1 | /////////////////////////////////////////////////////// |
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2 | version="$Id$"; |
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3 | category="Noncommutative"; |
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4 | info=" |
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5 | LIBRARY: fpadim.lib Algorithms for quotient algebras in the letterplace case |
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6 | AUTHORS: Grischa Studzinski, grischa.studzinski@rwth-aachen.de |
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7 | |
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8 | Support: Joint projects LE 2697/2-1 and KR 1907/3-1 of the Priority Programme SPP 1489: |
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9 | @* 'Algorithmische und Experimentelle Methoden in Algebra, Geometrie und Zahlentheorie' |
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10 | @* of the German DFG |
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11 | |
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12 | OVERVIEW: Given the free algebra A = K<x_1,...,x_n> and a (finite) Groebner basis |
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13 | @* GB = {g_1,..,g_w}, one is interested in the K-dimension and in the |
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14 | @* explicit K-basis of A/<GB>. |
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15 | @* Therefore one is interested in the following data: |
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16 | @* - the Ufnarovskij graph induced by GB |
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17 | @* - the mistletoes of A/<GB> |
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18 | @* - the K-dimension of A/<GB> |
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19 | @* - the Hilbert series of A/<GB> |
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20 | @* |
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21 | @* The Ufnarovskij graph is used to determine whether A/<GB> has finite |
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22 | @* K-dimension. One has to check if the graph contains cycles. |
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23 | @* For the whole theory we refer to [ufna]. Given a |
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24 | @* reduced set of monomials GB one can define the basis tree, whose vertex |
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25 | @* set V consists of all normal monomials w.r.t. GB. For every two |
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26 | @* monomials m_1, m_2 in V there is a direct edge from m_1 to m_2, if and |
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27 | @* only if there exists x_k in {x_1,..,x_n}, such that m_1*x_k = m_2. The |
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28 | @* set M = {m in V | there is no edge from m to another monomial in V} is |
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29 | @* called the set of mistletoes. As one can easily see it consists of |
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30 | @* the endpoints of the graph. Since there is a unique path to every |
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31 | @* monomial in V the whole graph can be described only from the knowledge |
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32 | @* of the mistletoes. Note that V corresponds to a basis of A/<GB>, so |
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33 | @* knowing the mistletoes we know a K-basis. For more details see |
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34 | @* [studzins]. This package uses the Letterplace format introduced by |
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35 | @* [lls]. The algebra can either be represented as a Letterplace ring or |
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36 | @* via integer vectors: Every variable will only be represented by its |
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37 | @* number, so variable one is represented as 1, variable two as 2 and so |
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38 | @* on. The monomial x_1*x_3*x_2 for example will be stored as (1,3,2). |
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39 | @* Multiplication is concatenation. Note that there is no algorithm for |
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40 | @* computing the normal form yet, but for our case it is not needed. |
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41 | @* |
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42 | |
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43 | References: |
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44 | |
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45 | @* [ufna] Ufnarovskij: Combinatorical and asymptotic methods in algebra, 1990 |
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46 | @* [lls] Levandovskyy, La Scala: Letterplace ideals and non-commutative |
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47 | Groebner bases, 2009 |
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48 | @* [studzins] Studzinski: Dimension computations in non-commutative, |
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49 | associative algebras, Diploma thesis, RWTH Aachen, 2010 |
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50 | |
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51 | Assumptions: |
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52 | @* - basering is always a Letterplace ring |
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53 | @* - all intvecs correspond to Letterplace monomials |
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54 | @* - if you specify a different degree bound d, |
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55 | d <= attrib(basering,uptodeg) holds |
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56 | @* In the procedures below, 'iv' stands for intvec representation |
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57 | and 'lp' for the letterplace representation of monomials |
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58 | |
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59 | PROCEDURES: |
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60 | |
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61 | lpGkDim(G); computes the Gelfand Kirillov dimension of A/<G> |
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62 | ivDHilbert(L,n[,d]); computes the K-dimension and the Hilbert series |
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63 | ivDHilbertSickle(L,n[,d]); computes mistletoes, K-dimension and Hilbert series |
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64 | ivDimCheck(L,n); checks if the K-dimension of A/<L> is infinite |
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65 | lpGlDimBound(G); computes upper bound of global dimension of A/<G> |
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66 | ivHilbert(L,n[,d]); computes the Hilbert series of A/<L> in intvec format |
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67 | ivKDim(L,n[,d]); computes the K-dimension of A/<L> in intvec format |
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68 | ivMis2Dim(M); computes the K-dimension of the factor algebra |
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69 | ivOrdMisLex(M); orders a list of intvecs lexicographically |
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70 | ivSickle(L,n[,d]); computes the mistletoes of A/<L> in intvec format |
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71 | ivSickleHil(L,n[,d]); computes the mistletoes and Hilbert series of A/<L> |
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72 | ivSickleDim(L,n[,d]); computes the mistletoes and the K-dimension of A/<L> |
