | list G = I1,I2; // ideal, which is already a Groebner basis |
---|
| 1536 | list I = J1,J2; // ideal, which is already a Groebner basis |
---|
| 1537 | ivKDim(G,2); // invokes the procedure without any degree bound |
---|
| 1538 | ivKDim(I,2,5); // invokes the procedure with degree bound 5 |
---|
| 1539 | } |
---|
| 1540 | |
---|
| 1541 | static proc ivMis2Base(list M) |
---|
| 1542 | "USAGE: ivMis2Base(M); M a list of intvecs |
---|
| 1543 | RETURN: ideal, a K-base of the given algebra |
---|
| 1544 | PURPOSE:Compute the K-base out of given mistletoes |
---|
| 1545 | ASSUME: - The mistletoes have to be ordered lexicographically -> OrdMisLex. |
---|
| 1546 | @* Otherwise there might some elements missing. |
---|
| 1547 | @* - basering is a Letterplace ring. |
---|
| 1548 | @* - mistletoes are stored as intvecs, as described in the overview |
---|
| 1549 | EXAMPLE: example ivMis2Base; shows examples |
---|
| 1550 | " |
---|
| 1551 | { |
---|
| 1552 | //checkAssumptions(0,M); |
---|
| 1553 | intvec L,A; |
---|
| 1554 | if (size(M) == 0){ERROR("There are no mistletoes, so it appears your dimension is infinite!");} |
---|
| 1555 | if (isInList(L,M) > 0) {print("1 is a mistletoe, therefore 1 is the only basis element"); return(list(intvec(0)));} |
---|
| 1556 | int i,j,d,s; |
---|
| 1557 | list Rt; |
---|
| 1558 | Rt[1] = intvec(0); |
---|
| 1559 | L = M[1]; |
---|
| 1560 | for (i = size(L); 1 <= i; i--) {Rt = insert(Rt,intvec(L[1..i]));} |
---|
| 1561 | for (i = 2; i <= size(M); i++) |
---|
| 1562 | {A = M[i]; L = M[i-1]; |
---|
| 1563 | s = size(A); |
---|
| 1564 | if (s > size(L)) |
---|
| 1565 | {d = size(L); |
---|
| 1566 | for (j = s; j > d; j--) {Rt = insert(Rt,intvec(A[1..j]));} |
---|
| 1567 | A = A[1..d]; |
---|
| 1568 | } |
---|
| 1569 | if (size(L) > s){L = L[1..s];} |
---|
| 1570 | while (A <> L) |
---|
| 1571 | {Rt = insert(Rt, intvec(A)); |
---|
| 1572 | if (size(A) > 1) |
---|
| 1573 | {A = A[1..(size(A)-1)]; |
---|
| 1574 | L = L[1..(size(L)-1)]; |
---|
| 1575 | } |
---|
| 1576 | else {break;} |
---|
| 1577 | } |
---|
| 1578 | } |
---|
| 1579 | return(Rt); |
---|
| 1580 | } |
---|
| 1581 | example |
---|
| 1582 | { |
---|
| 1583 | "EXAMPLE:"; echo = 2; |
---|
| 1584 | ring r = 0,(x,y),dp; |
---|
| 1585 | def R = makeLetterplaceRing(5); // constructs a Letterplace ring |
---|
| 1586 | R; |
---|
| 1587 | setring R; // sets basering to Letterplace ring |
---|
| 1588 | intvec i1 = 1,2; intvec i2 = 2,1,2; |
---|
| 1589 | // the mistletoes are xy and yxy, which are already ordered lexicographically |
---|
| 1590 | list L = i1,i2; |
---|
| 1591 | ivMis2Base(L); // returns the basis of the factor algebra |
---|
| 1592 | } |
---|
| 1593 | |
---|
| 1594 | |
---|
| 1595 | static proc ivMis2Dim(list M) |
---|
| 1596 | "USAGE: ivMis2Dim(M); M a list of intvecs |
---|
| 1597 | RETURN: int, the K-dimension of the given algebra |
---|
| 1598 | PURPOSE:Compute the K-dimension out of given mistletoes |
---|
| 1599 | ASSUME: - The mistletoes have to be ordered lexicographically -> OrdMisLex. |
---|
| 1600 | @* Otherwise the returned value may differ from the K-dimension. |
---|
| 1601 | @* - basering is a Letterplace ring. |
---|
| 1602 | EXAMPLE: example ivMis2Dim; shows examples |
---|
| 1603 | " |
---|
| 1604 | {checkAssumptions(0,M); |
---|
| 1605 | intvec L; |
---|
| 1606 | if (size(M) == 0){ERROR("There are no mistletoes, so it appears your dimension is infinite!");} |
---|
| 1607 | if (isInList(L,M) > 0) {print("1 is a mistletoe, therefore dim = 1"); return(1);} |
---|
| 1608 | int i,j,d,s; |
---|
| 1609 | j = 1; |
---|
| 1610 | d = 1 + size(M[1]); |
---|
| 1611 | for (i = 1; i < size(M); i++) |
---|
| 1612 | {s = size(M[i]); if (s > size(M[i+1])){s = size(M[i+1]);} |
---|
| 1613 | while ((M[i][j] == M[i+1][j]) && (j <= s)){j = j + 1;} |
---|
| 1614 | d = d + size(M[i+1])- j + 1; |
---|
| 1615 | } |
---|
| 1616 | return(d); |
---|
| 1617 | } |
---|
| 1618 | example |
---|
| 1619 | { |
---|
| 1620 | "EXAMPLE:"; echo = 2; |
---|
| 1621 | ring r = 0,(x,y),dp; |
---|
| 1622 | def R = makeLetterplaceRing(5); // constructs a Letterplace ring |
---|
| 1623 | R; |
---|
| 1624 | setring R; // sets basering to Letterplace ring |
---|
| 1625 | intvec i1 = 1,2; intvec i2 = 2,1,2; |
---|
| 1626 | // the mistletoes are xy and yxy, which are already ordered lexicographically |
---|
| 1627 | list L = i1,i2; |
---|
| 1628 | ivMis2Dim(L); // returns the dimension of the factor algebra |
---|
| 1629 | } |
---|
| 1630 | |
---|
| 1631 | static proc ivOrdMisLex(list M) |
---|
| 1632 | "USAGE: ivOrdMisLex(M); M a list of intvecs |
---|
| 1633 | RETURN: list, containing the ordered intvecs of M |
---|
| 1634 | PURPOSE:Orders a given set of mistletoes lexicographically |
---|
| 1635 | ASSUME: - basering is a Letterplace ring. |
---|
| 1636 | - intvecs correspond to monomials |
---|
| 1637 | NOTE: - This is preprocessing, it's not needed if the mistletoes are returned |
---|
| 1638 | @* from the sickle algorithm. |
---|
| 1639 | @* - Each entry of the list returned is an intvec. |
---|
| 1640 | EXAMPLE: example ivOrdMisLex; shows examples |
---|
| 1641 | " |
---|
| 1642 | {checkAssumptions(0,M); |
---|
| 1643 | return(sort(M)[1]); |
---|
| 1644 | } |
---|
| 1645 | example |
---|
| 1646 | { |
---|
| 1647 | "EXAMPLE:"; echo = 2; |
---|
| 1648 | ring r = 0,(x,y),dp; |
---|
| 1649 | def R = makeLetterplaceRing(5); // constructs a Letterplace ring |
---|
| 1650 | setring R; // sets basering to Letterplace ring |
---|
| 1651 | intvec i1 = 1,2,1; intvec i2 = 2,2,1; intvec i3 = 1,1; intvec i4 = 2,1,1,1; |
---|
| 1652 | // the corresponding monomials are xyx,y^2x,x^2,yx^3 |
---|
| 1653 | list M = i1,i2,i3,i4; |
---|
| 1654 | M; |
---|
| 1655 | ivOrdMisLex(M);// orders the list of monomials |
---|
| 1656 | } |
---|
| 1657 | |
---|
| 1658 | static proc ivSickle(list L, int n, list #) |
---|
| 1659 | "USAGE: ivSickle(L,n,[degbound]); L a list of intmats, n an int, degbound an |
---|
| 1660 | @* optional integer |
---|
| 1661 | RETURN: list, containing intvecs, the mistletoes of A/<L> |
---|
| 1662 | PURPOSE:Compute the mistletoes for a given Groebner basis L |
---|
| 1663 | ASSUME: - basering is a Letterplace ring. |
---|
| 1664 | @* - all rows of each intmat correspond to a Letterplace monomial |
---|
| 1665 | @* - if you specify a different degree bound degbound, |
---|
| 1666 | @* degbound <= attrib(basering,uptodeg) holds. |
---|
| 1667 | NOTE: - If degbound is set, a degree bound will be added. By default there |
---|
| 1668 | @* is no degree bound. |
---|
| 1669 | @* - n is the number of variables. |
---|
| 1670 | @* - If the K-dimension is known to be infinite, a degree bound is needed |
---|
| 1671 | EXAMPLE: example ivSickle; shows examples |
---|
| 1672 | " |
---|
| 1673 | {list M; |
---|
| 1674 | int degbound = 0; |
---|
| 1675 | if (size(#) > 0){if (typeof(#[1])=="int"){if (#[1] > 0){degbound = #[1];}}} |
---|
| 1676 | int i; |
---|
| 1677 | intvec P,H; |
---|
| 1678 | for (i = 1; i <= size(L); i++) |
---|
| 1679 | {P[i] = ncols(L[i]); |
---|
| 1680 | if (P[i] == 1) {if (isInMat(H,L[i]) > 0) {ERROR("Quotient algebra is trivial");}} |
---|
| 1681 | } |
---|
| 1682 | if (size(L) == 0) {ERROR("GB is empty, quotient algebra corresponds to free algebra");} |
---|
| 1683 | kill H; |
---|
| 1684 | checkAssumptions(degbound,L); |
---|
| 1685 | if (degbound == 0) |
---|
| 1686 | {intmat S; int sd; |
---|
| 1687 | sd = P[1]; |
---|
