[380a17b] | 1 | //////////////////////////////////////////////////////////////////////////// |
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[c3e2eaa] | 2 | version="version resolve.lib 4.3.1.3 Feb_2023 "; // $Id$ |
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[6237df] | 3 | category="Algebraic Geometry"; |
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[2e6eac2] | 4 | info=" |
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[4cfbb0] | 5 | LIBRARY: resolve.lib Resolution of singularities (Desingularization) |
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| 6 | Algorithm of Villamayor |
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[2e6eac2] | 7 | AUTHORS: A. Fruehbis-Krueger, anne@mathematik.uni-kl.de, |
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| 8 | @* G. Pfister, pfister@mathematik.uni-kl.de |
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| 9 | |
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[66d68c] | 10 | REFERENCES: |
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[c3cc38] | 11 | [1] J.Kollar: Lectures on Resolution of Singularities, Princeton University Press (2007)@* |
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| 12 | (contains large overview over various known methods for curves and surfaces as well as@* |
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| 13 | a detailed description of the approach in the general case)@* |
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| 14 | [2] A.Bravo, S.Encinas, O.Villamayor: A Simplified Proof of Desingularisation and@* |
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| 15 | Applications, Rev. Math. Iberoamericana 21 (2005), 349-458@* |
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| 16 | (description of the algorithmic proof of desingularization in characteristic zero |
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| 17 | which underlies this implementation)@* |
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[66d68c] | 18 | [3] A.Fruehbis-Krueger: Computational Aspects of Singularities, in J.-P. Brasselet, |
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[c3cc38] | 19 | J.Damon et al.: Singularities in Geometry and Topology, World Scientific |
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| 20 | Publishing, 253--327 (2007)@* |
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| 21 | (chapter 4 contains a detailed discussion on algorithmic desingularization and |
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| 22 | efficiency aspects thereof) |
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[6237df] | 23 | |
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[268a430] | 24 | PROCEDURES: |
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[453bc66] | 25 | blowUp(J,C[,W,E]) computes the blowing up of the variety V(J) (considered |
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| 26 | as embedded in V(W)) in the (smooth) center V(C), |
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| 27 | blowUp2(J,C) computes the blowing up of the variety V(J) in the |
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| 28 | (possibly singular) center V(C) |
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[2e6eac2] | 29 | Center(J[,W,E]) computes 'Villamayor'-center for blow up |
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| 30 | resolve(J) computes the desingularization of the variety V(J) |
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| 31 | |
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[453bc66] | 32 | showBO(BO) prints the content of a BO in more human readable form |
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| 33 | presentTree(L) prints the final charts in more human readable form |
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| 34 | showDataTypes() prints help text for output data types |
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| 35 | |
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[2e6eac2] | 36 | blowUpBO(BO,C) computes the blowing up of the variety V(BO[1]) in the |
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| 37 | center V(C). BO is a list (basic object), C is an ideal |
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| 38 | createBO(J,W,E) creates basic object from input data |
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| 39 | CenterBO(BO) computes the center for the next blow-up of the |
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| 40 | given basic object |
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| 41 | Delta(BO) apply the Delta-operator of [Bravo,Encinas,Villamayor] |
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| 42 | DeltaList(BO) list of results of Delta^0 to Delta^bmax |
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| 43 | "; |
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| 44 | LIB "elim.lib"; |
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| 45 | LIB "primdec.lib"; |
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| 46 | LIB "presolve.lib"; |
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| 47 | LIB "linalg.lib"; |
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| 48 | LIB "sing.lib"; |
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| 49 | /////////////////////////////////////////////////////////////////////////////// |
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| 50 | // Tasks: |
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| 51 | // 1) optimization of the local case |
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| 52 | // 2) optimization in Coeff |
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| 53 | // 3) change invariant to represent coeff=1 case |
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| 54 | /////////////////////////////////////////////////////////////////////////////// |
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| 55 | |
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[453bc66] | 56 | proc showDataTypes() |
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| 57 | "USAGE: showDataTypes(); |
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| 58 | RETURN: nothing, only pretty printing of extended version of help text |
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| 59 | EXAMPLE: none |
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| 60 | " |
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| 61 | { |
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| 62 | " "; |
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| 63 | "//////////////// Short description of data type BO ///////////////////"; |
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| 64 | "BO[1] an ideal, say Wi, defining the ambient space of the i-th chart"; |
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| 65 | " of the blowing up"; |
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| 66 | "BO[2] an ideal defining the strict transform"; |
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| 67 | "BO[3] intvec, the first integer b such that in the original object"; |
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| 68 | " (Delta^b(BO[2]))==1"; |
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| 69 | " the subsequent integers have the same property for Coeff-Objects"; |
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| 70 | " of BO[2] used when determining the center"; |
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| 71 | "BO[4] the list of exceptional divisors"; |
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| 72 | "BO[5] an ideal defining the map K[W] ----> K[Wi]"; |
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| 73 | "BO[6] an intvec BO[6][j]=1 indicates that <BO[4][j],BO[2]>=1, i.e. the"; |
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| 74 | " strict transform does not meet the j-th exceptional divisor"; |
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| 75 | "BO[7] intvec,"; |
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| 76 | " the index of the first blown-up object in the resolution process"; |
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| 77 | " leading to this object for which the value of b was BO[3]"; |
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| 78 | " the subsequent ones are the indices for the Coeff-Objects"; |
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| 79 | " of BO[2] used when determining the center"; |
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| 80 | "BO[i], i>7: internal data"; |
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| 81 | " "; |
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| 82 | pause(); |
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| 83 | " "; |
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| 84 | "///////////// Short description of data in a chart ///////////////////"; |
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| 85 | "All chart data is stored in an object of type ring, the following "; |
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| 86 | "variables are always present in such a ring:"; |
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| 87 | "BO: list of type basic object containing the variety, ambient space,"; |
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| 88 | " exceptional divisors and further data (see previous page)"; |
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| 89 | "cent: ideal, describing the upcoming center determined by the algorithm"; |
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| 90 | "path: module (autoconverted to matrix)"; |
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| 91 | " path[1][i]= (i-1)st chart in resolution history of this chart"; |
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| 92 | " path[2][i]= index of chart of the blow up leading to i-th chart"; |
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| 93 | " in resolution history"; |
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| 94 | "lastMap: ideal, describing the preceding blow up leading to this chart"; |
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| 95 | " "; |
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| 96 | pause(); |
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| 97 | " "; |
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| 98 | "///////////// Short description of type resolution data //////////////"; |
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| 99 | "list L containing two lists:"; |
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| 100 | "L[1]: list of rings, containing the final charts"; |
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| 101 | "L[2]: list of rings, containing all charts created in the resolution"; |
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| 102 | " process"; |
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| 103 | "The most convenient way to view these data is the procedure presentTree()"; |
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| 104 | "from this library. Alternatively, it can be digested using tools from"; |
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| 105 | "the libraries reszeta.lib and resgraph.lib"; |
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| 106 | "//////////////////////////////////////////////////////////////////////"; |
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| 107 | } |
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| 108 | /////////////////////////////////////////////////////////////////////////////// |
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[2e6eac2] | 109 | proc createBO(ideal J,list #) |
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| 110 | "USAGE: createBO(J[,W][,E]); |
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| 111 | @* J,W = ideals |
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| 112 | @* E = list |
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| 113 | ASSUME: J = ideal containing W ( W = 0 if not specified) |
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| 114 | @* E = list of smooth hypersurfaces (e.g. exceptional divisors) |
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| 115 | RETURN: list BO representing a basic object : |
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| 116 | BO[1] ideal W, if W has been specified; ideal(0) otherwise |
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| 117 | BO[2] ideal J |
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| 118 | BO[3] intvec |
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| 119 | BO[4] the list E of exceptional divisors if specified; |
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| 120 | empty list otherwise |
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| 121 | BO[5] an ideal defining the identity map |
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| 122 | BO[6] an intvec |
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| 123 | BO[7] intvec |
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| 124 | BO[8] a matrix |
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| 125 | entries 3,5,6,7,8 are initialized appropriately for use of CenterBO |
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| 126 | and blowUpBO |
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| 127 | EXAMPLE: example createBO; shows an example |
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| 128 | " |
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| 129 | { |
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| 130 | ideal W; |
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| 131 | list E; |
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| 132 | ideal abb=maxideal(1); |
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| 133 | intvec v; |
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| 134 | intvec bvec; |
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| 135 | intvec w=-1; |
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| 136 | matrix intE; |
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| 137 | if(size(#)>0) |
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| 138 | { |
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| 139 | if(typeof(#[1])=="ideal") |
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| 140 | { |
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| 141 | W=#[1]; |
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| 142 | } |
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| 143 | if(typeof(#[1])=="list") |
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| 144 | { |
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| 145 | E=#[1]; |
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| 146 | } |
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| 147 | if(size(#)>1) |
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| 148 | { |
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| 149 | if((typeof(#[2])=="list") && (size(E)==0)) |
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| 150 | { |
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| 151 | E=#[2]; |
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| 152 | } |
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| 153 | if((typeof(#[2])=="ideal") && (size(W)==0)) |
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| 154 | { |
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| 155 | W=#[2]; |
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| 156 | } |
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| 157 | } |
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| 158 | } |
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| 159 | list BO=W,J,bvec,E,abb,v,w,intE; |
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| 160 | return(BO); |
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| 161 | } |
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| 162 | example |
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| 163 | {"EXAMPLE:"; |
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| 164 | echo = 2; |
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| 165 | ring R=0,(x,y,z),dp; |
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| 166 | ideal J=x2-y3; |
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| 167 | createBO(J,ideal(z)); |
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| 168 | } |
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| 169 | /////////////////////////////////////////////////////////////////////////////// |
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| 170 | |
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| 171 | proc blowUp(ideal J,ideal C,list #) |
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| 172 | "USAGE: blowUp(J,C[,W][,E]); |
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[2e5802] | 173 | W,J,C = ideals, |
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| 174 | E = list |
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[2e6eac2] | 175 | ASSUME: J = ideal containing W ( W = 0 if not specified) |
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| 176 | @* C = ideal containing J |
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| 177 | @* E = list of smooth hypersurfaces (e.g. exceptional divisors) |
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[2e5802] | 178 | NOTE: W the ideal of the ambient space, C the ideal of the center of |
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| 179 | the blowup and J the ideal of the variety |
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[453bc66] | 180 | Important difference to blowUp2: |
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| 181 | - the ambient space V(W) is blown up and V(J) transformed in it |
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| 182 | @* - V(C) is assumed to be non-singular |
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[2e5802] | 183 | COMPUTE: the blowing up of W in C, the exceptional locus, the strict |
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| 184 | transform of J and the blowup map |
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| 185 | RETURN: list, say l, of size at most size(C), |
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[453bc66] | 186 | |
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[2e5802] | 187 | l[i] is the affine ring corresponding to the i-th chart |
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| 188 | each l[i] contains the ideals |
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| 189 | - aS, ideal of the blownup ambient space |
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| 190 | - sT, ideal of the strict transform |
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| 191 | - eD, ideal of the exceptional divisor |
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| 192 | - bM, ideal corresponding to the blowup map |
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[453bc66] | 193 | |
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| 194 | l[i] also contains a list BO, which can best be viewed with showBO(BO) |
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| 195 | detailed information on the data type BO can be viewed via the |
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| 196 | command showDataTypes(); |
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[2e6eac2] | 197 | EXAMPLE: example blowUp; shows an example |
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| 198 | " |
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| 199 | { |
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[2e5802] | 200 | def S=basering; |
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[2e6eac2] | 201 | ideal W; |
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| 202 | list E; |
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| 203 | ideal abb=maxideal(1); |
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| 204 | intvec v; |
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| 205 | intvec bvec; |
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| 206 | intvec w=-1; |
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| 207 | matrix intE; |
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| 208 | if(size(#)>0) |
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| 209 | { |
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| 210 | if(typeof(#[1])=="ideal") |
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| 211 | { |
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| 212 | W=#[1]; |
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| 213 | } |
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| 214 | if(typeof(#[1])=="list") |
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| 215 | { |
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| 216 | E=#[1]; |
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| 217 | } |
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| 218 | if(size(#)>1) |
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| 219 | { |
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| 220 | if((typeof(#[2])=="list") && (size(E)==0)) |
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| 221 | { |
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| 222 | E=#[2]; |
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| 223 | } |
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| 224 | if((typeof(#[2])=="ideal") && (size(W)==0)) |
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| 225 | { |
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| 226 | W=#[2]; |
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| 227 | } |
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| 228 | } |
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| 229 | } |
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| 230 | list BO=W,J,bvec,E,abb,v,w,intE; |
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| 231 | int locaT; |
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| 232 | export locaT; |
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| 233 | list blow=blowUpBO(BO,C,0); |
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| 234 | kill locaT; |
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[2e5802] | 235 | int i; |
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| 236 | for(i=1;i<=size(blow);i++) |
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| 237 | { |
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| 238 | def Q=blow[i]; |
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| 239 | setring Q; |
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| 240 | ideal aS=BO[1]; |
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| 241 | ideal sT=BO[2]; |
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| 242 | ideal eD=BO[4][size(BO[4])]; |
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| 243 | ideal bM=BO[5]; |
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| 244 | kill lastMap; |
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| 245 | kill thisChart; |
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| 246 | export(aS); |
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| 247 | export(sT); |
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| 248 | export(eD); |
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| 249 | export(bM); |
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| 250 | blow[i]=Q; |
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| 251 | setring S; |
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| 252 | kill Q; |
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| 253 | } |
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[2e6eac2] | 254 | return(blow); |
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| 255 | } |
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| 256 | example |
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| 257 | {"EXAMPLE:"; |
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| 258 | echo = 2; |
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| 259 | ring R=0,(x,y),dp; |
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| 260 | ideal J=x2-y3; |
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| 261 | ideal C=x,y; |
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| 262 | list blow=blowUp(J,C); |
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| 263 | def Q=blow[1]; |
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| 264 | setring Q; |
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[2e5802] | 265 | aS; |
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| 266 | sT; |
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| 267 | eD; |
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| 268 | bM; |
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[2e6eac2] | 269 | } |
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| 270 | /////////////////////////////////////////////////////////////////////////////// |
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| 271 | |
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[453bc66] | 272 | proc blowUp2(ideal J,ideal C) |
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| 273 | "USAGE: blowUp2(J,C); |
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| 274 | J,C = ideals, |
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| 275 | ASSUME: C = ideal containing J |
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| 276 | NOTE: C the ideal of the center of the blowup and J the ideal |
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| 277 | of the variety |
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| 278 | Important differences to blowUp: |
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| 279 | - V(J) itself is blown up, not the ambient space |
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| 280 | - C is not assumed to be non-singular |
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| 281 | COMPUTE: the blowing up of J in C, the exceptional locus and the blow-up |
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| 282 | map |
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| 283 | RETURN: list, say l, of size at most size(C), |
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| 284 | |
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| 285 | l[i] is the affine ring corresponding to the i-th chart |