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73 | lpDHilbert(G[,d,n]); computes the K-dimension and Hilbert series of A/<G> |
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74 | lpDHilbertSickle(G[,d,n]); computes mistletoes, K-dimension and Hilbert series |
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75 | lpHilbert(G[,d,n]); computes the Hilbert series of A/<G> in lp format |
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76 | lpDimCheck(G); checks if the K-dimension of A/<G> is infinite |
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77 | lpKDim(G[,d,n]); computes the K-dimension of A/<G> in lp format |
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78 | lpMis2Dim(M); computes the K-dimension of the factor algebra |
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79 | lpOrdMisLex(M); orders an ideal of lp-monomials lexicographically |
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80 | lpSickle(G[,d,n]); computes the mistletoes of A/<G> in lp format |
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81 | lpSickleHil(G[,d,n]); computes the mistletoes and Hilbert series of A/<G> |
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82 | lpSickleDim(G[,d,n]); computes the mistletoes and the K-dimension of A/<G> |
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83 | sickle(G[,m,d,h]); can be used to access all lp main procedures |
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84 | |
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85 | |
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86 | ivL2lpI(L); transforms a list of intvecs into an ideal of lp monomials |
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87 | iv2lp(I); transforms an intvec into the corresponding monomial |
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88 | iv2lpList(L); transforms a list of intmats into an ideal of lp monomials |
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89 | iv2lpMat(M); transforms an intmat into an ideal of lp monomials |
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90 | lp2iv(p); transforms a polynomial into the corresponding intvec |
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91 | lp2ivId(G); transforms an ideal into the corresponding list of intmats |
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92 | lpId2ivLi(G); transforms a lp-ideal into the corresponding list of intvecs |
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93 | |
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94 | SEE ALSO: freegb_lib |
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95 | "; |
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96 | |
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97 | LIB "freegb.lib"; //for letterplace rings |
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98 | LIB "general.lib";//for sorting mistletoes |
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99 | |
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100 | ///////////////////////////////////////////////////////// |
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101 | |
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102 | |
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103 | //--------------- auxiliary procedures ------------------ |
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104 | |
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105 | static proc allVars(list L, intvec P, int n) |
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106 | "USAGE: allVars(L,P,n); L a list of intmats, P an intvec, n an integer |
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107 | RETURN: int, 0 if all variables are contained in the quotient algebra, 1 otherwise |
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108 | " |
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109 | {int i,j,r; |
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110 | intvec V; |
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111 | for (i = 1; i <= size(P); i++) {if (P[i] == 1){ j = i; break;}} |
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112 | V = L[j][1..nrows(L[j]),1]; |
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113 | for (i = 1; i <= n; i++) {if (isInVec(i,V) == 0) {r = 1; break;}} |
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114 | if (r == 0) {return(1);} |
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115 | else {return(0);} |
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116 | } |
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117 | |
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118 | static proc checkAssumptions(int d, list L) |
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119 | "PURPOSE: Checks, if all the Assumptions are holding |
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120 | " |
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121 | {if (typeof(attrib(basering,"isLetterplaceRing"))=="string") {ERROR("Basering is not a Letterplace ring!");} |
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122 | if (d > attrib(basering,"uptodeg")) {ERROR("Specified degree bound exceeds ring parameter!");} |
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123 | int i; |
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124 | for (i = 1; i <= size(L); i++) |
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125 | {if (entryViolation(L[i], attrib(basering,"lV"))) |
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126 | {ERROR("Not allowed monomial/intvec found!");} |
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127 | } |
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128 | return(); |
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129 | } |
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130 | |
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131 | static proc createStartMat(int d, int n) |
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132 | "USAGE: createStartMat(d,n); d, n integers |
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133 | RETURN: intmat |
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134 | PURPOSE:Creating the intmat with all normal monomials in n variables and of degree d to start with |
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135 | NOTE: d has to be > 0 |
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136 | " |
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137 | {intmat M[(n^d)][d]; |
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138 | int i1,i2,i3,i4; |
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139 | for (i1 = 1; i1 <= d; i1++) //Spalten |
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140 | {i2 = 1; //durchlaeuft Zeilen |
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141 | while (i2 <= (n^d)) |
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142 | {for (i3 = 1; i3 <= n; i3++) |
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143 | {for (i4 = 1; i4 <= (n^(i1-1)); i4++) |
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144 | {M[i2,i1] = i3; |
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145 | i2 = i2 + 1; |
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146 | } |
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147 | } |
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148 | } |
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149 | } |
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150 | return(M); |
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151 | } |
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152 | |
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153 | static proc createStartMat1(int n, intmat M) |
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154 | "USAGE: createStartMat1(n,M); n an integer, M an intmat |