| 1688 | for (i = 2; i <= size(P); i++) {if (P[i] < sd) {sd = P[i];}} |
---|
| 1689 | sd = (sd - 1); |
---|
| 1690 | if (sd == 0) { for (i = 1; i <= size(L); i++){if (ncols(L[i]) == 1){S = createStartMat1(n,L[i]); break;}}} |
---|
| 1691 | else {S = createStartMat(sd,n);} |
---|
| 1692 | if (intvec(S) == 0) {return(list (intvec(0)));} |
---|
| 1693 | for (i = 1; i <= nrows(S); i++) |
---|
| 1694 | {intvec St = S[i,1..ncols(S)]; |
---|
| 1695 | M = M + findmistletoes(St,n,L,P); |
---|
| 1696 | kill St; |
---|
| 1697 | } |
---|
| 1698 | return(M); |
---|
| 1699 | } |
---|
| 1700 | else |
---|
| 1701 | {for (i = 1; i <= size(P); i++) |
---|
| 1702 | {if (P[i] > degbound) {ERROR("degreebound is too small, GB contains elements of higher degree");}} |
---|
| 1703 | intmat S; int sd; |
---|
| 1704 | sd = P[1]; |
---|
| 1705 | for (i = 2; i <= size(P); i++) {if (P[i] < sd) {sd = P[i];}} |
---|
| 1706 | sd = (sd - 1); |
---|
| 1707 | if (sd == 0) { for (i = 1; i <= size(L); i++){if (ncols(L[i]) == 1){S = createStartMat1(n,L[i]); break;}}} |
---|
| 1708 | else {S = createStartMat(sd,n);} |
---|
| 1709 | if (intvec(S) == 0) {return(list (intvec(0)));} |
---|
| 1710 | for (i = 1; i <= nrows(S); i++) |
---|
| 1711 | {intvec St = S[i,1..ncols(S)]; |
---|
| 1712 | M = M + findmistletoes(St,n,L,P,degbound); |
---|
| 1713 | kill St; |
---|
| 1714 | } |
---|
| 1715 | return(M); |
---|
| 1716 | } |
---|
| 1717 | } |
---|
| 1718 | example |
---|
| 1719 | { |
---|
| 1720 | "EXAMPLE:"; echo = 2; |
---|
| 1721 | ring r = 0,(x,y),dp; |
---|
| 1722 | def R = makeLetterplaceRing(5); // constructs a Letterplace ring |
---|
| 1723 | setring R; // sets basering to Letterplace ring |
---|
| 1724 | //some intmats, which contain monomials in intvec representation as rows |
---|
| 1725 | intmat I1 [2][2] = 1,1,2,2; intmat I2 [1][3] = 1,2,1; |
---|
| 1726 | intmat J1 [1][2] = 1,1; intmat J2 [2][3] = 2,1,2,1,2,1; |
---|
| 1727 | print(I1); |
---|
| 1728 | print(I2); |
---|
| 1729 | print(J1); |
---|
| 1730 | print(J2); |
---|
| 1731 | list G = I1,I2; // ideal, which is already a Groebner basis |
---|
| 1732 | list I = J1,J2; // ideal, which is already a Groebner basis |
---|
| 1733 | ivSickle(G,2); // invokes the procedure without any degree bound |
---|
| 1734 | ivSickle(I,2,5); // invokes the procedure with degree bound 5 |
---|
| 1735 | } |
---|
| 1736 | |
---|
| 1737 | static proc ivSickleDim(list L, int n, list #) |
---|
| 1738 | "USAGE: ivSickleDim(L,n[,degbound]); L a list of intmats, n an integer, degbound |
---|
| 1739 | @* an optional integer |
---|
| 1740 | RETURN: list |
---|
| 1741 | PURPOSE:Compute mistletoes and the K-dimension |
---|
| 1742 | ASSUME: - basering is a Letterplace ring. |
---|
| 1743 | @* - all rows of each intmat correspond to a Letterplace monomial |
---|
| 1744 | @* - if you specify a different degree bound degbound, |
---|
| 1745 | @* degbound <= attrib(basering,uptodeg) holds. |
---|
| 1746 | NOTE: - If L is the list returned, then L[1] is an integer, L[2] is a list, |
---|
| 1747 | @* containing the mistletoes as intvecs. |
---|
| 1748 | @* - If degbound is set, a degree bound will be added. By default there |
---|
| 1749 | @* is no degree bound. |
---|
| 1750 | @* - n is the number of variables. |
---|
| 1751 | @* - If the K-dimension is known to be infinite, a degree bound is needed |
---|
| 1752 | EXAMPLE: example ivSickleDim; shows examples |
---|
| 1753 | " |
---|
| 1754 | {list M; |
---|
| 1755 | int degbound = 0; |
---|
| 1756 | if (size(#) > 0){if (typeof(#[1])=="int"){if (#[1] > 0){degbound = #[1];}}} |
---|
| 1757 | int i,dimen; list R; |
---|
| 1758 | intvec P,H; |
---|
| 1759 | for (i = 1; i <= size(L); i++) |
---|
| 1760 | {P[i] = ncols(L[i]); |
---|
| 1761 | if (P[i] == 1) {if (isInMat(H,L[i]) > 0) {ERROR("Quotient algebra is trivial, dimension equals zero");}} |
---|
| 1762 | } |
---|
| 1763 | if (size(L) == 0) {ERROR("GB is empty, quotient algebra corresponds to free algebra");} |
---|
| 1764 | kill H; |
---|
| 1765 | checkAssumptions(degbound,L); |
---|
| 1766 | if (degbound == 0) |
---|
| 1767 | {int sd; dimen = 1; |
---|
| 1768 | intmat S; |
---|
| 1769 | sd = P[1]; |
---|
| 1770 | for (i = 2; i <= size(P); i++) {if (P[i] < sd) {sd = P[i];}} |
---|
| 1771 | sd = (sd - 1); |
---|
| 1772 | if (sd == 0) { for (i = 1; i <= size(L); i++){if (ncols(L[i]) == 1){S = createStartMat1(n,L[i]); break;}}} |
---|
| 1773 | else {S = createStartMat(sd,n);} |
---|
| 1774 | if (intvec(S) == 0) {return(list(dimen,list(intvec(0))));} |
---|
| 1775 | for (i = 1; i <= sd; i++) {dimen = dimen +(n^i);} |
---|
| 1776 | R[1] = dimen; |
---|
| 1777 | for (i = 1; i <= nrows(S); i++) |
---|
| 1778 | {intvec St = S[i,1..ncols(S)]; |
---|
| 1779 | R = findMisDim(St,n,L,P,R); |
---|
| 1780 | kill St; |
---|
| 1781 | } |
---|
| 1782 | return(R); |
---|
| 1783 | } |
---|
| 1784 | else |
---|
| 1785 | {for (i = 1; i <= size(P); i++) |
---|
| 1786 | {if (P[i] > degbound) {ERROR("degreebound is too small, GB contains elements of higher degree");}} |
---|
| 1787 | int sd; dimen = 1; |
---|
| 1788 | intmat S; |
---|
| 1789 | sd = P[1]; |
---|
| 1790 | for (i = 2; i <= size(P); i++) {if (P[i] < sd) {sd = P[i];}} |
---|
| 1791 | sd = (sd - 1); |
---|
| 1792 | if (sd == 0) { for (i = 1; i <= size(L); i++){if (ncols(L[i]) == 1){S = createStartMat1(n,L[i]); break;}}} |
---|
| 1793 | else {S = createStartMat(sd,n);} |
---|
| 1794 | if (intvec(S) == 0) {return(list(dimen,list(intvec(0))));} |
---|
| 1795 | for (i = 1; i <= sd; i++) {dimen = dimen +(n^i);} |
---|
| 1796 | R[1] = dimen; |
---|
| 1797 | for (i = 1; i <= nrows(S); i++) |
---|
| 1798 | {intvec St = S[i,1..ncols(S)]; |
---|
| 1799 | R = findMisDim(St,n,L,P,R,degbound); |
---|
| 1800 | kill St; |
---|
| 1801 | } |
---|
| 1802 | return(R); |
---|
| 1803 | } |
---|
| 1804 | } |
---|
| 1805 | example |
---|
| 1806 | { |
---|
| 1807 | "EXAMPLE:"; echo = 2; |
---|
| 1808 | ring r = 0,(x,y),dp; |
---|
| 1809 | def R = makeLetterplaceRing(5); // constructs a Letterplace ring |
---|
| 1810 | setring R; // sets basering to Letterplace ring |
---|
| 1811 | //some intmats, which contain monomials in intvec representation as rows |
---|
| 1812 | intmat I1 [2][2] = 1,1,2,2; intmat I2 [1][3] = 1,2,1; |
---|
| 1813 | intmat J1 [1][2] = 1,1; intmat J2 [2][3] = 2,1,2,1,2,1; |
---|
| 1814 | print(I1); |
---|
| 1815 | print(I2); |
---|
| 1816 | print(J1); |
---|
| 1817 | print(J2); |
---|
| 1818 | list G = I1,I2;// ideal, which is already a Groebner basis |
---|
| 1819 | list I = J1,J2; // ideal, which is already a Groebner basis |
---|
| 1820 | ivSickleDim(G,2); // invokes the procedure without any degree bound |
---|
| 1821 | ivSickleDim(I,2,5); // invokes the procedure with degree bound 5 |
---|
| 1822 | } |
---|
| 1823 | |
---|
| 1824 | static proc ivSickleHil(list L, int n, list #) |
---|
| 1825 | "USAGE:ivSickleHil(L,n[,degbound]); L a list of intmats, n an integer, |
---|
| 1826 | @* degbound an optional integer |
---|
| 1827 | RETURN: list |
---|
| 1828 | PURPOSE:Compute the mistletoes and the Hilbert series |
---|
| 1829 | ASSUME: - basering is a Letterplace ring. |
---|
| 1830 | @* - all rows of each intmat correspond to a Letterplace monomial |
---|
| 1831 | @* - if you specify a different degree bound degbound, |
---|
| 1832 | @* degbound <= attrib(basering,uptodeg) holds. |
---|
| 1833 | NOTE: - If L is the list returned, then L[1] is an intvec, L[2] is a list, |
---|
| 1834 | @* containing the mistletoes as intvecs. |