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| 286 | each l[i] contains the ideals |
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| 287 | - Jnew, ideal of the blownup J |
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| 288 | - eD, ideal of the new exceptional divisor |
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| 289 | - bM, ideal corresponding to the blowup map |
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| 290 | |
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| 291 | EXAMPLE: example blowUp2; shows an example |
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| 292 | " |
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| 293 | { |
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| 294 | //---------------------------------------------------------------------------- |
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| 295 | // Initialization and basic sanity checks of the input data |
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| 296 | //---------------------------------------------------------------------------- |
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| 297 | int i,j; |
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| 298 | def S=basering; |
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| 299 | def laM=maxideal(1); |
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| 300 | //--- number of generators of C should be as small as possible |
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| 301 | def mstdC=mstd(C); |
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| 302 | if(ncols(mstdC[1])<ncols(mstdC[2])) |
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| 303 | { |
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| 304 | C=mstdC[1]; |
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| 305 | } |
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| 306 | else |
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| 307 | { |
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| 308 | C=mstdC[2]; |
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| 309 | } |
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| 310 | C=simplify(interred(C),2); |
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| 311 | //--- the empty set is not a good center ;-) |
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| 312 | if(deg(lead(C[1]))==0) |
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| 313 | { |
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| 314 | ERROR("Your chosen center was the empty set. Exiting."); |
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| 315 | } |
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| 316 | //--- V(C) should be a subset of V(J) |
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[6518eba] | 317 | if(size(reduce(J,std(C),5))>0) |
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[453bc66] | 318 | { |
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| 319 | ERROR("V(J) does not contain V(C). Exiting."); |
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| 320 | } |
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| 321 | //--------------------------------------------------------------------------- |
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| 322 | // To compute the blowing up, we need to consider |
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| 323 | // (K[x_1,\dots,x_n]/J)[t*C[1],...,t*C[m]] |
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| 324 | // which we want to represent as a quotient of K[x_1,...,x_n,y_1,..,y_m] |
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| 325 | // by an ideal obtained by elimination of the extra variable t. |
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| 326 | // (in the comments n=nvars(S) and m=ncols(C)) |
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| 327 | //--------------------------------------------------------------------------- |
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| 328 | //--- set up rings for the elimination |
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[13814f0] | 329 | int ii; |
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| 330 | list l5; |
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| 331 | for (ii = 1; ii<= nvars(S); ii++) |
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| 332 | { |
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| 333 | l5[ii] = "x("+string(ii)+")"; |
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| 334 | } |
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| 335 | for (ii = 1; ii<= ncols(C); ii++) |
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| 336 | { |
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| 337 | l5[nvars(S)+ii] = "y("+string(ii)+")"; |
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| 338 | } |
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[62de185] | 339 | ring R = create_ring(ring_list(basering)[1], l5, "dp", "no_minpoly"); // ring for describing the transforms |
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[13814f0] | 340 | def C=fetch(S,C); |
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| 341 | list l6; |
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| 342 | for (ii = 1; ii<= nvars(S); ii++) |
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| 343 | { |
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| 344 | l6[ii] = "x("+string(ii)+")"; |
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| 345 | } |
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| 346 | for (ii = 1; ii<= ncols(C); ii++) |
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| 347 | { |
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| 348 | l6[nvars(S)+ii] = "y("+string(ii)+")"; |
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| 349 | } |
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| 350 | l6[size(l6)+1] = "t"; |
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[62de185] | 351 | ring Rt = create_ring(ring_list(basering)[1], l6, "(dp("+string(ncols(C)+nvars(S))+"),dp(1))", "no_minpoly"); |
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[453bc66] | 352 | def J=fetch(S,J); |
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| 353 | def C=fetch(S,C); |
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| 354 | //--- we need to eliminate t from J,y(1)-t*C[1],...y(m)-t*C[m] |
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| 355 | ideal elId=J; |
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| 356 | for(i=1;i<=ncols(C);i++) |
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| 357 | { |
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| 358 | elId=elId,y(i)-t*C[i]; |
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| 359 | } |
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| 360 | elId=eliminate(elId,t); // ideal describing the transform of J |
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| 361 | setring R; // get rid of t |
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| 362 | def elId=fetch(Rt,elId); |
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| 363 | def E=fetch(S,C); // determine exceptional divisor |
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| 364 | E=E+elId; |
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| 365 | def laM0=fetch(S,laM); // the blowup map |
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| 366 | //----------------------------------------------------------------------------- |
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| 367 | // The result is now represented in an A^n \times P^{m-1}, hence |
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| 368 | // involving n+m variables. For further computations we would like |
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| 369 | // to pass to charts to keep the total number of variables as low as |
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| 370 | // possible |
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| 371 | //---------------------------------------------------------------------------- |
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| 372 | list resList; |
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| 373 | ideal Jsub; |
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| 374 | ideal Esub; |
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| 375 | ideal laM; |
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| 376 | ideal testId; |
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| 377 | map phi; |
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| 378 | list templist; |
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| 379 | for(i=1;i<=nvars(R)-nvars(S);i++) |
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| 380 | { |
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| 381 | //--- first pass elId and E on to the i-th chart |
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| 382 | Jsub=std(subst(elId,y(i),1)); |
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| 383 | if(deg(Jsub[1])==0) |
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| 384 | { |
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| 385 | //--- transform does not meet this chart ==> ignore it |
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| 386 | i++; |
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| 387 | continue; |
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| 388 | } |
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| 389 | Esub=std(subst(E,y(i),1)); |
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| 390 | //--- now get rid of unnecessary variables |
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| 391 | //--- first by appropriate coordinate changes |
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| 392 | templist=elimpart(Jsub); |
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| 393 | if(size(templist[2])>0) |
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| 394 | { |
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| 395 | phi=R,templist[5]; |
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| 396 | Jsub=phi(Jsub); |
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| 397 | Jsub=simplify(interred(Jsub),2); |
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| 398 | Esub=phi(Esub); |
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| 399 | Esub=simplify(interred(Esub),2); |
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| 400 | laM=phi(laM0); |
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| 401 | } |
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| 402 | else |
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| 403 | { |
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| 404 | laM=laM0; |
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| 405 | } |
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| 406 | //--- then by dropping non-occuring variables |
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| 407 | testId=Jsub,Esub,laM; |
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[f999689] | 408 | templist=findvars(testId); |
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[453bc66] | 409 | if(size(templist[1])<nvars(R)) |
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| 410 | { |
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| 411 | for(j=1;j<=size(templist[4]);j++) |
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| 412 | { |
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| 413 | laM=subst(laM,var(templist[4][j]),0); |
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| 414 | } |
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[62de185] | 415 | ring Rnew = create_ring(ring_list(basering)[1], "("+string(templist[1])+")", "dp", "no_minpoly"); |
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[453bc66] | 416 | ideal Jnew=imap(R,Jsub); |
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| 417 | ideal eD=imap(R,Esub); |
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| 418 | ideal bM=imap(R,laM); |
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| 419 | } |
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| 420 | else |
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| 421 | { |
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| 422 | def Rnew=basering; |
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| 423 | ideal Jnew=Jsub; |
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| 424 | ideal eD=Esub; |
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| 425 | ideal bM=laM; |
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| 426 | } |
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| 427 | //--- export the relevant data of this ring and add the ring to the list |
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| 428 | export Jnew; |
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| 429 | export eD; |
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| 430 | export bM; |
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| 431 | resList[size(resList)+1]=Rnew; |
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| 432 | setring R; |
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| 433 | kill Rnew; |
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| 434 | } |
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| 435 | setring S; |
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| 436 | return(resList); |
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| 437 | } |
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| 438 | example |
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| 439 | { "EXAMPLE:"; |
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| 440 | echo = 2; |
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| 441 | ring r=0,(x,y,z),dp; |
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| 442 | ideal I=z2-x^3*y^2; |
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| 443 | ideal C=z,xy; |
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| 444 | list li=blowUp2(I,C); |
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| 445 | size(li); // number of charts |
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| 446 | def S1=li[1]; |
---|
| 447 | setring S1; // chart 1 |
---|
| 448 | basering; |
---|
| 449 | Jnew; |
---|
| 450 | eD; |
---|
| 451 | bM; |
---|
| 452 | def S2=li[2]; |
---|
| 453 | setring S2; // chart 2 |
---|
| 454 | basering; |
---|
| 455 | Jnew; |
---|
| 456 | eD; |
---|
| 457 | bM; |
---|
| 458 | } |
---|
| 459 | /////////////////////////////////////////////////////////////////////////////// |
---|
| 460 | |
---|
[2e6eac2] | 461 | proc Center(ideal J,list #) |
---|
| 462 | "USAGE: Center(J[,W][,E]) |
---|
| 463 | @* J,W = ideals |
---|
| 464 | @* E = list |
---|
| 465 | ASSUME: J = ideal containing W ( W = 0 if not specified) |
---|
| 466 | @* E = list of smooth hypersurfaces (e.g. exceptional divisors) |
---|
| 467 | COMPUTE: the center of the blow-up of J for the resolution algorithm |
---|
| 468 | of [Bravo,Encinas,Villamayor] |
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[453bc66] | 469 | RETURN: ideal, describing the center |
---|
[2e6eac2] | 470 | EXAMPLE: example Center; shows an example |
---|
| 471 | " |
---|
| 472 | { |
---|
| 473 | ideal W; |
---|
| 474 | list E; |
---|
| 475 | ideal abb=maxideal(1); |
---|
| 476 | intvec v; |
---|
| 477 | intvec bvec; |
---|
| 478 | intvec w=-1; |
---|
| 479 | matrix intE; |
---|
| 480 | if(size(#)>0) |
---|
| 481 | { |
---|
| 482 | if(typeof(#[1])=="ideal") |
---|
| 483 | { |
---|
| 484 | W=#[1]; |
---|
| 485 | } |
---|
| 486 | if(typeof(#[1])=="list") |
---|
| 487 | { |
---|
| 488 | E=#[1]; |
---|
| 489 | } |
---|
| 490 | if(size(#)>1) |
---|
| 491 | { |
---|
| 492 | if((typeof(#[2])=="list") && (size(E)==0)) |
---|
| 493 | { |
---|
| 494 | E=#[2]; |
---|
| 495 | } |
---|
| 496 | if((typeof(#[2])=="ideal") && (size(W)==0)) |
---|
| 497 | { |
---|
| 498 | W=#[2]; |
---|
| 499 | } |
---|
| 500 | if(size(#)==3){bvec=#[3];} |
---|
| 501 | } |
---|
| 502 | } |
---|
| 503 | |
---|
| 504 | list BO=W,J,bvec,E,abb,v,w,intE,intvec(0); |
---|
| 505 | if(defined(invSat)){kill invSat;} |
---|
| 506 | list invSat=ideal(0),intvec(0); |
---|
| 507 | export(invSat); |
---|
| 508 | |
---|
| 509 | list re=CenterBO(BO); |
---|
| 510 | ideal cent=re[1]; |
---|
| 511 | return(cent); |
---|
| 512 | } |
---|
| 513 | example |
---|
| 514 | { "EXAMPLE:"; |
---|
| 515 | echo = 2; |
---|
| 516 | ring R=0,(x,y),dp; |
---|
| 517 | ideal J=x2-y3; |
---|
| 518 | Center(J); |
---|
| 519 | } |
---|
| 520 | /////////////////////////////////////////////////////////////////////////////// |
---|
| 521 | |
---|
| 522 | proc blowUpBO(list BO, ideal C,int e) |
---|
| 523 | "USAGE: blowUpBO (BO,C,e); |
---|
| 524 | @* BO = basic object, a list: ideal W, |
---|
| 525 | @* ideal J, |
---|
| 526 | @* intvec b, |
---|
| 527 | @* list Ex, |
---|
| 528 | @* ideal ab, |
---|
| 529 | @* intvec v, |
---|
| 530 | @* intvec w, |
---|
| 531 | @* matrix M |
---|
| 532 | @* C = ideal |
---|
| 533 | @* e = integer (0 usual blowing up, 1 deleting extra charts, 2 deleting |
---|
| 534 | @* no charts ) |
---|
| 535 | ASSUME: R = basering, a polynomial ring, W an ideal of R, |
---|
| 536 | @* J = ideal containing W, |
---|
| 537 | @* C = ideal containing J |
---|
[4bde6b] | 538 | COMPUTE: the blowing up of BO[1] in C, the exceptional locus, the strict |
---|
[2e6eac2] | 539 | transform of BO[2] |
---|
| 540 | NOTE: blowUpBO may be applied to basic objects in the sense of |
---|
| 541 | @* [Bravo, Encinas, Villamayor] in the following referred to as BO and |
---|
| 542 | @* to presentations in the sense of [Bierstone, Milman] in the following |
---|
| 543 | @* referred to as BM. |
---|
| 544 | RETURN: a list l of length at most size(C), |
---|
| 545 | l[i] is a ring containing an object BO resp. BM: |
---|
[453bc66] | 546 | @* BO[1]=BM[1] an ideal, say Wi, defining the ambient space of the |
---|
[2e6eac2] | 547 | i-th chart of the blowing up |
---|
[453bc66] | 548 | @* BO[2]=BM[2] an ideal defining the strict transform |
---|
| 549 | @* BO[3] intvec, the first integer b such that in the original object |
---|
[2e6eac2] | 550 | (Delta^b(BO[2]))==1 |
---|
| 551 | the subsequent integers have the same property for Coeff-Objects |
---|
| 552 | of BO[2] used when determining the center |
---|
[453bc66] | 553 | @* BM[3] intvec, BM[3][i] is the assigned multiplicity of BM[2][i] |
---|
| 554 | @* BO[4]=BM[4] the list of exceptional divisors |
---|
| 555 | @* BO[5]=BM[5] an ideal defining the map K[W] ----> K[Wi] |
---|
| 556 | @* BO[6]=BM[6] an intvec BO[6][j]=1 indicates that <BO[4][j],BO[2]>=1, |
---|
[2e6eac2] | 557 | i.e. the strict transform does not meet the j-th exceptional |
---|
| 558 | divisor |
---|
[453bc66] | 559 | @* BO[7] intvec, |
---|
[2e6eac2] | 560 | the index of the first blown-up object in the resolution process |
---|
| 561 | leading to this object for which the value of b was BO[3] |
---|
| 562 | the subsequent ones are the indices for the Coeff-Objects |
---|
| 563 | of BO[2] used when determining the center |
---|
[453bc66] | 564 | @* BM[7] intvec, BM[7][i] is the index at which the (2i-1)st entry |
---|
[2e6eac2] | 565 | of the invariant first reached its current maximal value |
---|
[453bc66] | 566 | @* BO[8]=BM[8] a matrix indicating that BO[4][i] meets BO[4][j] by |
---|
[2e6eac2] | 567 | BO[8][i,j]=1 for i < j |
---|
[453bc66] | 568 | @* BO[9] empty |
---|
| 569 | @* BM[9] the invariant |
---|
[2e6eac2] | 570 | |
---|
| 571 | EXAMPLE: example blowUpBO; shows an example |
---|
| 572 | " |
---|
| 573 | { |
---|
| 574 | //--------------------------------------------------------------------------- |
---|
| 575 | // Initialization and sanity checks |
---|
| 576 | //--------------------------------------------------------------------------- |
---|
| 577 | def R0=basering; |
---|
| 578 | if(!defined(locaT)){int locaT;} |
---|
| 579 | if(locaT){poly pp=@p;} |
---|
| 580 | intvec v; |
---|
| 581 | int shortC=defined(shortcut); |
---|
| 582 | int invS=defined(invSat); |
---|
[13814f0] | 583 | int eq,hy,ii; |
---|
[2e6eac2] | 584 | int extra,noDel,keepDiv; |
---|
| 585 | if(e==1){extra=1;} |
---|
| 586 | //---keeps all charts |
---|
| 587 | if(e==2){noDel=1;} |
---|
| 588 | //---this is only for curves and surfaces |
---|
[4bde6b] | 589 | //---keeps all charts with relevant information on the exceptional divisors |
---|
[2e6eac2] | 590 | if(e==3){keepDiv=1;} |
---|
| 591 | if( typeof(attrib(BO[2],"isEqui"))=="int" ) |
---|
| 592 | { |
---|
| 593 | eq=attrib(BO[2],"isEqui"); |
---|
| 594 | } |
---|
| 595 | if( typeof(attrib(BO[2],"isHy"))=="int" ) |
---|
| 596 | { |
---|
| 597 | hy=attrib(BO[2],"isHy"); |
---|
| 598 | } |
---|
| 599 | string newvar; |
---|
| 600 | int n=nvars(R0); |
---|
| 601 | int i,j,l,m,x,jj,ll,haveCenters,co; |
---|
| 602 | //---the center should neither be the whole space nor empty |
---|
| 603 | if((size(C)==0)||(deg(C[1])==0)) |
---|
| 604 | { |
---|
| 605 | list result=R0; |
---|
| 606 | return(result); |
---|
| 607 | } |
---|
| 608 | if(!defined(debugBlowUp)) |
---|
| 609 | { |
---|
| 610 | int debugBlowUp=0; |
---|
| 611 | } |
---|
| 612 | //--------------------------------------------------------------------------- |
---|
| 613 | // Drop unnecessary variables |
---|
| 614 | //--------------------------------------------------------------------------- |
---|
| 615 | //---step 1: substitution |
---|
| 616 | if(!((keepDiv)||(noDel))) |
---|
| 617 | { |
---|
| 618 | //!!! in case keepDiv and noDel: |
---|
| 619 | //!!! maybe simplify the situation by an appropriate coordinate change |
---|
| 620 | //!!! of this kind -- without dropping variables? |
---|
| 621 | list L=elimpart(BO[1]); |
---|
| 622 | if(size(L[2])!=0) |
---|
| 623 | { |
---|
| 624 | map psi=R0,L[5]; |
---|
| 625 | C=psi(C); |
---|
| 626 | BO=psi(BO); |
---|
| 627 | } |
---|
| 628 | |
---|
| 629 | if(size(BO[1])==0) |
---|
| 630 | { |
---|
| 631 | ideal LL; |
---|
| 632 | for(j=1;j<=size(BO[4]);j++) |
---|
| 633 | { |
---|
| 634 | LL=LL,BO[4][j]; |
---|
| 635 | } |
---|
[f999689] | 636 | LL=variables(LL); |
---|
[2e6eac2] | 637 | L=elimpart(BO[2]); |
---|
| 638 | if((size(L[2])!=0)&&(size(std(LL+L[2]))==size(L[2])+size(LL))) |
---|
| 639 | { |
---|
| 640 | map chi=R0,L[5]; |
---|
| 641 | C=chi(C); |
---|
| 642 | BO=chi(BO); |
---|
| 643 | } |
---|
| 644 | } |
---|
| 645 | //---step 2: dropping non-occurring variables |
---|
| 646 | int s=size(C); |
---|
| 647 | ideal K=BO[1],BO[2],C; |
---|
| 648 | for(j=1;j<=size(BO[4]);j++) |
---|
| 649 | { |
---|
| 650 | K=K,BO[4][j]; |
---|
| 651 | } |
---|
[f999689] | 652 | list N=findvars(K); |
---|
[2e6eac2] | 653 | if(size(N[1])<n) |
---|
| 654 | { |
---|
| 655 | newvar=string(N[1]); |
---|
| 656 | v=N[4]; |
---|
| 657 | for(j=1;j<=size(v);j++){BO[5]=subst(BO[5],var(v[j]),0);} |
---|
[62de185] | 658 | ring R1 = create_ring(ring_list(R0)[1], "("+newvar+")", "dp", "no_minpoly"); |
---|
[2e6eac2] | 659 | list BO=imap(R0,BO); |
---|
| 660 | ideal C=imap(R0,C); |
---|
| 661 | n=nvars(R1); |
---|
| 662 | } |
---|
| 663 | else |
---|
| 664 | { |
---|
| 665 | def R1=basering; |
---|
| 666 | } |
---|
| 667 | } |
---|
| 668 | else |
---|
| 669 | { |
---|
| 670 | int s=size(C); |
---|
| 671 | def R1=basering; |
---|
| 672 | } |
---|
| 673 | if(debugBlowUp) |
---|
| 674 | { |
---|
| 675 | "---> In BlowUp: After dropping unnecessary variables"; |
---|
| 676 | "BO:"; |
---|
| 677 | BO; |
---|
| 678 | "C:"; |
---|
| 679 | C; |
---|
| 680 | } |
---|
| 681 | //--------------------------------------------------------------------------- |
---|
| 682 | // Do the actual blow-up |
---|
| 683 | //--------------------------------------------------------------------------- |
---|
| 684 | //--- control the names of the variables |
---|
[309b8a] | 685 | list l2; |
---|
[f7a540] | 686 | for (ii = 1; ii <= n; ii++) |
---|
[309b8a] | 687 | { |
---|
| 688 | l2[ii] = "x("+string(ii)+")"; |
---|
| 689 | } |
---|
[62de185] | 690 | ring R = create_ring(ring_list(R0)[1], l2, "dp", "no_minpoly"); |
---|
[c3e2eaa] | 691 | ideal R_maxid=maxideal(1); |
---|
[2e6eac2] | 692 | list BO=fetch(R1,BO); |
---|
| 693 | ideal C=fetch(R1,C); |
---|
| 694 | list Cmstd=mstd(C); |
---|
| 695 | C=Cmstd[2]; |
---|
| 696 | if(size(Cmstd[1])<=size(Cmstd[2])) |
---|
| 697 | { |
---|
| 698 | C=Cmstd[1]; |
---|
| 699 | } |
---|
| 700 | else |
---|
| 701 | { |
---|
| 702 | C=interred(C); |
---|
| 703 | } |
---|
| 704 | list result; |
---|
| 705 | //--- the blow-up process |
---|
| 706 | ideal W =BO[1]; |
---|
| 707 | ideal J =BO[2]; |
---|
| 708 | intvec bvec =BO[3]; |
---|
| 709 | list Ex =BO[4]; |
---|
| 710 | ideal abb=BO[5]; |
---|
| 711 | intvec wvec=BO[7]; |
---|
| 712 | ideal laM=maxideal(1); |
---|
| 713 | if((typeof(BO[9])=="intmat")||(typeof(BO[9])=="intvec")) |
---|
| 714 | { |
---|
| 715 | def @invmat=BO[9]; |
---|
| 716 | } |
---|
| 717 | if(size(BO)>9) |
---|
| 718 | { |
---|
| 719 | //--- check whether a previous center had been split into connected components |
---|
| 720 | if(size(BO[10])>0) |
---|
| 721 | { |
---|
| 722 | list knownCenters=BO[10]; |
---|
| 723 | haveCenters=1; |
---|
| 724 | } |
---|
| 725 | } |
---|
| 726 | |
---|
| 727 | matrix intE=BO[8]; |
---|
| 728 | Ex[size(Ex)+1]=var(1); |
---|
| 729 | //to have the list depending on R in case BO[4] is empty |
---|
| 730 | |
---|
[13814f0] | 731 | list l3 = ringlist(R)[2]; |
---|
[f7a540] | 732 | for (ii = 0; ii <= s-1; ii++) |
---|
[309b8a] | 733 | { |
---|
[13814f0] | 734 | l3[size(l3)+1] = "y("+string(ii)+")"; |
---|
[309b8a] | 735 | } |
---|
[e78236] | 736 | ring S = create_ring(ring_list(R)[1], l3, "dp", "no_minpoly"); |
---|
[2e6eac2] | 737 | list resu; |
---|
| 738 | list B; |
---|
[13814f0] | 739 | list l4 = ringlist(R)[2]; |
---|
[f7a540] | 740 | for (ii = 0; ii <= s-1; ii++) |
---|
[309b8a] | 741 | { |
---|
[13814f0] | 742 | l4[size(l4)+1] = "y("+string(ii)+")"; |
---|
[309b8a] | 743 | } |
---|
[13814f0] | 744 | l4[size(l4)+1] = "t"; |
---|
[62de185] | 745 | ring T = create_ring(ring_list(R)[1], l4, "dp", "no_minpoly"); |
---|
[2e6eac2] | 746 | ideal C=imap(R,C); |
---|
| 747 | ideal W=imap(R,W); |
---|
[c3e2eaa] | 748 | ideal R_im=imap(R,R_maxid); |
---|
| 749 | R_im=R_im,t*C; |
---|
| 750 | map phi=S,R_im; |
---|
[2e6eac2] | 751 | setring S; |
---|
| 752 | |