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155 | RETURN: intmat, with all variables except those in M |
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156 | " |
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157 | {int i; |
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158 | intvec V,Vt; |
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159 | V = M[(1..nrows(M)),1]; |
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160 | for (i = 1; i <= size(V); i++) {if (isInVec(i,V) == 0) {Vt = Vt,i;}} |
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161 | if (Vt == 0) {intmat S; return(S);} |
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162 | else {Vt = Vt[2..size(Vt)]; intmat S [size(Vt)][1]; S[1..size(Vt),1] = Vt; return(S);} |
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163 | } |
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164 | |
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165 | static proc entryViolation(intmat M, int n) |
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166 | "PURPOSE:checks, if all entries in M are variable-related |
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167 | " |
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168 | {int i,j; |
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169 | for (i = 1; i <= nrows(M); i++) |
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170 | {for (j = 1; j <= ncols(M); j++) |
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171 | {if(!((1<=M[i,j])&&(M[i,j]<=n))) {return(1);}} |
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172 | } |
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173 | return(0); |
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174 | } |
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175 | |
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176 | static proc findDimen(intvec V, int n, list L, intvec P, list #) |
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177 | "USAGE: findDimen(V,n,L,P,degbound); V,P intvecs, n, an integer, L a list, |
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178 | @* degbound an optional integer |
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179 | RETURN: int |
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180 | PURPOSE:Computing the K-dimension of the quotient algebra |
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181 | " |
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182 | {int degbound = 0; |
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183 | if (size(#) > 0) {if (#[1] > 0) {degbound = #[1];}} |
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184 | int dimen,i,j,w,it; |
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185 | intvec Vt,Vt2; |
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186 | module M; |
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187 | if (degbound == 0) |
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188 | {for (i = 1; i <= n; i++) |
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189 | {Vt = V, i; w = 0; |
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190 | for (j = 1; j<= size(P); j++) |
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191 | {if (P[j] <= size(Vt)) |
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192 | {Vt2 = Vt[(size(Vt)-P[j]+1)..size(Vt)]; |
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193 | if (isInMat(Vt2,L[j]) > 0) {w = 1; break;} |
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194 | } |
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195 | } |
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196 | if (w == 0) |
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197 | {vector Vtt; |
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198 | for (it = 1; it <= size(Vt); it++){Vtt = Vtt + Vt[it]*gen(it);} |
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199 | M = M,Vtt; |
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200 | kill Vtt; |
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201 | } |
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202 | } |
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203 | if (size(M) == 0) {return(0);} |
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204 | else |
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205 | {M = simplify(M,2); |
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206 | for (i = 1; i <= size(M); i++) |
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207 | {kill Vt; intvec Vt; |
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208 | for (j =1; j <= size(M[i]); j++){Vt[j] = int(leadcoef(M[i][j]));} |
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209 | dimen = dimen + 1 + findDimen(Vt,n,L,P); |
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210 | } |
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211 | return(dimen); |
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212 | } |
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213 | } |
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214 | else |
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215 | {if (size(V) > degbound) {ERROR("monomial exceeds degreebound");} |
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216 | if (size(V) == degbound) {return(0);} |
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217 | for (i = 1; i <= n; i++) |
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218 | {Vt = V, i; w = 0; |
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219 | for (j = 1; j<= size(P); j++) |
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220 | {if (P[j] <= size(Vt)) |
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221 | {Vt2 = Vt[(size(Vt)-P[j]+1)..size(Vt)]; |
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222 | if (isInMat(Vt2,L[j]) > 0) {w = 1; break;} |
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223 | } |
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224 | } |
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225 | if (w == 0) {vector Vtt; |
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226 | for (it = 1; it <= size(Vt); it++){Vtt = Vtt + Vt[it]*gen(it);} |
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227 | M = M,Vtt; |
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228 | kill Vtt; |
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229 | } |
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230 | } |
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231 | if (size(M) == 0) {return(0);} |
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232 | else |
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233 | {M = simplify(M,2); |
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234 | for (i = 1; i <= size(M); i++) |
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235 | {kill Vt; intvec Vt; |
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236 | for (j =1; j <= size(M[i]); j++){Vt[j] = int(leadcoef(M[i][j]));} |
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237 | dimen = dimen + 1 + findDimen(Vt,n,L,P,degbound); |
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238 | } |
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239 | return(dimen); |
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240 | } |
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241 | } |
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242 | } |
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243 | |
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244 | static proc findCycle(intvec V, list L, intvec P, int n, int ld, module M) |
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245 | "USAGE: |
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246 | RETURN: int, 1 if Ufn-graph contains a cycle, or 0 otherwise |
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247 | PURPOSE:Searching the Ufnarovskij graph for cycles |