---|
| 1835 | @* - If degbound is set, a degree bound will be added. By default there |
---|
| 1836 | @* is no degree bound. |
---|
| 1837 | @* - n is the number of variables. |
---|
| 1838 | @* - If I = L[1] is the intvec returned, then I[k] is the (k-1)-th |
---|
| 1839 | @* coefficient of the Hilbert series. |
---|
| 1840 | @* - If the K-dimension is known to be infinite, a degree bound is needed |
---|
| 1841 | EXAMPLE: example ivSickleHil; shows examples |
---|
| 1842 | " |
---|
| 1843 | {int degbound = 0; |
---|
| 1844 | if (size(#) > 0) {if (typeof(#[1])=="int"){if (#[1] > 0) {degbound = #[1];}}} |
---|
| 1845 | intvec P,H; int i; list R; |
---|
| 1846 | for (i = 1; i <= size(L); i++) |
---|
| 1847 | {P[i] = ncols(L[i]); |
---|
| 1848 | if (P[i] == 1) {if ( isInMat(H,L[i]) > 0) {ERROR("Quotient algebra is trivial");}} |
---|
| 1849 | } |
---|
| 1850 | if (size(L) == 0) {ERROR("GB is empty, quotient algebra corresponds to free algebra");} |
---|
| 1851 | H[1] = 1; |
---|
| 1852 | checkAssumptions(degbound,L); |
---|
| 1853 | if (degbound == 0) |
---|
| 1854 | {int sd; |
---|
| 1855 | intmat S; |
---|
| 1856 | sd = P[1]; |
---|
| 1857 | for (i = 2; i <= size(P); i++) {if (P[i] < sd) {sd = P[i];}} |
---|
| 1858 | sd = (sd - 1); |
---|
| 1859 | if (sd == 0) { for (i = 1; i <= size(L); i++){if (ncols(L[i]) == 1){S = createStartMat1(n,L[i]); break;}}} |
---|
| 1860 | else {S = createStartMat(sd,n);} |
---|
| 1861 | if (intvec(S) == 0) {return(list(H,list(intvec (0))));} |
---|
| 1862 | for (i = 1; i <= sd; i++) {H = H,(n^i);} |
---|
| 1863 | R[1] = H; kill H; |
---|
| 1864 | for (i = 1; i <= nrows(S); i++) |
---|
| 1865 | {intvec St = S[i,1..ncols(S)]; |
---|
| 1866 | R = findHCoeffMis(St,n,L,P,R); |
---|
| 1867 | kill St; |
---|
| 1868 | } |
---|
| 1869 | return(R); |
---|
| 1870 | } |
---|
| 1871 | else |
---|
| 1872 | {for (i = 1; i <= size(P); i++) |
---|
| 1873 | {if (P[i] > degbound) {ERROR("degreebound is too small, GB contains elements of higher degree");}} |
---|
| 1874 | int sd; |
---|
| 1875 | intmat S; |
---|
| 1876 | sd = P[1]; |
---|
| 1877 | for (i = 2; i <= size(P); i++) {if (P[i] < sd) {sd = P[i];}} |
---|
| 1878 | sd = (sd - 1); |
---|
| 1879 | if (sd == 0) { for (i = 1; i <= size(L); i++){if (ncols(L[i]) == 1){S = createStartMat1(n,L[i]); break;}}} |
---|
| 1880 | else {S = createStartMat(sd,n);} |
---|
| 1881 | if (intvec(S) == 0) {return(list(H,list(intvec(0))));} |
---|
| 1882 | for (i = 1; i <= sd; i++) {H = H,(n^i);} |
---|
| 1883 | R[1] = H; kill H; |
---|
| 1884 | for (i = 1; i <= nrows(S); i++) |
---|
| 1885 | {intvec St = S[i,1..ncols(S)]; |
---|
| 1886 | R = findHCoeffMis(St,n,L,P,R,degbound); |
---|
| 1887 | kill St; |
---|
| 1888 | } |
---|
| 1889 | return(R); |
---|
| 1890 | } |
---|
| 1891 | } |
---|
| 1892 | example |
---|
| 1893 | { |
---|
| 1894 | "EXAMPLE:"; echo = 2; |
---|
| 1895 | ring r = 0,(x,y),dp; |
---|
| 1896 | def R = makeLetterplaceRing(5); // constructs a Letterplace ring |
---|
| 1897 | setring R; // sets basering to Letterplace ring |
---|
| 1898 | //some intmats, which contain monomials in intvec representation as rows |
---|
| 1899 | intmat I1[2][2] = 1,1,2,2; intmat I2[1][3] = 1,2,1; |
---|
| 1900 | intmat J1[1][2] = 1,1; intmat J2[2][3] = 2,1,2,1,2,1; |
---|
| 1901 | print(I1); |
---|
| 1902 | print(I2); |
---|
| 1903 | print(J1); |
---|
| 1904 | print(J2); |
---|
| 1905 | list G = I1,I2;// ideal, which is already a Groebner basis |
---|
| 1906 | list I = J1,J2; // ideal, which is already a Groebner basis |
---|
| 1907 | ivSickleHil(G,2); // invokes the procedure without any degree bound |
---|
| 1908 | ivSickleHil(I,2,5); // invokes the procedure with degree bound 5 |
---|
| 1909 | } |
---|
| 1910 | |
---|
| 1911 | static proc lpDHilbert(ideal G, list #) |
---|
| 1912 | "USAGE: lpDHilbert(G[,degbound,n]); G an ideal, degbound, n optional integers |
---|
| 1913 | RETURN: list |
---|
| 1914 | PURPOSE:Compute K-dimension and Hilbert series, starting with a lp-ideal |
---|
| 1915 | ASSUME: - basering is a Letterplace ring. |
---|
| 1916 | @* - if you specify a different degree bound degbound, |
---|
| 1917 | @* degbound <= attrib(basering,uptodeg) holds. |
---|
| 1918 | NOTE: - If L is the list returned, then L[1] is an integer corresponding to the |
---|
| 1919 | @* dimension, L[2] is an intvec which contains the coefficients of the |
---|
| 1920 | @* Hilbert series |
---|
| 1921 | @* - If degbound is set, there will be a degree bound added. 0 means no |
---|
| 1922 | @* degree bound. Default: attrib(basering,uptodeg). |
---|
| 1923 | @* - n can be set to a different number of variables. |
---|
| 1924 | @* Default: n = attrib(basering, lV). |
---|
| 1925 | @* - If I = L[2] is the intvec returned, then I[k] is the (k-1)-th |
---|
| 1926 | @* coefficient of the Hilbert series. |
---|
| 1927 | @* - If the K-dimension is known to be infinite, a degree bound is needed |
---|
| 1928 | EXAMPLE: example lpDHilbert; shows examples |
---|
| 1929 | " |
---|
| 1930 | {int degbound = attrib(basering,"uptodeg");int n = attrib(basering, "isLetterplaceRing"); |
---|
| 1931 | if (size(#) > 0){if (typeof(#[1])=="int"){if (#[1] >= 0){degbound = #[1];}}} |
---|
| 1932 | if (size(#) > 1){if (typeof(#[1])=="int"){if (#[2] > 0){n = #[2];}}} |
---|
| 1933 | list L; |
---|
| 1934 | L = lp2ivId(normalize(lead(G))); |
---|
| 1935 | return(ivDHilbert(L,n,degbound)); |
---|
| 1936 | } |
---|
| 1937 | example |
---|
| 1938 | { |
---|
| 1939 | "EXAMPLE:"; echo = 2; |
---|
| 1940 | ring r = 0,(x,y),dp; |
---|
| 1941 | def R = makeLetterplaceRing(5); // constructs a Letterplace ring |
---|
| 1942 | setring R; // sets basering to Letterplace ring |
---|
| 1943 | ideal G = x*x, y*y,x*y*x; // ideal G contains a |
---|
| 1944 | //Groebner basis |
---|
| 1945 | lpDHilbert(G,5,2); // invokes procedure with degree bound 5 and 2 variables |
---|
| 1946 | // note that the optional parameters are not necessary, due to the finiteness |
---|
| 1947 | // of the K-dimension of the factor algebra |
---|
| 1948 | lpDHilbert(G); // procedure with ring parameters |
---|
| 1949 | lpDHilbert(G,0); // procedure without degreebound |
---|
| 1950 | } |
---|
| 1951 | |
---|
| 1952 | static proc lpDHilbertSickle(ideal G, list #) |
---|
| 1953 | "USAGE: lpDHilbertSickle(G[,degbound,n]); G an ideal, degbound, n optional |
---|
| 1954 | @* integers |
---|
| 1955 | RETURN: list |
---|
| 1956 | PURPOSE:Compute K-dimension, Hilbert series and mistletoes at once |
---|
| 1957 | ASSUME: - basering is a Letterplace ring. |
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| 1958 | @* - if you specify a different degree bound degbound, |
---|
| 1959 | @* degbound <= attrib(basering,uptodeg) holds. |
---|
| 1960 | NOTE: - If L is the list returned, then L[1] is an integer, the K-dimension, |
---|
| 1961 | @* L[2] is an intvec, the Hilbert series and L[3] is an ideal, |
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| 1962 | @* the mistletoes |
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| 1963 | @* - If degbound is set, there will be a degree bound added. 0 means no |
---|
| 1964 | @* degree bound. Default: attrib(basering,uptodeg). |
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| 1965 | @* - n can be set to a different number of variables. |
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| 1966 | @* Default: n = attrib(basering, lV). |