---|
| 753 | //--- the ideal describing the blow-up map |
---|
| 754 | ideal abb=imap(R,abb); |
---|
| 755 | ideal laM0=imap(R,laM); |
---|
| 756 | //--- the ideal of the blowing up of the ambient space |
---|
| 757 | ideal W=preimage(T,phi,W); |
---|
| 758 | //--- the ideal of the exceptional locus |
---|
| 759 | ideal E=imap(R,C); |
---|
| 760 | |
---|
| 761 | list E1=imap(R,Ex); |
---|
| 762 | E1[size(E1)]=E; |
---|
| 763 | ideal J=imap(R,J)+W; |
---|
| 764 | |
---|
| 765 | if(haveCenters){list kN=imap(R,knownCenters);} |
---|
| 766 | |
---|
| 767 | //--- the strict transform of the exceptional divisors |
---|
[3f7e01a] | 768 | for(j=1;j<size(E1);j++){E1[j]=sat(E1[j]+W,E);} |
---|
[2e6eac2] | 769 | //--- the intersection matrix of the exceptional divisors |
---|
| 770 | matrix intEold=imap(R,intE); |
---|
| 771 | matrix intE[size(E1)][size(E1)]; |
---|
| 772 | ideal U,Jsub,sLstd; |
---|
| 773 | |
---|
| 774 | for(j=1;j<size(E1);j++) |
---|
| 775 | { |
---|
| 776 | for(l=j+1;l<=size(E1);l++) |
---|
| 777 | { |
---|
| 778 | if(deg(E1[j][1])==0) |
---|
| 779 | { |
---|
| 780 | if(l<size(E1)){intE[j,l]=intEold[j,l];} |
---|
| 781 | else {intE[j,l]=0;} |
---|
| 782 | } |
---|
| 783 | else |
---|
| 784 | { |
---|
| 785 | if(deg(E1[l][1])==0) |
---|
| 786 | { |
---|
| 787 | if(l<size(E1)){intE[j,l]=intEold[j,l];} |
---|
| 788 | } |
---|
| 789 | else |
---|
| 790 | { |
---|
| 791 | U= std(E1[l]+E1[j]); |
---|
| 792 | if(dim(U)>0){intE[j,l]=1;} |
---|
| 793 | else {intE[j,l]=0;} |
---|
| 794 | } |
---|
| 795 | } |
---|
| 796 | } |
---|
| 797 | } |
---|
| 798 | if(debugBlowUp) |
---|
| 799 | { |
---|
| 800 | "----> In BlowUp: After Blowing-up, before Clean-Up"; |
---|
| 801 | "W:"; |
---|
| 802 | W; |
---|
| 803 | "J:"; |
---|
| 804 | J; |
---|
| 805 | } |
---|
| 806 | //---------------------------------------------------------------------------- |
---|
| 807 | // generating and cleaning up the different charts |
---|
| 808 | //---------------------------------------------------------------------------- |
---|
| 809 | list M; |
---|
| 810 | map psi; |
---|
| 811 | list E2; |
---|
| 812 | ideal K,JJ,laM,LL,MM; |
---|
| 813 | n=nvars(S); |
---|
| 814 | list N; |
---|
| 815 | list Bstd; |
---|
| 816 | intvec delCharts,extraCharts; |
---|
| 817 | delCharts[s]=0; |
---|
| 818 | extraCharts[s]=0; |
---|
| 819 | ideal MA=y(0..s-1); |
---|
| 820 | list ZRes,ZlaM,ZsLstd; |
---|
| 821 | for(i=0;i<=s-1;i++) |
---|
| 822 | { |
---|
| 823 | if(haveCenters) |
---|
| 824 | { |
---|
| 825 | B[10]=kN; |
---|
| 826 | for(j=1;j<=size(kN);j++) |
---|
| 827 | { |
---|
| 828 | B[10][j][1]=subst(B[10][j][1],y(i),1); |
---|
| 829 | } |
---|
| 830 | } |
---|
| 831 | B[8]=intE; |
---|
| 832 | B[1]=std(subst(W,y(i),1)); |
---|
| 833 | if(deg(B[1][1])==0) |
---|
| 834 | { |
---|
| 835 | //--- subsets of the empty set are not really interesting! |
---|
| 836 | delCharts[i+1]=1; |
---|
| 837 | ZRes[i+1]=B; |
---|
| 838 | ZlaM[i+1]=laM; |
---|
| 839 | i++; |
---|
| 840 | continue; |
---|
| 841 | } |
---|
| 842 | Jsub=subst(J,y(i),1); |
---|
| 843 | attrib(Jsub,"isEqui",eq); |
---|
| 844 | attrib(Jsub,"isHy",hy); |
---|
| 845 | B[2]=Jsub; |
---|
| 846 | B[3]=bvec; |
---|
| 847 | for(j=1;j<size(E1);j++){E2[j]=subst(E1[j],y(i),1);} |
---|
| 848 | E2[size(E1)]=E+B[1]; |
---|
| 849 | B[4]=E2; |
---|
| 850 | M=elimpart(B[1]); |
---|
| 851 | B[5]=abb; |
---|
| 852 | laM=laM0; |
---|
| 853 | psi=S,maxideal(1); |
---|
| 854 | if(size(M[2])!=0) |
---|
| 855 | { |
---|
| 856 | psi=S,M[5]; |
---|
| 857 | B=psi(B); |
---|
| 858 | laM=psi(laM); |
---|
| 859 | } |
---|
| 860 | Jsub=B[2]; |
---|
[3f7e01a] | 861 | B[2]=sat(Jsub,std(psi(E))); |
---|
[2e6eac2] | 862 | if(!defined(MAtmp)){ideal MAtmp=MA;} |
---|
| 863 | MAtmp[i+1]=0; |
---|
| 864 | JJ=std(B[2]+MAtmp); |
---|
| 865 | if(deg(JJ[1])==0) |
---|
| 866 | { |
---|
| 867 | delCharts[i+1]=1; |
---|
[4bde6b] | 868 | //--- the i-th chart will be marked for deleting because all information |
---|
[2e6eac2] | 869 | //--- are already contained in the union of the remaining charts |
---|
| 870 | } |
---|
| 871 | else |
---|
| 872 | { |
---|
| 873 | if((eq)&&(dim(JJ)<dim(std(B[2])))) |
---|
| 874 | { |
---|
| 875 | extraCharts[i+1]=1; |
---|
| 876 | //--- compute the singular locus |
---|
| 877 | if((B[3][1]<=1)&&(hy)) |
---|
| 878 | //--- B[2] is a smooth hypersurface |
---|
| 879 | { |
---|
| 880 | sLstd=ideal(1); |
---|
| 881 | } |
---|
| 882 | else |
---|
| 883 | { |
---|
| 884 | Bstd=mstd(B[2]); |
---|
| 885 | if(n-dim(Bstd[1])>4) |
---|
| 886 | { |
---|
| 887 | //--- in this case the singular locus is too complicated |
---|
| 888 | sLstd=ideal(0); |
---|
| 889 | } |
---|
| 890 | JJ=Bstd[2]; |
---|
| 891 | attrib(JJ,"isEqui",eq); |
---|
| 892 | B[2]=JJ; |
---|
| 893 | sLstd=slocusE(B[2]); |
---|
| 894 | } |
---|
| 895 | m=0; |
---|
| 896 | if(deg(std(sLstd+MAtmp)[1])==0) |
---|
| 897 | { |
---|
| 898 | //--- the singular locus of B[2] is in the union of the remaining charts |
---|
| 899 | m=1; |
---|
| 900 | for(l=1;l<=size(B[4]);l++) |
---|
| 901 | { |
---|
| 902 | if(deg(std(B[2]+B[4][l]+MAtmp)[1])!=0) |
---|
| 903 | { |
---|
| 904 | //--- the exceptional divisor meets B[2] at the locus of MAtmp |
---|
| 905 | //--- we continue only if the option extra=1 and we have transversal |
---|
| 906 | //--- intersection |
---|
| 907 | m=0; |
---|
| 908 | break; |
---|
| 909 | } |
---|
| 910 | } |
---|
| 911 | } |
---|
| 912 | if(m) |
---|
| 913 | { |
---|
[4bde6b] | 914 | //--- the i-th chart will be marked for deleting because all information |
---|
[2e6eac2] | 915 | //--- are already contained in the union of the remaining charts |
---|
| 916 | delCharts[i+1]=1; |
---|
| 917 | } |
---|
| 918 | } |
---|
| 919 | } |
---|
| 920 | if(delCharts[i+1]==0) |
---|
| 921 | { |
---|
| 922 | MAtmp[i+1]=MA[i+1]; |
---|
| 923 | ZsLstd[i+1]=sLstd; |
---|
| 924 | } |
---|
| 925 | ZRes[i+1]=B; |
---|
| 926 | ZlaM[i+1]=laM; |
---|
| 927 | } |
---|
| 928 | //--------------------------------------------------------------------------- |
---|
| 929 | // extra = ignore uninteresting charts even if there is a normal |
---|
| 930 | // crosssing intersection in it |
---|
| 931 | //--------------------------------------------------------------------------- |
---|
| 932 | if(extra) |
---|
| 933 | { |
---|
| 934 | for(i=0;i<=s-1;i++) |
---|
| 935 | { |
---|
| 936 | if((delCharts[i+1]==0)&&(extraCharts[i+1])) |
---|
| 937 | { |
---|
| 938 | MAtmp[i+1]=0; |
---|
| 939 | B=ZRes[i+1]; |
---|
| 940 | sLstd=ZsLstd[i+1]; |
---|
| 941 | m=0; |
---|
| 942 | if(deg(std(sLstd+MAtmp)[1])==0) |
---|
| 943 | { |
---|
| 944 | //--- the singular locus of B[2] is in the union of the remaining charts |
---|
| 945 | m=1; |
---|
| 946 | for(l=1;l<=size(B[4]);l++) |
---|
| 947 | { |
---|
| 948 | if(deg(std(B[2]+B[4][l]+MAtmp)[1])!=0) |
---|
| 949 | { |
---|
| 950 | //--- the exceptional divisor meets B[2] at the locus of MAtmp |
---|
| 951 | //--- we continue only if the option extra=1 and we have transversal |
---|
| 952 | //--- intersection |
---|
| 953 | m=2; |
---|
| 954 | if(!transversalTB(B[2],list(B[4][l]),MAtmp)) |
---|
| 955 | { |
---|
| 956 | m=0;break; |
---|
| 957 | } |
---|
| 958 | } |
---|
| 959 | } |
---|
| 960 | } |
---|
| 961 | if(m) |
---|
| 962 | { |
---|
| 963 | if(m==1) |
---|
| 964 | { |
---|
[4bde6b] | 965 | //--- the i-th chart will be marked for deleting because all information |
---|
[2e6eac2] | 966 | //--- are already contained in the union of the remaining charts |
---|
| 967 | delCharts[i+1]=1; |
---|
| 968 | } |
---|
| 969 | else |
---|
| 970 | { |
---|
| 971 | //--- the option extra=1 and we have transversal intersection |
---|
| 972 | //--- we delete the chart in case of normal crossings |
---|
| 973 | if(normalCrossB(B[2],B[4],MAtmp)) |
---|
| 974 | { |
---|
| 975 | //--- in case of the option extra |
---|
[4bde6b] | 976 | //--- the i-th chart will be marked for deleting because all information |
---|
[2e6eac2] | 977 | //--- are already contained in the union of the remaining charts |
---|
| 978 | |
---|
| 979 | delCharts[i+1]=1; |
---|
| 980 | } |
---|
| 981 | } |
---|
| 982 | } |
---|
| 983 | if(delCharts[i+1]==0) |
---|
| 984 | { |
---|
| 985 | MAtmp[i+1]=MA[i+1]; |
---|
| 986 | } |
---|
| 987 | } |
---|
| 988 | } |
---|
| 989 | for(i=0;i<=s-1;i++) |
---|
| 990 | { |
---|
| 991 | if(!delCharts[i+1]){break;} |
---|
| 992 | } |
---|
| 993 | if(i==s){delCharts[s]=0;} |
---|
| 994 | } |
---|
| 995 | for(i=0;i<=s-1;i++) |
---|
| 996 | { |
---|
| 997 | B=ZRes[i+1]; |
---|
| 998 | laM=ZlaM[i+1]; |
---|
| 999 | if(noDel){delCharts[i+1]=0;} |
---|
| 1000 | //---keeps chart if the exceptional divisor is not in any other chart |
---|
| 1001 | if((delCharts[i+1])&&(keepDiv)) |
---|
| 1002 | { |
---|
| 1003 | for(j=1;j<=size(B[4]);j++) |
---|
| 1004 | { |
---|
| 1005 | if(deg(std(B[4][j])[1])>0) |
---|
| 1006 | { |
---|
| 1007 | x=0; |
---|
| 1008 | for(l=0;l<=s-1;l++) |
---|
| 1009 | { |
---|
| 1010 | if((l!=i)&&(!delCharts[l+1])&&(deg(std(ZRes[l+1][4][j])[1])>0)) |
---|
| 1011 | { |
---|
| 1012 | x=1; |
---|
| 1013 | break; |
---|
| 1014 | } |
---|
| 1015 | } |
---|
| 1016 | if(!x) |
---|
| 1017 | { |
---|
| 1018 | delCharts[i+1]=0; |
---|
| 1019 | //!!!evtl. diese Karten markieren und nicht weiter aufblasen??? |
---|
| 1020 | break; |
---|
| 1021 | } |
---|
| 1022 | } |
---|
| 1023 | } |
---|
| 1024 | } |
---|
| 1025 | //---keeps charts if the intersection of 2 divisors is not in any other chart |
---|
| 1026 | if((delCharts[i+1])&&(keepDiv)) |
---|
| 1027 | { |
---|
| 1028 | for(j=1;j<=size(B[4])-1;j++) |
---|
| 1029 | { |
---|
| 1030 | for(l=j+1;l<=size(B[4]);l++) |
---|
| 1031 | { |
---|
| 1032 | if(deg(std(B[4][j]+B[4][l])[1])>0) |
---|
| 1033 | { |
---|
| 1034 | x=0; |
---|
| 1035 | for(ll=0;ll<=s-1;ll++) |
---|
| 1036 | { |
---|
| 1037 | if((ll!=i)&&(!delCharts[ll+1])) |
---|
| 1038 | { |
---|
| 1039 | if(deg(std(ZRes[ll+1][4][j]+ZRes[ll+1][4][l])[1])>0) |
---|
| 1040 | { |
---|
| 1041 | x=1; |
---|
| 1042 | break; |
---|
| 1043 | } |
---|
| 1044 | } |
---|
| 1045 | } |
---|
| 1046 | if(!x) |
---|
| 1047 | { |
---|
| 1048 | delCharts[i+1]=0; |
---|
| 1049 | break; |
---|
| 1050 | } |
---|
| 1051 | } |
---|
| 1052 | } |
---|
| 1053 | if(!delCharts[i+1]){break;} |
---|
| 1054 | } |
---|
| 1055 | } |
---|
| 1056 | //---keeps charts if the intersection of 3 divisors is not in any other chart |
---|
| 1057 | if((delCharts[i+1])&&(keepDiv)) |
---|
| 1058 | { |
---|
| 1059 | for(j=1;j<=size(B[4])-2;j++) |
---|
| 1060 | { |
---|
| 1061 | for(l=j+1;l<=size(B[4])-1;l++) |
---|
| 1062 | { |
---|
| 1063 | for(ll=l+1;ll<=size(B[4]);ll++) |
---|
| 1064 | { |
---|
| 1065 | if(deg(std(B[4][j]+B[4][l]+B[4][ll])[1])>0) |
---|
| 1066 | { |
---|
| 1067 | x=0; |
---|
| 1068 | for(jj=0;jj<=s-1;jj++) |
---|
| 1069 | { |
---|
| 1070 | if((jj!=i)&&(!delCharts[jj+1])) |
---|
| 1071 | { |
---|
| 1072 | if(deg(std(ZRes[jj+1][4][j] |
---|
| 1073 | +ZRes[jj+1][4][l]+ZRes[jj+1][4][ll])[1])>0) |
---|
| 1074 | { |
---|
| 1075 | x=1; |
---|
| 1076 | break; |
---|
| 1077 | } |
---|
| 1078 | } |
---|
| 1079 | } |
---|
| 1080 | if(!x) |
---|
| 1081 | { |
---|
| 1082 | delCharts[i+1]=0; |
---|
| 1083 | break; |
---|
| 1084 | } |
---|
| 1085 | } |
---|
| 1086 | } |
---|
| 1087 | if(!delCharts[i+1]){break;} |
---|
| 1088 | } |
---|
| 1089 | if(!delCharts[i+1]){break;} |
---|
| 1090 | } |
---|
| 1091 | } |
---|
| 1092 | if(delCharts[i+1]==0) |
---|
| 1093 | { |
---|
| 1094 | //--- try to decrease the number of variables by substitution |
---|
| 1095 | if((!keepDiv)&&(!noDel)) |
---|
| 1096 | { |
---|
| 1097 | list WW=elimpart(B[1]); |
---|
| 1098 | map phiW=basering,WW[5]; |
---|
| 1099 | B=phiW(B); |
---|
| 1100 | laM=phiW(laM); |
---|
| 1101 | kill WW; |
---|
| 1102 | kill phiW; |
---|
| 1103 | if(size(B[1])==0) |
---|
| 1104 | { |
---|
| 1105 | LL=0; |
---|
| 1106 | for(j=1;j<=size(B[4]);j++) |
---|
| 1107 | { |
---|
| 1108 | MM=std(B[4][j]); |
---|
| 1109 | if(deg(MM[1])>0){LL=LL,MM;} |
---|
| 1110 | } |
---|
[f999689] | 1111 | LL=variables(LL); |
---|
[2e6eac2] | 1112 | M=elimpart(B[2]); |
---|
| 1113 | if((size(M[2])!=0)&&(size(std(LL+M[2]))==size(M[2])+size(LL))) |
---|
| 1114 | { |
---|
| 1115 | psi=S,M[5]; |
---|
| 1116 | B=psi(B); |
---|
| 1117 | laM=psi(laM); |
---|
| 1118 | } |
---|
| 1119 | } |
---|
| 1120 | } |
---|
| 1121 | //---- interreduce B[1],B[2] and all B[4][j] |
---|
| 1122 | B[1]=interred(B[1]); |
---|
| 1123 | B[2]=interred(B[2]); |
---|
| 1124 | E2=B[4]; |
---|
| 1125 | for(j=1;j<=size(E2);j++){E2[j]=interred(E2[j]);} |
---|
| 1126 | B[4]=E2; |
---|
| 1127 | v=0;v[size(E2)]=0; |
---|
| 1128 | //--- mark those j for which B[4] does not meet B[2] |
---|
| 1129 | for(j=1;j<=size(E2);j++) |
---|
| 1130 | { |
---|
| 1131 | K=E2[j],B[2]; |
---|
| 1132 | K=std(K); |
---|
| 1133 | if(deg(K[1])==0) |
---|
| 1134 | { |
---|
| 1135 | v[j]=1; |
---|
| 1136 | } |
---|
| 1137 | } |
---|
| 1138 | B[6]=v; |
---|
| 1139 | B[7]=wvec; |
---|
| 1140 | //--- throw away variables which do not occur |
---|
| 1141 | K=B[1],B[2],B[5]; //Aenderung!!! |
---|
| 1142 | for(j=1;j<=size(B[4]);j++){K=K,B[4][j];} |
---|
[f999689] | 1143 | N=findvars(K); |
---|
[2e6eac2] | 1144 | if(size(N[1])<n) |
---|
| 1145 | { |
---|
| 1146 | newvar=string(N[1]); |
---|
| 1147 | v=N[4]; |
---|
| 1148 | for(j=1;j<=size(v);j++) |
---|
| 1149 | { |
---|
| 1150 | B[5]=subst(B[5],var(v[j]),0); |
---|
| 1151 | laM=subst(laM,var(v[j]),0); |
---|
| 1152 | } |
---|
[62de185] | 1153 | ring R2 = create_ring(ring_list(S)[1], "("+newvar+")", "dp", "no_minpoly"); |
---|
[2e6eac2] | 1154 | list BO=imap(S,B); |
---|
| 1155 | ideal laM=imap(S,laM); |
---|
| 1156 | } |
---|
| 1157 | else |
---|
| 1158 | { |
---|
| 1159 | def R2=basering; |
---|
| 1160 | list BO=B; |
---|
| 1161 | } |
---|
| 1162 | ideal JJ=BO[2]; |
---|
| 1163 | attrib(JJ,"isEqui",eq); |
---|
| 1164 | attrib(JJ,"isHy",hy); |
---|
| 1165 | BO[2]=JJ; |
---|
| 1166 | //--- strict transforms of the known centers |
---|
| 1167 | if(haveCenters) |
---|
| 1168 | { |
---|
| 1169 | ideal tt; |
---|
| 1170 | list tList; |
---|
| 1171 | for(j=1;j<=size(BO[10]);j++) |
---|
| 1172 | { |
---|
| 1173 | tt=std(BO[10][j][1]); |
---|
| 1174 | if(deg(tt[1])>0) |
---|
| 1175 | { |
---|
[3f7e01a] | 1176 | tt=sat(tt,BO[4][size(BO[4])]); |
---|
[2e6eac2] | 1177 | } |
---|
| 1178 | if((deg(tt[1])>0)&&(deg(std(tt+BO[2]+BO[1])[1])>0)) |
---|
| 1179 | { |
---|
| 1180 | tList[size(tList)+1]= |
---|
| 1181 | list(tt,BO[10][j][2],BO[10][j][3],BO[10][j][4]); |
---|
| 1182 | } |
---|
| 1183 | } |
---|
| 1184 | BO[10]=tList; |
---|
| 1185 | kill tList; |
---|
| 1186 | } |
---|
| 1187 | //--- marking variables which do not occur in BO[1] and BO[2] |
---|
| 1188 | //--- and occur in exactly one BO[4][j], which is the hyperplane given by |
---|
| 1189 | //--- this variable |
---|
| 1190 | //!!!! not necessarily in exactly one BO[4][j] |
---|
[f999689] | 1191 | list N=findvars(BO[1]+BO[2]); |
---|
[2e6eac2] | 1192 | if(size(N[1])<nvars(basering)) |
---|
| 1193 | { |
---|
| 1194 | v=N[4]; |
---|
| 1195 | if(defined(H)){kill H;} |
---|
| 1196 | if(defined(EE)){kill EE;} |
---|
| 1197 | if(defined(vv)){kill vv;} |
---|
| 1198 | list EE; |
---|
| 1199 | intvec vv; |
---|
| 1200 | ideal H=maxideal(1); |
---|
| 1201 | for(j=1;j<=size(v);j++) |
---|
| 1202 | { |
---|
| 1203 | H[v[j]]=0; |
---|
| 1204 | } |
---|
| 1205 | H=std(H); |
---|
| 1206 | for(l=1;l<=size(BO[4]);l++) |
---|
| 1207 | { |
---|
| 1208 | if(BO[6][l]==1){l++;continue;} |
---|
[867e1a3] | 1209 | if(size(ideal(matrix(reduce(BO[4][l],H))-BO[4][l]))==0) |
---|
[2e6eac2] | 1210 | { |
---|
| 1211 | //!!! need further cleanup: |
---|
| 1212 | //!!! this part of the code is no longer used since it did not glue properly |
---|
| 1213 | // BO[6][l]=2; |
---|
| 1214 | EE[size(EE)+1]=BO[4][l]; |
---|
| 1215 | vv[size(vv)+1]=l; |
---|
| 1216 | } |
---|
| 1217 | } |
---|
| 1218 | if((size(vv)>dim(std(BO[2])))&&(deg(BO[2][1])>0)) |
---|
| 1219 | { |
---|
| 1220 | list BOtemp3=BO; |
---|
| 1221 | BOtemp3[4]=EE; |
---|
| 1222 | intvec ivtemp3; |
---|
| 1223 | ivtemp3[size(BOtemp3[4])]=0; |
---|
| 1224 | BOtemp3[6]=ivtemp3; |
---|
| 1225 | BOtemp3[7][1]=-1; |
---|
| 1226 | list iEtemp3=inters_E(BOtemp3); |
---|
| 1227 | if(iEtemp3[2]>=dim(std(BOtemp3[2]))) |
---|
| 1228 | { |
---|
| 1229 | for(l=2;l<=size(vv);l++) |
---|
| 1230 | { |
---|
| 1231 | BO[6][vv[l]]=0; |
---|
| 1232 | } |
---|
| 1233 | } |
---|
| 1234 | kill BOtemp3,ivtemp3,iEtemp3; |
---|
| 1235 | } |
---|
| 1236 | } |
---|
| 1237 | list thisChart=ideal(0),i; |
---|
| 1238 | export thisChart; |
---|
| 1239 | |
---|
| 1240 | //---------------------------------------------------------------------------- |
---|
| 1241 | // export the basic object and append the ring to the list of rings |
---|
| 1242 | //---------------------------------------------------------------------------- |
---|
| 1243 | if(debugBlowUp) |
---|
| 1244 | { |
---|
| 1245 | "----> In BlowUp: Adding a single chart"; |
---|
| 1246 | "BO:"; |
---|
| 1247 | BO; |
---|
| 1248 | } |
---|
| 1249 | if(locaT) |
---|
| 1250 | { |
---|
| 1251 | map locaPhi=R0,laM; |
---|
| 1252 | poly @p=locaPhi(pp); |
---|
| 1253 | export(@p); |
---|
| 1254 | } |
---|
| 1255 | ideal lastMap=laM; |
---|
| 1256 | export lastMap; |
---|
| 1257 | if(invS){list invSat=imap(R0,invSat);export invSat;} |
---|
| 1258 | if(defined(@invmat)){BO[9]=@invmat;} |
---|
| 1259 | if(shortC){list shortcut=imap(R0,shortcut);export(shortcut);} |
---|
| 1260 | export BO; |
---|
| 1261 | result[size(result)+1]=R2; |
---|
| 1262 | setring S; |
---|
| 1263 | kill R2; |
---|
| 1264 | } |
---|
| 1265 | } |
---|
| 1266 | setring R0; |
---|
| 1267 | return(result); |
---|
| 1268 | } |
---|
| 1269 | example |
---|
| 1270 | {"EXAMPLE:"; |
---|
| 1271 | echo = 2; |
---|
| 1272 | ring R=0,(x,y),dp; |
---|
| 1273 | |
---|
| 1274 | ideal W; |
---|
| 1275 | ideal J=x2-y3; |
---|
| 1276 | intvec b=1; |
---|
| 1277 | list E; |
---|
| 1278 | ideal abb=maxideal(1); |
---|
| 1279 | intvec v; |
---|
| 1280 | intvec w=-1; |
---|
| 1281 | matrix M; |
---|
| 1282 | intvec ma; |
---|
| 1283 | list BO=W,J,b,E,abb,v,w,M,ma; |
---|
| 1284 | |
---|
| 1285 | ideal C=CenterBO(BO)[1]; |
---|
| 1286 | |
---|
| 1287 | list blow=blowUpBO(BO,C,0); |
---|
| 1288 | def Q=blow[1]; |
---|
| 1289 | setring Q; |
---|
| 1290 | BO; |
---|
| 1291 | } |
---|
| 1292 | ////////////////////////////////////////////////////////////////////////////// |
---|
[268a430] | 1293 | static proc slocusE(ideal i) |
---|
[2e6eac2] | 1294 | "Internal procedure - no help and no example available |
---|
| 1295 | " |
---|
| 1296 | { |
---|
| 1297 | //--- do slocus in equidimensional case directly -- speed up |
---|
| 1298 | if(size(i)==0){return(ideal(1));} |
---|
| 1299 | if( typeof(attrib(i,"isEqui"))=="int" ) |
---|
| 1300 | { |
---|
| 1301 | if(attrib(i,"isEqui")==1) |
---|
| 1302 | { |
---|
| 1303 | ideal j=std(i); |
---|
| 1304 | if(deg(j[1])==0){return(j);} |
---|
| 1305 | int cod = nvars(basering)-dim(j); |
---|
| 1306 | i = i+minor(jacob(i),cod); |
---|
| 1307 | return(i); |
---|
| 1308 | } |
---|
| 1309 | } |
---|
| 1310 | return(slocus(i)); |
---|
| 1311 | } |
---|
| 1312 | /////////////////////////////////////////////////////////////////////////////// |
---|
[268a430] | 1313 | static proc inters_E(list BO) |
---|
[2e6eac2] | 1314 | "USAGE: inters_E(BO); |
---|
| 1315 | @* BO = basic object, a list: ideal W, |
---|
| 1316 | @* ideal J, |
---|
| 1317 | @* intvec b, |
---|
| 1318 | @* list Ex, |
---|
| 1319 | @* ideal ab, |
---|
| 1320 | @* intvec v, |
---|
| 1321 | @* intvec w, |
---|
| 1322 | @* matrix M |
---|
| 1323 | ASSUME: R = basering, a polynomial ring, W an ideal of R, |
---|
| 1324 | @* J = ideal containing W, |
---|
| 1325 | @* BO in the setting of case 2 of [Bravo,Encinas,Villamayor] |
---|
| 1326 | @* BO[4]=E, BO[4][1..count]=E^- |
---|
| 1327 | @* BO[7][1]=count |
---|
| 1328 | COMPUTE: (W,(P,1),E^+) in the notation of [Bravo,Encinas,Villamayor] |
---|
| 1329 | RETURN: a list l , |
---|
| 1330 | l[1]: P = product of ideals I(H_i1)+..+I(H_in) over all |
---|
| 1331 | n-tuples of indices i1..in from 1..count |
---|
| 1332 | l[2]: n = maximal number of H_i from E^- meeting J simultaneously |
---|
| 1333 | l[3]: maximal number of H_i from E meeting J simultaneously |
---|
| 1334 | EXAMPLE: internal procedure - no example available |
---|
| 1335 | " |
---|
| 1336 | { |
---|
| 1337 | //--------------------------------------------------------------------- |
---|
| 1338 | // Initialization |
---|
| 1339 | //--------------------------------------------------------------------- |
---|
| 1340 | int kk,jj,ii,updated,n,count2,kkdiff; |
---|
| 1341 | def rb=basering; |
---|
| 1342 | def W=BO[1]; |
---|
| 1343 | ideal J=BO[1],BO[2]; |
---|
| 1344 | int nonnormal; |
---|
| 1345 | int maxkk=dim(std(J)); |
---|
| 1346 | int dimJ=maxkk; |
---|
| 1347 | ideal test2; |
---|
| 1348 | list merklist1,merklist2; |
---|
| 1349 | if(size(BO[4])==0) |
---|
| 1350 | { |
---|
| 1351 | list retlist=BO[2],n; |
---|
| 1352 | return(retlist); |
---|
| 1353 | } |
---|
| 1354 | def E=BO[4]; |
---|
| 1355 | intvec stoplist=BO[6]; |
---|
| 1356 | //--- fill in all known information about exceptional divisors not meeting |
---|
| 1357 | //--- current chart |
---|
| 1358 | for(ii=1;ii<=size(E);ii++) |
---|
| 1359 | { |
---|
| 1360 | if(deg(std(E[ii])[1])==0) |
---|
| 1361 | { |
---|
| 1362 | stoplist[ii]=1; |
---|
| 1363 | } |
---|
| 1364 | } |
---|
| 1365 | |
---|
| 1366 | int count=BO[7][1]; |
---|
| 1367 | if(!defined(debug_Inters_E)) |
---|
| 1368 | { |
---|
| 1369 | int debug_Inters_E=0; |
---|
| 1370 | } |
---|
| 1371 | //--------------------------------------------------------------------- |
---|
| 1372 | // we only want to look at E^-, not at all of E |
---|
| 1373 | //--------------------------------------------------------------------- |
---|
| 1374 | if (count>-1) |
---|
| 1375 | { |
---|
| 1376 | if (count>0) |
---|
| 1377 | { |
---|
| 1378 | list E_new=E[1..count]; |
---|
| 1379 | count2=size(E); |
---|
| 1380 | } |
---|
| 1381 | else |
---|
| 1382 | { |
---|
| 1383 | list E_new; |
---|
| 1384 | count2=size(E); |
---|
| 1385 | } |
---|
| 1386 | } |
---|
| 1387 | else |
---|
| 1388 | { |
---|
| 1389 | list E_new=E; |
---|
| 1390 | count=size(E_new); |
---|
| 1391 | count2=count; |
---|
| 1392 | } |
---|
| 1393 | //--------------------------------------------------------------------- |
---|
| 1394 | // combinatorics is expensive in an interpreted language, |
---|
| 1395 | // we leave it to the kernel by translating it into monomial |
---|
| 1396 | // ideals in a new ring with variables t(i) |
---|
| 1397 | //--------------------------------------------------------------------- |
---|
[13814f0] | 1398 | list l7; |
---|
| 1399 | for (int zz = 1; zz<= size(E); zz++) |
---|
| 1400 | { |
---|
| 1401 | l7[zz] = "t("+string(zz)+")"; |
---|
| 1402 | } |
---|
| 1403 | ring rcomb = create_ring(0, l7, "dp"); |
---|
[2e6eac2] | 1404 | ideal potid,potid2; |
---|
| 1405 | list monlist,comblist,merkmon; |
---|
| 1406 | for(kk=1;kk<=count;kk++) |
---|
| 1407 | { |
---|
| 1408 | if(stoplist[kk]==0) |
---|
| 1409 | { |
---|
| 1410 | //**************************************************************************/ |
---|
| 1411 | // it does not make sense to intersect twice by the same E_i |
---|
| 1412 | // ===> reduce by t(i)^2 |
---|
| 1413 | //**************************************************************************/ |
---|
| 1414 | potid=potid,t(kk)^2; |
---|
| 1415 | } |
---|
| 1416 | else |
---|
| 1417 | { |
---|
| 1418 | //**************************************************************************/ |
---|
| 1419 | // it does not make sense to consider E_i with v[i]==1 ===> reduce by t(i) |
---|
| 1420 | //**************************************************************************/ |
---|
| 1421 | potid=potid,t(kk); |
---|
| 1422 | //**************************************************************************/ |
---|
| 1423 | // if stoplist[kk]==2 then J and all E_i automatically intersect E_kk |
---|
| 1424 | // hence we need not test it, but we have to lower maxkk by one |
---|
| 1425 | //**************************************************************************/ |
---|
| 1426 | if(stoplist[kk]==2) |
---|
| 1427 | { |
---|
| 1428 | maxkk=maxkk-1; |
---|
| 1429 | kkdiff++; // count these for dimension check later on |
---|
| 1430 | |
---|
| 1431 | } |
---|
| 1432 | } |
---|
| 1433 | } |
---|
| 1434 | |
---|
| 1435 | potid2=std(potid); |
---|
| 1436 | if(count2>count) |
---|
| 1437 | { |
---|
| 1438 | potid=potid,t((count+1)..count2); |
---|
| 1439 | for(kk=max(1,count);kk<=count2;kk++) |
---|
| 1440 | { |
---|
| 1441 | potid2=potid2,t(kk)^2; |
---|
| 1442 | } |
---|
| 1443 | } |
---|
| 1444 | potid=std(potid); |
---|
[594560] | 1445 | if(!attrib(potid2,"isSB")) { potid2=std(potid2); } |
---|
[2e6eac2] | 1446 | for(kk=1;(((kk<=count)||(kk<=maxkk+1))&&(kk<=count2));kk++) |
---|
| 1447 | { |
---|
| 1448 | //------------------------------------------------------------------------- |
---|
| 1449 | // monlist[kk]=lists of kk entries of E_new, not containing an E_i twice, |
---|
| 1450 | // not containing an E_i where v[i]==1 |
---|
| 1451 | //------------------------------------------------------------------------- |
---|
| 1452 | monlist[kk]=redMax(kk,potid); |
---|
| 1453 | //*************************************************************************/ |
---|
| 1454 | // in the case of n<=maxkk we also need to know whether n would still be |
---|
| 1455 | // below this bound if we considered all of E instead of E_new |
---|
| 1456 | // ===> merkmon contains previously ignored tuples E_i1,..,E_im |
---|
| 1457 | //*************************************************************************/ |
---|
| 1458 | if(kk<=maxkk+1) |
---|
| 1459 | { |
---|
| 1460 | merkmon[kk]=redMax(kk,potid2); |
---|
| 1461 | merkmon[kk]=simplify(reduce(merkmon[kk],std(monlist[kk])),2); |
---|
| 1462 | } |
---|
| 1463 | } |
---|
| 1464 | if(debug_Inters_E) |
---|
| 1465 | { |
---|
| 1466 | "----> In Inters_E: the tuples"; |
---|
| 1467 | "tuples of E^-:"; |
---|
| 1468 | monlist; |
---|
| 1469 | "the remaining tuples:"; |
---|
| 1470 | merkmon; |
---|
| 1471 | } |
---|
| 1472 | //------------------------------------------------------------------------- |
---|
| 1473 | // check whether there is a kk-tuple of E_i intersecting J, |
---|
| 1474 | // kk running from 1 to count |
---|
| 1475 | //------------------------------------------------------------------------- |
---|
| 1476 | for(kk=1;kk<=count;kk++) |
---|