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248 | " |
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249 | {int i,j,w,r;intvec Vt,Vt2; |
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250 | int it, it2; |
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251 | if (size(V) < ld) |
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252 | {for (i = 1; i <= n; i++) |
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253 | {Vt = V,i; w = 0; |
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254 | for (j = 1; j <= size(P); j++) |
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255 | {if (P[j] <= size(Vt)) |
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256 | {Vt2 = Vt[(size(Vt)-P[j]+1)..size(Vt)]; |
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257 | if (isInMat(Vt2,L[j]) > 0) |
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258 | {w = 1; break;} |
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259 | } |
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260 | } |
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261 | if (w == 0) {r = findCycle(Vt,L,P,n,ld,M);} |
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262 | if (r == 1) {break;} |
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263 | } |
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264 | return(r); |
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265 | } |
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266 | else |
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267 | {j = size(M); |
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268 | if (j > 0) |
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269 | { |
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270 | intmat Mt[j][nrows(M)]; |
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271 | for (it = 1; it <= j; it++) |
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272 | { for(it2 = 1; it2 <= nrows(M);it2++) |
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273 | {Mt[it,it2] = int(leadcoef(M[it2,it]));} |
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274 | } |
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275 | Vt = V[(size(V)-ld+1)..size(V)]; |
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276 | //Mt; type(Mt);Vt;type(Vt); |
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277 | if (isInMat(Vt,Mt) > 0) {return(1);} |
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278 | else |
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279 | {vector Vtt; |
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280 | for (it =1; it <= size(Vt); it++) |
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281 | {Vtt = Vtt + Vt[it]*gen(it);} |
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282 | M = M,Vtt; |
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283 | kill Vtt; |
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284 | for (i = 1; i <= n; i++) |
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285 | {Vt = V,i; w = 0; |
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286 | for (j = 1; j <= size(P); j++) |
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287 | {if (P[j] <= size(Vt)) |
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288 | {Vt2 = Vt[(size(Vt)-P[j]+1)..size(Vt)]; |
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289 | //L[j]; type(L[j]);Vt2;type(Vt2); |
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290 | if (isInMat(Vt2,L[j]) > 0) |
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291 | {w = 1; break;} |
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292 | } |
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293 | } |
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294 | if (w == 0) {r = findCycle(Vt,L,P,n,ld,M);} |
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295 | if (r == 1) {break;} |
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296 | } |
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297 | return(r); |
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298 | } |
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299 | } |
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300 | else |
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301 | { Vt = V[(size(V)-ld+1)..size(V)]; |
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302 | vector Vtt; |
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303 | for (it = 1; it <= size(Vt); it++) |
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304 | {Vtt = Vtt + Vt[it]*gen(it);} |
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305 | M = Vtt; |
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306 | kill Vtt; |
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307 | for (i = 1; i <= n; i++) |
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308 | {Vt = V,i; w = 0; |
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309 | for (j = 1; j <= size(P); j++) |
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310 | {if (P[j] <= size(Vt)) |
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311 | {Vt2 = Vt[(size(Vt)-P[j]+1)..size(Vt)]; |
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312 | //L[j]; type(L[j]);Vt2;type(Vt2); |
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313 | if (isInMat(Vt2,L[j]) > 0) |
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314 | {w = 1; break;} |
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315 | } |
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316 | } |
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317 | if (w == 0) {r = findCycle(Vt,L,P,n,ld,M);} |
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318 | if (r == 1) {break;} |
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319 | } |
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320 | return(r); |
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321 | } |
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322 | } |
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323 | } |
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324 | |
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325 | |
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326 | static proc findCycleDFS(int i, intmat T, intvec V) |
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327 | " |
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328 | PURPOSE: |
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329 | this is a classical deep-first search for cycles contained in a graph given by an intmat |
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330 | " |
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331 | { |
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332 | intvec rV; |
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333 | int k,k1,t; |
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334 | int j = V[size(V)]; |
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335 | if (T[j,i] > 0) {return(V);} |
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336 | else |
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337 | { |
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338 | for (k = 1; k <= ncols(T); k++) |
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339 | { |
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340 | t = 0; |
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341 | if (T[j,k] > 0) |
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342 | { |
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343 | for (k1 = 1; k1 <= size(V); k1++) {if (V[k1] == k) {t = 1; break;}} |
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344 | if (t == 0) |
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345 | { |
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346 | rV = V; |
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347 | rV[size(rV)+1] = k; |