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| 1967 | @* - If I = L[1] is the intvec returned, then I[k] is the (k-1)-th |
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| 1968 | @* coefficient of the Hilbert series. |
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| 1969 | @* - If the K-dimension is known to be infinite, a degree bound is needed |
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| 1970 | EXAMPLE: example lpDHilbertSickle; shows examples |
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| 1971 | " |
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| 1972 | {int degbound = attrib(basering,"uptodeg");int n = attrib(basering, "isLetterplaceRing"); |
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| 1973 | if (size(#) > 0){if (typeof(#[1])=="int"){if (#[1] >= 0){degbound = #[1];}}} |
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| 1974 | if (size(#) > 1){if (typeof(#[1])=="int"){if (#[2] > 0){n = #[2];}}} |
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| 1975 | list L; |
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| 1976 | L = lp2ivId(normalize(lead(G))); |
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| 1977 | L = ivDHilbertSickle(L,n,degbound); |
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| 1978 | L[3] = ivL2lpI(L[3]); |
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| 1979 | return(L); |
---|
| 1980 | } |
---|
| 1981 | example |
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| 1982 | { |
---|
| 1983 | "EXAMPLE:"; echo = 2; |
---|
| 1984 | ring r = 0,(x,y),dp; |
---|
| 1985 | def R = makeLetterplaceRing(5); // constructs a Letterplace ring |
---|
| 1986 | setring R; // sets basering to Letterplace ring |
---|
| 1987 | ideal G = x*x, y*y,x*y*x; // ideal G contains a |
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| 1988 | //Groebner basis |
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| 1989 | lpDHilbertSickle(G,5,2); //invokes procedure with degree bound 5 and 2 variables |
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| 1990 | // note that the optional parameters are not necessary, due to the finiteness |
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| 1991 | // of the K-dimension of the factor algebra |
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| 1992 | lpDHilbertSickle(G); // procedure with ring parameters |
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| 1993 | lpDHilbertSickle(G,0); // procedure without degreebound |
---|
| 1994 | } |
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| 1995 | |
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| 1996 | proc lpHilbert(ideal G, list #) |
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| 1997 | "USAGE: lpHilbert(G[,degbound,n]); G an ideal, degbound, n optional integers |
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| 1998 | RETURN: intvec, containing the coefficients of the Hilbert series |
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| 1999 | PURPOSE: Compute the truncated Hilbert series of K<X>/<G> up to a degree bound |
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| 2000 | ASSUME: - basering is a Letterplace ring. |
---|
| 2001 | @* - if you specify a different degree bound degbound, |
---|
| 2002 | @* degbound <= attrib(basering,uptodeg) holds. |
---|
| 2003 | THEORY: Hilbert series of an algebra K<X>/<G> is @code{\sum_{i=0} h_i t^i}, |
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| 2004 | where @code{h_i} is the K-dimension of the space of monomials of degree i, |
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| 2005 | not contained in <G>. For finitely presented algebras Hilbert series NEED |
---|
| 2006 | NOT be a rational function, though it happens often. Therefore in general |
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| 2007 | there is no notion of a Hilbert polynomial. |
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| 2008 | NOTE: - If degbound is set, there will be a degree bound added. 0 means no |
---|
| 2009 | @* degree bound. Default: attrib(basering,uptodeg). |
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| 2010 | @* - n is the number of variables, which can be set to a different number. |
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| 2011 | @* Default: attrib(basering, lV). |
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| 2012 | @* - In the output intvec I, I[k] is the (k-1)-th coefficient of the Hilbert |
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| 2013 | @* series, i.e. @code{h_{k-1}} as above. |
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| 2014 | EXAMPLE: example lpHilbert; shows examples |
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| 2015 | SEE ALSO: ncHilb_lib |
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| 2016 | " |
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| 2017 | {int degbound = attrib(basering,"uptodeg");int n = attrib(basering, "isLetterplaceRing"); |
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| 2018 | if (size(#) > 0){if (typeof(#[1])=="int"){if (#[1] >= 0){degbound = #[1];}}} |
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| 2019 | if (size(#) > 1){if (typeof(#[1])=="int"){if (#[2] > 0){n = #[2];}}} |
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| 2020 | list L; |
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| 2021 | L = lp2ivId(normalize(lead(G))); |
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| 2022 | return(ivHilbert(L,n,degbound)); |
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| 2023 | } |
---|
| 2024 | example |
---|
| 2025 | { |
---|
| 2026 | "EXAMPLE:"; echo = 2; |
---|
| 2027 | ring r = 0,(x,y),dp; |
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| 2028 | def R = makeLetterplaceRing(5); // constructs a Letterplace ring |
---|
| 2029 | setring R; // sets basering to Letterplace ring |
---|
| 2030 | ideal G = y*y,x*y*x; // G is a Groebner basis |
---|
| 2031 | lpHilbert(G); // procedure with default parameters |
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| 2032 | lpHilbert(G,3,2); // invokes procedure with degree bound 3 and (same) 2 variables |
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| 2033 | } |
---|
| 2034 | |
---|
| 2035 | // compatibiltiy, do not put in header |
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| 2036 | proc lpDimCheck(ideal G) |
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| 2037 | { |
---|
| 2038 | return(lpKDimCheck(G)); |
---|
| 2039 | } |
---|
| 2040 | |
---|
| 2041 | proc lpKDimCheck(ideal G) |
---|
| 2042 | "USAGE: lpKDimCheck(G); |
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| 2043 | RETURN: int, 1 if K-dimension of the factor algebra is infinite, 0 otherwise |
---|
| 2044 | PURPOSE:Checking a factor algebra for finiteness of the K-dimension |
---|
| 2045 | ASSUME: - basering is a Letterplace ring. |
---|
| 2046 | EXAMPLE: example lpKDimCheck; shows examples |
---|
| 2047 | " |
---|
| 2048 | {int n = attrib(basering,"isLetterplaceRing"); |
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| 2049 | list L; |
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| 2050 | ideal R; |
---|
| 2051 | R = normalize(lead(G)); |
---|
| 2052 | L = lp2ivId(R); |
---|
| 2053 | return(ivKDimCheck(L,n)); |
---|
| 2054 | } |
---|
| 2055 | example |
---|
| 2056 | { |
---|
| 2057 | "EXAMPLE:"; echo = 2; |
---|
| 2058 | ring r = 0,(x,y),dp; |
---|
| 2059 | def R = makeLetterplaceRing(5); // constructs a Letterplace ring |