| 1477 | { |
---|
[0f0b30] | 1478 | if(size(monlist[kk])==0) break; |
---|
[2e6eac2] | 1479 | kill comblist; |
---|
| 1480 | list comblist; |
---|
| 1481 | //--- transscribe the tuples from monomial notation to intvec notation |
---|
| 1482 | for(jj=1;jj<=ncols(monlist[kk]);jj++) |
---|
| 1483 | { |
---|
| 1484 | comblist[jj]=leadexp(monlist[kk][jj]); |
---|
| 1485 | } |
---|
| 1486 | setring rb; |
---|
| 1487 | updated=0; |
---|
| 1488 | //------------------------------------------------------------------------ |
---|
| 1489 | // Do the intersections |
---|
| 1490 | //------------------------------------------------------------------------ |
---|
| 1491 | for(jj=1;jj<=size(comblist);jj++) |
---|
| 1492 | { |
---|
| 1493 | //--- jj-th tuple from list of tuples of kk E_i |
---|
| 1494 | test2=J; |
---|
| 1495 | for(ii=1;ii<=count;ii++) |
---|
| 1496 | { |
---|
| 1497 | if(comblist[jj][ii]==1) |
---|
| 1498 | { |
---|
| 1499 | test2=test2,E_new[ii]; |
---|
| 1500 | } |
---|
| 1501 | } |
---|
| 1502 | test2=std(test2); |
---|
| 1503 | //--- check whether this intersection is non-empty and store it accordingly |
---|
| 1504 | if(deg(test2[1])!=0) |
---|
| 1505 | { |
---|
| 1506 | //--- it is non-empty |
---|
| 1507 | if(updated!=0) |
---|
| 1508 | { |
---|
| 1509 | merklist1[size(merklist1)+1]=comblist[jj]; |
---|
| 1510 | } |
---|
| 1511 | else |
---|
| 1512 | { |
---|
| 1513 | kill merklist1; |
---|
| 1514 | list merklist1; |
---|
| 1515 | merklist1[1]=comblist[jj]; |
---|
| 1516 | updated=1; |
---|
| 1517 | n=kk; |
---|
| 1518 | } |
---|
| 1519 | if(dim(test2)!=maxkk-kk+kkdiff) |
---|
| 1520 | { |
---|
| 1521 | nonnormal=1; |
---|
| 1522 | } |
---|
| 1523 | } |
---|
| 1524 | else |
---|
| 1525 | { |
---|
| 1526 | //--- it is empty |
---|
| 1527 | merklist2[size(merklist2)+1]=jj; |
---|
| 1528 | } |
---|
| 1529 | } |
---|
| 1530 | setring rcomb; |
---|
| 1531 | ideal redid; |
---|
| 1532 | //--------------------------------------------------------------------- |
---|
| 1533 | // update monlist and merkmon by the knowledge what intersections are |
---|
| 1534 | // empty in the kk-th step |
---|
| 1535 | //--------------------------------------------------------------------- |
---|
| 1536 | for(jj=1;jj<=size(merklist2);jj++) |
---|
| 1537 | { |
---|
| 1538 | redid=redid,monlist[kk][merklist2[jj]]; |
---|
| 1539 | } |
---|
| 1540 | for(jj=kk+1;jj<=count;jj++) |
---|
| 1541 | { |
---|
| 1542 | monlist[jj]=simplify(reduce(monlist[jj],std(redid)),2); |
---|
| 1543 | if(jj<=maxkk+1) |
---|
| 1544 | { |
---|
| 1545 | merkmon[jj]=simplify(reduce(merkmon[jj],std(redid)),2); |
---|
| 1546 | } |
---|
| 1547 | } |
---|
| 1548 | kill redid; |
---|
| 1549 | kill merklist2; |
---|
| 1550 | list merklist2; |
---|
| 1551 | } |
---|
| 1552 | if(debug_Inters_E) |
---|
| 1553 | { |
---|
| 1554 | "----> In Inters_E: intersections found:"; |
---|
| 1555 | merklist1; |
---|
| 1556 | } |
---|
| 1557 | //--------------------------------------------------------------------- |
---|
| 1558 | // form the union of the intersections of the appropriate E_i |
---|
| 1559 | //--------------------------------------------------------------------- |
---|
| 1560 | setring rb; |
---|
| 1561 | ideal center,dummy; |
---|
| 1562 | list centlist; |
---|
| 1563 | for(kk=1;kk<=size(merklist1);kk++) |
---|
| 1564 | { |
---|
| 1565 | for(jj=1;jj<=size(merklist1[kk]);jj++) |
---|
| 1566 | { |
---|
| 1567 | if(merklist1[kk][jj]==1) |
---|
| 1568 | { |
---|
| 1569 | dummy=dummy,E_new[jj]; |
---|
| 1570 | } |
---|
| 1571 | } |
---|
| 1572 | if(size(center)==0) |
---|
| 1573 | { |
---|
| 1574 | center=dummy; |
---|
| 1575 | centlist[1]=dummy; |
---|
| 1576 | } |
---|
| 1577 | else |
---|
| 1578 | { |
---|
| 1579 | center=intersect(center,dummy); |
---|
| 1580 | centlist[size(centlist)+1]=dummy; |
---|
| 1581 | } |
---|
| 1582 | dummy=0; |
---|
| 1583 | } |
---|
| 1584 | if(debug_Inters_E) |
---|
| 1585 | { |
---|
| 1586 | "----> In Inters_E: intersection of E_i"; |
---|
| 1587 | "maximal number of E_i encountered in:"; |
---|
| 1588 | center; |
---|
| 1589 | "the components of this locus:"; |
---|
| 1590 | centlist; |
---|
| 1591 | "maximal number of E_i from E^- intersecting simultaneously:",n; |
---|
| 1592 | if(nonnormal) |
---|
| 1593 | { |
---|
| 1594 | "flag nonnormal is set"; |
---|
| 1595 | } |
---|
| 1596 | } |
---|
| 1597 | list retlist=center,n; |
---|
| 1598 | //------------------------------------------------------------------------- |
---|
| 1599 | // If n<=maxkk, then test if this is the case for all of E not just E_new |
---|
| 1600 | // using the pairs indicated by merkmon |
---|
| 1601 | //------------------------------------------------------------------------- |
---|
| 1602 | int ntotal=n; |
---|
| 1603 | if((n<=maxkk)&&(n<count2)&&(!nonnormal)) |
---|
| 1604 | { |
---|
| 1605 | //--- check kk-tuples |
---|
| 1606 | for(kk=n+1;kk<=maxkk+1;kk++) |
---|
| 1607 | { |
---|
| 1608 | setring rcomb; |
---|
| 1609 | //--- check if there are combinations to be checked |
---|
| 1610 | if(kk>size(merkmon)) |
---|
| 1611 | { |
---|
| 1612 | setring rb; |
---|
| 1613 | break; |
---|
| 1614 | } |
---|
| 1615 | if(size(merkmon[kk])!=0) |
---|
| 1616 | { |
---|
| 1617 | kill comblist; |
---|
| 1618 | list comblist; |
---|
| 1619 | //--- transscribe tuples from monomial notation to intvec notation |
---|
| 1620 | for(jj=1;jj<=size(merkmon[kk]);jj++) |
---|
| 1621 | { |
---|
| 1622 | comblist[jj]=leadexp(merkmon[kk][jj]); |
---|
| 1623 | } |
---|
| 1624 | setring rb; |
---|
| 1625 | //--- check jj-th tuple from the list of kk-tuples |
---|
| 1626 | for(jj=1;jj<=size(comblist);jj++) |
---|
| 1627 | { |
---|
| 1628 | test2=J; |
---|
| 1629 | for(ii=1;ii<=nvars(rcomb);ii++) |
---|
| 1630 | { |
---|
| 1631 | if(comblist[jj][ii]==1) |
---|
| 1632 | { |
---|
| 1633 | test2=test2,E[ii]; |
---|
| 1634 | } |
---|
| 1635 | } |
---|
| 1636 | test2=std(test2); |
---|
| 1637 | //--- as soon as we found one we can proceed to the subsequent kk |
---|
| 1638 | if(deg(test2[1])!=0) |
---|
| 1639 | { |
---|
| 1640 | ntotal=kk; |
---|
| 1641 | if(dim(test2)-kkdiff!=maxkk-kk) |
---|
| 1642 | { |
---|
| 1643 | nonnormal=2; |
---|
| 1644 | break; |
---|
| 1645 | } |
---|
| 1646 | } |
---|
| 1647 | } |
---|
| 1648 | //--- if we already know that too many E_i intersect simultaneously, |
---|
| 1649 | //--- we need not proceed any further |
---|
| 1650 | if(nonnormal) |
---|
| 1651 | { |
---|
| 1652 | break; |
---|
| 1653 | } |
---|
| 1654 | } |
---|
| 1655 | else |
---|
| 1656 | { |
---|
| 1657 | setring rb; |
---|
| 1658 | break; |
---|
| 1659 | } |
---|
| 1660 | } |
---|
| 1661 | } |
---|
| 1662 | //------------------------------------------------------------------------- |
---|
| 1663 | // update the result accordingly and return it |
---|
| 1664 | //------------------------------------------------------------------------- |
---|
| 1665 | if(maxkk<dimJ) |
---|
| 1666 | { |
---|
| 1667 | n=n+dimJ-maxkk; |
---|
| 1668 | ntotal=ntotal+dimJ-maxkk; |
---|
| 1669 | } |
---|
| 1670 | retlist[2]=n; |
---|
| 1671 | retlist[3]=ntotal; |
---|
| 1672 | if(n<=dimJ) |
---|
| 1673 | { |
---|
| 1674 | retlist[4]=centlist; |
---|
| 1675 | retlist[5]=merklist1; |
---|
| 1676 | if(nonnormal) |
---|
| 1677 | { |
---|
| 1678 | retlist[6]=nonnormal; |
---|
| 1679 | } |
---|
| 1680 | } |
---|
| 1681 | return(retlist); |
---|
| 1682 | } |
---|
| 1683 | /////////////////////////////////////////////////////////////////////////// |
---|
| 1684 | |
---|
| 1685 | proc Delta(list BO) |
---|
| 1686 | "USAGE: Delta (BO); |
---|
| 1687 | @* BO = basic object, a list: ideal W, |
---|
| 1688 | @* ideal J, |
---|
| 1689 | @* intvec b, |
---|
| 1690 | @* list Ex, |
---|
| 1691 | @* ideal ab, |
---|
| 1692 | @* intvec v, |
---|
| 1693 | @* intvec w, |
---|
| 1694 | @* matrix M |
---|
| 1695 | ASSUME: R = basering, a polynomial ring, W an ideal of R, |
---|
| 1696 | @* J = ideal containing W |
---|
| 1697 | COMPUTE: Delta-operator applied to J in the notation of |
---|
| 1698 | [Bravo,Encinas,Villamayor] |
---|
| 1699 | RETURN: ideal |
---|
| 1700 | EXAMPLE: example Delta; shows an example |
---|
| 1701 | " |
---|
| 1702 | { |
---|
| 1703 | //--------------------------------------------------------------------------- |
---|
| 1704 | // Initialization and sanity checks |
---|
| 1705 | //--------------------------------------------------------------------------- |
---|
| 1706 | ideal W=BO[1]; |
---|
| 1707 | ideal J=BO[2]; |
---|
| 1708 | ideal C=simplify(reduce(J,std(W)),2); |
---|
| 1709 | list LC; |
---|
| 1710 | int n=nvars(basering); |
---|
| 1711 | //--------------------------------------------------------------------------- |
---|
| 1712 | // Simple case: W is the empty set |
---|
| 1713 | //--------------------------------------------------------------------------- |
---|
| 1714 | if(size(W)==0) |
---|
| 1715 | { |
---|
| 1716 | C=C,jacob(J); |
---|
| 1717 | C=std(C); |
---|
| 1718 | return(C); |
---|
| 1719 | } |
---|
| 1720 | //--------------------------------------------------------------------------- |
---|
| 1721 | // General case: W is non-empty |
---|
| 1722 | // Step 1: Find a minor of the Jacobian of W which is not identically zero |
---|
| 1723 | // and look at the complement of the zero-set given by this minor; |
---|
| 1724 | // this leads to the system of local parameters |
---|
| 1725 | // Step 2: Form the derivatives w.r.t. this system of parameters |
---|
| 1726 | //--------------------------------------------------------------------------- |
---|
| 1727 | //--- Step 1 |
---|
| 1728 | list re=findMinor(W); |
---|
| 1729 | list L; |
---|
| 1730 | int ii,i,j,l,k; |
---|
| 1731 | J=C; |
---|
| 1732 | ideal D=ideal(1); |
---|
| 1733 | intvec v,w; |
---|
| 1734 | ideal V; |
---|
| 1735 | poly m; |
---|
| 1736 | |
---|
| 1737 | for(ii=1;ii<=size(re);ii++) |
---|
| 1738 | { |
---|
| 1739 | C=J; |
---|
| 1740 | L=re[ii]; |
---|
| 1741 | matrix A=L[1]; //(1/m)*A is the inverse matrix of the Jacobian of W |
---|
| 1742 | //corresponding to m |
---|
| 1743 | m=L[2]; //a k- minor of jacob(W), not identically zero |
---|
| 1744 | //k=n-dim(W) |
---|
| 1745 | V=L[3]; //the elements of W corresponding to m |
---|
| 1746 | v=L[4]; //the indices of variables corresponding to m |
---|
| 1747 | w=L[5]; //the indices of variables not corresponding to m |
---|
| 1748 | |
---|
| 1749 | //--- Step 2 |
---|
| 1750 | //--- first some initialization depending on results of step 1 |
---|
| 1751 | k=size(V); |
---|
| 1752 | matrix dg[1][k]; |
---|
| 1753 | matrix df[k][1]; |
---|
| 1754 | //--- derivatives of the generators of J w.r.t. system of parameters |
---|
| 1755 | for(i=1;i<=size(J);i++) |
---|
| 1756 | { |
---|
| 1757 | for(j=1;j<=n-k;j++) |
---|
| 1758 | { |
---|
| 1759 | for(l=1;l<=k;l++) |
---|
| 1760 | { |
---|
| 1761 | dg[1,l]=diff(V[l],var(w[j])); |
---|
| 1762 | df[l,1]=diff(J[i],var(v[l])); |
---|
| 1763 | } |
---|
| 1764 | C=C,m*diff(J[i],var(w[j]))-dg*A*df; |
---|
| 1765 | } |
---|
| 1766 | } |
---|
| 1767 | //--- everything should live in W, not just in the intersection of |
---|
| 1768 | //--- D(m) with W |
---|
| 1769 | C=C+W; |
---|
[3f7e01a] | 1770 | C=sat(C,m); |
---|
[2e6eac2] | 1771 | //--- intersect ideal with previously computed ones to make sure that no |
---|
| 1772 | //--- components are lost |
---|
| 1773 | D=intersect(D,C); |
---|
| 1774 | kill dg,df,A; |
---|
| 1775 | } |
---|
| 1776 | //--- return minimal set of generators of the result |
---|
| 1777 | list li=mstd(D); |
---|
| 1778 | D=li[2]; |
---|
| 1779 | if(size(li[1])<=size(D)){D=li[1];} |
---|
| 1780 | return(D); |
---|
| 1781 | } |
---|
| 1782 | example |
---|
| 1783 | { "EXAMPLE:"; |
---|
| 1784 | echo = 2; |
---|
| 1785 | ring R=0,(x,y,z),dp; |
---|
| 1786 | |
---|
| 1787 | ideal W=z^2-x; |
---|
| 1788 | ideal J=x*y^2+x^3; |
---|
| 1789 | intvec b=1; |
---|
| 1790 | list E; |
---|
| 1791 | ideal abb=maxideal(1); |
---|
| 1792 | intvec v; |
---|
| 1793 | intvec w=-1; |
---|
| 1794 | matrix M; |
---|
| 1795 | |
---|
| 1796 | list BO=W,J,b,E,abb,v,w,M; |
---|
| 1797 | |
---|
| 1798 | Delta(BO); |
---|
| 1799 | } |
---|
| 1800 | |
---|
| 1801 | ////////////////////////////////////////////////////////////////////////////// |
---|
[268a430] | 1802 | static proc redMax(int k,ideal J) |
---|
[2e6eac2] | 1803 | "Internal procedure - no help and no example available |
---|
| 1804 | " |
---|
| 1805 | { |
---|
| 1806 | //--- reduce maxideal(k) by J, more efficient approach |
---|
| 1807 | int i; |
---|
| 1808 | ideal K=simplify(reduce(maxideal(1),J),2); |
---|
| 1809 | for(i=2;i<=k;i++) |
---|
| 1810 | { |
---|
| 1811 | K=simplify(reduce(K*maxideal(1),J),2); |
---|
| 1812 | } |
---|
| 1813 | return(K); |
---|
| 1814 | } |
---|
| 1815 | |
---|
| 1816 | ////////////////////////////////////////////////////////////////////////////// |
---|
[268a430] | 1817 | static proc findMinor(ideal W) |
---|
[2e6eac2] | 1818 | "Internal procedure - no help and no example available |
---|
| 1819 | " |
---|
| 1820 | { |
---|
| 1821 | //--------------------------------------------------------------------------- |
---|
| 1822 | // Initialization and sanity checks |
---|
| 1823 | //--------------------------------------------------------------------------- |
---|
| 1824 | list L; |
---|
| 1825 | intvec v,w; |
---|
| 1826 | ideal Wstd=std(W); |
---|
| 1827 | int n=nvars(basering); // total number of columns of Jacobian |
---|
| 1828 | int k=n-dim(Wstd); // size of minors of Jacobian |
---|
| 1829 | int a=size(W); // total number of rows of Jacobian |
---|
| 1830 | matrix A[k][k]; |
---|
| 1831 | list LW=indexSet(a,k); // set of tuples of k rows |
---|
| 1832 | list LV=indexSet(n,k); // set of tuples of k columns |
---|
| 1833 | ideal IW,IV; |
---|
| 1834 | int i,j,l,e; |
---|
| 1835 | list re; |
---|
| 1836 | //--------------------------------------------------------------------------- |
---|
| 1837 | // We need to know which minor corresponds to which variable and to which |
---|
| 1838 | // generator of W - therefore we cannot use the function minor()! |
---|
| 1839 | //--------------------------------------------------------------------------- |
---|
| 1840 | //--- choose the generators which we want to differentiate |
---|
| 1841 | for(i=1;i<=size(LW);i++) |
---|
| 1842 | { |
---|
| 1843 | IW=0; |
---|
| 1844 | for(l=1;l<=a;l++) |
---|
| 1845 | { |
---|
| 1846 | if(LW[i][l]!=0){IW[size(IW)+1]=W[l];} |
---|
| 1847 | } |
---|
| 1848 | //--- choose the variables by which to differentiate and apply diff |
---|
| 1849 | for(j=1;j<=size(LV);j++) |
---|
| 1850 | { |
---|
| 1851 | IV=0;v=0;w=0; |
---|
| 1852 | for(l=1;l<=n;l++) |
---|
| 1853 | { |
---|
| 1854 | if(LV[j][l]!=0) |
---|
| 1855 | { |
---|
| 1856 | v[size(v)+1]=l; |
---|
| 1857 | IV[size(IV)+1]=var(l); |
---|
| 1858 | } |
---|
| 1859 | else |
---|
| 1860 | { |
---|
| 1861 | w[size(w)+1]=l; |
---|
| 1862 | |
---|
| 1863 | } |
---|
| 1864 | } |
---|
| 1865 | A=diff(IV,IW); // appropriate submatrix of Jacobian |
---|
| 1866 | //--- if the minor is non-zero, then it might be the one we need |
---|
| 1867 | //--- ==> put it in the list of candidates |
---|
| 1868 | if(det(A)!=0) |
---|
| 1869 | { |
---|
| 1870 | v=v[2..size(v)]; // first entry is zero for technical reasons |
---|
| 1871 | w=w[2..size(w)]; // first entry is zero for technical reasons |
---|
| 1872 | L=inverse_L(A); |
---|
| 1873 | L[3]=IW; |
---|
| 1874 | L[4]=v; |
---|
| 1875 | L[5]=w; |
---|
| 1876 | re[size(re)+1]=L; |
---|
| 1877 | } |
---|
| 1878 | } |
---|
| 1879 | } |
---|
| 1880 | //--------------------------------------------------------------------------- |
---|
| 1881 | // return the result |
---|
| 1882 | //--------------------------------------------------------------------------- |
---|
| 1883 | return(re); |
---|
| 1884 | } |
---|
| 1885 | |
---|
| 1886 | ///////////////////////////////////////////////////////////////////////////// |
---|
[268a430] | 1887 | static proc indexSet(int a, int b) |
---|
[2e6eac2] | 1888 | "Internal procedure - no help and no example available |
---|
| 1889 | " |
---|
| 1890 | { |
---|
| 1891 | //--------------------------------------------------------------------------- |
---|
| 1892 | // Find all tuples of size b containing pairwise distict elements from a |
---|
| 1893 | // list of a elements |
---|
| 1894 | //--------------------------------------------------------------------------- |
---|
| 1895 | //**************************************************************************/ |
---|
| 1896 | // Combinatorics is expensive in an interpreted language |
---|
| 1897 | // ==> shift it into the kernel |
---|
| 1898 | //**************************************************************************/ |
---|
| 1899 | def R=basering; |
---|
| 1900 | list L; |
---|
| 1901 | ring S=2,x(1..a),dp; |
---|
| 1902 | ideal I=maxideal(b); |
---|
| 1903 | int i; |
---|
| 1904 | ideal J=x(1)^2; |
---|
| 1905 | for(i=2;i<=a;i++){J=J,x(i)^2;} |
---|
| 1906 | attrib(J,"isSB",1); |
---|
| 1907 | I=reduce(I,J); |
---|
| 1908 | I=simplify(I,2); |
---|
| 1909 | for(i=1;i<=size(I);i++){L[i]=leadexp(I[i]);} |
---|
| 1910 | setring R; |
---|
| 1911 | return(L); |
---|
| 1912 | } |
---|
| 1913 | |
---|
| 1914 | ///////////////////////////////////////////////////////////////////////////// |
---|
| 1915 | |
---|
| 1916 | proc DeltaList(list BO) |
---|
| 1917 | "USAGE: DeltaList (BO); |
---|
| 1918 | @* BO = basic object, a list: ideal W, |
---|
| 1919 | @* ideal J, |
---|
| 1920 | @* intvec b, |
---|
| 1921 | @* list Ex, |
---|
| 1922 | @* ideal ab, |
---|
| 1923 | @* intvec v, |
---|
| 1924 | @* intvec w, |
---|
| 1925 | @* matrix M |
---|
| 1926 | ASSUME: R = basering, a polynomial ring, W an ideal of R, |
---|
| 1927 | @* J = ideal containing W |
---|
| 1928 | COMPUTE: Delta-operator iteratively applied to J in the notation of |
---|
| 1929 | [Bravo,Encinas,Villamayor] |
---|
| 1930 | RETURN: list l of length ((max w-ord) * b), |
---|
| 1931 | l[i+1]=Delta^i(J) |
---|
| 1932 | EXAMPLE: example DeltaList; shows an example |
---|
| 1933 | " |
---|
| 1934 | { |
---|
| 1935 | //---------------------------------------------------------------------------- |
---|
| 1936 | // Iteratively apply proc Delta |
---|
| 1937 | //---------------------------------------------------------------------------- |
---|
| 1938 | int i; |
---|
| 1939 | list L; |
---|
| 1940 | ideal C=BO[2]; |
---|
| 1941 | while(deg(C[1])!=0) |
---|
| 1942 | { |
---|
| 1943 | L[size(L)+1]=C; |
---|
| 1944 | C=Delta(BO); |
---|
| 1945 | BO[2]=C; |
---|
| 1946 | } |
---|
| 1947 | return(L); |
---|
| 1948 | } |
---|
| 1949 | example |
---|
| 1950 | { |
---|
| 1951 | "EXAMPLE:"; |
---|
| 1952 | echo = 2; |
---|
| 1953 | ring R=0,(x,y,z),dp; |
---|
| 1954 | |
---|
| 1955 | ideal W=z^2-x; |
---|
| 1956 | ideal J=x*y^2+x^3; |
---|
| 1957 | intvec b=1; |
---|
| 1958 | list E; |
---|
| 1959 | ideal abb=maxideal(1); |
---|
| 1960 | intvec v; |
---|
| 1961 | intvec w=-1; |
---|
| 1962 | matrix M; |
---|
| 1963 | |
---|
| 1964 | list BO=W,J,b,E,abb,v,w,M; |
---|
| 1965 | |
---|
| 1966 | DeltaList(BO); |
---|
| 1967 | } |
---|
| 1968 | ///////////////////////////////////////////////////////////////////////////// |
---|
| 1969 | proc CenterBM(list BM) |
---|
| 1970 | "USAGE: CenterBM(BM); |
---|
| 1971 | @* BM = object related to a presentation, |
---|
| 1972 | a list: ideal W, |
---|
| 1973 | @* ideal J, |
---|
| 1974 | @* intvec b, |
---|
| 1975 | @* list Ex, |
---|
| 1976 | @* ideal ab, |
---|
| 1977 | @* intvec v, |
---|
| 1978 | @* intvec w, |
---|
| 1979 | @* matrix M |
---|
| 1980 | ASSUME: R = basering, a polynomial ring, W an ideal of R, |
---|
| 1981 | @* J = ideal containing W |
---|
| 1982 | COMPUTE: the center of the next blow-up of BM in the resolution algorithm |
---|
| 1983 | of [Bierstone, Milman] |
---|
| 1984 | RETURN: list l, |
---|
| 1985 | l[1]: ideal describing the center |
---|
| 1986 | l[2]: intvec w obtained in the process of determining l[1] |
---|
| 1987 | l[3]: intvec b obtained in the process of determining l[1] |
---|
| 1988 | l[4]: intmat invmat obtained in the process of determining l[1] |
---|
| 1989 | EXAMPLE: example CenterBM; shows an example |
---|
| 1990 | " |
---|
| 1991 | { |
---|
| 1992 | //!!! NOCH NICHT IN BETRIEB |
---|
| 1993 | ERROR("Not implemented yet"); |
---|
| 1994 | int i,j; |
---|
| 1995 | intmat tmat[2][1]=0,-1; |
---|
| 1996 | //--- re=center,E^- indices, b vector, n vector |
---|
| 1997 | list re=ideal(1),BM[7],BM[3],tmat; |
---|
| 1998 | ideal J=BM[2]; |
---|
| 1999 | if(size(J)==0) |
---|
| 2000 | { |
---|
| 2001 | re[1]=ideal(0); |
---|
| 2002 | return(re); |
---|
| 2003 | } |
---|
| 2004 | //--- find Delta^(b-1)(J) |
---|
[6518eba] | 2005 | if(size(reduce(J,std(BM[1]),5))!=0) |
---|
[2e6eac2] | 2006 | { |
---|
| 2007 | list L=DeltaList(BM); |
---|
| 2008 | } |
---|
| 2009 | else |
---|
| 2010 | { |
---|
| 2011 | list L; |
---|
| 2012 | L[1]=J; |
---|
| 2013 | } |
---|
| 2014 | if(!defined(debugCenter)) |
---|
| 2015 | { |
---|
| 2016 | int debugCenter; |
---|
| 2017 | } |
---|
| 2018 | if(debugCenter) |
---|
| 2019 | { |
---|
| 2020 | "----> In Center: after DeltaList"; |
---|
| 2021 | "W"; |
---|
| 2022 | BM[1]; |
---|
| 2023 | "J"; |
---|
| 2024 | BM[2]; |
---|
| 2025 | "The Delta List:"; |
---|
| 2026 | L; |
---|
| 2027 | } |
---|
| 2028 | int b=size(L); |
---|
| 2029 | if(b==0) |
---|
| 2030 | { |
---|
| 2031 | //--- if J=W, we do not need to do anything |
---|
| 2032 | //--- returning center=1 marks this chart as completed |
---|
| 2033 | return(re); |
---|
| 2034 | } |
---|
| 2035 | //--------------------------------------------------------------------------- |
---|
| 2036 | // check whether max ord is constant |
---|
| 2037 | //--------------------------------------------------------------------------- |
---|
| 2038 | if((BM[9][2,1]<0)||(BM[9][1,1]>b)) |
---|
| 2039 | { |
---|
| 2040 | //--- we are either at the beginning or the invariant has dropped |
---|
| 2041 | intvec tempvec=size(BM[4]); |
---|
| 2042 | BM[7]=tempvec; |
---|
| 2043 | //!!!! nur fuer hyperflaechen!!!!!!!! |
---|
| 2044 | tempvec=b; |
---|
| 2045 | BM[3]=tempvec; |
---|
| 2046 | //!!!! Ende !!!!!!!!!!!! |
---|
| 2047 | kill tempvec; |
---|
| 2048 | BM[9][1,1]=b; |
---|
| 2049 | BM[9][2,1]=1; |
---|
| 2050 | } |
---|
| 2051 | //--------------------------------------------------------------------------- |
---|
| 2052 | // prepare for intersection with E_i |
---|
| 2053 | //--------------------------------------------------------------------------- |
---|
| 2054 | ideal C=L[b]; |
---|
| 2055 | re[2]=BM[7]; |
---|
| 2056 | re[3]=BM[3]; |
---|
| 2057 | BM[2]=C; |
---|
| 2058 | if(debugCenter) |
---|
| 2059 | { |
---|
| 2060 | "----> In Center: before intersection with E_i:"; |
---|
| 2061 | "bmax:",b; |
---|
| 2062 | "Sing(J,bmax):"; |
---|
| 2063 | C; |
---|
| 2064 | "E:"; |
---|
| 2065 | BO[4]; |
---|
| 2066 | "list marking a priori known intersection properties:",BO[6]; |
---|
| 2067 | "index of last element of E^- in E:",BO[7][1]; |
---|
| 2068 | } |
---|
| 2069 | list E=inters_E(BM); |
---|
| 2070 | // !!!!!!!!! Drop Redundant fehlt noch!!!!!!!! |
---|
| 2071 | //--------------------------------------------------------------------------- |
---|
| 2072 | // Check whether it is a single point |
---|
| 2073 | //--------------------------------------------------------------------------- |
---|
| 2074 | ideal C1=std(ideal(L[b])+E[1]); |
---|
| 2075 | if(dim(C1)==0) |
---|
| 2076 | { |
---|
| 2077 | if(size(E[4])==1) |
---|
| 2078 | { |
---|
| 2079 | tmat[1,1]=BM[9][1,1]; |
---|
| 2080 | tmat[2,1]=BM[9][2,1]; |
---|
| 2081 | re[4]=tmat; |
---|
| 2082 | re[1]=radical(C1); |
---|
| 2083 | return(re); |
---|
| 2084 | } |
---|
| 2085 | } |
---|
| 2086 | if(size(BM[9])>2) |
---|
| 2087 | { |
---|
| 2088 | BM[9][1,2]=E[2]; |
---|
| 2089 | BM[9][2,2]=1; |
---|
| 2090 | } |
---|
| 2091 | else |
---|
| 2092 | { |
---|
| 2093 | intmat tempInt[2][1]=E[2],1; |
---|
| 2094 | BM[9]=concatInt(BM[9],tempInt); |
---|
| 2095 | kill tempInt; |
---|
| 2096 | } |
---|
| 2097 | BM[2]=J; |
---|
| 2098 | list BM1=dropDim(BM); |
---|
| 2099 | list BMlist,hilfList; |
---|
| 2100 | ideal hilf; |
---|
| 2101 | intvec tempvec; |
---|
| 2102 | |
---|
| 2103 | if(!attrib(BM1[1],"isSB")){BM1[1]=std(BM1[1]);} |
---|
| 2104 | for(i=1;i<=size(BM1[4]);i++) |
---|
| 2105 | { |
---|
| 2106 | hilf=simplify(reduce(BM1[4][i],BM1[1]),2); |
---|
| 2107 | if(size(hilf)>1){"Problem with BM1[4]in CenterBM";~;} |
---|
| 2108 | hilfList[i]=hilf[1]; |
---|
| 2109 | } |
---|
| 2110 | for(i=1;i<=size(E[4]);i++) |
---|
| 2111 | { |
---|
| 2112 | BMlist[i]=BM1; |
---|
| 2113 | for(j=size(E[5][i]);j>=1;j--) |
---|
| 2114 | { |
---|
| 2115 | if(E[5][i][j]!=0) |
---|
| 2116 | { |
---|
| 2117 | BMlist[i][2][size(BMlist[i][2])+1]=hilfList[j]; |
---|
| 2118 | BMlist[i][3][size(BMlist[i][3])+1]=1; |
---|
| 2119 | } |
---|
| 2120 | BMlist[i][4]=delete(BMlist[i][4],j); |
---|
| 2121 | BMlist[i][6]=deleteInt(BMlist[i][6],j,0); |
---|
| 2122 | } |
---|
| 2123 | BMlist[i][7]=deleteInt(BMlist[i][7],1,-1); |
---|
| 2124 | if(size(BMlist[i][9])>4) |
---|
| 2125 | { |
---|
| 2126 | intmat tempInt[2][ncols(BMlist[i][9])]=BMlist[i][9]; |
---|
| 2127 | intmat tempInt2[2][ncols(BMlist[i][9])-2]= |
---|
| 2128 | tempInt[1..2,3..ncols(BMlist[i][9])]; |
---|
| 2129 | BMlist[i][9]=tempInt2; |
---|
| 2130 | kill tempInt,tempInt2; |
---|
| 2131 | } |
---|
| 2132 | else |
---|
| 2133 | { |
---|
| 2134 | BMlist[i][9]=tmat; |
---|
| 2135 | } |
---|
| 2136 | } |
---|
| 2137 | kill hilfList;list hilfList; |
---|
| 2138 | hilfList[1]=CenterTail(BMlist[1],C); |
---|
| 2139 | intmat maxmat=hilfList[1][9]; |
---|
| 2140 | intvec maxiv=E[4][1]; |
---|
| 2141 | int pos=1; |
---|
| 2142 | for(i=2;i<=size(E[4]);i++) |
---|
| 2143 | { |
---|
| 2144 | hilfList[i]=CenterTail(BMlist[i],C); |
---|
| 2145 | if(invGreater(hilfList[i][4]),maxmat,E[4][i],maxiv) |
---|
| 2146 | { |
---|
| 2147 | maxmat=hilfList[i][4]; |
---|
| 2148 | maxiv=E[4][i]; |
---|
| 2149 | pos=i; |
---|
| 2150 | } |
---|
| 2151 | } |
---|
| 2152 | re[1]=hilfList[pos][1]; |
---|
| 2153 | intmat tempint=BM[9]; |
---|
| 2154 | intmat tempint1[2][2]=tempint[1..2,1..2]; |
---|
| 2155 | re[4]=concatInt(tempint1,maxmat); |
---|
| 2156 | re[2]=re[2][1],hilfList[pos][2]; |
---|
| 2157 | re[3]=b; |
---|
| 2158 | ~; |
---|
| 2159 | return(re); |
---|
| 2160 | } |
---|
| 2161 | example |
---|
| 2162 | { "EXAMPLE:"; |
---|
| 2163 | echo = 2; |
---|
| 2164 | ring R=0,(x,y),dp; |
---|
| 2165 | |
---|
| 2166 | ideal W; |
---|