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348 | rV = findCycleDFS(i,T,rV); |
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349 | if (rV[1] > -1) {return(rV);} |
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350 | } |
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351 | } |
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352 | } |
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353 | } |
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354 | return(intvec(-1)); |
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355 | } |
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356 | |
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357 | |
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358 | |
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359 | static proc findHCoeff(intvec V,int n,list L,intvec P,intvec H,list #) |
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360 | "USAGE: findHCoeff(V,n,L,P,H,degbound); L a list of intmats, degbound an integer |
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361 | RETURN: intvec |
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362 | PURPOSE:Computing the coefficient of the Hilbert series (upto degree degbound) |
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363 | NOTE: Starting with a part of the Hilbert series we change the coefficient |
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364 | @* depending on how many basis elements we found on the actual branch |
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365 | " |
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366 | {int degbound = 0; |
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367 | if (size(#) > 0){if (#[1] > 0){degbound = #[1];}} |
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368 | int i,w,j,it; |
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369 | int h1 = 0; |
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370 | intvec Vt,Vt2,H1; |
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371 | module M; |
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372 | if (degbound == 0) |
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373 | {for (i = 1; i <= n; i++) |
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374 | {Vt = V, i; w = 0; |
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375 | for (j = 1; j<= size(P); j++) |
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376 | {if (P[j] <= size(Vt)) |
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377 | {Vt2 = Vt[(size(Vt)-P[j]+1)..size(Vt)]; |
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378 | if (isInMat(Vt2,L[j]) > 0) {w = 1; break;} |
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379 | } |
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380 | } |
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381 | if (w == 0) |
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382 | {vector Vtt; |
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383 | for (it = 1; it <= size(Vt); it++){Vtt = Vtt + Vt[it]*gen(it);} |
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384 | M = M,Vtt; |
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385 | kill Vtt; |
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386 | } |
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387 | } |
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388 | if (size(M) == 0) {return(H);} |
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389 | else |
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390 | {M = simplify(M,2); |
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391 | for (i = 1; i <= size(M); i++) |
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392 | {kill Vt; intvec Vt; |
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393 | for (j =1; j <= size(M[i]); j++) {Vt[j] = int(leadcoef(M[i][j]));} |
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394 | h1 = h1 + 1; H1 = findHCoeff(Vt,n,L,P,H1); |
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395 | } |
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396 | if (size(H1) < (size(V)+2)) {H1[(size(V)+2)] = h1;} |
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397 | else {H1[(size(V)+2)] = H1[(size(V)+2)] + h1;} |
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398 | H1 = H1 + H; |
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399 | return(H1); |
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400 | } |
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401 | } |
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402 | else |
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403 | {if (size(V) > degbound) {ERROR("monomial exceeds degreebound");} |
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404 | if (size(V) == degbound) {return(H);} |
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405 | for (i = 1; i <= n; i++) |
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406 | {Vt = V, i; w = 0; |
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407 | for (j = 1; j<= size(P); j++) |
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408 | {if (P[j] <= size(Vt)) |
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409 | {Vt2 = Vt[(size(Vt)-P[j]+1)..size(Vt)]; |
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410 | if (isInMat(Vt2,L[j]) > 0) {w = 1; break;} |
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411 | } |
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412 | } |
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413 | if (w == 0) |
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414 | {vector Vtt; |
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415 | for (it = 1; it <= size(Vt); it++){Vtt = Vtt + Vt[it]*gen(it);} |
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416 | M = M,Vtt; |
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417 | kill Vtt; |
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418 | } |
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419 | } |
---|
420 | if (size(M) == 0) {return(H);} |
---|
421 | else |
---|
422 | {M = simplify(M,2); |
---|
423 | for (i = 1; i <= size(M); i++) |
---|
424 | {kill Vt; intvec Vt; |
---|
425 | for (j =1; j <= size(M[i]); j++) |
---|
426 | {Vt[j] = int(leadcoef(M[i][j]));} |
---|
427 | h1 = h1 + 1; H1 = findHCoeff(Vt,n,L,P,H1,degbound); |
---|
428 | } |
---|
429 | if (size(H1) < (size(V)+2)) { H1[(size(V)+2)] = h1;} |
---|
430 | else {H1[(size(V)+2)] = H1[(size(V)+2)] + h1;} |
---|
431 | H1 = H1 + H; |
---|
432 | return(H1); |
---|
433 | } |
---|
434 | } |
---|
435 | } |
---|
436 | |
---|
437 | static proc findHCoeffMis(intvec V, int n, list L, intvec P, list R,list #) |
---|
438 | "USAGE: findHCoeffMis(V,n,L,P,R,degbound); degbound an optional integer, L a |
---|
439 | @* list of Intmats, R |
---|
440 | RETURN: list |
---|
441 | PURPOSE:Computing the coefficients of the Hilbert series and the Mistletoes all |
---|
442 | @* at once |
---|
443 | " |
---|
444 | {int degbound = 0; |
---|
445 | if (size(#) > 0) {if (#[1] > 0) {degbound = #[1];}} |
---|
446 | int i,w,j,h1; |
---|
447 | intvec Vt,Vt2,H1; int it; |
---|
448 | module M; |
---|
449 | if (degbound == 0) |
---|
450 | {for (i = 1; i <= n; i++) |
---|
451 | {Vt = V, i; w = 0; |
---|
452 | for (j = 1; j<= size(P); j++) |
---|
453 | {if (P[j] <= size(Vt)) |
---|
454 | {Vt2 = Vt[(size(Vt)-P[j]+1)..size(Vt)]; |
---|
455 | if (isInMat(Vt2,L[j]) > 0) {w = 1; break;} |
---|
456 | } |
---|
457 | } |
---|
458 | if (w == 0) |
---|
459 | {vector Vtt; |
---|
460 | for (it = 1; it <= size(Vt); it++){Vtt = Vtt + Vt[it]*gen(it);} |
---|
461 | M = M,Vtt; |
---|
462 | kill Vtt; |
---|
463 | } |
---|
464 | } |
---|
465 | if (size(M) == 0) {if (size(R) < 2){R[2] = list(V);} else {R[2] = R[2] + list(V);} return(R);} |
---|
466 | else |
---|
467 | {M = simplify(M,2); |
---|