---|
| 2060 | setring R; // sets basering to Letterplace ring |
---|
| 2061 | ideal G = x*x, y*y,x*y*x; |
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| 2062 | // Groebner basis |
---|
| 2063 | ideal I = x*x, y*x*y, x*y*x; |
---|
| 2064 | // Groebner basis |
---|
| 2065 | lpKDimCheck(G); // invokes procedure, factor algebra is of finite K-dimension |
---|
| 2066 | lpKDimCheck(I); // invokes procedure, factor algebra is of infinite Kdimension |
---|
| 2067 | } |
---|
| 2068 | |
---|
| 2069 | proc lpKDim(ideal G, list #) |
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| 2070 | "USAGE: lpKDim(G[,degbound, n]); G an ideal, degbound, n optional integers |
---|
| 2071 | RETURN: int, the K-dimension of the factor algebra |
---|
| 2072 | PURPOSE:Compute the K-dimension of a factor algebra, given via an ideal |
---|
| 2073 | ASSUME: - basering is a Letterplace ring |
---|
| 2074 | @* - if you specify a different degree bound degbound, |
---|
| 2075 | @* degbound <= attrib(basering,uptodeg) holds. |
---|
| 2076 | NOTE: - If degbound is set, there will be a degree bound added. 0 means no |
---|
| 2077 | @* degree bound. Default: attrib(basering, uptodeg). |
---|
| 2078 | @* - n is the number of variables, which can be set to a different number. |
---|
| 2079 | @* Default: attrib(basering, lV). |
---|
| 2080 | @* - If the K-dimension is known to be infinite, a degree bound is needed |
---|
| 2081 | EXAMPLE: example lpKDim; shows examples |
---|
| 2082 | " |
---|
| 2083 | {int degbound = attrib(basering, "uptodeg");int n = attrib(basering, "isLetterplaceRing"); |
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| 2084 | if (size(#) > 0){if (typeof(#[1])=="int"){if (#[1] >= 0){degbound = #[1];}}} |
---|
| 2085 | if (size(#) > 1){if (typeof(#[1])=="int"){if (#[2] > 0){n = #[2];}}} |
---|
| 2086 | list L; |
---|
| 2087 | L = lp2ivId(normalize(lead(G))); |
---|
| 2088 | return(ivKDim(L,n,degbound)); |
---|
| 2089 | } |
---|
| 2090 | example |
---|
| 2091 | { |
---|
| 2092 | "EXAMPLE:"; echo = 2; |
---|
| 2093 | ring r = 0,(x,y),dp; |
---|
| 2094 | def R = makeLetterplaceRing(5); // constructs a Letterplace ring |
---|
| 2095 | setring R; // sets basering to Letterplace ring |
---|
| 2096 | ideal G = x*x, y*y,x*y*x; |
---|
| 2097 | // ideal G contains a Groebner basis |
---|
| 2098 | lpKDim(G); //procedure invoked with ring parameters |
---|
| 2099 | // the factor algebra is finite, so the degree bound given by the Letterplace |
---|
| 2100 | // ring is not necessary |
---|
| 2101 | lpKDim(G,0); // procedure without any degree bound |
---|
| 2102 | } |
---|
| 2103 | |
---|
| 2104 | static proc lpMis2Base(ideal M) |
---|
| 2105 | "USAGE: lpMis2Base(M); M an ideal |
---|
| 2106 | RETURN: ideal, a K-basis of the factor algebra |
---|
| 2107 | PURPOSE:Compute a K-basis out of given mistletoes |
---|
| 2108 | ASSUME: - basering is a Letterplace ring. G is a Letterplace ideal. |
---|
| 2109 | @* - M contains only monomials |
---|
| 2110 | NOTE: - The mistletoes have to be ordered lexicographically -> OrdMisLex. |
---|
| 2111 | EXAMPLE: example lpMis2Base; shows examples |
---|
| 2112 | " |
---|
| 2113 | {list L; |
---|
| 2114 | L = lpId2ivLi(M); |
---|
| 2115 | return(ivL2lpI(ivMis2Base(L))); |
---|
| 2116 | } |
---|
| 2117 | example |
---|
| 2118 | { |
---|
| 2119 | "EXAMPLE:"; echo = 2; |
---|
| 2120 | ring r = 0,(x,y),dp; |
---|
| 2121 | def R = makeLetterplaceRing(5); // constructs a Letterplace ring |
---|
| 2122 | setring R; // sets basering to Letterplace ring |
---|
| 2123 | ideal L = x*y,y*x*y; |
---|
| 2124 | // ideal containing the mistletoes |
---|
| 2125 | lpMis2Base(L); // returns the K-basis of the factor algebra |
---|
| 2126 | } |
---|
| 2127 | |
---|
| 2128 | static proc lpMis2Dim(ideal M) |
---|
| 2129 | "USAGE: lpMis2Dim(M); M an ideal |
---|
| 2130 | RETURN: int, the K-dimension of the factor algebra |
---|
| 2131 | PURPOSE:Compute the K-dimension out of given mistletoes |
---|
| 2132 | ASSUME: - basering is a Letterplace ring. |
---|
| 2133 | @* - M contains only monomials |
---|
| 2134 | NOTE: - The mistletoes have to be ordered lexicographically -> OrdMisLex. |
---|
| 2135 | EXAMPLE: example lpMis2Dim; shows examples |
---|
| 2136 | " |
---|
| 2137 | {list L; |
---|
| 2138 | L = lpId2ivLi(M); |
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| 2139 | return(ivMis2Dim(L)); |
---|
| 2140 | } |
---|
| 2141 | example |
---|
| 2142 | { |
---|
| 2143 | "EXAMPLE:"; echo = 2; |
---|
| 2144 | ring r = 0,(x,y),dp; |
---|
| 2145 | def R = makeLetterplaceRing(5); // constructs a Letterplace ring |
---|
| 2146 | setring R; // sets basering to Letterplace ring |
---|
| 2147 | ideal L = x*y,y*x*y; |
---|
| 2148 | // ideal containing the mistletoes |
---|
| 2149 | lpMis2Dim(L); // returns the K-dimension of the factor algebra |
---|
| 2150 | } |
---|
| 2151 | |
---|
| 2152 | static proc lpOrdMisLex(ideal M) |
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| 2153 | "USAGE: lpOrdMisLex(M); M an ideal of mistletoes |
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| 2154 | RETURN: ideal, containing the mistletoes, ordered lexicographically |
---|
| 2155 | PURPOSE:A given set of mistletoes is ordered lexicographically |
---|
| 2156 | ASSUME: - basering is a Letterplace ring. |
---|
| 2157 | NOTE: This is preprocessing, it is not needed if the mistletoes are returned |
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| 2158 | @* from the sickle algorithm. |
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| 2159 | EXAMPLE: example lpOrdMisLex; shows examples |
---|
| 2160 | " |
---|
| 2161 | {return(ivL2lpI(sort(lpId2ivLi(M))[1]));} |
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| 2162 | example |
---|
| 2163 | { |
---|
| 2164 | "EXAMPLE:"; echo = 2; |
---|
| 2165 | ring r = 0,(x,y),dp; |
---|
| 2166 | def R = makeLetterplaceRing(5); // constructs a Letterplace ring |
---|
| 2167 | setring R; // sets basering to Letterplace ring |
---|
| 2168 | ideal M = x*y*x, y*y*x, x*x, y*x*x*x; |
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| 2169 | // some monomials |
---|
| 2170 | lpOrdMisLex(M); // orders the monomials lexicographically |
---|
| 2171 | } |
---|
| 2172 | |
---|
| 2173 | static proc lpSickle(ideal G, list #) |
---|
| 2174 | "USAGE: lpSickle(G[,degbound,n]); G an ideal, degbound, n optional integers |
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| 2175 | RETURN: ideal |
---|
| 2176 | PURPOSE:Compute the mistletoes of K[X]/<G> |
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| 2177 | ASSUME: - basering is a Letterplace ring. |
---|
| 2178 | @* - if you specify a different degree bound degbound, |
---|
| 2179 | @* degbound <= attrib(basering,uptodeg) holds. |
---|
| 2180 | NOTE: - If degbound is set, there will be a degree bound added. 0 means no |
---|
| 2181 | @* degree bound. Default: attrib(basering,uptodeg). |
---|
| 2182 | @* - n is the number of variables, which can be set to a different number. |
---|
| 2183 | @* Default: attrib(basering, lV). |
---|
| 2184 | @* - If the K-dimension is known to be infinite, a degree bound is needed |
---|
| 2185 | EXAMPLE: example lpSickle; shows examples |
---|
| 2186 | " |
---|
| 2187 | {int degbound = attrib(basering,"uptodeg"); int n = attrib(basering, "isLetterplaceRing"); |
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| 2188 | if (size(#) > 0){if (typeof(#[1])=="int"){if (#[1] >= 0){degbound = #[1];}}} |