| 2167 | ideal J=x2-y3; |
---|
| 2168 | intvec b=1; |
---|
| 2169 | list E; |
---|
| 2170 | ideal abb=maxideal(1); |
---|
| 2171 | intvec v; |
---|
| 2172 | intvec w=-1; |
---|
| 2173 | matrix M; |
---|
| 2174 | intmat invmat[2][1]=0,-1; |
---|
| 2175 | |
---|
| 2176 | list BM=W,J,b,E,abb,v,w,M,invmat; |
---|
| 2177 | |
---|
| 2178 | CenterBM(BM); |
---|
| 2179 | } |
---|
| 2180 | |
---|
| 2181 | ///////////////////////////////////////////////////////////////////////////// |
---|
[268a430] | 2182 | static proc invGreater(intmat M1, intmat M2, intvec iv1, intvec iv2) |
---|
[2e6eac2] | 2183 | { |
---|
[4bde6b] | 2184 | // Auxiliary procedure, BM-algorithm |
---|
[2e6eac2] | 2185 | int i; |
---|
| 2186 | for(i=1;i<=min(ncols(M1),ncols(M2));i++) |
---|
| 2187 | { |
---|
| 2188 | if(M1[2,i]==-1) |
---|
| 2189 | { |
---|
| 2190 | if(M1[1,i]==0){ERROR("Invariant not set");} |
---|
| 2191 | if(M2[2,i]!=-1){return(1);} |
---|
| 2192 | if(M2[1,i]==0){ERROR("Invariant not set");} |
---|
| 2193 | break; |
---|
| 2194 | } |
---|
| 2195 | else |
---|
| 2196 | { |
---|
| 2197 | if(M2[2,i]==-1) |
---|
| 2198 | { |
---|
| 2199 | if(M2[1,i]==0){ERROR("Invariant not set");} |
---|
| 2200 | return(0); |
---|
| 2201 | } |
---|
| 2202 | if(M1[1,i]*M2[2,i]!= M2[1,i]*M1[2,i]) |
---|
| 2203 | { |
---|
| 2204 | return(M1[1,i]*M2[2,i]> M2[1,i]*M1[2,i]); |
---|
| 2205 | } |
---|
| 2206 | } |
---|
| 2207 | } |
---|
| 2208 | return(iv1>iv2); |
---|
| 2209 | } |
---|
| 2210 | ///////////////////////////////////////////////////////////////////////////// |
---|
| 2211 | proc CenterTail(list BM, ideal C) |
---|
| 2212 | { |
---|
[4bde6b] | 2213 | //!!! Auxiliary procedure, BM-algorithm |
---|
[2e6eac2] | 2214 | //!!!!!!!!Rueckgabe im Zentrumsformat |
---|
| 2215 | int i,j,bmin; |
---|
| 2216 | int alpha=lcm(BM[3]); |
---|
| 2217 | vector w; |
---|
| 2218 | list re; |
---|
| 2219 | if(size(BM[2])==0) |
---|
| 2220 | { |
---|
| 2221 | re[1]=C+BM[1]; |
---|
| 2222 | intvec tvec; |
---|
| 2223 | re[3]=tvec; |
---|
| 2224 | intmat tmat[2][1]=-1,-1; |
---|
| 2225 | re[4]=tmat; |
---|
| 2226 | tvec=size(BM[4]); |
---|
| 2227 | re[2]=tvec; |
---|
| 2228 | return(re); |
---|
| 2229 | } |
---|
| 2230 | for(i=1;i<=size(BM[3]);i++) |
---|
| 2231 | { |
---|
| 2232 | if(BM[2][i]!=0) |
---|
| 2233 | { |
---|
| 2234 | w[size(w)+1]=BM[2][i]^(alpha/BM[3][i]); |
---|
| 2235 | } |
---|
| 2236 | } |
---|
| 2237 | module M=w; |
---|
| 2238 | intvec satex; |
---|
| 2239 | list satList; |
---|
| 2240 | for(i=1;i<=size(BM[4]);i++) |
---|
| 2241 | { |
---|
[3f7e01a] | 2242 | satList=sat_with_exp(M,BM[4][i]); |
---|
[2e6eac2] | 2243 | satex[i]=satList[2]; |
---|
| 2244 | M=satList[1]; |
---|
| 2245 | } |
---|
| 2246 | //!!!!Hilfsobjekt G bilden!!!!!!!!!!!!!!!! |
---|
| 2247 | //!!!!Hilfsobjekt H,codim -1 bilden!!!!!!!!!!!!!!!! |
---|
| 2248 | //!!!! ???? an welcher stelle????????? |
---|
| 2249 | list deltaL; |
---|
| 2250 | list BMtemp=BM; |
---|
| 2251 | for(i=1;i<=nrows(M[1]);i++) |
---|
| 2252 | { |
---|
| 2253 | BMtemp[2]=ideal(M[1][i])+BMtemp[1]; |
---|
| 2254 | deltaL[i]=DeltaList(BMtemp); |
---|
| 2255 | if(i==1) |
---|
| 2256 | { |
---|
| 2257 | bmin=size(deltaL[i]); |
---|
| 2258 | } |
---|
| 2259 | else |
---|
| 2260 | { |
---|
| 2261 | bmin=min(size(deltaL[i]),bmin); |
---|
| 2262 | } |
---|
| 2263 | } |
---|
| 2264 | if(bmin==0) |
---|
| 2265 | { |
---|
| 2266 | re[1]=C+BM[1]; |
---|
| 2267 | intvec tvec; |
---|
| 2268 | re[3]=tvec; |
---|
| 2269 | if((BM[9][2,1]==-1)||(BM[9][1,1]!=0)) |
---|
| 2270 | { |
---|
| 2271 | tvec=size(BM[4]); |
---|
| 2272 | re[2]=tvec; |
---|
| 2273 | intmat tmat[2][1]=0,1; |
---|
| 2274 | re[4]=tmat; |
---|
| 2275 | } |
---|
| 2276 | else |
---|
| 2277 | { |
---|
| 2278 | re[2]=BM[7]; |
---|
| 2279 | re[4]=BM[9]; |
---|
| 2280 | } |
---|
| 2281 | return(re); |
---|
| 2282 | } |
---|
| 2283 | ideal Ctemp=ideal(1); |
---|
| 2284 | while(deg(Ctemp[1])==0) |
---|
| 2285 | { |
---|
| 2286 | Ctemp=C; |
---|
| 2287 | for(i=1;i<=nrows(M[1]);i++) |
---|
| 2288 | { |
---|
| 2289 | Ctemp=Ctemp,deltaL[i][bmin]; |
---|
| 2290 | } |
---|
| 2291 | Ctemp=std(Ctemp); |
---|
| 2292 | bmin--; |
---|
| 2293 | if(bmin==0){ERROR("empty set");} |
---|
| 2294 | } |
---|
| 2295 | //!!!!!!!!!!!!!Invariante ist bmin/alpha |
---|
| 2296 | // naechster Eintrag s_i wie in CenterBM |
---|
| 2297 | // dann dropDim .... |
---|
| 2298 | } |
---|
| 2299 | |
---|
| 2300 | ///////////////////////////////////////////////////////////////////////////// |
---|
[268a430] | 2301 | static proc deleteInt(intvec v,int i,int ini) |
---|
[2e6eac2] | 2302 | { |
---|
| 2303 | //!!! Should be in kernel of Singular |
---|
| 2304 | //--- delete i-th entry in intvec v, |
---|
| 2305 | //--- if necessary reinitializing v with value ini |
---|
| 2306 | int s=size(v); |
---|
| 2307 | intvec w; |
---|
| 2308 | if((i<s)&&(i>1)){w=v[1..i-1],v[i+1..s];} |
---|
| 2309 | if(s==1){w=ini;return(w);} |
---|
| 2310 | if(i==1){w=v[2..s];} |
---|
| 2311 | if(i==s){w=v[1..s-1];} |
---|
| 2312 | return(w); |
---|
| 2313 | } |
---|
| 2314 | ///////////////////////////////////////////////////////////////////////////// |
---|
[268a430] | 2315 | static proc concatInt(intmat A, intmat B) |
---|
[2e6eac2] | 2316 | { |
---|
| 2317 | //!!! Should be in kernel of Singular |
---|
| 2318 | //--- concatenate two intmats |
---|
| 2319 | if(nrows(A)!=nrows(B)){ERROR("could not concat, wrong number of rows");} |
---|
| 2320 | intmat tempmat[nrows(A)][ncols(A)+ncols(B)]; |
---|
| 2321 | tempmat[1..nrows(A),1..ncols(A)]=A[1..nrows(A),1..ncols(A)]; |
---|
| 2322 | tempmat[1..nrows(A),ncols(A)+1..ncols(tempmat)]=B[1..nrows(A),1..ncols(B)]; |
---|
| 2323 | return(tempmat); |
---|
| 2324 | } |
---|
| 2325 | ///////////////////////////////////////////////////////////////////////////// |
---|
| 2326 | proc dropDim(list BM) |
---|
| 2327 | { |
---|
| 2328 | ERROR("Not implemented yet"); |
---|
| 2329 | } |
---|
| 2330 | ///////////////////////////////////////////////////////////////////////////// |
---|
| 2331 | proc CenterBO(list BO,list #) |
---|
| 2332 | "USAGE: CenterBO(BO); |
---|
| 2333 | @* BO = basic object, a list: ideal W, |
---|
| 2334 | @* ideal J, |
---|
| 2335 | @* intvec b, |
---|
| 2336 | @* list Ex, |
---|
| 2337 | @* ideal ab, |
---|
| 2338 | @* intvec v, |
---|
| 2339 | @* intvec w, |
---|
| 2340 | @* matrix M |
---|
| 2341 | ASSUME: R = basering, a polynomial ring, W an ideal of R, |
---|
| 2342 | @* J = ideal containing W |
---|
| 2343 | COMPUTE: the center of the next blow-up of BO in the resolution algorithm |
---|
| 2344 | of [Bravo,Encinas,Villamayor] |
---|
| 2345 | RETURN: list l, |
---|
[906458] | 2346 | l[1]: ideal describing the center@* |
---|
| 2347 | l[2]: intvec w obtained in the process of determining l[1]@* |
---|
| 2348 | l[3]: intvec b obtained in the process of determining l[1]@* |
---|
[2e6eac2] | 2349 | l[4]: intvec inv obtained in the process of determining l[1] |
---|
| 2350 | EXAMPLE: example CenterBO; shows an example |
---|
| 2351 | " |
---|
| 2352 | { |
---|
| 2353 | //--------------------------------------------------------------------------- |
---|
| 2354 | // Initialization and sanity checks |
---|
| 2355 | //--------------------------------------------------------------------------- |
---|
| 2356 | int i,bo7save; |
---|
| 2357 | intvec tvec; |
---|
| 2358 | //--- re=center,E^- indices, b vector, n vector |
---|
| 2359 | list re=ideal(1),BO[7],BO[3],tvec; |
---|
| 2360 | ideal J=BO[2]; |
---|
| 2361 | if(size(J)==0) |
---|
| 2362 | { |
---|
| 2363 | re[1]=ideal(0); |
---|
| 2364 | return(re); |
---|
| 2365 | } |
---|
| 2366 | //--- find Delta^(b-1)(J) |
---|
| 2367 | if(size(reduce(J,std(BO[1])))!=0) |
---|
| 2368 | { |
---|
| 2369 | list L=DeltaList(BO); |
---|
| 2370 | } |
---|
| 2371 | else |
---|
| 2372 | { |
---|
| 2373 | list L; |
---|
| 2374 | L[1]=J; |
---|
| 2375 | } |
---|
| 2376 | if(!defined(debugCenter)) |
---|
| 2377 | { |
---|
| 2378 | int debugCenter; |
---|
| 2379 | } |
---|
| 2380 | if(debugCenter) |
---|
| 2381 | { |
---|
| 2382 | "----> In Center: after DeltaList"; |
---|
| 2383 | "W"; |
---|
| 2384 | BO[1]; |
---|
| 2385 | "J"; |
---|
| 2386 | BO[2]; |
---|
| 2387 | "The Delta List:"; |
---|
| 2388 | L; |
---|
| 2389 | } |
---|
| 2390 | int b=size(L); |
---|
| 2391 | if(b==0) |
---|
| 2392 | { |
---|
| 2393 | //--- if J=W, we do not need to do anything |
---|
| 2394 | //--- returning center=1 marks this chart as completed |
---|
| 2395 | return(re); |
---|
| 2396 | } |
---|
| 2397 | //--------------------------------------------------------------------------- |
---|
| 2398 | // check whether max w-ord is constant |
---|
| 2399 | //--------------------------------------------------------------------------- |
---|
| 2400 | if(b==BO[3][1]) |
---|
| 2401 | { |
---|
| 2402 | //--- max w-ord is constant |
---|
| 2403 | if(BO[7][1]==-1) |
---|
| 2404 | { |
---|
| 2405 | //--- first got its value in the previous step ==> initialize BO[7] |
---|
| 2406 | tvec[1]=size(BO[4])-1; |
---|
| 2407 | for(i=2;i<=size(BO[7]);i++) |
---|
| 2408 | { |
---|
| 2409 | tvec[i]=BO[7][i]; |
---|
| 2410 | } |
---|
| 2411 | re[2]=tvec; |
---|
| 2412 | BO[7]=tvec; |
---|
| 2413 | } |
---|
| 2414 | } |
---|
| 2415 | else |
---|
| 2416 | { |
---|
| 2417 | //--- max w-ord changed ==> reset BO[7], correct BO[3] |
---|
| 2418 | tvec[1]=-1; |
---|
| 2419 | re[2]=tvec; |
---|
| 2420 | BO[7]=tvec; |
---|
| 2421 | tvec[1]=b; |
---|
| 2422 | BO[3]=tvec; |
---|
| 2423 | if(defined(invSat)) |
---|
| 2424 | { |
---|
| 2425 | invSat[2]=intvec(0); |
---|
| 2426 | } |
---|
| 2427 | } |
---|
| 2428 | re[3]=BO[3]; |
---|
| 2429 | //--------------------------------------------------------------------------- |
---|
| 2430 | // reduce from case 2 to case 1 of [Bravo, Encinas, Villamayor] |
---|
| 2431 | //--------------------------------------------------------------------------- |
---|
| 2432 | ideal C=L[b]; |
---|
| 2433 | BO[2]=C; |
---|
| 2434 | if(debugCenter) |
---|
| 2435 | { |
---|
| 2436 | "----> In Center: before intersection with E_i:"; |
---|
| 2437 | "bmax:",b; |
---|
| 2438 | "Sing(J,bmax):"; |
---|
| 2439 | C; |
---|
| 2440 | "E:"; |
---|
| 2441 | BO[4]; |
---|
| 2442 | "list marking a priori known intersection properties:",BO[6]; |
---|
| 2443 | "index of last element of E^- in E:",BO[7][1]; |
---|
| 2444 | } |
---|
| 2445 | //--- is intermediate result in iteration already good? |
---|
| 2446 | //--- return it to calling proc CenterBO |
---|
| 2447 | if(size(#)>0) |
---|
| 2448 | { |
---|
| 2449 | if(#[1]==2) |
---|
| 2450 | { |
---|
| 2451 | re[1]=C; |
---|
| 2452 | kill tvec; |
---|
| 2453 | intvec tvec=re[2][1]; |
---|
| 2454 | re[2]=tvec; |
---|
| 2455 | kill tvec; |
---|
| 2456 | intvec tvec=re[3][1]; |
---|
| 2457 | re[3]=tvec; |
---|
| 2458 | kill tvec; |
---|
| 2459 | intvec tvec; |
---|
| 2460 | re[4]=tvec; |
---|
| 2461 | return(re); |
---|
| 2462 | } |
---|
| 2463 | } |
---|
| 2464 | //--- do the reduction to case 1 |
---|
| 2465 | list E=inters_E(BO); |
---|
| 2466 | //--- if J is smooth and not too many E_i intersect simultaneously, let us |
---|
| 2467 | //--- try to drop redundant components of the candidate for the center |
---|
| 2468 | if((b==1)&&(size(E)>3)) |
---|
| 2469 | { |
---|
| 2470 | //--- if J is not smooth we do not want to drop any information |
---|
| 2471 | if((size(E[4])>0) && (dim(std(slocusE(BO[2])))<0)) |
---|
| 2472 | { |
---|
| 2473 | //--- BO[2]==J because b==1 |
---|
| 2474 | //--- DropRedundant is the counterpart to DropCoeff |
---|
| 2475 | //--- do not leave out one of them separately!!! |
---|
| 2476 | E=DropRedundant(BO,E); |
---|
| 2477 | if(size(E)==1) |
---|
| 2478 | { |
---|
| 2479 | kill tvec; |
---|
| 2480 | intvec tvec=re[2][1]; |
---|
| 2481 | re[2]=tvec; |
---|
| 2482 | tvec[1]=re[3][1]; |
---|
| 2483 | re[3]=tvec; |
---|
| 2484 | tvec[1]=re[4][1]; |
---|
| 2485 | re[4]=tvec; |
---|
| 2486 | re[1]=E[1]; |
---|
| 2487 | return(re); |
---|
| 2488 | } |
---|
| 2489 | } |
---|
| 2490 | } |
---|
| 2491 | //--- set n correctly |
---|
| 2492 | if(E[2]<BO[9][1]) |
---|
| 2493 | { |
---|
| 2494 | //--- if n dropped, the subsequent Coeff-object will not be the same |
---|
| 2495 | //--- ===> set BO[3][2] to zero to make sure that no previous data is used |
---|
| 2496 | if(defined(tvec)) {kill tvec;} |
---|
| 2497 | intvec tvec=BO[7][1],-1; |
---|
| 2498 | BO[7]=tvec; |
---|
| 2499 | tvec=BO[3][1],0; |
---|
| 2500 | BO[3]=tvec; |
---|
| 2501 | if(defined(invSat)) |
---|
| 2502 | { |
---|
| 2503 | invSat[2]=intvec(0); |
---|
| 2504 | } |
---|
| 2505 | } |
---|
| 2506 | re[4][1]=E[2]; |
---|
| 2507 | C=E[1]^b+J; |
---|
| 2508 | C=mstd(C)[2]; |
---|
| 2509 | ideal C1=std(ideal(L[b])+E[1]); |
---|
| 2510 | if(debugCenter) |
---|
| 2511 | { |
---|
| 2512 | "----> In Center: reduction of case 2 to case 1"; |
---|
| 2513 | "Output of inters_E, after dropping redundant components:"; |
---|
| 2514 | E; |
---|
| 2515 | "result of intersection with E^-, i.e.(E^-)^b+J:"; |
---|
| 2516 | C; |
---|
| 2517 | "candidate for center:"; |
---|
| 2518 | C1; |
---|
| 2519 | } |
---|
| 2520 | //--------------------------------------------------------------------------- |
---|
| 2521 | // Check whether we have a hypersurface component |
---|
| 2522 | //--------------------------------------------------------------------------- |
---|
| 2523 | if(dim(C1)==dim(std(BO[1]))-1) |
---|
| 2524 | { |
---|
[6518eba] | 2525 | if((size(reduce(J,C1,5))==0)&&(size(reduce(C1,std(J),5))==0)) |
---|
[2e6eac2] | 2526 | { |
---|
| 2527 | //--- C1 equals J and is of codimension 1 in W |
---|
| 2528 | re[1]=C1; |
---|
| 2529 | } |
---|
| 2530 | else |
---|
| 2531 | { |
---|
| 2532 | //--- C1 has a codimension 1 (in W) component |
---|
| 2533 | re[1]=std(equiRadical(C1)); |
---|
| 2534 | } |
---|
| 2535 | kill tvec; |
---|
| 2536 | intvec tvec=re[2][1]; |
---|
| 2537 | re[2]=tvec; |
---|
| 2538 | tvec[1]=re[3][1]; |
---|
| 2539 | re[3]=tvec; |
---|
| 2540 | tvec[1]=re[4][1]; |
---|
| 2541 | re[4]=tvec; |
---|
| 2542 | //--- is the codimension 1 component a good choice or do we need to reset |
---|
| 2543 | //--- the information from the previous steps |
---|
| 2544 | if(transversalT(re[1],BO[4])) |
---|
| 2545 | { |
---|
| 2546 | if(size(E)>2) |
---|
| 2547 | { |
---|
| 2548 | if(E[3]>E[2]) |
---|
| 2549 | { |
---|
| 2550 | if(defined(shortcut)){kill shortcut;} |
---|
| 2551 | list shortcut=ideal(0),size(BO[4]),BO[7]; |
---|
| 2552 | export(shortcut); |
---|
| 2553 | } |
---|
| 2554 | } |
---|
| 2555 | return(re); |
---|
| 2556 | } |
---|
| 2557 | |
---|
| 2558 | ERROR("reset in Center, please send the example to the authors."); |
---|
| 2559 | } |
---|
| 2560 | //--------------------------------------------------------------------------- |
---|
| 2561 | // Check whether it is a single point |
---|
| 2562 | //--------------------------------------------------------------------------- |
---|
| 2563 | if(dim(C1)==0) |
---|
| 2564 | { |
---|
| 2565 | C1=std(radical(C1)); |
---|
| 2566 | if(vdim(C1)==1) |
---|
| 2567 | { |
---|
| 2568 | //--- C1 is one point |
---|
| 2569 | re[1]=C1; |
---|
| 2570 | kill tvec; |
---|
| 2571 | intvec tvec=re[2][1]; |
---|
| 2572 | re[2]=tvec; |
---|
| 2573 | kill tvec; |
---|
| 2574 | intvec tvec=re[3][1]; |
---|
| 2575 | re[3]=tvec; |
---|
| 2576 | return(re); |
---|
| 2577 | } |
---|
| 2578 | } |
---|
| 2579 | //--------------------------------------------------------------------------- |
---|
| 2580 | // Prepare input for forming the Coeff-Ideal |
---|
| 2581 | //--------------------------------------------------------------------------- |
---|
| 2582 | BO[2]=C; |
---|
| 2583 | if(size(BO[2])>5) |
---|
| 2584 | { |
---|
| 2585 | BO[2]=mstd(BO[2])[2]; |
---|
| 2586 | } |
---|
| 2587 | //--- drop leading entry of BO[3] |
---|
| 2588 | tvec=BO[3]; |
---|
| 2589 | if(size(tvec)>1) |
---|
| 2590 | { |
---|
| 2591 | tvec=tvec[2..size(tvec)]; |
---|
| 2592 | BO[3]=tvec; |
---|
| 2593 | } |
---|
| 2594 | else |
---|
| 2595 | { |
---|
| 2596 | BO[3][1]=0; |
---|
| 2597 | } |
---|
| 2598 | tvec=BO[9]; |
---|
| 2599 | if(size(tvec)>1) |
---|
| 2600 | { |
---|
| 2601 | tvec=tvec[2..size(tvec)]; |
---|
| 2602 | BO[9]=tvec; |
---|
| 2603 | } |
---|
| 2604 | else |
---|
| 2605 | { |
---|
| 2606 | BO[9][1]=0; |
---|
| 2607 | } |
---|
| 2608 | bo7save=BO[7][1]; // original value needed for result |
---|
| 2609 | if(defined(shortcut)) |
---|
| 2610 | { |
---|
| 2611 | if((bo7save!=shortcut[3][1])&&(size(shortcut[3])!=1)) |
---|
| 2612 | { |
---|
| 2613 | kill shortcut; |
---|
| 2614 | } |
---|
| 2615 | else |
---|
| 2616 | { |
---|
| 2617 | shortcut[2]=shortcut[2]-bo7save; |
---|
| 2618 | tvec=shortcut[3]; |
---|
| 2619 | if(size(tvec)>1) |
---|
| 2620 | { |
---|
| 2621 | tvec=tvec[2..size(tvec)]; |
---|
| 2622 | shortcut[3]=tvec; |
---|
| 2623 | } |
---|
| 2624 | else |
---|
| 2625 | { |
---|
| 2626 | shortcut[3]=intvec(shortcut[2]); |
---|
| 2627 | } |
---|
| 2628 | } |
---|
| 2629 | } |
---|
| 2630 | if(BO[7][1]>-1) |
---|
| 2631 | { |
---|
| 2632 | //--- drop E^- and the corresponding information from BO[6] |
---|
| 2633 | for(i=1;i<=BO[7][1];i++) |
---|
| 2634 | { |
---|
| 2635 | BO[4]=delete(BO[4],1); |
---|
| 2636 | intvec bla1=BO[6]; |
---|
| 2637 | BO[6]=intvec(bla1[2..size(bla1)]); |
---|
| 2638 | kill bla1; |
---|
| 2639 | } |
---|
| 2640 | //--- drop leading entry of BO[7] |
---|
| 2641 | tvec=BO[7]; |
---|
| 2642 | if(size(tvec)>1) |
---|
| 2643 | { |
---|
| 2644 | tvec=tvec[2..size(tvec)]; |
---|
| 2645 | BO[7]=tvec; |
---|
| 2646 | } |
---|
| 2647 | else |
---|
| 2648 | { |
---|
| 2649 | BO[7][1]=-1; |
---|
| 2650 | } |
---|
| 2651 | } |
---|
| 2652 | else |
---|
| 2653 | { |
---|
| 2654 | if(BO[7][1]==-1) |
---|
| 2655 | { |
---|
| 2656 | list tplist; |
---|
| 2657 | BO[4]=tplist; |
---|
| 2658 | kill tplist; |
---|
| 2659 | } |
---|
| 2660 | } |
---|
| 2661 | if(debugCenter) |
---|
| 2662 | { |
---|
| 2663 | "----> In Center: Input to Coeff"; |
---|
| 2664 | "b:",b; |
---|
| 2665 | "BO:"; |
---|
| 2666 | BO; |
---|
| 2667 | } |
---|
| 2668 | //--- prepare the third entry of the invariant tuple |
---|
| 2669 | int invSatSave=invSat[2][1]; |
---|
| 2670 | tvec=invSat[2]; |
---|
| 2671 | if(size(tvec)>1) |
---|
| 2672 | { |
---|
| 2673 | tvec=tvec[2..size(tvec)]; |
---|
| 2674 | invSat[2]=tvec; |
---|
| 2675 | } |
---|
| 2676 | else |
---|
| 2677 | { |
---|
| 2678 | invSat[2][1]=0; |
---|
| 2679 | } |
---|
| 2680 | //--------------------------------------------------------------------------- |
---|
| 2681 | // Form the Coeff-ideal, if possible and useful; otherwise use the previous |
---|
| 2682 | // candidate for the center |
---|
| 2683 | //--------------------------------------------------------------------------- |
---|
| 2684 | list BO1=Coeff(BO,b); |
---|
| 2685 | if(debugCenter) |
---|
| 2686 | { |
---|
| 2687 | "----> In Center: Output of Coeff"; |
---|
| 2688 | BO1; |
---|
| 2689 | } |
---|
| 2690 | //--- Coeff returns int if something went wrong |
---|
| 2691 | if(typeof(BO1[1])=="int") |
---|
| 2692 | { |
---|
| 2693 | if(BO1[1]==0) |
---|
| 2694 | { |
---|
| 2695 | //--- Coeff ideal was already resolved |
---|
| 2696 | re[1]=C1; |
---|
| 2697 | return(re); |
---|
| 2698 | } |
---|
| 2699 | else |
---|
| 2700 | { |
---|
| 2701 | //--- no global hypersurface found |
---|
| 2702 | re=CoverCenter(BO,b,BO1[2]); |
---|
| 2703 | kill tvec; |
---|
| 2704 | intvec tvec=invSatSave; |
---|
| 2705 | for(i=1;i<=size(invSat[2]);i++) |
---|
| 2706 | { |
---|
| 2707 | tvec[i+1]=invSat[2][i]; |
---|
| 2708 | } |
---|
| 2709 | invSat[2]=tvec; |
---|
| 2710 | return(re); |
---|
| 2711 | } |
---|
| 2712 | } |
---|
| 2713 | int coeff_invar; |
---|
| 2714 | ideal Idropped=1; |
---|
| 2715 | //--- if b=1 drop redundant components of the Coeff-ideal |
---|
| 2716 | if(b==1) |
---|
| 2717 | { |
---|
| 2718 | //--- Counterpart to DropRedundant -- do not leave out one of them separately |
---|
| 2719 | Idropped=DropCoeff(BO1); // blow-up in these components |
---|
| 2720 | // is unnecessary |
---|
| 2721 | } |
---|
| 2722 | //--- to switch off DropCoeff, set Idropped=1; |
---|
[3f7e01a] | 2723 | BO1[2]=sat(BO1[2],Idropped); |
---|
[2e6eac2] | 2724 | if(deg(BO1[2][1])==0) |
---|
| 2725 | { |
---|
| 2726 | //--- Coeff ideal is trivial |
---|
| 2727 | C1=radical(C1); |
---|
[3f7e01a] | 2728 | ideal C2=sat(C1,Idropped); |
---|
[2e6eac2] | 2729 | if(deg(std(C2)[1])!=0) |
---|
| 2730 | { |
---|
| 2731 | C1=C2; |
---|
| 2732 | } |
---|
| 2733 | //Aenderung: Strategie: nur im Notfall ganze except. Divisoren |
---|
| 2734 | if(deg(std(BO1[2])[1])==0) |
---|
| 2735 | { |
---|
| 2736 | list BOtemp=BO; |
---|
| 2737 | int bo17save=BO1[7][1]; |
---|
| 2738 | BOtemp[7]=0; |
---|
| 2739 | BO1=Coeff(BOtemp,b,int(0)); |
---|
[3f7e01a] | 2740 | BO1[2]=sat(BO1[2],Idropped); |
---|
[2e6eac2] | 2741 | if(deg(std(BO1[2])[1])==0) |
---|
| 2742 | { |
---|
| 2743 | //--- there is really nothing left to do for the Coeff ideal |
---|
| 2744 | //--- the whole original BO1[2], i.e. Idropped, is the upcoming center |
---|
| 2745 | re[1]=Idropped; |
---|
| 2746 | re[2]=intvec(bo7save); |
---|
| 2747 | re[3]=intvec(b); |
---|
| 2748 | re[4]=intvec(E[2]); |
---|
| 2749 | return(re); |
---|
| 2750 | } |
---|
| 2751 | if(deg(std(slocus(radical(BO1[2])))[1])==0) |
---|
| 2752 | { |
---|
| 2753 | re[1]=BO1[2]; |
---|
| 2754 | // re[2]=intvec(bo7save,BO1[7][1]); |
---|
| 2755 | re[2]=intvec(bo7save,bo17save); |
---|
| 2756 | re[3]=intvec(b,1); |
---|
| 2757 | re[4]=intvec(E[2],1); |
---|
| 2758 | invSat[2]=intvec(1,0); |
---|
| 2759 | return(re); |
---|
| 2760 | } |
---|
| 2761 | //!!! effizienter machen??? |
---|
| 2762 | list pr=primdecGTZ(BO1[2]); |
---|
| 2763 | ideal Itemp1=1; |
---|
| 2764 | int aa,bb; |
---|
| 2765 | for(aa=1;aa<=size(pr);aa++) |
---|
| 2766 | { |
---|
| 2767 | if(dim(std(pr[aa][2])) < (dim(std(BO1[1]))-1)) |
---|
| 2768 | { |
---|
| 2769 | //--- drop components which are themselves exceptional diviosrs |
---|
| 2770 | Itemp1=intersect(Itemp1,pr[aa][1]); |
---|
| 2771 | } |
---|
| 2772 | } |
---|
| 2773 | if(deg(std(Itemp1)[1])!=0) |
---|
| 2774 | { |
---|
| 2775 | //--- treat the remaining components of the weak Coeff ideal |
---|
| 2776 | BO1[2]=Itemp1; |
---|
| 2777 | } |
---|
| 2778 | BO1[7]=BO[7]; |
---|
| 2779 | for(aa=1;aa<=size(BO1[4]);aa++) |
---|
| 2780 | { |
---|
| 2781 | if(deg(std(BO1[4][aa])[1])==0){aa++;continue;} |
---|
| 2782 | if(defined(satlist)){kill satlist;} |
---|
[3f7e01a] | 2783 | list satlist=sat_with_exp(BO1[2],BO1[4][aa]+BO1[1]); |
---|
[2e6eac2] | 2784 | if(deg(std(satlist[1])[1])==0) |
---|
| 2785 | { |
---|
| 2786 | coeff_invar++; |
---|
| 2787 | if(satlist[2]!=0) |
---|
| 2788 | { |
---|
| 2789 | for(bb=1;bb<=satlist[2]-1;bb++) |
---|
| 2790 | { |
---|
| 2791 | BO1[2]=quotient(BO1[2],BO1[4][aa]+BO1[1]); |
---|
| 2792 | } |
---|
| 2793 | } |
---|
| 2794 | else |
---|
| 2795 | { |
---|
| 2796 | ERROR("J of temporary object had unexpected value; |
---|
| 2797 | please send this example to the authors."); |
---|
| 2798 | } |
---|
| 2799 | } |
---|
| 2800 | else |
---|
| 2801 | { |
---|
| 2802 | BO1[2]=satlist[1]; |
---|
| 2803 | } |
---|
| 2804 | } |
---|
| 2805 | if(deg(std(Itemp1)[1])==0) |
---|
| 2806 | { |
---|
| 2807 | re[1]=BO1[2]; |
---|
| 2808 | re[2]=intvec(bo7save,BO1[7][1]); |
---|
| 2809 | re[3]=intvec(b,1); |
---|
| 2810 | re[4]=intvec(E[2],1); |
---|
| 2811 | invSat[2]=intvec(1,0); |
---|
| 2812 | return(re); |
---|
| 2813 | } |
---|
| 2814 | kill aa,bb; |
---|
| 2815 | } |
---|
| 2816 | } |
---|
| 2817 | if(invSatSave<coeff_invar) |
---|
| 2818 | { |
---|
| 2819 | invSatSave=coeff_invar; |
---|
| 2820 | } |
---|
| 2821 | //--------------------------------------------------------------------------- |
---|
| 2822 | // Determine Center of Coeff-ideal and use it as the new center |
---|
| 2823 | //--------------------------------------------------------------------------- |
---|
| 2824 | if(!defined(templist)) |
---|
| 2825 | { |
---|
| 2826 | if(size(BO1[2])>5) |
---|
| 2827 | { |
---|
| 2828 | BO1[2]=mstd(BO1[2])[2]; |
---|
| 2829 | } |
---|
| 2830 | list templist=CenterBO(BO1,2); |
---|
| 2831 | //--- only a sophisticated guess of a good center computed by |
---|
| 2832 | //--- leaving center before intersection with the E_i. |
---|
| 2833 | //--- whether the guess was good, is stored in 'good'. |
---|
| 2834 | //--- (this variant saves charts in cases like the Whitney umbrella) |
---|
| 2835 | list E0,E1; |
---|
| 2836 | |
---|
| 2837 | ideal Cstd=std(radical(templist[1])); |
---|
| 2838 | int good=((deg(std(slocusE(Cstd))[1])==0)&&(dim(std(BO1[2]))<=2)); |
---|
| 2839 | //if(defined(satlist)){good=0;} |
---|
| 2840 | if(good) |
---|
| 2841 | { |
---|
| 2842 | for(i=1;i<=size(BO[4]);i++) |
---|
| 2843 | { |
---|
[6518eba] | 2844 | if((deg(BO[4][i][1])>0)&&(size(reduce(BO[4][i],Cstd,5))!=0)) |
---|
[2e6eac2] | 2845 | { |
---|
| 2846 | E0[size(E0)+1]=BO[4][i]+Cstd; |
---|
| 2847 | E1[size(E1)+1]=BO[4][i]; |
---|
| 2848 | } |
---|
| 2849 | } |
---|
| 2850 | good=transversalT(Cstd,E1); |
---|
| 2851 | if(good) |
---|
| 2852 | { |
---|
| 2853 | good=normalCross(E0); |
---|
| 2854 | } |
---|
| 2855 | } |
---|
| 2856 | if(good) |
---|
| 2857 | { |
---|
| 2858 | list templist2=CenterBO(BO1,1); |
---|
| 2859 | if(dim(std(templist2[1]))!=dim(Cstd)) |
---|
| 2860 | { |
---|
| 2861 | templist[1]=Cstd; |
---|
| 2862 | if(defined(shortcut)){kill shortcut;} |
---|
| 2863 | list shortcut=ideal(0),size(BO1[4]),templist[2]; |
---|
| 2864 | export(shortcut); |
---|
| 2865 | } |
---|
| 2866 | else |
---|
| 2867 | { |
---|
| 2868 | templist=templist2; |
---|
| 2869 | } |
---|
| 2870 | kill templist2; |
---|
| 2871 | } |
---|
| 2872 | else |
---|
| 2873 | { |
---|
| 2874 | //--- sophisticated guess was wrong, follow Villamayor's approach |
---|
| 2875 | kill templist; |
---|
| 2876 | list templist=CenterBO(BO1,1); |
---|
| 2877 | } |
---|
| 2878 | } |
---|
| 2879 | if((dim(std(templist[1]))==dim(std(BO1[1]))-1) |
---|
| 2880 | &&(size(templist[4])==1)) |
---|
| 2881 | { |
---|
| 2882 | if(templist[4][1]==0) |
---|
| 2883 | { |
---|
| 2884 | for(i=1;i<=size(BO1[4]);i++) |
---|
| 2885 | { |
---|
[6518eba] | 2886 | if(size(reduce(templist[1],std(BO1[4][i]),5))==0) |
---|
[2e6eac2] | 2887 | { |
---|
| 2888 | templist[4][1]=1; |
---|
| 2889 | break; |
---|
| 2890 | } |
---|
| 2891 | } |
---|
| 2892 | } |
---|
| 2893 | } |
---|
| 2894 | //!!! subsequent line should be deleted |
---|
| 2895 | //if(defined(satlist)){templist[3][1]=BO[3][1];} |
---|
| 2896 | if(debugCenter) |
---|
| 2897 | { |
---|