468 | for (i = 1; i <= size(M); i++) |
---|
469 | {kill Vt; intvec Vt; |
---|
470 | for (j =1; j <= size(M[i]); j++) |
---|
471 | {Vt[j] = int(leadcoef(M[i][j]));} |
---|
472 | if (size(R[1]) < (size(V)+2)) { R[1][(size(V)+2)] = 1;} |
---|
473 | else |
---|
474 | {R[1][(size(V)+2)] = R[1][(size(V)+2)] + 1;} |
---|
475 | R = findHCoeffMis(Vt,n,L,P,R); |
---|
476 | } |
---|
477 | return(R); |
---|
478 | } |
---|
479 | } |
---|
480 | else |
---|
481 | {if (size(V) > degbound) {ERROR("monomial exceeds degreebound");} |
---|
482 | if (size(V) == degbound) |
---|
483 | {if (size(R) < 2){R[2] = list (V);} |
---|
484 | else{R[2] = R[2] + list (V);} |
---|
485 | return(R); |
---|
486 | } |
---|
487 | for (i = 1; i <= n; i++) |
---|
488 | {Vt = V, i; w = 0; |
---|
489 | for (j = 1; j<= size(P); j++) |
---|
490 | {if (P[j] <= size(Vt)) |
---|
491 | {Vt2 = Vt[(size(Vt)-P[j]+1)..size(Vt)]; |
---|
492 | if (isInMat(Vt2,L[j]) > 0) {w = 1; break;} |
---|
493 | } |
---|
494 | } |
---|
495 | if (w == 0) |
---|
496 | {vector Vtt; |
---|
497 | for (it = 1; it <= size(Vt); it++){Vtt = Vtt + Vt[it]*gen(it);} |
---|
498 | M = M,Vtt; |
---|
499 | kill Vtt; |
---|
500 | } |
---|
501 | } |
---|
502 | if (size(M) == 0) {if (size(R) < 2){R[2] = list(V);} else {R[2] = R[2] + list(V);} return(R);} |
---|
503 | else |
---|
504 | {M = simplify(M,2); |
---|
505 | for (i = 1; i <= ncols(M); i++) |
---|
506 | {kill Vt; intvec Vt; |
---|
507 | for (j =1; j <= size(M[i]); j++) |
---|
508 | {Vt[j] = int(leadcoef(M[i][j]));} |
---|
509 | if (size(R[1]) < (size(V)+2)) { R[1][(size(V)+2)] = 1;} |
---|
510 | else |
---|
511 | {R[1][(size(V)+2)] = R[1][(size(V)+2)] + 1;} |
---|
512 | R = findHCoeffMis(Vt,n,L,P,R,degbound); |
---|
513 | } |
---|
514 | return(R); |
---|
515 | } |
---|
516 | } |
---|
517 | } |
---|
518 | |
---|
519 | |
---|
520 | static proc findMisDim(intvec V,int n,list L,intvec P,list R,list #) |
---|
521 | "USAGE: |
---|
522 | RETURN: list |
---|
523 | PURPOSE:Computing the K-dimension and the Mistletoes all at once |
---|
524 | " |
---|
525 | {int degbound = 0; |
---|
526 | if (size(#) > 0) {if (#[1] > 0) {degbound = #[1];}} |
---|
527 | int dimen,i,j,w; |
---|
528 | intvec Vt,Vt2; int it; |
---|
529 | module M; |
---|
530 | if (degbound == 0) |
---|
531 | {for (i = 1; i <= n; i++) |
---|
532 | {Vt = V, i; w = 0; |
---|
533 | for (j = 1; j<= size(P); j++) |
---|
534 | {if (P[j] <= size(Vt)) |
---|
535 | {Vt2 = Vt[(size(Vt)-P[j]+1)..size(Vt)]; |
---|
536 | if (isInMat(Vt2,L[j]) > 0) {w = 1; break;} |
---|
537 | } |
---|
538 | } |
---|
539 | if (w == 0) |
---|
540 | {vector Vtt; |
---|
541 | for (it = 1; it <= size(Vt); it++){Vtt = Vtt + Vt[it]*gen(it);} |
---|
542 | M = M,Vtt; |
---|
543 | kill Vtt; |
---|
544 | } |
---|
545 | } |
---|
546 | if (size(M) == 0) |
---|
547 | {if (size(R) < 2){R[2] = list (V);} |
---|
548 | else{R[2] = R[2] + list(V);} |
---|
549 | return(R); |
---|
550 | } |
---|
551 | else |
---|
552 | {M = simplify(M,2); |
---|
553 | for (i = 1; i <= size(M); i++) |
---|
554 | {kill Vt; intvec Vt; |
---|
555 | for (j =1; j <= size(M[i]); j++){Vt[j] = int(leadcoef(M[i][j]));} |
---|
556 | R[1] = R[1] + 1; R = findMisDim(Vt,n,L,P,R); |
---|
557 | } |
---|
558 | return(R); |
---|
559 | } |
---|
560 | } |
---|
561 | else |
---|
562 | {if (size(V) > degbound) {ERROR("monomial exceeds degreebound");} |
---|
563 | if (size(V) == degbound) |
---|
564 | {if (size(R) < 2){R[2] = list (V);} |
---|
565 | else{R[2] = R[2] + list (V);} |
---|
566 | return(R); |
---|
567 | } |
---|
568 | for (i = 1; i <= n; i++) |
---|
569 | {Vt = V, i; w = 0; |
---|
570 | for (j = 1; j<= size(P); j++) |
---|
571 | {if (P[j] <= size(Vt)) |
---|
572 | {Vt2 = Vt[(size(Vt)-P[j]+1)..size(Vt)]; |
---|
573 | if (isInMat(Vt2,L[j]) > 0) {w = 1; break;} |
---|
574 | } |
---|
575 | } |
---|
576 | if (w == 0) |
---|
577 | {vector Vtt; |
---|
578 | for (it = 1; it <= size(Vt); it++){Vtt = Vtt + Vt[it]*gen(it);} |
---|
579 | M = M,Vtt; |
---|
580 | kill Vtt; |
---|
581 | } |
---|
582 | } |
---|
583 | if (size(M) == 0) |
---|
584 | {if (size(R) < 2){R[2] = list (V);} |
---|
585 | else{R[2] = R[2] + list(V);} |
---|
586 | return(R); |
---|
587 | } |
---|
588 | else |
---|
589 | {M = simplify(M,2); |
---|
590 | for (i = 1; i <= size(M); i++) |
---|
591 | {kill Vt; intvec Vt; |
---|
592 | for (j =1; j <= size(M[i]); j++){Vt[j] = int(leadcoef(M[i][j]));} |
---|
593 | R[1] = R[1] + 1; R = findMisDim(Vt,n,L,P,R,degbound); |
---|
594 | } |
---|
595 | return(R); |
---|
596 | } |
---|
597 | } |
---|
598 | } |
---|
599 | |
---|
600 | |
---|
601 | static proc findmistletoes(intvec V, int n, list L, intvec P, list #) |
---|
602 | "USAGE: findmistletoes(V,n,L,P,degbound); V a normal word, n the number of |
---|
603 | @* variables, L the GB, P the occuring degrees, |
---|
604 | @* and degbound the (optional) degreebound |
---|
605 | RETURN: list |
---|
606 | PURPOSE:Computing mistletoes starting in V |
---|
607 | NOTE: V has to be normal w.r.t. L, it will not be checked for being so |
---|
608 | " |
---|
609 | {int degbound = 0; |
---|
610 | if (size(#) > 0) {if (#[1] > 0) {degbound = #[1];}} |
---|
611 | list R; intvec Vt,Vt2; int it; |
---|
612 | int i,j; |
---|
613 | module M; |
---|
614 | if (degbound == 0) |
---|
615 | {int w; |
---|
616 | for (i = 1; i <= n; i++) |
---|
617 | {Vt = V,i; w = 0; |
---|
618 | for (j = 1; j <= size(P); j++) |
---|
619 | {if (P[j] <= size(Vt)) |
---|
620 | {Vt2 = Vt[(size(Vt)-P[j]+1)..size(Vt)]; |
---|
621 | if (isInMat(Vt2,L[j]) > 0) |
---|
622 | {w = 1; break;} |
---|
623 | } |
---|
624 | } |
---|
625 | if (w == 0) |
---|
626 | {vector Vtt; |
---|
627 | for (it = 1; it <= size(Vt); it++){Vtt = Vtt + Vt[it]*gen(it);} |
---|
628 | M = M,Vtt; |
---|
629 | kill Vtt; |
---|
630 | } |
---|
631 | } |
---|
632 | if (size(M)==0) {R = V; return(R);} |
---|
633 | else |
---|
634 | {M = simplify(M,2); |
---|
635 | for (i = 1; i <= size(M); i++) |
---|
636 | {kill Vt; intvec Vt; |
---|
637 | for (j =1; j <= size(M[i]); j++){Vt[j] = int(leadcoef(M[i][j]));} |
---|
638 | R = R + findmistletoes(Vt,n,L,P); |
---|
639 | } |
---|
640 | return(R); |
---|
641 | } |
---|
642 | } |
---|
643 | else |
---|
644 | {if (size(V) > degbound) {ERROR("monomial exceeds degreebound");} |
---|
645 | if (size(V) == degbound) {R = V; return(R);} |
---|
646 | int w; |
---|
647 | for (i = 1; i <= n; i++) |
---|
648 | {Vt = V,i; w = 0; |
---|
649 | for (j = 1; j <= size(P); j++) |
---|
650 | {if (P[j] <= size(Vt)) |
---|
651 | {Vt2 = Vt[(size(Vt)-P[j]+1)..size(Vt)]; |
---|
652 | if (isInMat(Vt2,L[j]) > 0){w = 1; break;} |
---|
653 | } |
---|
654 | } |
---|
655 | if (w == 0) |
---|
656 | {vector Vtt; |
---|
657 | for (it = 1; it <= size(Vt); it++){Vtt = Vtt + Vt[it]*gen(it);} |
---|
658 | M = M,Vtt; |
---|
659 | kill Vtt; |
---|
660 | } |
---|
661 | } |
---|
662 | if (size(M) == 0) {R = V; return(R);} |
---|
663 | else |
---|
664 | {M = simplify(M,2); |
---|
665 | for (i = 1; i <= ncols(M); i++) |
---|
666 | {kill Vt; intvec Vt; |
---|
667 | for (j =1; j <= size(M[i]); j++) |
---|
668 | {Vt[j] = int(leadcoef(M[i][j]));} |
---|
669 | //Vt; typeof(Vt); size(Vt); |
---|
670 | R = R + findmistletoes(Vt,n,L,P,degbound); |
---|
671 | } |
---|
672 | return(R); |
---|
673 | } |
---|
674 | } |
---|
675 | } |
---|
676 | |
---|
677 | static proc growthAlg(intmat T, list #) |
---|
678 | " |
---|
679 | real algorithm for checking the growth of an algebra |
---|
680 | " |
---|
681 | { |
---|
682 | int s = 1; |
---|
683 | if (size(#) > 0) { s = #[1];} |
---|
684 | int j; |
---|
685 | int n = ncols(T); |
---|
686 | intvec NV,C; NV[n] = 0; int m,i; |
---|
687 | intmat T2[n][n] = T[1..n,1..n]; intmat N[n][n]; |
---|
688 | if (T2 == N) |
---|
689 | { |
---|
690 | for (i = 1; i <= n; i++) |
---|
691 | { |
---|
692 | if (m < T[n+1,i]) { m = T[n+1,i];} |
---|
693 | } |
---|
694 | return(m); |
---|
695 | } |
---|
696 | |
---|
697 | //first part: the diagonals |
---|
698 | for (i = s; i <= n; i++) |