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| 2189 | if (size(#) > 1){if (typeof(#[1])=="int"){if (#[2] > 0){n = #[2];}}} |
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| 2190 | list L; ideal R; |
---|
| 2191 | R = normalize(lead(G)); |
---|
| 2192 | L = lp2ivId(R); |
---|
| 2193 | L = ivSickle(L,n,degbound); |
---|
| 2194 | R = ivL2lpI(L); |
---|
| 2195 | return(R); |
---|
| 2196 | } |
---|
| 2197 | example |
---|
| 2198 | { |
---|
| 2199 | "EXAMPLE:"; echo = 2; |
---|
| 2200 | ring r = 0,(x,y),dp; |
---|
| 2201 | def R = makeLetterplaceRing(5); // constructs a Letterplace ring |
---|
| 2202 | setring R; // sets basering to Letterplace ring |
---|
| 2203 | ideal G = x*x, y*y,x*y*x; // ideal G contains a |
---|
| 2204 | //Groebner basis |
---|
| 2205 | lpSickle(G); //invokes the procedure with ring parameters |
---|
| 2206 | // the factor algebra is finite, so the degree bound given by the Letterplace |
---|
| 2207 | // ring is not necessary |
---|
| 2208 | lpSickle(G,0); // procedure without any degree bound |
---|
| 2209 | } |
---|
| 2210 | |
---|
| 2211 | proc lpSickleDim(ideal G, list #) |
---|
| 2212 | "USAGE: lpSickleDim(G[,degbound,n]); G an ideal, degbound, n optional integers |
---|
| 2213 | RETURN: list |
---|
| 2214 | PURPOSE:Compute the K-dimension and the mistletoes |
---|
| 2215 | ASSUME: - basering is a Letterplace ring. |
---|
| 2216 | @* - if you specify a different degree bound degbound, |
---|
| 2217 | @* degbound <= attrib(basering,uptodeg) holds. |
---|
| 2218 | NOTE: - If L is the list returned, then L[1] is an integer, the K-dimension, |
---|
| 2219 | @* L[2] is an ideal, the mistletoes. |
---|
| 2220 | @* - If degbound is set, there will be a degree bound added. 0 means no |
---|
| 2221 | @* degree bound. Default: attrib(basering,uptodeg). |
---|
| 2222 | @* - n is the number of variables, which can be set to a different number. |
---|
| 2223 | @* Default: attrib(basering, lV). |
---|
| 2224 | @* - If the K-dimension is known to be infinite, a degree bound is needed |
---|
| 2225 | EXAMPLE: example lpSickleDim; shows examples |
---|
| 2226 | " |
---|
| 2227 | {int degbound = attrib(basering,"uptodeg");int n = attrib(basering, "isLetterplaceRing"); |
---|
| 2228 | if (size(#) > 0){if (typeof(#[1])=="int"){if (#[1] >= 0){degbound = #[1];}}} |
---|
| 2229 | if (size(#) > 1){if (typeof(#[1])=="int"){if (#[2] > 0){n = #[2];}}} |
---|
| 2230 | list L; |
---|
| 2231 | L = lp2ivId(normalize(lead(G))); |
---|
| 2232 | L = ivSickleDim(L,n,degbound); |
---|
| 2233 | L[2] = ivL2lpI(L[2]); |
---|
| 2234 | return(L); |
---|
| 2235 | } |
---|
| 2236 | example |
---|
| 2237 | { |
---|
| 2238 | "EXAMPLE:"; echo = 2; |
---|
| 2239 | ring r = 0,(x,y),dp; |
---|
| 2240 | def R = makeLetterplaceRing(5); // constructs a Letterplace ring |
---|
| 2241 | setring R; // sets basering to Letterplace ring |
---|
| 2242 | ideal G = x*x, y*y,x*y*x; // ideal G contains a |
---|
| 2243 | //Groebner basis |
---|
| 2244 | lpSickleDim(G); // invokes the procedure with ring parameters |
---|
| 2245 | // the factor algebra is finite, so the degree bound given by the Letterplace |
---|
| 2246 | // ring is not necessary |
---|
| 2247 | lpSickleDim(G,0); // procedure without any degree bound |
---|
| 2248 | } |
---|
| 2249 | |
---|
| 2250 | static proc lpSickleHil(ideal G, list #) |
---|
| 2251 | "USAGE: lpSickleHil(G); |
---|
| 2252 | RETURN: list |
---|
| 2253 | PURPOSE:Compute the Hilbert series and the mistletoes |
---|
| 2254 | ASSUME: - basering is a Letterplace ring. |
---|
| 2255 | @* - if you specify a different degree bound degbound, |
---|
| 2256 | @* degbound <= attrib(basering,uptodeg) holds. |
---|
| 2257 | NOTE: - If L is the list returned, then L[1] is an intvec, corresponding to the |
---|
| 2258 | @* Hilbert series, L[2] is an ideal, the mistletoes. |
---|
| 2259 | @* - If degbound is set, there will be a degree bound added. 0 means no |
---|
| 2260 | @* degree bound. Default: attrib(basering,uptodeg). |
---|
| 2261 | @* - n is the number of variables, which can be set to a different number. |
---|
| 2262 | @* Default: attrib(basering, lV). |
---|
| 2263 | @* - If I = L[1] is the intvec returned, then I[k] is the (k-1)-th |
---|
| 2264 | @* coefficient of the Hilbert series. |
---|
| 2265 | @* - If the K-dimension is known to be infinite, a degree bound is needed |
---|
| 2266 | EXAMPLE: example lpSickleHil; shows examples |
---|
| 2267 | " |
---|
| 2268 | {int degbound = attrib(basering,"uptodeg");int n = attrib(basering, "isLetterplaceRing"); |
---|
| 2269 | if (size(#) > 0){if (typeof(#[1])=="int"){if (#[1] >= 0){degbound = #[1];}}} |
---|
| 2270 | if (size(#) > 1){if (typeof(#[1])=="int"){if (#[2] > 0){n = #[2];}}} |
---|
| 2271 | list L; |
---|
| 2272 | L = lp2ivId(normalize(lead(G))); |
---|
| 2273 | L = ivSickleHil(L,n,degbound); |
---|
| 2274 | L[2] = ivL2lpI(L[2]); |
---|
| 2275 | return(L); |
---|
| 2276 | } |
---|
| 2277 | example |
---|
| 2278 | { |
---|
| 2279 | "EXAMPLE:"; echo = 2; |
---|
| 2280 | ring r = 0,(x,y),dp; |
---|
| 2281 | def R = makeLetterplaceRing(5); // constructs a Letterplace ring |
---|
| 2282 | setring R; // sets basering to Letterplace ring |
---|
| 2283 | ideal G = x*x, y*y,x*y*x; // ideal G contains a |
---|
| 2284 | //Groebner basis |
---|
| 2285 | lpSickleHil(G); // invokes the procedure with ring parameters |
---|
| 2286 | // the factor algebra is finite, so the degree bound given by the Letterplace |
---|
| 2287 | // ring is not necessary |
---|
| 2288 | lpSickleHil(G,0); // procedure without any degree bound |
---|
| 2289 | } |
---|
| 2290 | |
---|
| 2291 | static proc sickle(ideal G, list #) |
---|
| 2292 | "USAGE: sickle(G[,m, d, h, degbound]); G an ideal; m,d,h,degbound optional |
---|
| 2293 | @* integers |
---|
| 2294 | RETURN: list |
---|
| 2295 | PURPOSE:Allowing the user to access all procs with one command |
---|
| 2296 | ASSUME: - basering is a Letterplace ring. |
---|
| 2297 | @* - if you specify a different degree bound degbound, |
---|
| 2298 | @* degbound <= attrib(basering,uptodeg) holds. |
---|
| 2299 | NOTE: The returned object will always be a list, but the entries of the |
---|
| 2300 | @* returned list may be very different |
---|
| 2301 | @* case m=1,d=1,h=1: see lpDHilbertSickle |
---|
| 2302 | @* case m=1,d=1,h=0: see lpSickleDim |
---|
| 2303 | @* case m=1,d=0,h=1: see lpSickleHil |
---|
| 2304 | @* case m=1,d=0,h=0: see lpSickle (this is the default case) |
---|
| 2305 | @* case m=0,d=1,h=1: see lpDHilbert |
---|
| 2306 | @* case m=0,d=1,h=0: see lpKDim |
---|
| 2307 | @* case m=0,d=0,h=1: see lpHilbert |
---|
| 2308 | @* case m=0,d=0,h=0: returns an error |
---|
| 2309 | @* - If degbound is set, there will be a degree bound added. 0 means no |
---|
| 2310 | @* degree bound. Default: attrib(basering,uptodeg). |
---|
| 2311 | @* - If the K-dimension is known to be infinite, a degree bound is needed |
---|
| 2312 | EXAMPLE: example sickle; shows examples |
---|
| 2313 | " |
---|
| 2314 | {int m,d,h,degbound; |
---|
| 2315 | m = 1; d = 0; h = 0; degbound = attrib(basering,"uptodeg"); |
---|
| 2316 | if (size(#) > 0) {if (typeof(#[1])=="int"){if (#[1] < 1) {m = 0;}}} |
---|
| 2317 | if (size(#) > 1) {if (typeof(#[1])=="int"){if (#[2] > 0) {d = 1;}}} |
---|
| 2318 | if (size(#) > 2) {if (typeof(#[1])=="int"){if (#[3] > 0) {h = 1;}}} |
---|
| 2319 | if (size(#) > 3) {if (typeof(#[1])=="int"){if (#[4] >= 0) {degbound = #[4];}}} |