| 2898 | "----> In Center: Iterated Center returned:"; |
---|
| 2899 | templist; |
---|
| 2900 | } |
---|
| 2901 | //-------------------------------------------------------------------------- |
---|
| 2902 | // set up the result and return it |
---|
| 2903 | //-------------------------------------------------------------------------- |
---|
| 2904 | re[1]=templist[1]; |
---|
| 2905 | kill tvec; |
---|
| 2906 | intvec tvec; |
---|
| 2907 | tvec[1]=bo7save; |
---|
| 2908 | for(i=1;i<=size(templist[2]);i++) |
---|
| 2909 | { |
---|
| 2910 | tvec[i+1]=templist[2][i]; |
---|
| 2911 | } |
---|
| 2912 | re[2]=tvec; |
---|
| 2913 | if(defined(shortcut)) |
---|
| 2914 | { |
---|
| 2915 | shortcut[2]=shortcut[2]+bo7save; |
---|
| 2916 | shortcut[3]=tvec; |
---|
| 2917 | } |
---|
| 2918 | kill tvec; |
---|
| 2919 | intvec tvec; |
---|
| 2920 | tvec[1]=invSatSave; |
---|
| 2921 | for(i=1;i<=size(invSat[2]);i++) |
---|
| 2922 | { |
---|
| 2923 | tvec[i+1]=invSat[2][i]; |
---|
| 2924 | } |
---|
| 2925 | invSat[2]=tvec; |
---|
| 2926 | kill tvec; |
---|
| 2927 | intvec tvec; |
---|
| 2928 | tvec[1]=b; |
---|
| 2929 | for(i=1;i<=size(templist[3]);i++) |
---|
| 2930 | { |
---|
| 2931 | tvec[i+1]=templist[3][i]; |
---|
| 2932 | } |
---|
| 2933 | re[3]=tvec; |
---|
| 2934 | kill tvec; |
---|
| 2935 | intvec tvec; |
---|
| 2936 | tvec[1]=E[2]; |
---|
| 2937 | for(i=1;i<=size(templist[4]);i++) |
---|
| 2938 | { |
---|
| 2939 | tvec[i+1]=templist[4][i]; |
---|
| 2940 | } |
---|
| 2941 | re[4]=tvec; |
---|
| 2942 | |
---|
| 2943 | return(re); |
---|
| 2944 | } |
---|
| 2945 | example |
---|
| 2946 | { "EXAMPLE:"; |
---|
| 2947 | echo = 2; |
---|
| 2948 | ring R=0,(x,y),dp; |
---|
| 2949 | |
---|
| 2950 | ideal W; |
---|
| 2951 | ideal J=x2-y3; |
---|
| 2952 | intvec b=1; |
---|
| 2953 | list E; |
---|
| 2954 | ideal abb=maxideal(1); |
---|
| 2955 | intvec v; |
---|
| 2956 | intvec w=-1; |
---|
| 2957 | matrix M; |
---|
| 2958 | |
---|
| 2959 | list BO=W,J,b,E,abb,v,w,M,v; |
---|
| 2960 | |
---|
| 2961 | CenterBO(BO); |
---|
| 2962 | } |
---|
| 2963 | ////////////////////////////////////////////////////////////////////////////// |
---|
[268a430] | 2964 | static proc CoverCenter(list BO,int b, ideal Jb) |
---|
[2e6eac2] | 2965 | { |
---|
| 2966 | //---------------------------------------------------------------------------- |
---|
| 2967 | // Initialization |
---|
| 2968 | //---------------------------------------------------------------------------- |
---|
| 2969 | def R=basering; |
---|
| 2970 | int i,j,k; |
---|
| 2971 | intvec merk,merk2,maxv,fvec; |
---|
| 2972 | list L,ceList,re; |
---|
| 2973 | ceList[1]=ideal(0); |
---|
| 2974 | poly @p,@f; |
---|
| 2975 | ideal K,dummy; |
---|
| 2976 | if(!attrib(BO[2],"isSB")) |
---|
| 2977 | { |
---|
| 2978 | BO[2]=std(BO[2]); |
---|
| 2979 | } |
---|
| 2980 | for(i=1;i<=size(Jb);i++) |
---|
| 2981 | { |
---|
| 2982 | list tempmstd=mstd(slocus(Jb[i])); |
---|
| 2983 | if(size(tempmstd[1])>size(tempmstd[2])) |
---|
| 2984 | { |
---|
| 2985 | dummy=tempmstd[2]; |
---|
| 2986 | } |
---|
| 2987 | else |
---|
| 2988 | { |
---|
| 2989 | dummy=tempmstd[1]; |
---|
| 2990 | } |
---|
| 2991 | kill tempmstd; |
---|
| 2992 | L[i]=dummy; |
---|
| 2993 | K=K,dummy; |
---|
| 2994 | } |
---|
| 2995 | K=simplify(K,2); |
---|
| 2996 | //--------------------------------------------------------------------------- |
---|
| 2997 | // The intersection of the singular loci of the L[i] is empty. |
---|
| 2998 | // Find a suitable open covering of the affine chart, such that a global |
---|
| 2999 | // hypersurface can be found in each open set. |
---|
| 3000 | //--------------------------------------------------------------------------- |
---|
| 3001 | matrix M=lift(K,ideal(1)); |
---|
| 3002 | j=1; |
---|
| 3003 | for(i=1;i<=nrows(M);i++) |
---|
| 3004 | { |
---|
| 3005 | if(M[i,1]!=0) |
---|
| 3006 | { |
---|
| 3007 | merk[size(merk)+1]=i; |
---|
| 3008 | fvec[size(merk)]=j; |
---|
| 3009 | } |
---|
| 3010 | if((i-k)==size(L[j])) |
---|
| 3011 | { |
---|
| 3012 | k=i; |
---|
| 3013 | j++; |
---|
| 3014 | } |
---|
| 3015 | } |
---|
| 3016 | //-------------------------------------------------------------------------- |
---|
| 3017 | // Find a candidate for the center in each open set |
---|
| 3018 | //-------------------------------------------------------------------------- |
---|
| 3019 | //--- first entry of merk is 0 by construction of merk |
---|
| 3020 | for(i=2;i<=size(merk);i++) |
---|
| 3021 | { |
---|
| 3022 | //--- open set is D(@p) |
---|
| 3023 | @p=K[merk[i]]; |
---|
| 3024 | //--- hypersurface is V(@f) |
---|
| 3025 | @f=Jb[fvec[i]]; |
---|
[62de185] | 3026 | ring R1 = create_ring(ring_list(R)[1], "(@y,"+varstr(R)+")", "dp", "no_minpoly"); |
---|
[2e6eac2] | 3027 | poly p=imap(R,@p); |
---|
| 3028 | poly f=imap(R,@f); |
---|
| 3029 | list @ce; |
---|
| 3030 | list BO=imap(R,BO); |
---|
| 3031 | BO[1]=BO[1]+ideal(@y*p-1); |
---|
| 3032 | BO[2]=BO[2]+ideal(@y*p-1); |
---|
| 3033 | for(j=1;j<=size(BO[4]);j++) |
---|
| 3034 | { |
---|
| 3035 | BO[4][j]=BO[4][j]+ideal(@y*p-1); |
---|
| 3036 | } |
---|
| 3037 | //--- like usual Coeff, but hypersurface is already known |
---|
| 3038 | list BO1=SpecialCoeff(BO,b,f); |
---|
| 3039 | //--- special situation in SpecialCoeff are marked by an error code of |
---|
| 3040 | //--- type int |
---|
| 3041 | if(typeof(BO1[1])=="int") |
---|
| 3042 | { |
---|
| 3043 | if(BO1[1]==0) |
---|
| 3044 | { |
---|
| 3045 | //--- Coeff ideal was already resolved |
---|
| 3046 | @ce[1]=BO[2]; |
---|
| 3047 | @ce[2]=BO[7]; |
---|
| 3048 | @ce[3]=BO[3]; |
---|
| 3049 | } |
---|
| 3050 | else |
---|
| 3051 | { |
---|
| 3052 | if(BO[3]!=0) |
---|
| 3053 | { |
---|
| 3054 | //--- intersections with E do not meet conditions ==> reset |
---|
[4bde6b] | 3055 | ERROR("reset in Coeff, please send the example to the authors"); |
---|
[2e6eac2] | 3056 | } |
---|
| 3057 | } |
---|
| 3058 | } |
---|
| 3059 | else |
---|
| 3060 | { |
---|
| 3061 | //--- now do the recursion as usual |
---|
| 3062 | @ce=CenterBO(BO1); |
---|
| 3063 | } |
---|
| 3064 | //--------------------------------------------------------------------------- |
---|
| 3065 | // Go back to the whole affine chart and form a suitable union of the |
---|
| 3066 | // candidates |
---|
| 3067 | //--------------------------------------------------------------------------- |
---|
| 3068 | //--- pass from open set to the whole affine chart by taking the closure |
---|
| 3069 | @ce[1]=eliminate(@ce[1],@y); |
---|
| 3070 | setring R; |
---|
| 3071 | ceList[i]=imap(R1,@ce); |
---|
| 3072 | //--- set up invariant vector and determine maximum value of it |
---|
| 3073 | if(size(ceList[i][3])==size(ceList[i][4])) |
---|
| 3074 | { |
---|
| 3075 | kill merk2,maxv; |
---|
| 3076 | intvec merk2,maxv; |
---|
| 3077 | for(j=1;j<=size(ceList[i][3]);j++) |
---|
| 3078 | { |
---|
| 3079 | merk2[2*j-1]=ceList[i][3][j]; |
---|
| 3080 | merk2[2*j]=ceList[i][4][j]; |
---|
| 3081 | ceList[i][5]=merk2; |
---|
| 3082 | if(maxv<merk2) |
---|
| 3083 | { |
---|
| 3084 | maxv=merk2; |
---|
| 3085 | } |
---|
| 3086 | } |
---|
| 3087 | } |
---|
| 3088 | else |
---|
| 3089 | { |
---|
| 3090 | ERROR("This situation should not occur, please send the example |
---|
| 3091 | to the authors."); |
---|
| 3092 | } |
---|
| 3093 | kill R1; |
---|
| 3094 | } |
---|
| 3095 | kill merk2; |
---|
| 3096 | intvec merk2=-2; |
---|
| 3097 | //--- form the union of the components of the center with maximum invariant |
---|
| 3098 | for(i=1;i<=size(ceList);i++) |
---|
| 3099 | { |
---|
[6518eba] | 3100 | if(size(reduce(ceList[i][1],BO[2],5))==0) |
---|
[2e6eac2] | 3101 | { |
---|
| 3102 | //--- already resolved ==> ignore |
---|
| 3103 | i++; |
---|
| 3104 | continue; |
---|
| 3105 | } |
---|
| 3106 | if(ceList[i][5]==maxv) |
---|
| 3107 | { |
---|
| 3108 | if(merk2!=ceList[i][2]) |
---|
| 3109 | { |
---|
| 3110 | //--- E^- not of the same size as before resp. initialization |
---|
| 3111 | if(merk2[1]==-2) |
---|
| 3112 | { |
---|
| 3113 | //--- initialization: save size of E^- |
---|
| 3114 | merk2=ceList[i][2]; |
---|
| 3115 | re[1]=ceList[i][1]; |
---|
| 3116 | re[2]=ceList[i][2]; |
---|
| 3117 | re[3]=ceList[i][3]; |
---|
| 3118 | re[4]=ceList[i][4]; |
---|
| 3119 | } |
---|
| 3120 | else |
---|
| 3121 | { |
---|
| 3122 | //--- otherwise ignore |
---|
| 3123 | i++; |
---|
| 3124 | continue; |
---|
| 3125 | } |
---|
| 3126 | } |
---|
| 3127 | else |
---|
| 3128 | { |
---|
| 3129 | re[1]=intersect(re[1],ceList[i][1]); |
---|
| 3130 | } |
---|
| 3131 | } |
---|
| 3132 | } |
---|
| 3133 | //-------------------------------------------------------------------------- |
---|
| 3134 | // Perform last checks and return the result |
---|
| 3135 | //-------------------------------------------------------------------------- |
---|
| 3136 | if(size(re)!=4) |
---|
| 3137 | { |
---|
| 3138 | //--- oops: already resolved in all open sets |
---|
| 3139 | re[1]=BO[2]; |
---|
| 3140 | re[2]=-1; |
---|
| 3141 | re[3]=0; |
---|
| 3142 | re[4]=intvec(0); |
---|
| 3143 | } |
---|
| 3144 | return(re); |
---|
| 3145 | } |
---|
| 3146 | ////////////////////////////////////////////////////////////////////////////// |
---|
[268a430] | 3147 | static proc SpecialCoeff(list BO,int b,poly f) |
---|
[2e6eac2] | 3148 | { |
---|
| 3149 | //---------------------------------------------------------------------------- |
---|
| 3150 | // Coeff with given hypersurface -- no checks of the hypersurface performed |
---|
| 3151 | //---------------------------------------------------------------------------- |
---|
| 3152 | int i,ch; |
---|
[b154b3] | 3153 | int e=int(factorial(b)); |
---|
[2e6eac2] | 3154 | ideal C; |
---|
| 3155 | list L=DeltaList(BO); |
---|
| 3156 | int d=size(L); |
---|
| 3157 | //--- set up ideal |
---|
| 3158 | for(i=0;i<b;i++) |
---|
| 3159 | { |
---|
| 3160 | C=C+L[i+1]^(e/(b-i)); |
---|
| 3161 | } |
---|
| 3162 | //--- move to hypersurface V(Z) |
---|
| 3163 | ideal Z=f; |
---|
| 3164 | C=C,Z; |
---|
| 3165 | BO[1]=BO[1]+Z; |
---|
| 3166 | BO[2]=C; |
---|
| 3167 | for(i=1;i<=size(BO[4]);i++) |
---|
| 3168 | { |
---|
| 3169 | BO[6][i]=0; // reset intersection indicator |
---|
| 3170 | BO[4][i]=BO[4][i]+Z; // intersect the E_i |
---|
[3f7e01a] | 3171 | BO[2]=sat(BO[2],BO[4][i]+BO[1]); |
---|
[2e6eac2] | 3172 | // "strict transform" of J w.r.t E, not "total" |
---|
| 3173 | } |
---|
| 3174 | return(BO); |
---|
| 3175 | } |
---|
| 3176 | |
---|
| 3177 | ////////////////////////////////////////////////////////////////////////////// |
---|
[268a430] | 3178 | static proc DropCoeff(list BO) |
---|
[2e6eac2] | 3179 | "Internal procedure - no help and no example available |
---|
| 3180 | " |
---|
| 3181 | { |
---|
| 3182 | //--- Initialization |
---|
| 3183 | int i,j; |
---|
| 3184 | list pr=minAssGTZ(BO[2]); |
---|
| 3185 | ideal I=BO[2]; |
---|
| 3186 | ideal Itemp; |
---|
| 3187 | ideal Idropped=1; |
---|
| 3188 | //--- Tests |
---|
| 3189 | for(i=1;i<=size(pr);i++) |
---|
| 3190 | { |
---|
| 3191 | if(i>size(pr)) |
---|
| 3192 | { |
---|
| 3193 | //--- the continue statement does not test the loop condition *sigh* |
---|
| 3194 | break; |
---|
| 3195 | } |
---|
| 3196 | if(deg(std(slocus(pr[i]))[1])!=0) |
---|
| 3197 | { |
---|
| 3198 | //--- this component is singular ===> we still need it |
---|
| 3199 | i++; |
---|
| 3200 | continue; |
---|
| 3201 | } |
---|
[3f7e01a] | 3202 | Itemp=sat(I,pr[i]); |
---|
[2e6eac2] | 3203 | if(deg(std(Itemp+pr[i])[1])!=0) |
---|
| 3204 | { |
---|
| 3205 | //--- this component is not disjoint from the other ones ===> we still need it |
---|
| 3206 | i++; |
---|
| 3207 | continue; |
---|
| 3208 | } |
---|
| 3209 | if(!transversalT(pr[i],BO[4])) |
---|
| 3210 | { |
---|
| 3211 | //--- this component does not meet one of the remaining E_i transversally |
---|
| 3212 | //--- ===> we still need it |
---|
| 3213 | i++; |
---|
| 3214 | continue; |
---|
| 3215 | } |
---|
| 3216 | if(!normalCross(BO[4],pr[i])) |
---|
| 3217 | { |
---|
| 3218 | //--- this component is not normal crossing with the remaining E_i |
---|
| 3219 | //--- ===> we still need it |
---|
| 3220 | i++; |
---|
| 3221 | continue; |
---|
| 3222 | } |
---|
| 3223 | if(defined(EE)){kill EE;} |
---|
| 3224 | list EE; |
---|
| 3225 | for(j=1;j<=size(BO[4]);j++) |
---|
| 3226 | { |
---|
| 3227 | EE[j]=BO[4][j]+pr[i]; |
---|
| 3228 | } |
---|
| 3229 | if(!normalCross(EE)) |
---|
| 3230 | { |
---|
| 3231 | //--- we do not have a normal crossing situation for this component after all |
---|
| 3232 | //--- ===> we still need it |
---|
| 3233 | i++; |
---|
| 3234 | continue; |
---|
| 3235 | } |
---|
| 3236 | Idropped=intersect(Idropped,pr[i]); |
---|
| 3237 | I=Itemp; |
---|
| 3238 | } |
---|
| 3239 | return(Idropped); |
---|
| 3240 | } |
---|
| 3241 | |
---|
| 3242 | ////////////////////////////////////////////////////////////////////////////// |
---|
[268a430] | 3243 | static proc DropRedundant(list BO,list E) |
---|
[2e6eac2] | 3244 | "Internal procedure - no help and no example available |
---|
| 3245 | " |
---|
| 3246 | { |
---|
| 3247 | //--------------------------------------------------------------------------- |
---|
| 3248 | // Initialization and sanity checks |
---|
| 3249 | //--------------------------------------------------------------------------- |
---|
| 3250 | int ii,jj,kkdiff,nonnormal,ok; |
---|
| 3251 | ideal testid,dummy; |
---|
| 3252 | ideal center; |
---|
| 3253 | intvec transverse,dontdrop,zerovec; |
---|
| 3254 | transverse[size(BO[4])]=0; |
---|
| 3255 | dontdrop[size(E[4])]=0; |
---|
| 3256 | zerovec[size(E[4])]=0; |
---|
| 3257 | ideal J=BO[2]; |
---|
| 3258 | int dimJ=dim(std(BO[2])); |
---|
| 3259 | list templist; |
---|
| 3260 | if(size(E)<5) |
---|
| 3261 | { |
---|
| 3262 | //--- should not occur |
---|
| 3263 | return(E); |
---|
| 3264 | } |
---|
| 3265 | for(ii=1;ii<=BO[7][1];ii++) |
---|
| 3266 | { |
---|
| 3267 | if(BO[6][ii]==2) |
---|
| 3268 | { |
---|
| 3269 | kkdiff++; |
---|
| 3270 | } |
---|
| 3271 | } |
---|
| 3272 | int expDim=dimJ-E[2]+kkdiff; |
---|
| 3273 | if(size(E)==6) |
---|
| 3274 | { |
---|
| 3275 | nonnormal=E[6]; |
---|
| 3276 | } |
---|
| 3277 | //--------------------------------------------------------------------------- |
---|
| 3278 | // if dimJ were smaller than E[2], we would not have more than 3 entries in |
---|
| 3279 | // in the list E |
---|
| 3280 | // if dimJ is also at least E[3] and nonnormal is 0, we only need to test that |
---|
| 3281 | // * the intersection is of the expected dimension |
---|
| 3282 | // * the intersections of the BO[4][i] and J are normal crossing |
---|
| 3283 | // * the elements of E^+ have no influence (is done below) |
---|
| 3284 | //--------------------------------------------------------------------------- |
---|
| 3285 | if((E[3]<=dimJ)&&(!nonnormal)) |
---|
| 3286 | { |
---|
| 3287 | ideal bla=E[1]+BO[2]+BO[1]; |
---|
| 3288 | bla=radical(bla); |
---|
| 3289 | bla=mstd(bla)[2]; |
---|
| 3290 | |
---|
| 3291 | if(dim(std(slocusE(bla)))<0) |
---|
| 3292 | { |
---|
| 3293 | if(transversalT(J,BO[4])) |
---|
| 3294 | { |
---|
| 3295 | ok=1; |
---|
| 3296 | if(E[2]==E[3]) |
---|
| 3297 | { |
---|
| 3298 | //--- no further intersection with elements from E^+ |
---|
| 3299 | for(ii=1;ii<=size(E[4]);ii++) |
---|
| 3300 | { |
---|
| 3301 | if(dim_slocus(BO[2]+E[4][ii])!=-1) |
---|
| 3302 | { |
---|
| 3303 | dontdrop[ii]=1; |
---|
| 3304 | } |
---|
| 3305 | } |
---|
| 3306 | if(dontdrop==zerovec) |
---|
| 3307 | { |
---|
| 3308 | list relist; |
---|
| 3309 | relist[1]=std(J); |
---|
| 3310 | return(relist); |
---|
| 3311 | } |
---|
| 3312 | } |
---|
| 3313 | } |
---|
| 3314 | } |
---|
| 3315 | } |
---|
| 3316 | //--------------------------------------------------------------------------- |
---|
| 3317 | // now check whether the E_i actually occurring in the intersections meet |
---|
| 3318 | // J transversally (one by one) and mark those elements of E[4] where it is |
---|
| 3319 | // not the case |
---|
| 3320 | //--------------------------------------------------------------------------- |
---|
| 3321 | if(!ok) |
---|
| 3322 | { |
---|
| 3323 | for(ii=1;ii<=size(E[5]);ii++) |
---|
| 3324 | { |
---|
| 3325 | //--- test the ii-th tuple of E[4] resp. its indices E[5] |
---|
| 3326 | for(jj=1;jj<=size(E[5][ii]);jj++) |
---|
| 3327 | { |
---|
| 3328 | //--- if E[5][ii][jj]==1, E_jj is involved in E[4][ii] |
---|
| 3329 | if(E[5][ii][jj]==1) |
---|
| 3330 | { |
---|
| 3331 | //--- transversality not yet determined |
---|
| 3332 | if(transverse[jj]==0) |
---|
| 3333 | { |
---|
| 3334 | templist[1]=BO[4][jj]; |
---|
| 3335 | if(transversalT(BO[2],templist)) |
---|
| 3336 | { |
---|
| 3337 | transverse[jj]=1; |
---|
| 3338 | } |
---|
| 3339 | else |
---|
| 3340 | { |
---|
| 3341 | transverse[jj]=-1; |
---|
| 3342 | dontdrop[ii]=1; |
---|
| 3343 | } |
---|
| 3344 | } |
---|
| 3345 | else |
---|
| 3346 | { |
---|
| 3347 | //--- already computed transversality |
---|
| 3348 | if(transverse[jj]<0) |
---|
| 3349 | { |
---|
| 3350 | dontdrop[ii]=1; |
---|
| 3351 | } |
---|
| 3352 | } |
---|
| 3353 | } |
---|
| 3354 | } |
---|
| 3355 | } |
---|
| 3356 | } |
---|
| 3357 | //--------------------------------------------------------------------------- |
---|
| 3358 | // if one of the non-marked tuples from E^- in E[4] has an intersection |
---|
| 3359 | // of the expected dimension and does not meet any E_i from E^+ |
---|
| 3360 | // - except the ones which are met trivially - , it should be |
---|
| 3361 | // dropped from the list. |
---|
| 3362 | // it can also be dropped if an intersection occurs and normal crossing has |
---|
| 3363 | // been checked. |
---|
| 3364 | //--------------------------------------------------------------------------- |
---|
| 3365 | for(ii=1;ii<=size(E[4]);ii++) |
---|
| 3366 | { |
---|
| 3367 | //--- if E[4][ii] does not have transversal intersections, we cannot drop it |
---|
| 3368 | if(dontdrop[ii]==1) |
---|
| 3369 | { |
---|
| 3370 | ii++; |
---|
| 3371 | continue; |
---|
| 3372 | } |
---|
| 3373 | //--- testing ii-th tuple from E[4] |
---|
| 3374 | testid=BO[1]+BO[2]+E[4][ii]; |
---|
| 3375 | if(dim(std(testid))!=expDim) |
---|
| 3376 | { |
---|
| 3377 | //--- not expected dimension |
---|
| 3378 | dontdrop[ii]=1; |
---|
| 3379 | ii++; |
---|
| 3380 | continue; |
---|
| 3381 | } |
---|
| 3382 | testid=mstd(testid)[2]; |
---|
| 3383 | |
---|
| 3384 | if(dim(std(slocusE(testid)))>=0) |
---|
| 3385 | { |
---|
| 3386 | //--- not smooth, i.e. more than one component which intersect |
---|
| 3387 | dontdrop[ii]=1; |
---|
| 3388 | ii++; |
---|
| 3389 | continue; |
---|
| 3390 | } |
---|
| 3391 | //--- if E^+ is empty, we are done; otherwise check intersections with E^+ |
---|
| 3392 | if(BO[7][1]!=-1) |
---|
| 3393 | { |
---|
| 3394 | if(defined(pluslist)){kill pluslist;} |
---|
| 3395 | list pluslist; |
---|
| 3396 | for(jj=BO[7][1]+1;jj<=size(BO[4]);jj++) |
---|
| 3397 | { |
---|
| 3398 | dummy=BO[4][jj]+testid; |
---|
| 3399 | dummy=std(dummy); |
---|
| 3400 | if(expDim==dim(dummy)) |
---|
| 3401 | { |
---|
| 3402 | //--- intersection has wrong dimension |
---|
| 3403 | dontdrop[ii]=1; |
---|
| 3404 | break; |
---|
| 3405 | } |
---|
| 3406 | pluslist[jj-BO[7][1]]=BO[4][jj]+testid; |
---|
| 3407 | } |
---|
| 3408 | if(dontdrop[ii]==1) |
---|
| 3409 | { |
---|
| 3410 | ii++; |
---|
| 3411 | continue; |
---|
| 3412 | } |
---|
| 3413 | if(!normalCross(pluslist)) |
---|
| 3414 | { |
---|
| 3415 | //--- unfortunately, it is not normal crossing |
---|
| 3416 | dontdrop[ii]=1; |
---|
| 3417 | } |
---|
| 3418 | } |
---|
| 3419 | } |
---|
| 3420 | //--------------------------------------------------------------------------- |
---|
| 3421 | // The returned list should look like the truncated output of inters_E |
---|
| 3422 | //--------------------------------------------------------------------------- |
---|
| 3423 | list retlist; |
---|
| 3424 | for(ii=1;ii<=size(E[4]);ii++) |
---|
| 3425 | { |
---|
| 3426 | if(dontdrop[ii]==1) |
---|
| 3427 | { |
---|
| 3428 | if(size(center)>0) |
---|
| 3429 | { |
---|
| 3430 | center=intersect(center,E[4][ii]); |
---|
| 3431 | } |
---|
| 3432 | else |
---|
| 3433 | { |
---|
| 3434 | center=E[4][ii]; |
---|
| 3435 | } |
---|
| 3436 | } |
---|
| 3437 | } |
---|
| 3438 | retlist[1]=center; |
---|
| 3439 | retlist[2]=E[2]; |
---|
| 3440 | retlist[3]=E[3]; |
---|
| 3441 | return(retlist); |
---|
| 3442 | } |
---|
| 3443 | ////////////////////////////////////////////////////////////////////////////// |
---|
[268a430] | 3444 | static proc transversalT(ideal J, list E,list #) |
---|
[2e6eac2] | 3445 | "Internal procedure - no help and no example available |
---|
| 3446 | " |
---|
| 3447 | { |
---|
| 3448 | //---------------------------------------------------------------------------- |
---|
| 3449 | // check whether J and each element of the list E meet transversally |
---|
| 3450 | //---------------------------------------------------------------------------- |
---|
| 3451 | def R=basering; |
---|
| 3452 | if(size(#)>0) |
---|
| 3453 | { |
---|
| 3454 | ideal pp=#[1]; |
---|
| 3455 | } |
---|
| 3456 | int i; |
---|
| 3457 | ideal T,M; |
---|
| 3458 | ideal Jstd=std(J); |
---|
| 3459 | ideal Tstd; |
---|
| 3460 | int d=nvars(basering)-dim(Jstd)+1; // d=n-dim(V(J) \cap hypersurface) |
---|
| 3461 | for(i=1;i<=size(E);i++) |
---|
| 3462 | { |
---|
[6518eba] | 3463 | if(size(reduce(E[i],Jstd,5))==0) |
---|
[2e6eac2] | 3464 | { |
---|
| 3465 | //--- V(J) is contained in E[i] |
---|
| 3466 | return(0); |
---|
| 3467 | } |
---|
| 3468 | T=J,E[i]; |
---|
| 3469 | Tstd=std(T); |
---|
| 3470 | d=nvars(basering)-dim(Tstd); |
---|
| 3471 | if(deg(Tstd[1])!=0) |
---|
| 3472 | { |
---|
| 3473 | //--- intersection is non-empty |
---|
| 3474 | //!!! abgeklemmt, da es doch in der Praxis vorkommt und korrekt sein kann!!! |
---|
| 3475 | //!!! wenn ueberhaupt dann -1 zurueckgeben!!! |
---|
| 3476 | // if((d>=4)&&(size(T)>=10)){return(0);} |
---|
| 3477 | M=minor(jacob(T),d,Tstd)+T; |
---|
| 3478 | M=std(M); |
---|
| 3479 | if(deg(M[1])>0) |
---|
| 3480 | { |
---|
| 3481 | //--- intersection is not transversal |
---|
| 3482 | if(size(#)==0) |
---|
| 3483 | { |
---|
| 3484 | return(0); |
---|
| 3485 | } |
---|
| 3486 | M=std(radical(M)); |
---|
[6518eba] | 3487 | if(size(reduce(pp,M,5))>0){return(0);} |
---|
[2e6eac2] | 3488 | } |
---|
| 3489 | } |
---|
| 3490 | } |
---|
| 3491 | //--- passed all tests |
---|
| 3492 | return(1); |
---|
| 3493 | } |
---|
| 3494 | /////////////////////////////////////////////////////////////////////////////// |
---|
[268a430] | 3495 | static proc transversalTB(ideal J, list E,ideal V) |
---|
[2e6eac2] | 3496 | "Internal procedure - no help and no example available |
---|
| 3497 | " |
---|
| 3498 | { |
---|
| 3499 | //---------------------------------------------------------------------------- |
---|
| 3500 | // check whether J and each element of the list E meet transversally |
---|
| 3501 | //---------------------------------------------------------------------------- |
---|
| 3502 | def R=basering; |
---|
| 3503 | |
---|
| 3504 | int i; |
---|
| 3505 | ideal T,M; |
---|
| 3506 | ideal Jstd=std(J); |
---|
| 3507 | ideal Tstd; |
---|
| 3508 | int d=nvars(basering)-dim(Jstd)+1; // d=n-dim(V(J) \cap hypersurface) |
---|
| 3509 | for(i=1;i<=size(E);i++) |
---|
| 3510 | { |
---|
[6518eba] | 3511 | if(size(reduce(E[i],Jstd,5))==0) |
---|
[2e6eac2] | 3512 | { |
---|
| 3513 | //--- V(J) is contained in E[i] |
---|
| 3514 | return(0); |
---|
| 3515 | } |
---|
| 3516 | T=J,E[i]; |
---|
| 3517 | Tstd=std(T); |
---|
| 3518 | d=nvars(basering)-dim(Tstd); |
---|
| 3519 | if(deg(Tstd[1])!=0) |
---|
| 3520 | { |
---|
| 3521 | //--- intersection is non-empty |
---|
| 3522 | if((d>=4)&&(size(T)>=10)){return(0);} |
---|
| 3523 | M=minor(jacob(T),d,Tstd)+T; |
---|
| 3524 | M=std(M+V); |
---|
| 3525 | if(deg(M[1])>0) |
---|
| 3526 | { |
---|
| 3527 | return(0); |
---|
| 3528 | } |
---|
| 3529 | } |
---|
| 3530 | } |
---|
| 3531 | //--- passed all tests |
---|
| 3532 | return(1); |
---|
| 3533 | } |
---|
| 3534 | /////////////////////////////////////////////////////////////////////////////// |
---|
[268a430] | 3535 | static proc powerI(ideal I,int n,int m) |
---|
[2e6eac2] | 3536 | { |
---|
| 3537 | //--- compute (n!/m)-th power of I, more efficient variant |
---|
| 3538 | int i; |
---|
| 3539 | int mon=1; |
---|
| 3540 | for(i=1;i<=size(I);i++) |
---|
| 3541 | { |
---|
| 3542 | if(size(I[i])>1){mon=0;break;} |
---|
| 3543 | } |
---|
| 3544 | if(mon) |
---|
| 3545 | { |
---|
[6518eba] | 3546 | if(size(reduce(I,std(radical(I[1])),5))<size(I)-1){mon=0;} |
---|
[2e6eac2] | 3547 | } |
---|
| 3548 | if((mon)&&(size(I)>3)) |
---|
| 3549 | { |
---|
[b154b3] | 3550 | int e=int(factorial(n))/m; |
---|
[2e6eac2] | 3551 | ideal J=1; |
---|
| 3552 | poly p=I[1]; |
---|
| 3553 | I=I[2..size(I)]; |
---|
| 3554 | ideal K=p^e; |
---|
| 3555 | for(i=1;i<=e;i++) |
---|
| 3556 | { |
---|
| 3557 | J=interred(J*I); |
---|
| 3558 | K=K,(p^(e-i))*J; |
---|
| 3559 | } |
---|
| 3560 | return(K); |
---|
| 3561 | } |
---|
| 3562 | for(i=n;i>1;i--) |
---|
| 3563 | { |
---|
| 3564 | if(i!=m) |
---|
| 3565 | { |
---|
| 3566 | I=I^i; |
---|
| 3567 | } |
---|
| 3568 | } |
---|
| 3569 | return(I); |
---|
| 3570 | } |
---|
| 3571 | |
---|
| 3572 | /////////////////////////////////////////////////////////////////////////////// |
---|
[268a430] | 3573 | static proc Coeff(list BO, int b, list #) |
---|
[2e6eac2] | 3574 | "USAGE: Coeff (BO); |
---|
| 3575 | @* BO = basic object, a list: ideal W, |
---|
| 3576 | @* ideal J, |
---|
| 3577 | @* intvec b (already truncated for Coeff), |
---|
| 3578 | @* list Ex (already truncated for Coeff), |
---|
| 3579 | @* ideal ab, |
---|
| 3580 | @* intvec v, |
---|
| 3581 | @* intvec w (already truncated for Coeff), |
---|
| 3582 | @* matrix M |
---|
| 3583 | @* b = integer indication bmax(BO) |
---|
| 3584 | ASSUME: R = basering, a polynomial ring, W an ideal of R, |
---|
| 3585 | @* J = ideal containing W |
---|
| 3586 | COMPUTE: Coeff-Ideal of BO as defined in [Bravo,Encinas,Villamayor] |
---|
| 3587 | RETURN: basic object of the Coeff-Ideal |
---|
| 3588 | EXAMPLE: example Coeff; shows an example |
---|
| 3589 | " |
---|
| 3590 | { |
---|
| 3591 | //!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! |
---|
| 3592 | //!!! TASK: lower dimension by more than one in a single step if possible !!! |
---|
| 3593 | //!!! (improve bookkeeping of invariants in Coeff and Center) !!! |
---|
| 3594 | //!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! |