---|
699 | { |
---|
700 | if (T[i,i] > 0) |
---|
701 | { |
---|
702 | if ((T[i,i] >= 1) && (T[n+1,i] > 0)) {return(-1);} |
---|
703 | if ((T[i,i] == 1) && (T[n+1,i] == 0)) |
---|
704 | { |
---|
705 | T[i,i] = 0; |
---|
706 | T[n+1,i] = 1; |
---|
707 | return(growthAlg(T)); |
---|
708 | } |
---|
709 | } |
---|
710 | } |
---|
711 | |
---|
712 | //second part: searching for the last but one vertices |
---|
713 | T2 = T2*T2; |
---|
714 | for (i = s; i <= n; i++) |
---|
715 | { |
---|
716 | if ((intvec(T[i,1..n]) <> intvec(0)) && (intvec(T2[i,1..n]) == intvec(0))) |
---|
717 | { |
---|
718 | for (j = 1; j <= n; j++) |
---|
719 | { |
---|
720 | if ((T[i,j] > 0) && (m < T[n+1,j])) {m = T[n+1,j];} |
---|
721 | } |
---|
722 | T[n+1,i] = T[n+1,i] + m; |
---|
723 | T[i,1..n] = NV; |
---|
724 | return(growthAlg(T)); |
---|
725 | } |
---|
726 | } |
---|
727 | m = 0; |
---|
728 | |
---|
729 | //third part: searching for circles |
---|
730 | for (i = s; i <= n; i++) |
---|
731 | { |
---|
732 | T2 = T[1..n,1..n]; |
---|
733 | C = findCycleDFS(i,T2, intvec(i)); |
---|
734 | if (C[1] > 0) |
---|
735 | { |
---|
736 | for (j = 2; j <= size(C); j++) |
---|
737 | { |
---|
738 | T[i,1..n] = T[i,1..n] + T[C[j],1..n]; |
---|
739 | T[C[j],1..n] = NV; |
---|
740 | } |
---|
741 | for (j = 2; j <= size(C); j++) |
---|
742 | { |
---|
743 | T[1..n,i] = T[1..n,i] + T[1..n,C[j]]; |
---|
744 | T[1..n,C[j]] = NV; |
---|
745 | } |
---|
746 | T[i,i] = T[i,i] - size(C) + 1; |
---|
747 | m = 0; |
---|
748 | for (j = 1; j <= size(C); j++) |
---|
749 | { |
---|
750 | m = m + T[n+1,C[j]]; |
---|
751 | } |
---|
752 | for (j = 1; j <= size(C); j++) |
---|
753 | { |
---|
754 | T[n+1,C[j]] = m; |
---|
755 | } |
---|
756 | return(growthAlg(T,i)); |
---|
757 | } |
---|
758 | else {ERROR("No Cycle found, something seems wrong! Please contact the authors.");} |
---|
759 | } |
---|
760 | |
---|
761 | m = 0; |
---|
762 | for (i = 1; i <= n; i++) |
---|
763 | { |
---|
764 | if (m < T[n+1,i]) |
---|
765 | { |
---|
766 | m = T[n+1,i]; |
---|
767 | } |
---|
768 | } |
---|
769 | return(m); |
---|
770 | } |
---|
771 | |
---|
772 | static proc GlDimSuffix(intvec v, intvec g) |
---|
773 | { |
---|
774 | //Computes the shortest r such that g is a suffix for vr |
---|
775 | //only valid for lex orderings? |
---|
776 | intvec r,gt,vt,lt,g2; |
---|
777 | int lg,lv,l,i,c,f; |
---|
778 | lg = size(g); lv = size(v); |
---|
779 | if (lg <= lv) |
---|
780 | { |
---|
781 | l = lv-lg; |
---|
782 | } |
---|
783 | else |
---|
784 | { |
---|
785 | l = 0; g2 = g[(lv+1)..lg]; |
---|
786 | g = g[1..lv]; lg = size(g); |
---|
787 | c = 1; |
---|
788 | } |
---|
789 | while (l < lv) |
---|
790 | { |
---|
791 | vt = v[(l+1)..lv]; |
---|
792 | gt = g[1..(lv-l)]; |
---|
793 | lt = size(gt); |
---|
794 | for (i = 1; i <= lt; i++) |
---|
795 | { |
---|
796 | if (vt[i]<>gt[i]) {l++; break;} |
---|
797 | } |
---|
798 | if (lt <=i ) { f = 1; break;} |
---|
799 | } |
---|
800 | if (f == 0) {return(g);} |
---|
801 | r = g[(lv-l+1)..lg]; |
---|
802 | if (c == 1) {r = r,g2;} |
---|
803 | return(r); |
---|
804 | } |
---|
805 | |
---|
806 | static proc isNormal(intvec V, list G) |
---|
807 | { |
---|
808 | int i,j,k,l; |
---|
809 | k = 0; |
---|
810 | for (i = 1; i <= size(G); i++) |
---|
811 | { |
---|
812 | if ( size(G[i]) <= size(V) ) |
---|
813 | { |
---|
814 | while ( size(G[i])+k <= size(V) ) |
---|
815 | { |
---|
816 | if ( G[i] == V[(1+k)..size(V)] ) {return(1);} |
---|
817 | } |
---|
818 | } |
---|
819 | } |
---|
820 | return(0); |
---|
821 | } |
---|
822 | |
---|
823 | static proc findDChain(list L) |
---|
824 | { |
---|
825 | list Li; int i,j; |
---|
826 | for (i = 1; i <= size(L); i++) {Li[i] = size(L[i]);} |
---|
827 | Li = sort(Li); Li = Li[1]; |
---|
828 | return(Li[size(Li)]); |
---|
829 | } |
---|
830 | |
---|
831 | static proc isInList(intvec V, list L) |
---|
832 | "USAGE: isInList(V,L); V an intvec, L a list of intvecs |
---|
833 | RETURN: int |
---|
834 | PURPOSE:Finding the position of V in L, returns 0, if V is not in M |
---|
835 | " |
---|
836 | {int i,n; |
---|
837 | n = 0; |
---|
838 | for (i = 1; i <= size(L); i++) {if (L[i] == V) {n = i; break;}} |
---|
839 | return(n); |
---|
840 | } |
---|
841 | |
---|
842 | static proc isInMat(intvec V, intmat M) |
---|
843 | "USAGE: isInMat(V,M);V an intvec, M an intmat |
---|
844 | RETURN: int |
---|
845 | PURPOSE:Finding the position of V in M, returns 0, if V is not in M |
---|
846 | " |
---|
847 | {if (size(V) <> ncols(M)) {return(0);} |
---|
848 | int i; |
---|
849 | intvec Vt; |
---|
850 | for (i = 1; i <= nrows(M); i++) |
---|
851 | {Vt = M[i,1..ncols(M)]; |
---|
852 | if ((V-Vt) == 0){return(i);} |
---|
853 | } |
---|
854 | return(0); |
---|
855 | } |
---|
856 | |
---|
857 | static proc isInVec(int v,intvec V) |
---|
858 | "USAGE: isInVec(v,V); v an integer,V an intvec |
---|
859 | RETURN: int |
---|
860 | PURPOSE:Finding the position of v in V, returns 0, if v is not in V |
---|
861 | " |
---|
862 | {int i,n; |
---|
863 | n = 0; |
---|
864 | for (i = 1; i <= size(V); i++) {if (V[i] == v) {n = i; break;}} |
---|
865 | return(n); |
---|
866 | } |
---|
867 | |
---|
868 | |
---|
869 | static proc isPF(intvec P, intvec I) |
---|
870 | " |
---|
871 | PURPOSE: |
---|
872 | checks, if a word P is a proper praefix of another word I |
---|
873 | " |
---|
874 | { |
---|
875 | int n = size(P); |
---|
876 | if (n <= 0) {return(1);} |
---|
877 | if (size(I) < n) {return(0);} |
---|
878 | intvec IP = I[1..n]; |
---|
879 | if (IP == P) {return(1);} |
---|
880 | else {return(0);} |
---|
881 | } |
---|
882 | |
---|
883 | static proc isIF(intvec IF, intvec I) |
---|
884 | " |
---|
885 | PURPOSE: |
---|
886 | checks, if a word IF is an infix of another word I |
---|
887 | " |
---|
888 | { |
---|
889 | int n = size(IF); |
---|
890 | int m = size(I); |
---|
891 | |
---|
892 | if (n <= 0) { |
---|
893 | return(1); |
---|
894 | } |
---|
895 | if (m < n) { |
---|
896 | return(0); |
---|
897 | } |
---|
898 | |
---|
899 | for (int i = 0; (n + i) <= m; i++){ |
---|
900 | intvec IIF = I[(1 + i)..(n + i)]; |
---|
901 | if (IIF == IF) { |
---|
902 | return(1); |
---|
903 | } |
---|
904 | } |
---|
905 | return(0); |
---|
906 | } |
---|
907 | |
---|
908 | proc ivL2lpI(list L) |
---|
909 | "USAGE: ivL2lpI(L); L a list of intvecs |
---|
910 | RETURN: ideal |
---|
911 | PURPOSE:Transforming a list of intvecs into an ideal of Letterplace monomials |
---|
912 | ASSUME: - Intvec corresponds to a Letterplace monomial |
---|
913 | @* - basering has to be a Letterplace ring |
---|
914 | EXAMPLE: example ivL2lpI; shows examples |
---|
915 | " |
---|
916 | {checkAssumptions(0,L); |
---|
917 | int i; ideal G; |
---|
918 | poly p; |
---|
919 | for (i = 1; i <= size(L); i++) |
---|
920 | {p = iv2lp(L[i]); |
---|
921 | G[(size(G) + 1)] = p; |
---|
922 | } |
---|
923 | return(G); |
---|
924 | } |
---|
925 | example |
---|
926 | { |
---|
927 | "EXAMPLE:"; echo = 2; |
---|
928 | ring r = 0,(x,y,z),dp; |
---|
929 | def R = makeLetterplaceRing(5);// constructs a Letterplace ring |
---|
930 | setring R; //sets basering to Letterplace ring |
---|
931 | intvec u = 1,1,2; intvec v = 2,1,3; intvec w = 3,1,1; |
---|
932 | // u = x^2y, v = yxz, w = zx^2 in intvec representation |
---|
933 | list L = u,v,w; |
---|
934 | ivL2lpI(L);// invokes the procedure, returns the ideal containing u,v,w |
---|
935 | } |
---|
936 | |
---|
937 | proc iv2lp(intvec I) |
---|
938 | "USAGE: iv2lp(I); I an intvec |
---|
939 | RETURN: poly |
---|
940 | PURPOSE:Transforming an intvec into the corresponding Letterplace polynomial |
---|
941 | ASSUME: - Intvec corresponds to a Letterplace monomial |
---|
942 | @* - basering has to be a Letterplace ring |
---|
943 | NOTE: - Assumptions will not be checked! |
---|
944 | EXAMPLE: example iv2lp; shows examples |
---|
945 | " |
---|
946 | {if (I[1] == 0) {return(1);} |
---|
947 | int i = size(I); |
---|