---|
| 2320 | if (m == 1) |
---|
| 2321 | {if (d == 0) |
---|
| 2322 | {if (h == 0) {return(lpSickle(G,degbound,attrib(basering,"isLetterplaceRing")));} |
---|
| 2323 | else {return(lpSickleHil(G,degbound,attrib(basering,"isLetterplaceRing")));} |
---|
| 2324 | } |
---|
| 2325 | else |
---|
| 2326 | {if (h == 0) {return(lpSickleDim(G,degbound,attrib(basering,"isLetterplaceRing")));} |
---|
| 2327 | else {return(lpDHilbertSickle(G,degbound,attrib(basering,"isLetterplaceRing")));} |
---|
| 2328 | } |
---|
| 2329 | } |
---|
| 2330 | else |
---|
| 2331 | {if (d == 0) |
---|
| 2332 | {if (h == 0) {ERROR("You request to do nothing, so relax and do so");} |
---|
| 2333 | else {return(lpHilbert(G,degbound,attrib(basering,"isLetterplaceRing")));} |
---|
| 2334 | } |
---|
| 2335 | else |
---|
| 2336 | {if (h == 0) {return(lpKDim(G,degbound,attrib(basering,"isLetterplaceRing")));} |
---|
| 2337 | else {return(lpDHilbert(G,degbound,attrib(basering,"isLetterplaceRing")));} |
---|
| 2338 | } |
---|
| 2339 | } |
---|
| 2340 | } |
---|
| 2341 | example |
---|
| 2342 | { |
---|
| 2343 | "EXAMPLE:"; echo = 2; |
---|
| 2344 | ring r = 0,(x,y),dp; |
---|
| 2345 | def R = makeLetterplaceRing(5); // constructs a Letterplace ring |
---|
| 2346 | setring R; // sets basering to Letterplace ring |
---|
| 2347 | ideal G = x*x, y*y,x*y*x; |
---|
| 2348 | // G contains a Groebner basis |
---|
| 2349 | sickle(G,1,1,1); // computes mistletoes, K-dimension and the Hilbert series |
---|
| 2350 | sickle(G,1,0,0); // computes mistletoes only |
---|
| 2351 | sickle(G,0,1,0); // computes K-dimension only |
---|
| 2352 | sickle(G,0,0,1); // computes Hilbert series only |
---|
| 2353 | } |
---|
| 2354 | |
---|
| 2355 | proc lpMonomialBasis(int d, int donly, ideal J) |
---|
| 2356 | "USAGE: lpMonomialBasis(d, donly, J); d, donly integers, J an ideal |
---|
| 2357 | RETURN: ideal |
---|
| 2358 | PURPOSE: computes a list of free monomials in a Letterplace |
---|
| 2359 | @* basering R of degree at most d and not contained in <LM(J)> |
---|
| 2360 | @* if donly <> 0, only monomials of degree d are returned |
---|
| 2361 | ASSUME: - basering is a Letterplace ring. |
---|
| 2362 | @* - d <= attrib(basering,uptodeg) holds. |
---|
| 2363 | @* - J is a Groebner basis |
---|
| 2364 | " |
---|
| 2365 | { |
---|
| 2366 | int nv = attrib(basering,"uptodeg"); |
---|
| 2367 | if ((d>nv) || (d<0) ) |
---|
| 2368 | { |
---|
| 2369 | ERROR("incorrect degree"); |
---|
| 2370 | } |
---|
| 2371 | nv = attrib(basering,"isLetterplaceRing"); // nvars |
---|
| 2372 | if (d==0) |
---|
| 2373 | { |
---|
| 2374 | return(ideal(1)); |
---|
| 2375 | } |
---|
| 2376 | /* from now on d>=1 */ |
---|
| 2377 | ideal I; |
---|
| 2378 | if (size(J)==0) |
---|
| 2379 | { |
---|
| 2380 | I = maxideal(d); |
---|
| 2381 | if (!donly) |
---|
| 2382 | { |
---|
| 2383 | for (int i = d-1; i >= 0; i--) |
---|
| 2384 | { |
---|
| 2385 | I = maxideal(i), I; |
---|
| 2386 | } kill i; |
---|
| 2387 | } |
---|
| 2388 | return(I); |
---|
| 2389 | } |
---|
| 2390 | // ok, Sickle misbehaves: have to remove all |
---|
| 2391 | // elts from J of degree >d |
---|
| 2392 | ideal JJ; |
---|
| 2393 | int j; int sj = ncols(J); |
---|
| 2394 | int cnt=0; |
---|
| 2395 | for(j=1;j<=sj;j++) |
---|
| 2396 | { |
---|
| 2397 | if (deg(J[j]) <= d) |
---|
| 2398 | { |
---|
| 2399 | cnt++; |
---|
| 2400 | JJ[cnt]=lead(J[j]); // only LMs are needed |
---|
| 2401 | } |
---|
| 2402 | } |
---|
| 2403 | if (cnt==0) |
---|
| 2404 | { |
---|
| 2405 | // there are no elements in J of degree <= d |
---|
| 2406 | // return free stuff and the 1 |
---|
| 2407 | I = lpMonomialBasis(d, donly, ideal(0)); |
---|
| 2408 | if (!donly) |
---|
| 2409 | { |
---|
| 2410 | I = 1, I; |
---|
| 2411 | } |
---|
| 2412 | return(I); |
---|
| 2413 | } |
---|
| 2414 | // from here on, Ibase is not zero |
---|
| 2415 | ideal Ibase = lpMis2Base(lpSickle(JJ,d)); // the complete K-basis modulo J up to d |
---|
| 2416 | if (!donly) |
---|
| 2417 | { |
---|
| 2418 | // for not donly, give everything back |
---|
| 2419 | Ibase = sort(Ibase)[1]; |
---|
| 2420 | return(Ibase); |
---|
| 2421 | } |
---|
| 2422 | /* !donly: pick out only monomials of degree d */ |
---|
| 2423 | int i; int si = ncols(Ibase); |
---|
| 2424 | cnt=0; |
---|
| 2425 | I=0; |
---|
| 2426 | for(i=1;i<=si;i++) |
---|
| 2427 | { |
---|
| 2428 | if (deg(Ibase[i]) == d) |
---|
| 2429 | { |
---|
| 2430 | cnt++; |
---|
| 2431 | I[cnt]=Ibase[i]; |
---|
| 2432 | } |
---|
| 2433 | } |
---|
| 2434 | kill Ibase; |
---|
| 2435 | return(I); |
---|
| 2436 | } |
---|
| 2437 | example |
---|
| 2438 | { |
---|
| 2439 | "EXAMPLE:"; echo = 2; |
---|
| 2440 | ring r = 0,(x,y),dp; |
---|
| 2441 | def R = makeLetterplaceRing(7); setring R; |
---|
| 2442 | ideal J = x*y*x - y*x*y; |
---|
| 2443 | option(redSB); option(redTail); |
---|
| 2444 | J = letplaceGBasis(J); |
---|
| 2445 | J; |
---|
| 2446 | lpMonomialBasis(2,1,ideal(0)); |
---|
| 2447 | lpMonomialBasis(2,0,ideal(0)); |
---|
| 2448 | lpMonomialBasis(3,1,J); |
---|
| 2449 | lpMonomialBasis(3,0,J); |
---|
| 2450 | } |
---|
| 2451 | |
---|
| 2452 | /////////////////////////////////////////////////////////////////////////////// |
---|
| 2453 | |
---|
| 2454 | /* |
---|
| 2455 | Here are some examples one may try. Just copy them into your console. |
---|
| 2456 | These are relations for braid groups, up to degree d: |
---|
| 2457 | |
---|
| 2458 | LIB "fpadim.lib"; |
---|
| 2459 | ring r = 0,(x,y,z),dp; |
---|
| 2460 | int d =10; // degree |
---|
| 2461 | def R = makeLetterplaceRing(d); |
---|
| 2462 | setring R; |
---|
| 2463 | ideal I = y*x*y - z*y*z, x*y*x - z*x*y, |
---|
| 2464 | z*x*z - y*z*x, x*x*x + y*y*y + |
---|
| 2465 | z*z*z + x*y*z; |
---|
| 2466 | option(prot); |
---|
| 2467 | option(redSB);option(redTail);option(mem); |
---|
| 2468 | ideal J = system("freegb",I,d,3); |
---|
| 2469 | lpKDimCheck(J); |
---|
| 2470 | sickle(J,1,1,1,d);//Computes mistletoes, K-dimension and the Hilbert series |
---|
| 2471 | |
---|
| 2472 | |
---|
| 2473 | |
---|
| 2474 | LIB "fpadim.lib"; |
---|
| 2475 | ring r = 0,(x,y,z),dp; |
---|
| 2476 | int d =11; // degree |
---|
| 2477 | def R = makeLetterplaceRing(d); |
---|
| 2478 | setring R; |
---|
| 2479 | ideal I = y*x*y - z*y*z, x*y*z - z*x*y, |
---|
| 2480 | z*x*z - y*z*x, x*x*x + y*y*y + |
---|
| 2481 | z*z*z + x*y*z; |
---|
| 2482 | option(prot); |
---|
| 2483 | option(redSB);option(redTail);option(mem); |
---|
| 2484 | ideal J = system("freegb",I,d,3); |
---|
| 2485 | lpKDimCheck(J); |
---|
| 2486 | sickle(J,1,1,1,d); |
---|
| 2487 | |
---|
| 2488 | |
---|
| 2489 | |
---|
| 2490 | LIB "fpadim.lib"; |
---|
| 2491 | ring r = 0,(x,y,z),dp; |
---|
| 2492 | int d = 6; // degree |
---|
| 2493 | def R = makeLetterplaceRing(d); |
---|
| 2494 | setring R; |
---|
| 2495 | ideal I = y*x*y - z*y*z, x*y*x - z*x*y, |
---|
| 2496 | z*x*z - y*z*x, x*x*x -2*y*y*y + 3*z*z*z -4*x*y*z + 5*x*z*z- 6*x*y*y +7*x*x*z - 8*x*x*y; |
---|
| 2497 | option(prot); |
---|
| 2498 | option(redSB);option(redTail);option(mem); |
---|
| 2499 | ideal J = system("freegb",I,d,3); |
---|
| 2500 | lpKDimCheck(J); |
---|
| 2501 | sickle(J,1,1,1,d); |
---|
| 2502 | */ |
---|
| 2503 | |
---|
| 2504 | /* |
---|
| 2505 | Here are some examples, which can also be found in [studzins]: |
---|
| 2506 | |
---|
| 2507 | // takes up to 880Mb of memory |
---|
| 2508 | LIB "fpadim.lib"; |
---|
| 2509 | ring r = 0,(x,y,z),dp; |
---|
| 2510 | int d =10; // degree |