---|
| 3595 | //--------------------------------------------------------------------------- |
---|
| 3596 | // Initialization and sanity checks |
---|
| 3597 | //--------------------------------------------------------------------------- |
---|
| 3598 | int i,k,dummy,errtype; |
---|
| 3599 | int ma=size(BO[4]); |
---|
| 3600 | intvec merk; |
---|
| 3601 | if(!defined(debugCoeff)) |
---|
| 3602 | { |
---|
| 3603 | int debugCoeff; |
---|
| 3604 | } |
---|
| 3605 | ideal C; |
---|
| 3606 | list L; |
---|
| 3607 | if(size(#)!=0) |
---|
| 3608 | { |
---|
| 3609 | if(typeof(#[1])=="ideal") |
---|
| 3610 | { |
---|
| 3611 | L=#; |
---|
| 3612 | } |
---|
| 3613 | else |
---|
| 3614 | { |
---|
| 3615 | ma=#[1]; |
---|
| 3616 | L=DeltaList(BO); |
---|
| 3617 | } |
---|
| 3618 | } |
---|
| 3619 | else |
---|
| 3620 | { |
---|
| 3621 | L=DeltaList(BO); |
---|
| 3622 | } |
---|
| 3623 | |
---|
| 3624 | if(debugCoeff) |
---|
| 3625 | { |
---|
| 3626 | "----> In Coeff: result of DeltaList:"; |
---|
| 3627 | L; |
---|
| 3628 | } |
---|
| 3629 | int d=size(L); // bmax of BO |
---|
| 3630 | if((debugCoeff)&&(d!=b)) |
---|
| 3631 | { |
---|
| 3632 | "!!!!!! Length of DeltaList does not equal second argument !!!!!!"; |
---|
| 3633 | "!!!!!! BO might not have been ord ~ 1 or wrong b !!!!!!"; |
---|
| 3634 | } |
---|
| 3635 | if(b>=6){return(0);} // b is too big |
---|
[b154b3] | 3636 | int e=int(factorial(b)); // b of Coeff-Ideal |
---|
[2e6eac2] | 3637 | if(e==0) |
---|
| 3638 | { |
---|
[b154b3] | 3639 | ERROR( "// integer size too small for forming b! ."); |
---|
[2e6eac2] | 3640 | } |
---|
| 3641 | if(b==0) |
---|
| 3642 | { |
---|
| 3643 | ERROR( "// second argument to Coeff should never be zero." ); |
---|
| 3644 | } |
---|
| 3645 | //---------------------------------------------------------------------------- |
---|
| 3646 | // Form the Coeff-Ideal |
---|
| 3647 | // Step 1: choose hypersurface |
---|
| 3648 | // Step 2: sum over correct powers of Delta^i(BO[2]) |
---|
| 3649 | // Step 3: do the intersection |
---|
| 3650 | //---------------------------------------------------------------------------- |
---|
| 3651 | //--- Step 1 |
---|
| 3652 | ideal Z; |
---|
| 3653 | poly p; |
---|
| 3654 | for(i=1;i<=ncols(L[d]);i++) |
---|
| 3655 | { |
---|
| 3656 | //--- Look for smooth hypersurface in generators of Delta^(bmax-1)(BO[2]) |
---|
| 3657 | dummy=goodChoice(BO,L[d][i]); |
---|
| 3658 | if(!dummy) |
---|
| 3659 | { |
---|
| 3660 | Z= L[d][i]; |
---|
| 3661 | break; |
---|
| 3662 | } |
---|
| 3663 | else |
---|
| 3664 | { |
---|
| 3665 | if(dummy>1) |
---|
| 3666 | { |
---|
| 3667 | merk[size(merk)+1]=i; |
---|
| 3668 | } |
---|
| 3669 | if(dummy>errtype) |
---|
| 3670 | { |
---|
| 3671 | errtype=dummy; |
---|
| 3672 | } |
---|
| 3673 | } |
---|
| 3674 | } |
---|
| 3675 | if(size(Z)==0) |
---|
| 3676 | { |
---|
| 3677 | //--- no suitable element in generators of Delta^(bmax-1)(BO[2]) |
---|
| 3678 | //--- try random linear combination |
---|
| 3679 | for(k=1;k<=10;k++) |
---|
| 3680 | { |
---|
| 3681 | for(i=2;i<=size(merk);i++) |
---|
| 3682 | { |
---|
| 3683 | p=p+random(-100,100)*L[d][merk[i]]; |
---|
| 3684 | } |
---|
| 3685 | dummy=goodChoice(BO,p); |
---|
| 3686 | if(!dummy) |
---|
| 3687 | { |
---|
| 3688 | Z=p; |
---|
| 3689 | break; |
---|
| 3690 | } |
---|
| 3691 | else |
---|
| 3692 | { |
---|
| 3693 | p=0; |
---|
| 3694 | } |
---|
| 3695 | } |
---|
| 3696 | if(dummy) |
---|
| 3697 | { |
---|
| 3698 | for(i=1;i<=size(L[d]);i++) |
---|
| 3699 | { |
---|
| 3700 | p=p+random(-100,100)*L[d][i]; |
---|
| 3701 | } |
---|
| 3702 | dummy=goodChoice(BO,p); |
---|
| 3703 | if(!dummy) |
---|
| 3704 | { |
---|
| 3705 | //--- found a suitable one |
---|
| 3706 | Z=p; |
---|
| 3707 | } |
---|
| 3708 | } |
---|
| 3709 | if(dummy) |
---|
| 3710 | { |
---|
| 3711 | //--- did not find a suitable random linear combination either |
---|
| 3712 | if(dummy>errtype) |
---|
| 3713 | { |
---|
| 3714 | errtype=dummy; |
---|
| 3715 | } |
---|
| 3716 | list retlist=errtype,L[d]; |
---|
| 3717 | return(retlist); |
---|
| 3718 | } |
---|
| 3719 | } |
---|
| 3720 | if(debugCoeff) |
---|
| 3721 | { |
---|
| 3722 | "----> In Coeff: Chosen hypersurface"; |
---|
| 3723 | Z; |
---|
| 3724 | } |
---|
| 3725 | //--- Step 2 |
---|
| 3726 | C=Z; |
---|
| 3727 | for(i=0;i<b;i++) |
---|
| 3728 | { |
---|
| 3729 | C=C,powerI(simplify(reduce(L[i+1],std(Z)),2),b,b-i); |
---|
| 3730 | } |
---|
| 3731 | C=interred(C); |
---|
| 3732 | |
---|
| 3733 | if(debugCoeff) |
---|
| 3734 | { |
---|
| 3735 | "----> In Coeff: J before saturation"; |
---|
| 3736 | C; |
---|
| 3737 | } |
---|
| 3738 | |
---|
| 3739 | //--- Step 3 |
---|
| 3740 | BO[1]=BO[1]+Z; |
---|
| 3741 | BO[2]=C; |
---|
| 3742 | for(i=1;i<=size(BO[4]);i++) |
---|
| 3743 | { |
---|
| 3744 | BO[6][i]=0; // reset intersection indicator |
---|
| 3745 | BO[4][i]=BO[4][i]+Z; // intersect the E_i |
---|
| 3746 | if(i<=ma) |
---|
| 3747 | { |
---|
[3f7e01a] | 3748 | BO[2]=sat(BO[2],BO[4][i]+BO[1]); |
---|
[2e6eac2] | 3749 | // "strict transform" of J w.r.t E, not "total" |
---|
| 3750 | } |
---|
| 3751 | } |
---|
| 3752 | if(debugCoeff) |
---|
| 3753 | { |
---|
| 3754 | "----> In Coeff:"; |
---|
| 3755 | " J after saturation:"; |
---|
| 3756 | BO[2]; |
---|
| 3757 | } |
---|
| 3758 | return(BO); |
---|
| 3759 | } |
---|
| 3760 | example |
---|
| 3761 | {"EXAMPLE:"; |
---|
| 3762 | echo = 2; |
---|
| 3763 | ring R=0,(x,y,z),dp; |
---|
| 3764 | |
---|
| 3765 | ideal W; |
---|
| 3766 | ideal J=z^2+x^2*y^2; |
---|
| 3767 | intvec b=0; |
---|
| 3768 | list E; |
---|
| 3769 | ideal abb=maxideal(1); |
---|
| 3770 | intvec v; |
---|
| 3771 | intvec w=-1; |
---|
| 3772 | matrix M; |
---|
| 3773 | |
---|
| 3774 | list BO=W,J,b,E,abb,v,w,M; |
---|
| 3775 | |
---|
| 3776 | Coeff(BO,2); |
---|
| 3777 | } |
---|
| 3778 | ////////////////////////////////////////////////////////////////////////////// |
---|
[268a430] | 3779 | static proc goodChoice(list BO, poly p) |
---|
[2e6eac2] | 3780 | "Internal procedure - no help and no example available |
---|
| 3781 | " |
---|
| 3782 | { |
---|
| 3783 | //--------------------------------------------------------------------------- |
---|
| 3784 | // test whether new W is smooth |
---|
| 3785 | //--------------------------------------------------------------------------- |
---|
| 3786 | ideal W=BO[1]+ideal(p); |
---|
[6518eba] | 3787 | if(size(reduce(p,std(BO[1]),5))==0) |
---|
[2e6eac2] | 3788 | { |
---|
| 3789 | //--- p is already in BO[1], i.e. does not define a hypersurface in W |
---|
| 3790 | return(1); |
---|
| 3791 | } |
---|
| 3792 | if(dim(std(slocusE(W)))>=0) |
---|
| 3793 | // if(dim(timeStd(slocusE(W),20))>=0) |
---|
| 3794 | { |
---|
| 3795 | //--- new W would not be smooth |
---|
| 3796 | return(1); |
---|
| 3797 | } |
---|
| 3798 | if(size(BO[4])==0) |
---|
| 3799 | { |
---|
| 3800 | //--- E is empty, no further tests necessary |
---|
| 3801 | return(0); |
---|
| 3802 | } |
---|
| 3803 | //-------------------------------------------------------------------------- |
---|
| 3804 | // test whether the hypersurface meets the E_i transversally |
---|
| 3805 | //-------------------------------------------------------------------------- |
---|
| 3806 | list E=BO[4]; |
---|
| 3807 | int i,d; |
---|
| 3808 | ideal T=W; |
---|
| 3809 | ideal Tstd=std(T); |
---|
| 3810 | d=nvars(basering)-dim(Tstd)+1; |
---|
| 3811 | ideal M; |
---|
| 3812 | for(i=1;i<=size(E);i++) |
---|
| 3813 | { |
---|
| 3814 | T=W,E[i]; |
---|
| 3815 | M=minor(jacob(T),d,Tstd)+T; |
---|
| 3816 | M=std(M); |
---|
| 3817 | if(deg(M[1])>0) |
---|
| 3818 | { |
---|
| 3819 | //--- intersection not transversal |
---|
| 3820 | return(2); |
---|
| 3821 | } |
---|
| 3822 | } |
---|
| 3823 | //-------------------------------------------------------------------------- |
---|
| 3824 | // test whether the new E_i have normal crossings |
---|
| 3825 | //-------------------------------------------------------------------------- |
---|
| 3826 | for(i=1;i<=size(E);i++) |
---|
| 3827 | { |
---|
| 3828 | E[i]=E[i],p; |
---|
| 3829 | } |
---|
| 3830 | if(normalCross(E)) |
---|
| 3831 | { |
---|
| 3832 | return(0); |
---|
| 3833 | } |
---|
| 3834 | else |
---|
| 3835 | { |
---|
| 3836 | return(2); |
---|
| 3837 | } |
---|
| 3838 | } |
---|
| 3839 | ////////////////////////////////////////////////////////////////////////////// |
---|
| 3840 | |
---|
[453bc66] | 3841 | proc presentTree(list L) |
---|
| 3842 | "USAGE: presentTree(L); |
---|
| 3843 | L=list, output of resolve |
---|
| 3844 | RETURN: nothing, only pretty printing of the output data of resolve() |
---|
| 3845 | EXAMPLE: none |
---|
| 3846 | " |
---|
| 3847 | { |
---|
| 3848 | def r=basering; |
---|
| 3849 | int i,j,k; |
---|
| 3850 | if(size(L[2])==1) |
---|
| 3851 | { |
---|
| 3852 | "The object was already resolved or the list L does not"; |
---|
| 3853 | "have required input format. There is just one chart in"; |
---|
| 3854 | "the tree."; |
---|
| 3855 | return(); |
---|
| 3856 | } |
---|
| 3857 | for(i=1;i<=size(L[1]);i++) |
---|
| 3858 | { |
---|
| 3859 | " "; |
---|
| 3860 | "/////////////////////////// Final Chart",i,"/////////////////////////"; |
---|
| 3861 | def s=L[1][i]; |
---|
| 3862 | setring s; |
---|
| 3863 | "======================== History of this chart ======================"; |
---|
| 3864 | for(j=2;j<=ncols(path);j++) |
---|
| 3865 | { |
---|
| 3866 | " "; |
---|
| 3867 | "Blow Up",j-1,":"; |
---|
| 3868 | " Center determined in L[2]["+string(path[1,j])+"],"; |
---|
| 3869 | " Passing to chart ",path[2,j]," in resulting blow up."; |
---|
| 3870 | } |
---|
| 3871 | " "; |
---|
| 3872 | "======================== Data of this chart ========================"; |
---|
| 3873 | showBO(BO); |
---|
| 3874 | setring r; |
---|
| 3875 | kill s; |
---|
| 3876 | pause(); |
---|
| 3877 | } |
---|
| 3878 | "////////////////////////////////////////////////////////////////////"; |
---|
| 3879 | "For identification of exceptional divisors please use the tools"; |
---|
| 3880 | "provided by reszeta.lib, e.g. collectDiv."; |
---|
| 3881 | "For viewing an illustration of the tree of charts please use the"; |
---|
| 3882 | "procedure ResTree from resgraph.lib."; |
---|
| 3883 | "////////////////////////////////////////////////////////////////////"; |
---|
| 3884 | return(); |
---|
| 3885 | } |
---|
| 3886 | ////////////////////////////////////////////////////////////////////////////// |
---|
| 3887 | |
---|
[2e6eac2] | 3888 | proc showBO(list BO) |
---|
| 3889 | "USAGE: showBO(BO); |
---|
| 3890 | @* BO=basic object, a list: ideal W, |
---|
| 3891 | @* ideal J, |
---|
| 3892 | @* intvec b (already truncated for Coeff), |
---|
| 3893 | @* list Ex (already truncated for Coeff), |
---|
| 3894 | @* ideal ab, |
---|
| 3895 | @* intvec v, |
---|
| 3896 | @* intvec w (already truncated for Coeff), |
---|
| 3897 | @* matrix M |
---|
| 3898 | RETURN: nothing, only pretty printing |
---|
| 3899 | EXAMPLE: none |
---|
| 3900 | " |
---|
| 3901 | { |
---|
| 3902 | " "; |
---|
[453bc66] | 3903 | "==== Ambient Space: ";BO[1];" "; |
---|
| 3904 | "==== Ideal of Variety: ";BO[2];" "; |
---|
[2e6eac2] | 3905 | int i; |
---|
| 3906 | list M; |
---|
| 3907 | for(i=1;i<=size(BO[4]);i++) |
---|
| 3908 | { |
---|
| 3909 | M[i]=ideal(BO[4][i]); |
---|
| 3910 | } |
---|
[453bc66] | 3911 | "==== Exceptional Divisors: ";print(M);" "; |
---|
| 3912 | "==== Images of variables of original ring:";BO[5];" "; |
---|
[2e6eac2] | 3913 | } |
---|
| 3914 | ////////////////////////////////////////////////////////////////////////////// |
---|
| 3915 | //////////////////////// main procedure //////////////////////////////// |
---|
| 3916 | ////////////////////////////////////////////////////////////////////////////// |
---|
| 3917 | proc resolve(ideal J, list #) |
---|
| 3918 | "USAGE: resolve (J); or resolve (J,i[,k]); |
---|
| 3919 | @* J ideal |
---|
| 3920 | @* i,k int |
---|
| 3921 | COMPUTE: a resolution of J, |
---|
| 3922 | @* if i > 0 debugging is turned on according to the following switches: |
---|
| 3923 | @* j1: value 0 or 1; turn off or on correctness checks in all steps |
---|
| 3924 | @* j2: value 0 or 2; turn off or on debugCenter |
---|
| 3925 | @* j3: value 0 or 4; turn off or on debugBlowUp |
---|
| 3926 | @* j4: value 0 or 8; turn off or on debugCoeff |
---|
| 3927 | @* j5: value 0 or 16:turn off or on debugging of Intersection with E^- |
---|
[4bde6b] | 3928 | @* j6: value 0 or 32:turn off or on stop after pass through the loop |
---|
[2e6eac2] | 3929 | @* i=j1+j2+j3+j4+j5+j6 |
---|
| 3930 | RETURN: a list l of 2 lists of rings |
---|
| 3931 | l[1][i] is a ring containing a basic object BO, the result of the |
---|
| 3932 | resolution. |
---|
[80f8f6c] | 3933 | l[2] contains all rings which occurred during the resolution process |
---|
[453bc66] | 3934 | NOTE: result may be viewed in a human readable form using presentTree() |
---|
[2e6eac2] | 3935 | EXAMPLE: example resolve; shows an example |
---|
| 3936 | " |
---|
| 3937 | { |
---|
| 3938 | //---------------------------------------------------------------------------- |
---|
| 3939 | // Initialization and sanity checks |
---|
| 3940 | //---------------------------------------------------------------------------- |
---|
| 3941 | def R=basering; |
---|
| 3942 | list allRings; |
---|
| 3943 | allRings[1]=R; |
---|
| 3944 | list endRings; |
---|
| 3945 | module path=[0,-1]; |
---|
| 3946 | ideal W; |
---|
| 3947 | list E; |
---|
| 3948 | ideal abb=maxideal(1); |
---|
| 3949 | intvec v; |
---|
| 3950 | intvec bvec; |
---|
| 3951 | intvec w=-1; |
---|
| 3952 | matrix intE; |
---|
| 3953 | int extra,bm; |
---|
[3b77465] | 3954 | if(defined(BO)){kill BO;} |
---|
| 3955 | if(defined(cent)){kill cent;} |
---|
[2e6eac2] | 3956 | |
---|
| 3957 | ideal Jrad=equiRadical(J); |
---|
[6518eba] | 3958 | if(size(reduce(Jrad,std(J),5))!=0) |
---|
[2e6eac2] | 3959 | { |
---|
| 3960 | "WARNING! The input is not reduced or not equidimensional!"; |
---|
| 3961 | "We will continue with the reduced top-dimensional part of input"; |
---|
| 3962 | J=Jrad; |
---|
| 3963 | } |
---|
| 3964 | |
---|
| 3965 | int i,j,debu,loca,locaT,ftemp,debugResolve,smooth; |
---|
| 3966 | //--- switches for local and for debugging may occur in any order |
---|
| 3967 | i=size(#); |
---|
| 3968 | extra=3; |
---|
| 3969 | for(j=1;j<=i;j++) |
---|
| 3970 | { |
---|
| 3971 | if(typeof(#[j])=="int") |
---|
| 3972 | { |
---|
| 3973 | debugResolve=#[j]; |
---|
| 3974 | //--- debu: debug switch for resolve, smallest bit in debugResolve |
---|
| 3975 | debu=debugResolve mod 2; |
---|
| 3976 | } |
---|
| 3977 | else |
---|
| 3978 | { |
---|
| 3979 | if(#[j]=="M") |
---|
| 3980 | { |
---|
| 3981 | bm=1; |
---|
| 3982 | ERROR("Not implemented yet"); |
---|
| 3983 | } |
---|
| 3984 | if(#[j]=="E"){extra=0;} |
---|
| 3985 | if(#[j]=="A"){extra=2;} |
---|
| 3986 | if(#[j]=="K"){extra=3;} |
---|
| 3987 | if(#[j]=="L"){loca=1;} |
---|
| 3988 | } |
---|
| 3989 | } |
---|
| 3990 | if(loca) |
---|
| 3991 | { |
---|
| 3992 | list qs=minAssGTZ(J); |
---|
| 3993 | ideal K=ideal(1); |
---|
| 3994 | for(j=1;j<=size(qs);j++) |
---|
| 3995 | { |
---|
[6518eba] | 3996 | if(size(reduce(qs[j],maxideal(1),5))==0) |
---|
[2e6eac2] | 3997 | { |
---|
| 3998 | K=intersect(K,qs[j]); |
---|
| 3999 | } |
---|
| 4000 | } |
---|
| 4001 | J=K; |
---|
| 4002 | list qr=minAssGTZ(slocus(J)); |
---|
| 4003 | K=ideal(1); |
---|
| 4004 | for(j=1;j<=size(qr);j++) |
---|
| 4005 | { |
---|
[6518eba] | 4006 | if(size(reduce(qr[j],maxideal(1),5))!=0) |
---|
[2e6eac2] | 4007 | { |
---|
| 4008 | K=intersect(K,qr[j]); |
---|
| 4009 | smooth++; |
---|
| 4010 | } |
---|
| 4011 | else |
---|
| 4012 | { |
---|
| 4013 | if(dim(std(qr[j]))>0){loca=0;} |
---|
| 4014 | //---- test for isolated singularity at 0 |
---|
| 4015 | } |
---|
| 4016 | } |
---|
| 4017 | K=std(K); |
---|
| 4018 | //---- if deg(K[1])==0 the point 0 is on all components of the singular |
---|
| 4019 | //---- locus and we can work globally |
---|
| 4020 | if(smooth==size(qr)){smooth=-1;} |
---|
| 4021 | //---- the point 0 is not on the singular locus |
---|
| 4022 | if((deg(K[1])>0)&&(smooth>=0)&&(!loca)) |
---|
| 4023 | { |
---|
| 4024 | locaT=1; |
---|
| 4025 | poly @p; |
---|
| 4026 | for(j=1;j<=size(K);j++) |
---|
| 4027 | { |
---|
| 4028 | if(jet(K[j],0)!=0) |
---|
| 4029 | { |
---|
| 4030 | @p=K[j]; |
---|
| 4031 | break; |
---|
| 4032 | } |
---|
| 4033 | } |
---|
| 4034 | export(@p); |
---|
| 4035 | } |
---|
| 4036 | if((loca)&&(!smooth)){loca=0;} |
---|
| 4037 | //---- the case that 0 is isolated singularity and the only singular point |
---|
| 4038 | } |
---|
| 4039 | export(locaT); |
---|
| 4040 | //---In case of option "L" the following holds |
---|
| 4041 | //---loca=0 and locaT=0 we perform the global case |
---|
| 4042 | //---loca !=0: 0 is isolated singular point, but there are other singularities |
---|
| 4043 | //---locaT!=0: O is singular point, but not isolated, and there is a componente//--- of the singular locus not containing 0 |
---|
| 4044 | |
---|
| 4045 | //--- if necessary, set the corresponding debugFlags |
---|
| 4046 | if(defined(debugResolve)) |
---|
| 4047 | { |
---|
| 4048 | //--- 2nd bit from the right |
---|
| 4049 | int debugCenter=(debugResolve div 2) mod 2; |
---|
| 4050 | export debugCenter; |
---|
| 4051 | //--- 3rd bit from the right |
---|
| 4052 | int debugBlowUp=(debugResolve div 4) mod 2; |
---|
| 4053 | export debugBlowUp; |
---|
| 4054 | //--- 4th bit from the right |
---|
| 4055 | int debugCoeff=(debugResolve div 8) mod 2; |
---|
| 4056 | export debugCoeff; |
---|
| 4057 | //--- 5th bit from the right |
---|
| 4058 | int debug_Inters_E=(debugResolve div 16) mod 2; |
---|
| 4059 | export debug_Inters_E; |
---|
| 4060 | //--- 6th bit from the right |
---|
| 4061 | int praes_stop=(debugResolve div 32) mod 2; |
---|
| 4062 | } |
---|
| 4063 | //--- set the correct attributes to J for speed ups |
---|
| 4064 | if( typeof(attrib(J,"isEqui"))!="int" ) |
---|
| 4065 | { |
---|
| 4066 | if(size(J)==1) |
---|
| 4067 | { |
---|
| 4068 | attrib(J,"isEqui",1); |
---|
| 4069 | } |
---|
| 4070 | else |
---|
| 4071 | { |
---|
| 4072 | attrib(J,"isEqui",0); |
---|
| 4073 | } |
---|
| 4074 | } |
---|
| 4075 | if(size(J)==1) |
---|
| 4076 | { |
---|
| 4077 | attrib(J,"isHy",1); |
---|
| 4078 | } |
---|
| 4079 | else |
---|
| 4080 | { |
---|
| 4081 | attrib(J,"isHy",0); |
---|
| 4082 | } |
---|
| 4083 | //--- create the BO |
---|
| 4084 | list BO=W,J,bvec,E,abb,v,w,intE; |
---|
| 4085 | if(defined(invSat)){kill invSat;} |
---|
| 4086 | list invSat=ideal(0),intvec(0); |
---|
| 4087 | export(invSat); |
---|
| 4088 | if(bm) |
---|
| 4089 | { |
---|
| 4090 | intmat invmat[2][1]=0,-1; |
---|
| 4091 | BO[9]=invmat; |
---|
| 4092 | } |
---|
| 4093 | else |
---|
| 4094 | { |
---|
| 4095 | BO[9]=intvec(0); |
---|
| 4096 | } |
---|
| 4097 | export BO; |
---|
| 4098 | list tmpList; |
---|
| 4099 | int blo; |
---|
| 4100 | int k,Ecount,tmpPtr; |
---|
| 4101 | i=0; |
---|
| 4102 | if(smooth==-1) |
---|
| 4103 | { |
---|
| 4104 | endRings[1]=R; |
---|
| 4105 | list result=endRings,allRings; |
---|
| 4106 | if(debu) |
---|
| 4107 | { |
---|
| 4108 | "============= result will be tested =========="; |
---|
| 4109 | " "; |
---|
| 4110 | "the number of charts obtained:",size(endRings); |
---|
| 4111 | "============= result is o.k. =========="; |
---|
| 4112 | } |
---|
| 4113 | kill debugCenter,debugBlowUp,debugCoeff,debug_Inters_E; |
---|
| 4114 | return(result); |
---|
| 4115 | } |
---|
| 4116 | //----------------------------------------------------------------------------- |
---|
| 4117 | // While there are rings to be considered, determine center and blow up |
---|
| 4118 | //----------------------------------------------------------------------------- |
---|
| 4119 | while(i<size(allRings)) |
---|
| 4120 | { |
---|
| 4121 | i++; |
---|
| 4122 | def S=allRings[i]; |
---|
| 4123 | setring S; |
---|
| 4124 | list pr; |
---|
| 4125 | ideal Jstd=std(BO[2]); |
---|
| 4126 | //----------------------------------------------------------------------------- |
---|
| 4127 | // Determine Center |
---|
| 4128 | //----------------------------------------------------------------------------- |
---|
| 4129 | if(i==1) |
---|
| 4130 | { |
---|
| 4131 | list deltaL=DeltaList(BO); |
---|
| 4132 | ideal sL=radical(deltaL[size(deltaL)]); |
---|
| 4133 | if((deg(std(slocus(sL))[1])==0)&&(size(minAssGTZ(sL))==1)) |
---|
| 4134 | { |
---|
| 4135 | list @ce=sL,intvec(-1),intvec(0),intvec(0); |
---|
| 4136 | ideal cent=@ce[1]; |
---|
| 4137 | } |
---|
| 4138 | } |
---|
| 4139 | //--- before computing a center, check whether we have a stored one |
---|
| 4140 | if(size(BO)>9) |
---|
| 4141 | { |
---|
| 4142 | while(size(BO[10])>0) |
---|
| 4143 | { |
---|
| 4144 | list @ce=BO[10][1]; |
---|
| 4145 | //--- check of the center |
---|
| 4146 | // @ce=correctC(BO,@ce,bm); |
---|
| 4147 | //--- use stored center |
---|
| 4148 | BO[10]=delete(BO[10],1); |
---|
| 4149 | if(size(@ce[1])==0) |
---|
| 4150 | { |
---|
| 4151 | //--- stored center was not ok |
---|
| 4152 | continue; |
---|
| 4153 | } |
---|
| 4154 | tmpPtr=0; |
---|
| 4155 | for(Ecount=1;Ecount <= size(@ce[2]); Ecount++) |
---|
| 4156 | { |
---|
| 4157 | if(@ce[2][Ecount]>-1) |
---|
| 4158 | { |
---|
| 4159 | tmpPtr=tmpPtr+@ce[2][Ecount]; |
---|
| 4160 | } |
---|
| 4161 | else |
---|
| 4162 | { |
---|
| 4163 | @ce[2][Ecount]=size(BO[4])-tmpPtr-1; |
---|
| 4164 | for(int cnthlp=1;cnthlp<=size(BO[10]);cnthlp++) |
---|
| 4165 | { |
---|
| 4166 | BO[10][cnthlp][2][Ecount]=@ce[2][Ecount]; |
---|
| 4167 | } |
---|
| 4168 | kill cnthlp; |
---|
| 4169 | break; |
---|
| 4170 | } |
---|
| 4171 | } |
---|
| 4172 | if(Ecount<size(@ce[2])) |
---|
| 4173 | { |
---|
| 4174 | for(tmpPtr=Ecount+1;tmpPtr<=size(@ce[2]);tmpPtr++) |
---|
| 4175 | { |
---|
| 4176 | @ce[2][tmpPtr]=0; |
---|
| 4177 | for(int cnthlp=1;cnthlp<=size(BO[10]);cnthlp++) |
---|
| 4178 | { |
---|
| 4179 | BO[10][cnthlp][2][tmpPtr]=@ce[2][tmpPtr]; |
---|
| 4180 | } |
---|
| 4181 | kill cnthlp; |
---|
| 4182 | } |
---|
| 4183 | } |
---|
| 4184 | break; |
---|
| 4185 | } |
---|
| 4186 | if(defined(@ce)) |
---|
| 4187 | { |
---|
| 4188 | if(size(@ce[1])==0) |
---|
| 4189 | { |
---|
| 4190 | kill @ce; |
---|
| 4191 | } |
---|
| 4192 | else |
---|
| 4193 | { |
---|
| 4194 | ideal cent=@ce[1]; |
---|
| 4195 | } |
---|
| 4196 | } |
---|
| 4197 | if(size(BO[10])==0) |
---|
| 4198 | { |
---|
| 4199 | //--- previously had stored centers, all have been used; we need to clean up |
---|
| 4200 | BO=delete(BO,10); |
---|
| 4201 | } |
---|
| 4202 | } |
---|
| 4203 | if((loca)&&(i==1)) |
---|
| 4204 | { |
---|
| 4205 | //--- local case: initial step is blow-up in origin |
---|
| 4206 | if(defined(@ce)){kill @ce;} |
---|
| 4207 | if(defined(cent)){kill cent;} |
---|
[6518eba] | 4208 | if(size(reduce(slocusE(BO[2]),maxideal(1),5))==0) |
---|
[2e6eac2] | 4209 | { |
---|
| 4210 | list @ce=maxideal(1),intvec(-1),intvec(0),intvec(0); |
---|
| 4211 | } |
---|
| 4212 | else |
---|
| 4213 | { |
---|
| 4214 | list @ce=BO[2],intvec(-1),intvec(1),intvec(0); |
---|
| 4215 | } |
---|
| 4216 | ideal cent=@ce[1]; |
---|
| 4217 | } |
---|
| 4218 | if(((loca)||(locaT))&&(i!=1)) |
---|
| 4219 | { |
---|
| 4220 | int JmeetsE; |
---|
| 4221 | for(j=1;j<=size(BO[4]);j++) |
---|
| 4222 | { |
---|
| 4223 | if(deg(std(BO[2]+BO[4][j])[1])!=0) |
---|
| 4224 | { |
---|
| 4225 | JmeetsE=1; |
---|
| 4226 | break; |
---|
| 4227 | } |
---|
| 4228 | } |
---|
| 4229 | if(!JmeetsE) |
---|
| 4230 | { |
---|
| 4231 | list @ce=BO[2],intvec(-1),intvec(1),intvec(0); |
---|
| 4232 | ideal cent=@ce[1]; |
---|
| 4233 | } |
---|
| 4234 | kill JmeetsE; |
---|
| 4235 | } |
---|
| 4236 | if((locaT)&&(!defined(@ce))) |
---|
| 4237 | { |
---|
| 4238 | if(@p!=1) |
---|
| 4239 | { |
---|
| 4240 | list tr=minAssGTZ(slocusE(BO[2])); |
---|
| 4241 | ideal L=ideal(1); |
---|
| 4242 | for(j=1;j<=size(tr);j++) |
---|
| 4243 | { |
---|
[6518eba] | 4244 | if(size(reduce(ideal(@p),std(tr[j]),5))==0) |
---|
[2e6eac2] | 4245 | { |
---|
| 4246 | L=intersect(L,tr[j]); |
---|
| 4247 | } |
---|
| 4248 | } |
---|
| 4249 | L=std(L); |
---|
| 4250 | if(deg(L[1])==0) |
---|
| 4251 | { |
---|
| 4252 | @p=1; |
---|
| 4253 | } |
---|
| 4254 | else |
---|
| 4255 | { |
---|
| 4256 | ideal fac=factorize(@p,1); |
---|
| 4257 | if(size(fac)==1) |
---|
| 4258 | { |
---|
| 4259 | @p=fac[1]; |
---|
| 4260 | } |
---|
| 4261 | else |
---|
| 4262 | { |
---|
| 4263 | for(j=1;j<=size(fac);j++) |
---|
| 4264 | { |
---|
| 4265 | if(reduce(fac[j],L)==0) |
---|
| 4266 | { |
---|
| 4267 | @p=fac[j]; |
---|
| 4268 | break; |
---|
| 4269 | } |
---|
| 4270 | } |
---|
| 4271 | } |
---|
| 4272 | } |
---|
| 4273 | kill tr,L; |
---|
| 4274 | } |
---|
[62de185] | 4275 | ring R1 = create_ring(ring_list(S)[1], "(@z,"+varstr(S)+")", "dp", "no_minpoly"); |
---|
[2e6eac2] | 4276 | poly p=imap(S,@p); |
---|
| 4277 | list BO=imap(S,BO); |
---|
| 4278 | list invSat=imap(S,invSat); |
---|
| 4279 | export(invSat); |
---|
| 4280 | ideal te=std(BO[2]); |
---|
| 4281 | BO[1]=BO[1]+ideal(@z*p-1); |
---|
| 4282 | BO[2]=BO[2]+ideal(@z*p-1); |
---|
| 4283 | for(j=1;j<=size(BO[4]);j++) |
---|
| 4284 | { |
---|
| 4285 | BO[4][j]=BO[4][j]+ideal(@z*p-1); |
---|
| 4286 | } |
---|
| 4287 | //--- for computation of center: drop components not meeting the Ei |
---|
| 4288 | def BO2=BO; |
---|
| 4289 | list qs=minAssGTZ(BO2[2]); |
---|
| 4290 | ideal K=ideal(1); |
---|
| 4291 | for(j=1;j<=size(qs);j++) |
---|
| 4292 | { |
---|
| 4293 | if(CompMeetsE(qs[j],BO2[4])) |
---|
| 4294 | { |
---|
| 4295 | K=intersect(K,qs[j]); |
---|
| 4296 | } |
---|
| 4297 | } |
---|
| 4298 | BO2[2]=K; |
---|
| 4299 | //--- check whether we are done |
---|
| 4300 | if(deg(std(BO2[2])[1])==0) |
---|
| 4301 | { |
---|
| 4302 | list @ce=BO[2],intvec(-1),intvec(1),intvec(0); |
---|
| 4303 | } |
---|
| 4304 | if(!defined(@ce)) |
---|
| 4305 | { |
---|
| 4306 | if(bm) |
---|
| 4307 | { |
---|
| 4308 | list @ce=CenterBM(BO2); |
---|
| 4309 | } |
---|
| 4310 | else |
---|
| 4311 | { |
---|
| 4312 | list @ce=CenterBO(BO2); |
---|
| 4313 | } |
---|
| 4314 | } |
---|
| 4315 | //--- if computation of center returned BO2[2], we are done |
---|
| 4316 | //--- ==> set @ce to BO[2], because later checks work with BO instead of BO2 |
---|
[6518eba] | 4317 | if((size(reduce(@ce[1],std(BO2[2]),5))==0)&& |
---|
| 4318 | (size(reduce(BO2[2],std(@ce[1]),5))==0)) |
---|
[2e6eac2] | 4319 | { |
---|
| 4320 | @ce[1]=BO[2]; |
---|
| 4321 | } |
---|
| 4322 | if(size(specialReduce(@ce[1],te,p))==0) |
---|
| 4323 | { |
---|
| 4324 | BO=imap(S,BO); |
---|
| 4325 | @ce[1]=BO[2]; |
---|
| 4326 | } |
---|
| 4327 | else |
---|
| 4328 | { |
---|
| 4329 | //@ce=correctC(BO,@ce,bm); |
---|
| 4330 | @ce[1]=eliminate(@ce[1],@z); |
---|
| 4331 | } |
---|
| 4332 | setring S; |
---|
| 4333 | list @ce=imap(R1,@ce); |
---|
| 4334 | kill R1; |
---|
| 4335 | |
---|
[6518eba] | 4336 | if((size(reduce(BO[2],std(@ce[1]),5))==0) |
---|
| 4337 | &&(size(reduce(@ce[1],Jstd,5))==0)) |
---|
[2e6eac2] | 4338 | { |
---|
| 4339 | //--- J and center coincide |
---|
| 4340 | pr[1]=@ce[1]; |
---|
| 4341 | ideal cent=@ce[1]; |
---|
| 4342 | } |
---|
| 4343 | else |
---|
| 4344 | { |
---|
| 4345 | //--- decompose center and use first component |
---|
| 4346 | pr=minAssGTZ(@ce[1]); |
---|
[6518eba] | 4347 | if(size(reduce(@p,std(pr[1]),5))==0){"Achtung";~;} |
---|
[2e6eac2] | 4348 | if(deg(std(slocus(pr[1]))[1])>0){"singulaer";~;} |
---|
| 4349 | ideal cent=pr[1]; |
---|
| 4350 | } |
---|
| 4351 | if(size(pr)>1) |
---|
| 4352 | { |
---|
| 4353 | //--- store the other components |
---|
| 4354 | for(k=2;k<=size(pr);k++) |
---|
| 4355 | { |
---|
[6518eba] | 4356 | if(size(reduce(@p,std(pr[k]),5))==0){"Achtung";~;} |
---|
[2e6eac2] | 4357 | if(deg(std(slocus(pr[k]))[1])>0){"singulaer";~;} |
---|
[6518eba] | 4358 | if(size(reduce(@p,std(pr[k]),5))!=0) |
---|
[2e6eac2] | 4359 | { |
---|
| 4360 | tmpList[size(tmpList)+1]=list(pr[k],@ce[2],@ce[3],@ce[4]); |
---|
| 4361 | } |
---|
| 4362 | } |
---|
| 4363 | BO[10]=tmpList; |
---|
| 4364 | kill tmpList; |
---|
| 4365 | list tmpList; |
---|
| 4366 | } |
---|
| 4367 | } |
---|
| 4368 | if(!defined(@ce)) |
---|
| 4369 | { |
---|
| 4370 | //--- no previously determined center, we need to compute one |
---|
| 4371 | if(loca) |
---|
| 4372 | { |
---|
| 4373 | //--- local case: center should be inside exceptional locus |
---|
| 4374 | ideal Ex=ideal(1); |
---|
| 4375 | k=0; |
---|
| 4376 | for(j=1;j<=size(BO[4]);j++) |
---|
| 4377 | { |
---|
| 4378 | if(deg(BO[4][j][1])!=0) |
---|
| 4379 | { |
---|
| 4380 | Ex=Ex*BO[4][j]; //----!!!!hier evtl. Durchschnitt??? |
---|
| 4381 | k++; |
---|
| 4382 | } |
---|
| 4383 | } |
---|
| 4384 | //--- for computation of center: drop components not meeting the Ei |
---|
| 4385 | list BOloc=BO; |
---|
| 4386 | list qs=minAssGTZ(BOloc[2]); |
---|
| 4387 | ideal K=ideal(1); |
---|
| 4388 | for(j=1;j<=size(qs);j++) |
---|
| 4389 | { |
---|
| 4390 | if(CompMeetsE(qs[j],BOloc[4])) |
---|
| 4391 | { |
---|
| 4392 | K=intersect(K,qs[j]); |
---|
| 4393 | } |
---|
| 4394 | } |
---|
| 4395 | BOloc[2]=K; |
---|
| 4396 | //--- check whether we are done |
---|
| 4397 | if(deg(std(BOloc[2])[1])==0) |
---|
| 4398 | { |
---|
| 4399 | list @ce=BO[2],intvec(-1),intvec(1),intvec(0); |
---|
| 4400 | } |
---|
| 4401 | if(!defined(@ce)) |
---|
| 4402 | { |
---|
| 4403 | if(BO[3][1]!=0) |
---|
| 4404 | { |
---|
| 4405 | BOloc[2]=BO[2]+Ex^((BO[3][1] div k)+1);//!!!!Vereinfachen??? |
---|
| 4406 | } |
---|
| 4407 | else |
---|
| 4408 | { |
---|
| 4409 | BOloc[2]=BO[2]+Ex^((size(DeltaList(BO)) div k)+1); |
---|
| 4410 | } |
---|
| 4411 | if(bm) |
---|
| 4412 | { |
---|
| 4413 | list @ce=CenterBM(BOloc); |
---|
| 4414 | } |
---|
| 4415 | else |
---|
| 4416 | { |
---|
| 4417 | list @ce=CenterBO(BOloc); |
---|
| 4418 | } |
---|
[6518eba] | 4419 | if(size(reduce(Ex,std(@ce[1]),5))!=0) |
---|
[2e6eac2] | 4420 | { |
---|
| 4421 | list tempPr=minAssGTZ(@ce[1]); |
---|
| 4422 | for(k=size(tempPr);k>=1;k--) |
---|
| 4423 | { |
---|
[6518eba] | 4424 | if(size(reduce(Ex,std(tempPr[k]),5))!=0) |
---|
[2e6eac2] | 4425 | { |
---|
| 4426 | tempPr=delete(tempPr,k); |
---|
| 4427 | } |
---|
| 4428 | } |
---|
| 4429 | @ce[1]=1; |
---|
| 4430 | for(k=1;k<=size(tempPr);k++) |
---|
| 4431 | { |
---|
| 4432 | @ce[1]=intersect(@ce[1],tempPr[k]); |
---|
| 4433 | } |
---|
| 4434 | if(deg(std(@ce[1])[1])==0) |
---|
| 4435 | { |
---|
| 4436 | @ce[1]=BO[2]; |
---|
| 4437 | } |
---|
| 4438 | } |
---|
| 4439 | } |
---|
| 4440 | //--- test whether we are done |
---|
[6518eba] | 4441 | if(size(reduce(slocusE(BO[2]),std(@ce[1]),5))!=0) |
---|
[2e6eac2] | 4442 | { |
---|
| 4443 | if(transversalT(BO[2],BO[4])) |
---|
| 4444 | { |
---|
| 4445 | if(defined(E)){kill E;} |
---|
| 4446 | list E=BO[4]; |
---|
| 4447 | for(j=1;j<=size(E);j++){if(deg(E[j][1])>0){E[j]=E[j]+BO[2];}} |
---|
| 4448 | if(normalCross(E)) |
---|
| 4449 | { |
---|
| 4450 | @ce[1]=BO[2]; |
---|
| 4451 | } |
---|
| 4452 | kill E; |
---|
| 4453 | } |
---|
| 4454 | } |
---|
| 4455 | } |
---|
| 4456 | else |
---|
| 4457 | { |
---|
| 4458 | //--- non-local |
---|
| 4459 | if(bm) |
---|
| 4460 | { |
---|
| 4461 | list @ce=CenterBM(BO); |
---|
| 4462 | } |
---|
| 4463 | else |
---|
| 4464 | { |
---|
| 4465 | list @ce=CenterBO(BO); |
---|
| 4466 | } |
---|
| 4467 | //--- check of the center |
---|
| 4468 | //@ce=correctC(BO,@ce,bm); |
---|
| 4469 | if((size(@ce[1])==0)&&(size(@ce[4])<(size(@ce[3])-1))) |
---|
| 4470 | { |
---|
| 4471 | intvec xxx=@ce[3]; |
---|
| 4472 | xxx=xxx[1..size(@ce[4])]; |
---|
| 4473 | @ce[3]=xxx; |
---|
| 4474 | xxx=@ce[2]; |
---|
| 4475 | xxx=xxx[1..size(@ce[4])]; |
---|
| 4476 | @ce[2]=xxx; |
---|
| 4477 | kill xxx; |
---|
| 4478 | } |
---|
| 4479 | } |
---|
[6518eba] | 4480 | if((size(reduce(BO[2],std(@ce[1]),5))==0) |
---|
| 4481 | &&(size(reduce(@ce[1],Jstd,5))==0)) |
---|
[2e6eac2] | 4482 | { |
---|
| 4483 | //--- J and center coincide |
---|
| 4484 | pr[1]=@ce[1]; |
---|
| 4485 | ideal cent=@ce[1]; |
---|
| 4486 | } |
---|
| 4487 | else |
---|
| 4488 | { |
---|
| 4489 | //--- decompose center and use first component |
---|
| 4490 | pr=minAssGTZ(@ce[1]); |
---|
| 4491 | ideal cent=pr[1]; |
---|
| 4492 | } |
---|
| 4493 | if(size(pr)>1) |
---|
| 4494 | { |
---|
| 4495 | //--- store the other components |
---|
| 4496 | for(k=2;k<=size(pr);k++) |
---|
| 4497 | { |
---|
| 4498 | tmpList[k-1]=list(pr[k],@ce[2],@ce[3],@ce[4]); |
---|
| 4499 | } |
---|
| 4500 | BO[10]=tmpList; |
---|
| 4501 | kill tmpList; |
---|
| 4502 | list tmpList; |
---|
| 4503 | } |
---|
| 4504 | } |
---|
| 4505 | //--- do not forget to update BO[7] and BO[3] |
---|
| 4506 | export cent; |
---|
| 4507 | BO[7]=@ce[2]; |
---|
| 4508 | BO[3]=@ce[3]; |
---|
| 4509 | if((loca||locaT)&&(size(@ce)<4)){@ce[4]=0;} //Provisorium !!! |
---|
| 4510 | if((size(@ce[4])<size(@ce[2])-1)||(size(@ce[4])<size(@ce[3])-1)) |
---|
| 4511 | { |
---|
| 4512 | if((deg(std(@ce[1])[1])==0)&&(deg(std(BO[2])[1])==0)) |
---|
| 4513 | { |
---|
| 4514 | intvec nullvec; |
---|
| 4515 | nullvec[size(@ce[2])-1]=0; |
---|
| 4516 | @ce[4]=nullvec; |
---|
| 4517 | kill nullvec; |
---|
| 4518 | } |
---|
| 4519 | else |
---|
| 4520 | { |
---|
| 4521 | "ERROR:@ce[4] hat falsche Laenge - nicht-trivialer Fall"; |
---|
| 4522 | ~; |
---|
| 4523 | } |
---|
| 4524 | } |
---|
| 4525 | if((typeof(@ce[4])=="intvec") || (typeof(@ce[4])=="intmat")) |
---|
| 4526 | { |
---|
| 4527 | BO[9]=@ce[4]; |
---|
| 4528 | } |
---|
| 4529 | //--------------------------------------------------------------------------- |
---|
| 4530 | // various checks and debug output |
---|
| 4531 | //--------------------------------------------------------------------------- |
---|
| 4532 | if((debu) || (praes_stop)) |
---|
| 4533 | { |
---|
| 4534 | //--- Show BO of this step |
---|
[453bc66] | 4535 | "++++++++++++++ Overview of Current Chart +++++++++++++++++++++++"; |
---|
| 4536 | "Current number of final charts:",size(endRings); |
---|
| 4537 | "Total number of charts currently in chart-tree:",size(allRings); |
---|
| 4538 | "Index of the current chart in chart-tree:",i; |
---|
| 4539 | "++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++"; |
---|
[2e6eac2] | 4540 | showBO(BO); |
---|
[453bc66] | 4541 | "-------------------------- Upcoming Center ---------------------"; |
---|
[2e6eac2] | 4542 | interred(cent); |
---|
[453bc66] | 4543 | "----------------------------------------------------------------"; |
---|
[2e6eac2] | 4544 | } |
---|
| 4545 | if(praes_stop) |
---|
| 4546 | { |
---|
| 4547 | ~; |
---|
| 4548 | } |
---|
| 4549 | if(debu) |
---|
| 4550 | { |
---|
| 4551 | //--- various checks, see output for comments |
---|
| 4552 | if(size(BO[1])>0) |
---|
| 4553 | { |
---|
| 4554 | if(deg(BO[1][1])==0) |
---|
| 4555 | { |
---|
| 4556 | "!!! W is empty !!!"; |
---|
| 4557 | path; |
---|
| 4558 | setring R; |
---|
| 4559 | kill S; |
---|
| 4560 | list result=endRings,allRings; |
---|
| 4561 | return(result); |
---|
| 4562 | } |
---|
| 4563 | if(deg(std(slocusE(BO[1]))[1])>0) |
---|
| 4564 | { |
---|
| 4565 | "!!! W not smooth !!!"; |
---|
| 4566 | path; |
---|
| 4567 | setring R; |
---|
| 4568 | kill S; |
---|
| 4569 | list result=endRings,allRings; |
---|
| 4570 | return(result); |
---|
| 4571 | } |
---|
| 4572 | } |
---|
| 4573 | if((!loca)&&(!locaT)) |
---|
| 4574 | { |
---|
| 4575 | if(deg(std(slocusE(cent+BO[1]))[1])>0) |
---|
| 4576 | { |
---|
| 4577 | "!!! Center not smooth !!!"; |
---|
| 4578 | path; |
---|
| 4579 | std(cent+BO[1]); |
---|
| 4580 | ~; |
---|
| 4581 | setring R; |
---|
| 4582 | kill S; |
---|
| 4583 | list result=endRings,allRings; |
---|
| 4584 | return(result); |
---|
| 4585 | } |
---|
| 4586 | } |
---|
| 4587 | for(j=1;j<=size(BO[4]);j++) |
---|
| 4588 | { |
---|
| 4589 | if(deg(BO[4][j][1])>0) |
---|
| 4590 | { |
---|
| 4591 | if(deg(std(slocusE(BO[4][j]+BO[1]))[1])>0) |
---|
| 4592 | { |
---|
| 4593 | "!!! exceptional divisor is not smooth !!!"; |
---|
| 4594 | path; |
---|
| 4595 | setring R; |
---|
| 4596 | kill S; |
---|
| 4597 | list result=endRings,allRings; |
---|
| 4598 | return(result); |
---|
| 4599 | } |
---|
| 4600 | } |
---|
| 4601 | } |
---|
| 4602 | if((!loca)&&(!locaT)) |
---|
| 4603 | { |
---|
[6518eba] | 4604 | if((norC(BO,cent))&&(size(reduce(cent,Jstd,5))!=0)) |
---|
[2e6eac2] | 4605 | { |
---|
| 4606 | "!!! this chart is already finished !!!"; |
---|
| 4607 | cent=BO[2]; |
---|
| 4608 | ~; |
---|
| 4609 | } |
---|
| 4610 | } |
---|
| 4611 | } |
---|
| 4612 | //---------------------------------------------------------------------------- |
---|
| 4613 | // Do the blow up |
---|
| 4614 | //---------------------------------------------------------------------------- |
---|
| 4615 | //!!!! Change this as soon as there is time!!! |
---|
| 4616 | //!!!! quick and dirty bug fix for old shortcut which has not yet been killed |
---|
| 4617 | if((dim(std(cent))==0)&&defined(shortcut)) {kill shortcut;} |
---|
| 4618 | //!!! end of bugfix |
---|
[6518eba] | 4619 | if(size(reduce(cent,Jstd,5))!=0) |
---|
[2e6eac2] | 4620 | { |
---|
| 4621 | //--- center does not equal J |
---|
| 4622 | tmpList=blowUpBO(BO,cent,extra); |
---|
| 4623 | if((debu)&&(!loca)&&(!locaT)) |
---|
| 4624 | { |
---|
| 4625 | //--- test it, if debu is set |
---|
| 4626 | if(!testBlowUp(BO,cent,tmpList,i,extra)) |
---|
| 4627 | { |
---|
| 4628 | "non-redundant chart has been killed!"; |
---|
| 4629 | ~; |
---|
| 4630 | } |
---|
| 4631 | } |
---|
| 4632 | //--- extend the list of all rings |
---|
| 4633 | allRings[size(allRings)+1..size(allRings)+size(tmpList)]= |
---|
| 4634 | tmpList[1..size(tmpList)]; |
---|
| 4635 | for(j=1;j<=size(tmpList);j++) |
---|
| 4636 | { |
---|
| 4637 | def Q=allRings[size(allRings)-j+1]; |
---|
| 4638 | setring Q; |
---|
| 4639 | def path=imap(S,path); |
---|
| 4640 | path=path,[i,size(tmpList)-j+1]; |
---|
| 4641 | export path; |
---|
| 4642 | setring S; |
---|
| 4643 | kill Q; |
---|
| 4644 | } |
---|
| 4645 | kill tmpList; |
---|
| 4646 | list tmpList; |
---|
| 4647 | } |
---|
| 4648 | else |
---|
| 4649 | { |
---|
| 4650 | //--- center equals J |
---|
| 4651 | k=0; |
---|
| 4652 | for(j=1;j<=size(BO[6]);j++) |
---|
| 4653 | { |
---|
| 4654 | if(BO[6][j]!=1) |
---|
| 4655 | { |
---|
| 4656 | //--- there is an E_i which meets J in this chart |
---|
| 4657 | k=1; |
---|
| 4658 | break; |
---|
| 4659 | } |
---|
| 4660 | } |
---|
[f679fd2] | 4661 | if((k)||(extra==2)) |
---|
[2e6eac2] | 4662 | { |
---|
| 4663 | //--- chart finished, non-redundant |
---|
| 4664 | endRings[size(endRings)+1]=S; |
---|
| 4665 | } |
---|
| 4666 | } |
---|
| 4667 | kill pr; |
---|
| 4668 | setring R; |
---|
| 4669 | kill S; |
---|
| 4670 | } |
---|
| 4671 | //--------------------------------------------------------------------------- |
---|
| 4672 | // set up the result, test it (if debu is set) and return it |
---|
| 4673 | //--------------------------------------------------------------------------- |
---|
| 4674 | list result=endRings,allRings; |
---|
| 4675 | if(debu) |
---|
| 4676 | { |
---|
| 4677 | "============= result will be tested =========="; |
---|
| 4678 | " "; |
---|
| 4679 | "the number of charts obtained:",size(endRings); |
---|
| 4680 | if(locaT){loca=2;} |
---|
| 4681 | int tes=testRes(J,endRings,loca); |
---|
| 4682 | if(tes) |
---|
| 4683 | { |
---|
| 4684 | "============= result is o.k. =========="; |
---|
| 4685 | } |
---|
| 4686 | else |
---|
| 4687 | { |
---|
| 4688 | "============ result is wrong =========="; ~; |
---|
| 4689 | } |
---|
| 4690 | } |
---|
| 4691 | kill debugCenter,debugBlowUp,debugCoeff,debug_Inters_E; |
---|
| 4692 | if(locaT){kill @p;} |
---|
| 4693 | kill locaT; |
---|
| 4694 | return(result); |
---|
| 4695 | } |
---|
| 4696 | example |
---|
| 4697 | {"EXAMPLE:"; |
---|
| 4698 | echo = 2; |
---|
| 4699 | ring R=0,(x,y,z),dp; |
---|
| 4700 | ideal J=x3+y5+yz2+xy4; |
---|
[fa8122] | 4701 | list L=resolve(J,0); |
---|
[2e6eac2] | 4702 | def Q=L[1][7]; |
---|
| 4703 | setring Q; |
---|
| 4704 | showBO(BO); |
---|
| 4705 | } |
---|
| 4706 | ////////////////////////////////////////////////////////////////////////// |
---|
| 4707 | //static |
---|
| 4708 | proc CompMeetsE(ideal J, list E) |
---|
| 4709 | "Internal procedure - no help and no example available |
---|
| 4710 | " |
---|
| 4711 | { |
---|
| 4712 | int i; |
---|
| 4713 | for(i=1;i<=size(E);i++) |
---|
| 4714 | { |
---|
| 4715 | if(deg(std(E[i])[1])!=0) |
---|
| 4716 | { |
---|
| 4717 | if(deg(std(J+E[i])[1])!=0) |
---|
| 4718 | { |
---|
| 4719 | return(1); |
---|
| 4720 | } |
---|
| 4721 | } |
---|
| 4722 | } |
---|
| 4723 | return(0); |
---|
| 4724 | } |
---|
| 4725 | |
---|
| 4726 | //======================================================================== |
---|
| 4727 | //-------------- procedures for testing the result ---------------------- |
---|
| 4728 | // (not yet commented) |
---|
| 4729 | //======================================================================== |
---|
| 4730 | |
---|
| 4731 | ////////////////////////////////////////////////////////////////////////// |
---|
[268a430] | 4732 | static proc testRes(ideal J,list L,int loca) |
---|
[2e6eac2] | 4733 | "Internal procedure - no help and no example available |
---|
| 4734 | " |
---|
| 4735 | { |
---|
| 4736 | int loc; |
---|
| 4737 | if(defined(locaT)){loc=locaT;} |
---|
| 4738 | if(loc){loca=0;} |
---|
| 4739 | def R=basering; |
---|
| 4740 | ideal M=maxideal(1); |
---|
| 4741 | int i,j,tr; |
---|
| 4742 | for(i=1;i<=size(L);i++) |
---|
| 4743 | { |
---|
| 4744 | def Q=L[i]; |
---|
| 4745 | setring Q; |
---|
| 4746 | ideal J=BO[2]; |
---|
| 4747 | list E=BO[4]; |
---|
| 4748 | map phi=R,BO[5]; |
---|
| 4749 | ideal K=phi(J)+BO[1]; |
---|
| 4750 | ideal stTK=std(K); |
---|
| 4751 | |
---|
| 4752 | if(loca) |
---|
| 4753 | { |
---|
| 4754 | ideal M=phi(M)+BO[1]; |
---|
| 4755 | ideal stTM=std(M); |
---|
| 4756 | } |
---|
| 4757 | for(j=1;j<=size(E);j++) |
---|
| 4758 | { |
---|
| 4759 | if(deg(E[j][1])>0) |
---|
| 4760 | { |
---|
[3f7e01a] | 4761 | stTK=sat(stTK,E[j]); |
---|
[2e6eac2] | 4762 | } |
---|
| 4763 | if(loca) |
---|
| 4764 | { |
---|
[3f7e01a] | 4765 | stTM=sat(stTM,E[j]); |
---|
[2e6eac2] | 4766 | } |
---|
| 4767 | } |
---|
| 4768 | ideal sL=slocusE(J); |
---|
| 4769 | if(loca){sL=sL+stTM;} |
---|
| 4770 | ideal sLstd=std(sL); |
---|
| 4771 | if(deg(sLstd[1])>0) |
---|
| 4772 | { |
---|
| 4773 | if(!loc) |
---|
| 4774 | { |
---|
| 4775 | "J is not smooth";i; |
---|
| 4776 | setring R; |
---|
| 4777 | return(0); |
---|
| 4778 | } |
---|
[6518eba] | 4779 | if(size(reduce(@p,std(radical(sLstd)),5))>0) |
---|
[2e6eac2] | 4780 | { |
---|
| 4781 | "J is not smooth";i; |
---|
| 4782 | setring R; |
---|
| 4783 | return(0); |
---|
| 4784 | } |
---|
| 4785 | } |
---|
[6518eba] | 4786 | if(!((size(reduce(J,std(stTK),5))==0) |
---|
| 4787 | &&(size(reduce(stTK,std(J),5))==0))) |
---|
[2e6eac2] | 4788 | { |
---|
| 4789 | "map is wrong";i; |
---|
| 4790 | setring R; |
---|
| 4791 | return(0); |
---|
| 4792 | } |
---|
| 4793 | if(loc){tr=transversalT(J,E,@p);} |
---|
| 4794 | else{tr=transversalT(J,E);} |
---|
| 4795 | if(!tr) |
---|
| 4796 | { |
---|
| 4797 | "E not transversal with J";i; |
---|
| 4798 | setring R; |
---|
| 4799 | return(0); |
---|
| 4800 | } |
---|
| 4801 | if(!normalCross(E)) |
---|
| 4802 | { |
---|
| 4803 | "E not normal crossings";i; |
---|
| 4804 | setring R; |
---|
| 4805 | return(0); |
---|
| 4806 | } |
---|
| 4807 | for(j=1;j<=size(E);j++) |
---|
| 4808 | { |
---|
| 4809 | if(deg(E[j][1])>0){E[j]=E[j]+J;} |
---|
| 4810 | } |
---|
| 4811 | if(!normalCross(E)) |
---|
| 4812 | { |
---|
| 4813 | "E not normal crossings with J";i; |
---|
| 4814 | setring R; |
---|
| 4815 | return(0); |
---|
[3c4dcc] | 4816 | } |
---|
[2e6eac2] | 4817 | kill J,E,phi,K,stTK; |
---|
| 4818 | if(loca){kill M,stTM;} |
---|
| 4819 | setring R; |
---|
| 4820 | kill Q; |
---|
| 4821 | } |
---|
| 4822 | return(1); |
---|
| 4823 | } |
---|
| 4824 | ////////////////////////////////////////////////////////////////////////////// |
---|
[268a430] | 4825 | static proc testBlowUp(list BO,ideal cent,list tmpList, int j, int extra) |
---|
[2e6eac2] | 4826 | { |
---|
| 4827 | def R=basering; |
---|
| 4828 | int n=nvars(basering); |
---|
| 4829 | int i; |
---|
| 4830 | if((extra!=3)&&(extra!=2)) |
---|
| 4831 | { |
---|
| 4832 | ideal K=BO[1],BO[2],cent; |
---|
| 4833 | for(i=1;i<=size(BO[4]);i++) |
---|
| 4834 | { |
---|
| 4835 | K=K,BO[4][i]; |
---|
| 4836 | } |
---|
[f999689] | 4837 | list N=findvars(K); |
---|
| 4838 | //list N=findvars(BO[2]); |
---|
[2e6eac2] | 4839 | if(size(N[1])<n) |
---|
| 4840 | { |
---|
| 4841 | string newvar=string(N[1]); |
---|
[62de185] | 4842 | ring R1 = create_ring(ring_list(R)[1], "("+newvar+")", "dp", "no_minpoly"); |
---|
[2e6eac2] | 4843 | list BO=imap(R,BO); |
---|
| 4844 | ideal cent=imap(R,cent); |
---|
| 4845 | n=nvars(R1); |
---|
| 4846 | } |
---|
| 4847 | else |
---|
| 4848 | { |
---|
| 4849 | def R1=basering; |
---|
| 4850 | } |
---|
| 4851 | } |
---|
| 4852 | else |
---|
| 4853 | { |
---|
| 4854 | def R1=basering; |
---|
| 4855 | } |
---|
| 4856 | |
---|
| 4857 | i=0; |
---|
| 4858 | ideal T=cent; |
---|
| 4859 | ideal TW; |
---|
| 4860 | for(i=1;i<=size(tmpList);i++) |
---|
| 4861 | { |
---|
| 4862 | def Q=tmpList[i]; |
---|
| 4863 | setring Q; |
---|
| 4864 | map phi=R1,lastMap; |
---|
| 4865 | ideal TE=radical(slocusE(BO[2])); |
---|
| 4866 | setring R1; |
---|
| 4867 | TW=preimage(Q,phi,TE); |
---|
| 4868 | T=intersect(T,TW); |
---|
| 4869 | kill Q; |
---|
| 4870 | } |
---|
| 4871 | ideal sL=intersect(slocusE(BO[2]),cent); |
---|
[6518eba] | 4872 | if(size(reduce(sL,std(radical(T)),5))>0){setring R;return(0);} |
---|
| 4873 | if(size(reduce(T,std(radical(sL)),5))>0){setring R;return(0);} |
---|
[2e6eac2] | 4874 | setring R; |
---|
| 4875 | return(1); |
---|
| 4876 | } |
---|
| 4877 | ////////////////////////////////////////////////////////////////////////////// |
---|
[268a430] | 4878 | static proc normalCross(list E,list #) |
---|
[2e6eac2] | 4879 | "Internal procedure - no help and no example available |
---|
| 4880 | " |
---|
| 4881 | { |
---|
| 4882 | int loc; |
---|
| 4883 | if((defined(locaT))&&(defined(@p))) |
---|
| 4884 | { |
---|
| 4885 | loc=1; |
---|
| 4886 | ideal pp=@p; |
---|
| 4887 | } |
---|
| 4888 | int i,d,j; |
---|
| 4889 | int n=nvars(basering); |
---|
| 4890 | list E1,E2; |
---|
| 4891 | ideal K,M,Estd,cent; |
---|
| 4892 | intvec v,w; |
---|
| 4893 | if(size(#)>0){cent=#[1];} |
---|
| 4894 | |
---|
| 4895 | for(i=1;i<=size(E);i++) |
---|
| 4896 | { |
---|
| 4897 | Estd=std(E[i]); |
---|
| 4898 | if(deg(Estd[1])>0) |
---|
| 4899 | { |
---|
| 4900 | E1[size(E1)+1]=Estd; |
---|
| 4901 | } |
---|
| 4902 | } |
---|
| 4903 | E=E1; |
---|
| 4904 | for(i=1;i<=size(E);i++) |
---|
| 4905 | { |
---|
| 4906 | v=i; |
---|
| 4907 | E1[i]=list(E[i],v); |
---|
| 4908 | } |
---|
| 4909 | list ll; |
---|
| 4910 | int re=1; |
---|
| 4911 | int ok; |
---|
| 4912 | while(size(E1)>0) |
---|
| 4913 | { |
---|
| 4914 | K=E1[1][1]; |
---|
| 4915 | v=E1[1][2]; |
---|
| 4916 | attrib(K,"isSB",1); |
---|
| 4917 | E1=delete(E1,1); |
---|
| 4918 | d=n-dim(K); |
---|
| 4919 | M=minor(jacob(K),d)+K; |
---|
| 4920 | if(deg(std(M)[1])>0) |
---|
| 4921 | { |
---|
| 4922 | if(size(#)>0) |
---|
| 4923 | { |
---|
[6518eba] | 4924 | if(size(reduce(M,std(cent),5))>0) |
---|
[2e6eac2] | 4925 | { |
---|
| 4926 | ll[size(ll)+1]=std(M); |
---|
| 4927 | } |
---|
| 4928 | else |
---|
| 4929 | { |
---|
| 4930 | ok=1; |
---|
| 4931 | } |
---|
| 4932 | } |
---|
| 4933 | if(!loc) |
---|
| 4934 | { |
---|
| 4935 | re=0; |
---|
| 4936 | } |
---|
| 4937 | else |
---|
| 4938 | { |
---|
[6518eba] | 4939 | if(size(reduce(pp,std(radical(M)),5))>0){re=0;} |
---|
[2e6eac2] | 4940 | } |
---|
| 4941 | } |
---|
| 4942 | for(i=1;i<=size(E);i++) |
---|
| 4943 | { |
---|
| 4944 | for(j=1;j<=size(v);j++){if(v[j]==i){break;}} |
---|
| 4945 | if(j<=size(v)){if(v[j]==i){i++;continue;}} |
---|
| 4946 | Estd=std(K+E[i]); |
---|
| 4947 | w=v; |
---|
| 4948 | if(deg(Estd[1])==0){i++;continue;} |
---|
| 4949 | if(d==n-dim(Estd)) |
---|
| 4950 | { |
---|
| 4951 | if(size(#)>0) |
---|
| 4952 | { |
---|
[6518eba] | 4953 | if(size(reduce(Estd,std(cent),5))>0) |
---|
[2e6eac2] | 4954 | { |
---|
| 4955 | ll[size(ll)+1]=Estd; |
---|
| 4956 | } |
---|
| 4957 | else |
---|
| 4958 | { |
---|
| 4959 | ok=1; |
---|
| 4960 | } |
---|
| 4961 | } |
---|
| 4962 | if(!loc) |
---|
| 4963 | { |
---|
| 4964 | re=0; |
---|
| 4965 | } |
---|
| 4966 | else |
---|
| 4967 | { |
---|
[6518eba] | 4968 | if(size(reduce(pp,std(radical(M)),5))>0){re=0;} |
---|
[2e6eac2] | 4969 | } |
---|
| 4970 | } |
---|
| 4971 | w[size(w)+1]=i; |
---|
| 4972 | E2[size(E2)+1]=list(Estd,w); |
---|
| 4973 | } |
---|
| 4974 | if(size(E2)>0) |
---|
| 4975 | { |
---|
| 4976 | if(size(E1)>0) |
---|
| 4977 | { |
---|
| 4978 | E1[size(E1)+1..size(E1)+size(E2)]=E2[1..size(E2)]; |
---|
| 4979 | } |
---|
| 4980 | else |
---|
| 4981 | { |
---|
| 4982 | E1=E2; |
---|
| 4983 | } |
---|
| 4984 | } |
---|
| 4985 | kill E2; |
---|
| 4986 | list E2; |
---|
| 4987 | } |
---|
| 4988 | /* |
---|
| 4989 | if((!ok)&&(!re)&&(size(#)==1)) |
---|
| 4990 | { |
---|
| 4991 | |
---|
| 4992 | "the center is wrong"; |
---|
| 4993 | "it could be one of the following list"; |
---|
| 4994 | ll; |
---|
| 4995 | ~; |
---|
| 4996 | } |
---|
| 4997 | */ |
---|
| 4998 | if((!ok)&&(!re)&&(size(#)==2)) |
---|
| 4999 | { |
---|
| 5000 | return(2); //for Center correction |
---|
| 5001 | } |
---|
| 5002 | return(re); |
---|
| 5003 | } |
---|
| 5004 | ////////////////////////////////////////////////////////////////////////////// |
---|
[268a430] | 5005 | static proc normalCrossB(ideal J,list E,ideal V) |
---|
[2e6eac2] | 5006 | "Internal procedure - no help and no example available |
---|
| 5007 | " |
---|
| 5008 | { |
---|
| 5009 | int i,d,j; |
---|
| 5010 | int n=nvars(basering); |
---|
| 5011 | list E1,E2; |
---|
| 5012 | ideal K,M,Estd; |
---|
| 5013 | intvec v,w; |
---|
| 5014 | |
---|
| 5015 | for(i=1;i<=size(E);i++) |
---|
| 5016 | { |
---|
| 5017 | Estd=std(E[i]+J); |
---|
| 5018 | if(deg(Estd[1])>0) |
---|
| 5019 | { |
---|
| 5020 | E1[size(E1)+1]=Estd; |
---|
| 5021 | } |
---|
| 5022 | } |
---|
| 5023 | E=E1; |
---|
| 5024 | for(i=1;i<=size(E);i++) |
---|
| 5025 | { |
---|
| 5026 | v=i; |
---|
| 5027 | E1[i]=list(E[i],v); |
---|
| 5028 | } |
---|
| 5029 | list ll; |
---|
| 5030 | int re=1; |
---|
| 5031 | |
---|
| 5032 | while((size(E1)>0)&&(re==1)) |
---|
| 5033 | { |
---|
| 5034 | K=E1[1][1]; |
---|
| 5035 | v=E1[1][2]; |
---|
| 5036 | attrib(K,"isSB",1); |
---|
| 5037 | E1=delete(E1,1); |
---|
| 5038 | d=n-dim(K); |
---|
| 5039 | M=minor(jacob(K),d)+K; |
---|
| 5040 | if(deg(std(M+V)[1])>0) |
---|
| 5041 | { |
---|
| 5042 | re=0; |
---|
| 5043 | break; |
---|
| 5044 | } |
---|
| 5045 | for(i=1;i<=size(E);i++) |
---|
| 5046 | { |
---|
| 5047 | for(j=1;j<=size(v);j++){if(v[j]==i){break;}} |
---|
| 5048 | if(j<=size(v)){if(v[j]==i){i++;continue;}} |
---|
| 5049 | Estd=std(K+E[i]); |
---|
| 5050 | w=v; |
---|
| 5051 | if(deg(Estd[1])==0){i++;continue;} |
---|
| 5052 | if(d==n-dim(Estd)) |
---|
| 5053 | { |
---|
| 5054 | if(deg(std(Estd+V)[1])>0) |
---|
| 5055 | { |
---|
| 5056 | re=0; |
---|
| 5057 | break; |
---|
| 5058 | } |
---|
| 5059 | } |
---|
| 5060 | w[size(w)+1]=i; |
---|
| 5061 | E2[size(E2)+1]=list(Estd,w); |
---|
| 5062 | } |
---|
| 5063 | if(size(E2)>0) |
---|
| 5064 | { |
---|
| 5065 | if(size(E1)>0) |
---|
| 5066 | { |
---|
| 5067 | E1[size(E1)+1..size(E1)+size(E2)]=E2[1..size(E2)]; |
---|
| 5068 | } |
---|
| 5069 | else |
---|
| 5070 | { |
---|
| 5071 | E1=E2; |
---|
| 5072 | } |
---|
| 5073 | } |
---|
| 5074 | kill E2; |
---|
| 5075 | list E2; |
---|
| 5076 | } |
---|
| 5077 | return(re); |
---|
| 5078 | } |
---|
| 5079 | ////////////////////////////////////////////////////////////////////////////// |
---|
[268a430] | 5080 | static proc norC(list BO,ideal cent) |
---|
[2e6eac2] | 5081 | "Internal procedure - no help and no example available |
---|
| 5082 | " |
---|
| 5083 | { |
---|
| 5084 | int j; |
---|
| 5085 | list E=BO[4]; |
---|
| 5086 | ideal N=BO[2]; |
---|
| 5087 | if(BO[3][1]>1){return(0);} |
---|
| 5088 | if(deg(std(slocusE(BO[2]))[1])>0){return(0);} |
---|
| 5089 | if(!transversalT(N,E)){return(0);} |
---|
| 5090 | for(j=1;j<=size(E);j++){if(deg(E[j][1])>0){E[j]=E[j]+N;}} |
---|
| 5091 | if(!normalCross(E,cent)){return(0);} |
---|
| 5092 | return(1); |
---|
| 5093 | } |
---|
| 5094 | ////////////////////////////////////////////////////////////////////////////// |
---|
[268a430] | 5095 | static proc specialReduce(ideal I,ideal J,poly p) |
---|
[2e6eac2] | 5096 | { |
---|
| 5097 | matrix M; |
---|
| 5098 | int i,j; |
---|
| 5099 | for(i=1;i<=ncols(I);i++) |
---|
| 5100 | { |
---|
| 5101 | M=coeffs(I[i],@z); |
---|
| 5102 | I[i]=0; |
---|
| 5103 | for(j=1;j<=nrows(M);j++) |
---|
| 5104 | { |
---|
| 5105 | I[i]=I[i]+M[j,1]*p^(nrows(M)-j); |
---|
| 5106 | } |
---|
| 5107 | I[i]=reduce(I[i],J); |
---|
| 5108 | } |
---|
| 5109 | return(I); |
---|
| 5110 | } |
---|