---|
| 2511 | def R = makeLetterplaceRing(d); |
---|
| 2512 | setring R; |
---|
| 2513 | ideal I = |
---|
| 2514 | z*z*z*z + y*x*y*x - x*y*y*x - 3*z*y*x*z, x*x*x + y*x*y - x*y*x, z*y*x-x*y*z + z*x*z; |
---|
| 2515 | option(prot); |
---|
| 2516 | option(redSB);option(redTail);option(mem); |
---|
| 2517 | ideal J = system("freegb",I,d,nvars(r)); |
---|
| 2518 | lpKDimCheck(J); |
---|
| 2519 | sickle(J,1,1,1,d); // dimension is 24872 |
---|
| 2520 | |
---|
| 2521 | |
---|
| 2522 | LIB "fpadim.lib"; |
---|
| 2523 | ring r = 0,(x,y,z),dp; |
---|
| 2524 | int d =10; // degree |
---|
| 2525 | def R = makeLetterplaceRing(d); |
---|
| 2526 | setring R; |
---|
| 2527 | ideal I = x*y + y*z, x*x + x*y - y*x - y*y; |
---|
| 2528 | option(prot); |
---|
| 2529 | option(redSB);option(redTail);option(mem); |
---|
| 2530 | ideal J = system("freegb",I,d,3); |
---|
| 2531 | lpKDimCheck(J); |
---|
| 2532 | sickle(J,1,1,1,d); |
---|
| 2533 | */ |
---|
| 2534 | |
---|
| 2535 | |
---|
| 2536 | /* |
---|
| 2537 | Example for Compute GK dimension: |
---|
| 2538 | returns a ring which contains an ideal I |
---|
| 2539 | run gkDim(I) inside this ring and it should return 2n (the GK dimension |
---|
| 2540 | of n-th Weyl algebra including evaluation operators). |
---|
| 2541 | |
---|
| 2542 | static proc createWeylEx(int n, int d) |
---|
| 2543 | " |
---|
| 2544 | " |
---|
| 2545 | { |
---|
| 2546 | int baseringdef; |
---|
| 2547 | if (defined(basering)) // if a basering is defined, it should be saved for later use |
---|
| 2548 | { |
---|
| 2549 | def save = basering; |
---|
| 2550 | baseringdef = 1; |
---|
| 2551 | } |
---|
| 2552 | ring r = 0,(d(1..n),x(1..n),e(1..n)),dp; |
---|
| 2553 | def R = makeLetterplaceRing(d); |
---|
| 2554 | setring R; |
---|
| 2555 | ideal I; int i,j; |
---|
| 2556 | |
---|
| 2557 | for (i = 1; i <= n; i++) |
---|
| 2558 | { |
---|
| 2559 | for (j = i+1; j<= n; j++) |
---|
| 2560 | { |
---|
| 2561 | I[size(I)+1] = lpMult(var(i),var(j)); |
---|
| 2562 | } |
---|
| 2563 | } |
---|
| 2564 | |
---|
| 2565 | for (i = 1; i <= n; i++) |
---|
| 2566 | { |
---|
| 2567 | for (j = i+1; j<= n; j++) |
---|
| 2568 | { |
---|
| 2569 | I[size(I)+1] = lpMult(var(n+i),var(n+j)); |
---|
| 2570 | } |
---|
| 2571 | } |
---|
| 2572 | for (i = 1; i <= n; i++) |
---|
| 2573 | { |
---|
| 2574 | for (j = 1; j<= n; j++) |
---|
| 2575 | { |
---|
| 2576 | I[size(I)+1] = lpMult(var(i),var(n+j)); |
---|
| 2577 | } |
---|
| 2578 | } |
---|
| 2579 | for (i = 1; i <= n; i++) |
---|
| 2580 | { |
---|
| 2581 | for (j = 1; j<= n; j++) |
---|
| 2582 | { |
---|
| 2583 | I[size(I)+1] = lpMult(var(i),var(2*n+j)); |
---|
| 2584 | } |
---|
| 2585 | } |
---|
| 2586 | for (i = 1; i <= n; i++) |
---|
| 2587 | { |
---|
| 2588 | for (j = 1; j<= n; j++) |
---|
| 2589 | { |
---|
| 2590 | I[size(I)+1] = lpMult(var(2*n+i),var(n+j)); |
---|
| 2591 | } |
---|
| 2592 | } |
---|
| 2593 | for (i = 1; i <= n; i++) |
---|
| 2594 | { |
---|
| 2595 | for (j = 1; j<= n; j++) |
---|
| 2596 | { |
---|
| 2597 | I[size(I)+1] = lpMult(var(2*n+i),var(2*n+j)); |
---|
| 2598 | } |
---|
| 2599 | } |
---|
| 2600 | I = simplify(I,2+4); |
---|
| 2601 | I = letplaceGBasis(I); |
---|
| 2602 | export(I); |
---|
| 2603 | if (baseringdef == 1) {setring save;} |
---|
| 2604 | return(R); |
---|
| 2605 | } |
---|
| 2606 | |
---|
| 2607 | proc TestGKAuslander3() |
---|
| 2608 | { |
---|
| 2609 | ring r = (0,q),(z,x,y),(dp(2),dp(2)); |
---|
| 2610 | def R = makeLetterplaceRing(5); // constructs a Letterplace ring |
---|
| 2611 | R; setring R; // sets basering to Letterplace ring |
---|
| 2612 | ideal I; |
---|
| 2613 | I = q*x*y - y*x, z*y - y*z, z*x - x*z; |
---|
| 2614 | I = letplaceGBasis(I); |
---|
| 2615 | lpGkDim(I); // must be 3 |
---|
| 2616 | I = x*y*z - y*x, z*y - y*z, z*x - x*z;//gkDim = 2 |
---|
| 2617 | I = letplaceGBasis(I); // not finite BUT contains a poly in x,y only |
---|
| 2618 | lpGkDim(I); // must be 4 |
---|
| 2619 | |
---|
| 2620 | ring r = 0,(y,x,z),dp; |
---|
| 2621 | def R = makeLetterplaceRing(10); // constructs a Letterplace ring |
---|
| 2622 | R; setring R; // sets basering to Letterplace ring |
---|
| 2623 | ideal I; |
---|
| 2624 | I = x*y*z - y*x, z*y - y*z, z*x - x*z;//gkDim = 2 |
---|
| 2625 | I = letplaceGBasis(I); // computed as it would be homogenized; infinite |
---|
| 2626 | poly p = x*y*y*x-y*x*x*y; |
---|
| 2627 | lpNF(p, I); // 0 as expected |
---|
| 2628 | |
---|
| 2629 | // with inverse of z |
---|
| 2630 | ring r = 0,(iz,z,x,y),dp; |
---|
| 2631 | def R = makeLetterplaceRing(11); // constructs a Letterplace ring |
---|
| 2632 | R; setring R; // sets basering to Letterplace ring |
---|
| 2633 | ideal I; |
---|
| 2634 | I = x*y*z - y*x, z*y - y*z, z*x - x*z, |
---|
| 2635 | iz*y - y*iz, iz*x - x*iz, iz*z-1, z*iz -1; |
---|
| 2636 | I = letplaceGBasis(I); // |
---|
| 2637 | setring r; |
---|
| 2638 | def R2 = makeLetterplaceRing(23); // constructs a Letterplace ring |
---|
| 2639 | setring R2; // sets basering to Letterplace ring |
---|
| 2640 | ideal I = imap(R,I); |
---|
| 2641 | lpGkDim(I); |
---|
| 2642 | |
---|
| 2643 | |
---|
| 2644 | ring r = 0,(t,z,x,y),(dp,dp); |
---|
| 2645 | def R = makeLetterplaceRing(20); // constructs a Letterplace ring |
---|
| 2646 | R; setring R; // sets basering to Letterplace ring |
---|
| 2647 | ideal I; |
---|
| 2648 | I = x*y*z - y*x*t, z*y - y*z, z*x - x*z, |
---|
| 2649 | t*y - y*t, t*x - x*t, t*z - z*t;//gkDim = 2 |
---|
| 2650 | I = letplaceGBasis(I); // computed as it would be homogenized; infinite |
---|
| 2651 | LIB "elim.lib"; |
---|
| 2652 | ideal Inoz = nselect(I,intvec(2,6,10,14,18,22,26,30)); |
---|
| 2653 | for(int i=1; i<=20; i++) |
---|
| 2654 | { |
---|
| 2655 | Inoz=subst(Inoz,t(i),1); |
---|
| 2656 | } |
---|
| 2657 | ideal J = x*y*y*x-y*x*x*y; |
---|
| 2658 | J = letplaceGBasis(J); |
---|
| 2659 | |
---|
| 2660 | poly p = x*y*y*x-y*x*x*y; |
---|
| 2661 | lpNF(p, I); // 0 as expected |
---|
| 2662 | |
---|
| 2663 | ring r2 = 0,(x,y),dp; |
---|
| 2664 | def R2 = makeLetterplaceRing(50); // constructs a Letterplace ring |
---|
| 2665 | setring R2; |
---|
| 2666 | ideal J = x*y*y*x-y*x*x*y; |
---|
| 2667 | J = letplaceGBasis(J); |
---|
| 2668 | } |
---|
| 2669 | |
---|
| 2670 | */ |
---|
| 2671 | |
---|
| 2672 | |
---|
| 2673 | /* more tests : downup algebra A |
---|
| 2674 | LIB "fpadim.lib"; |
---|
| 2675 | ring r = (0,a,b,g),(x,y),Dp; |
---|
| 2676 | def R = makeLetterplaceRing(6); // constructs a Letterplace ring |
---|
| 2677 | setring R; |
---|
| 2678 | poly F1 = g*x; |
---|
| 2679 | poly F2 = g*y; |
---|
| 2680 | ideal J = x*x*y-a*x*y*x - b*y*x*x - F1, |
---|
| 2681 | x*y*y-a*y*x*y - b*y*y*x - F2; |
---|
| 2682 | J = letplaceGBasis(J); |
---|
| 2683 | lpGkDim(J); // 3 == correct |
---|
| 2684 | |
---|
| 2685 | // downup algebra B |
---|
| 2686 | LIB "fpadim.lib"; |
---|
| 2687 | ring r = (0,a,b,g, p(1..7),q(1..7)),(x,y),Dp; |
---|
| 2688 | def R = makeLetterplaceRing(6); // constructs a Letterplace ring |
---|
| 2689 | setring R; |
---|
| 2690 | ideal imn = 1, y*y*y, x*y, y*x, x*x, y*y, x, y; |
---|
| 2691 | int i; |
---|
| 2692 | poly F1, F2; |
---|
| 2693 | for(i=1;i<=7;i++) |
---|
| 2694 | { |
---|
| 2695 | F1 = F1 + p(i)*imn[i]; |
---|
| 2696 | F2 = F2 + q(i)*imn[i]; |
---|
| 2697 | } |
---|
| 2698 | ideal J = x*x*y-a*x*y*x - b*y*x*x - F1, |
---|
| 2699 | x*y*y-a*y*x*y - b*y*y*x - F2; |
---|
| 2700 | J = letplaceGBasis(J); |
---|
| 2701 | lpGkDim(J); // 3 == correct |
---|
| 2702 | */ |
---|