948 | if (i > attrib(basering,"uptodeg")) {ERROR("polynomial exceeds degreebound");} |
---|
949 | int j; poly p = 1; |
---|
950 | for (j = 1; j <= i; j++) {if (I[j] > 0) { p = lpMult(p,var(I[j]));}} //ignore zeroes, because they correspond to 1 |
---|
951 | return(p); |
---|
952 | } |
---|
953 | example |
---|
954 | { |
---|
955 | "EXAMPLE:"; echo = 2; |
---|
956 | ring r = 0,(x,y,z),dp; |
---|
957 | def R = makeLetterplaceRing(5); // constructs a Letterplace ring |
---|
958 | setring R; //sets basering to Letterplace ring |
---|
959 | intvec u = 1,1,2; intvec v = 2,1,3; intvec w = 3,1,1; |
---|
960 | // u = x^2y, v = yxz, w = zx^2 in intvec representation |
---|
961 | iv2lp(u); // invokes the procedure and returns the corresponding poly |
---|
962 | iv2lp(v); |
---|
963 | iv2lp(w); |
---|
964 | } |
---|
965 | |
---|
966 | proc iv2lpList(list L) |
---|
967 | "USAGE: iv2lpList(L); L a list of intmats |
---|
968 | RETURN: ideal |
---|
969 | PURPOSE:Converting a list of intmats into an ideal of corresponding monomials |
---|
970 | ASSUME: - The rows of each intmat in L must correspond to a Letterplace monomial |
---|
971 | @* - basering has to be a Letterplace ring |
---|
972 | EXAMPLE: example iv2lpList; shows examples |
---|
973 | " |
---|
974 | {checkAssumptions(0,L); |
---|
975 | ideal G; |
---|
976 | int i; |
---|
977 | for (i = 1; i <= size(L); i++){G = G + iv2lpMat(L[i]);} |
---|
978 | return(G); |
---|
979 | } |
---|
980 | example |
---|
981 | { |
---|
982 | "EXAMPLE:"; echo = 2; |
---|
983 | ring r = 0,(x,y,z),dp; |
---|
984 | def R = makeLetterplaceRing(5); // constructs a Letterplace ring |
---|
985 | setring R; // sets basering to Letterplace ring |
---|
986 | intmat u[3][1] = 1,1,2; intmat v[1][3] = 2,1,3; intmat w[2][3] = 3,1,1,2,3,1; |
---|
987 | // defines intmats of different size containing intvec representations of |
---|
988 | // monomials as rows |
---|
989 | list L = u,v,w; |
---|
990 | print(u); print(v); print(w); // shows the intmats contained in L |
---|
991 | iv2lpList(L); // returns the corresponding monomials as an ideal |
---|
992 | } |
---|
993 | |
---|
994 | |
---|
995 | proc iv2lpMat(intmat M) |
---|
996 | "USAGE: iv2lpMat(M); M an intmat |
---|
997 | RETURN: ideal |
---|
998 | PURPOSE:Converting an intmat into an ideal of the corresponding monomials |
---|
999 | ASSUME: - The rows of M must correspond to Letterplace monomials |
---|
1000 | @* - basering has to be a Letterplace ring |
---|
1001 | EXAMPLE: example iv2lpMat; shows examples |
---|
1002 | " |
---|
1003 | {list L = M; |
---|
1004 | checkAssumptions(0,L); |
---|
1005 | kill L; |
---|
1006 | ideal G; poly p; |
---|
1007 | int i; intvec I; |
---|
1008 | for (i = 1; i <= nrows(M); i++) |
---|
1009 | { I = M[i,1..ncols(M)]; |
---|
1010 | p = iv2lp(I); |
---|
1011 | G[size(G)+1] = p; |
---|
1012 | } |
---|
1013 | return(G); |
---|
1014 | } |
---|
1015 | example |
---|
1016 | { |
---|
1017 | "EXAMPLE:"; echo = 2; |
---|
1018 | ring r = 0,(x,y,z),dp; |
---|
1019 | def R = makeLetterplaceRing(5); // constructs a Letterplace ring |
---|
1020 | setring R; // sets basering to Letterplace ring |
---|
1021 | intmat u[3][1] = 1,1,2; intmat v[1][3] = 2,1,3; intmat w[2][3] = 3,1,1,2,3,1; |
---|
1022 | // defines intmats of different size containing intvec representations of |
---|
1023 | // monomials as rows |
---|
1024 | iv2lpMat(u); // returns the monomials contained in u |
---|
1025 | iv2lpMat(v); // returns the monomials contained in v |
---|
1026 | iv2lpMat(w); // returns the monomials contained in w |
---|
1027 | } |
---|
1028 | |
---|
1029 | proc lpId2ivLi(ideal G) |
---|
1030 | "USAGE: lpId2ivLi(G); G an ideal |
---|
1031 | RETURN: list |
---|
1032 | PURPOSE:Transforming an ideal into the corresponding list of intvecs |
---|
1033 | ASSUME: - basering has to be a Letterplace ring |
---|
1034 | EXAMPLE: example lpId2ivLi; shows examples |
---|
1035 | " |
---|
1036 | {int i,j,k; |
---|
1037 | list M; |
---|
1038 | checkAssumptions(0,M); |
---|
1039 | for (i = 1; i <= size(G); i++) {M[i] = lp2iv(G[i]);} |
---|
1040 | return(M); |
---|
1041 | } |
---|
1042 | example |
---|
1043 | { |
---|
1044 | "EXAMPLE:"; echo = 2; |
---|
1045 | ring r = 0,(x,y),dp; |
---|
1046 | def R = makeLetterplaceRing(5); // constructs a Letterplace ring |
---|
1047 | setring R; // sets basering to Letterplace ring |
---|
1048 | ideal L = x(1)*x(2),y(1)*y(2),x(1)*y(2)*x(3); |
---|
1049 | lpId2ivLi(L); // returns the corresponding intvecs as a list |
---|
1050 | } |
---|
1051 | |
---|
1052 | proc lp2iv(poly p) |
---|
1053 | "USAGE: lp2iv(p); p a poly |
---|
1054 | RETURN: intvec |
---|
1055 | PURPOSE:Transforming a monomial into the corresponding intvec |
---|
1056 | ASSUME: - basering has to be a Letterplace ring |
---|
1057 | NOTE: - Assumptions will not be checked! |
---|
1058 | EXAMPLE: example lp2iv; shows examples |
---|
1059 | " |
---|
1060 | {p = normalize(lead(p)); |
---|
1061 | intvec I; |
---|
1062 | int i,j; |
---|
1063 | if (deg(p) > attrib(basering,"uptodeg")) {ERROR("Monomial exceeds degreebound");} |
---|
1064 | if (p == 1) {return(I);} |
---|
1065 | if (p == 0) {ERROR("Monomial is not allowed to equal zero");} |
---|
1066 | intvec lep = leadexp(p); |
---|
1067 | for ( i = 1; i <= attrib(basering,"lV"); i++) {if (lep[i] == 1) {I = i; break;}} |
---|
1068 | for (i = (attrib(basering,"lV")+1); i <= size(lep); i++) |
---|
1069 | {if (lep[i] == 1) |
---|
1070 | { j = (i mod attrib(basering,"lV")); |
---|
1071 | if (j == 0) {I = I,attrib(basering,"lV");} |
---|
1072 | else {I = I,j;} |
---|
1073 | } |
---|
1074 | else { if (lep[i] > 1) {ERROR("monomial has a not allowed multidegree");}} |
---|
1075 | } |
---|
1076 | if (I[1] == 0) {ERROR("monomial has a not allowed multidegree");} |
---|
1077 | |
---|
1078 | return(I); |
---|
1079 | } |
---|
1080 | example |
---|
1081 | { |
---|
1082 | "EXAMPLE:"; echo = 2; |
---|
1083 | ring r = 0,(x,y,z),dp; |
---|
1084 | def R = makeLetterplaceRing(5); // constructs a Letterplace ring |
---|
1085 | setring R; // sets basering to Letterplace ring |
---|
1086 | poly p = x(1)*x(2)*z(3); poly q = y(1)*y(2)*x(3)*x(4); |
---|
1087 | poly w= z(1)*y(2)*x(3)*z(4)*z(5); |
---|
1088 | // p,q,w are some polynomials we want to transform into their |
---|
1089 | // intvec representation |
---|
1090 | lp2iv(p); lp2iv(q); lp2iv(w); |
---|
1091 | } |
---|
1092 | |
---|
1093 | proc lp2ivId(ideal G) |
---|
1094 | "USAGE: lp2ivId(G); G an ideal |
---|
1095 | RETURN: list |
---|
1096 | PURPOSE:Converting an ideal into an list of intmats, |
---|
1097 | @* the corresponding intvecs forming the rows |
---|
1098 | ASSUME: - basering has to be a Letterplace ring |
---|
1099 | EXAMPLE: example lp2ivId; shows examples |
---|
1100 | " |
---|
1101 | {G = normalize(lead(G)); |
---|
1102 | intvec I; list L; |
---|
1103 | checkAssumptions(0,L); |
---|
1104 | int i,md; |
---|
1105 | for (i = 1; i <= size(G); i++) { if (md <= deg(G[i])) {md = deg(G[i]);}} |
---|
1106 | while (size(G) > 0) |
---|
1107 | {ideal Gt; |
---|
1108 | for (i = 1; i <= ncols(G); i++) {if (md == deg(G[i])) {Gt = Gt + G[i]; G[i] = 0;}} |
---|
1109 | if (size(Gt) > 0) |
---|
1110 | {G = simplify(G,2); |
---|
1111 | intmat M [size(Gt)][md]; |
---|
1112 | for (i = 1; i <= size(Gt); i++) {M[i,1..md] = lp2iv(Gt[i]);} |
---|
1113 | L = insert(L,M); |
---|
1114 | kill M; kill Gt; |
---|
1115 | md = md - 1; |
---|
1116 | } |
---|
1117 | else {kill Gt; md = md - 1;} |
---|
1118 | } |
---|
1119 | return(L); |
---|
1120 | } |
---|
1121 | example |
---|
1122 | { |
---|
1123 | "EXAMPLE:"; echo = 2; |
---|
1124 | ring r = 0,(x,y,z),dp; |
---|
1125 | def R = makeLetterplaceRing(5); // constructs a Letterplace ring |
---|
1126 | setring R; // sets basering to Letterplace ring |
---|
1127 | poly p = x(1)*x(2)*z(3); poly q = y(1)*y(2)*x(3)*x(4); |
---|
1128 | poly w = z(1)*y(2)*x(3)*z(4); |
---|
1129 | // p,q,w are some polynomials we want to transform into their |
---|
1130 | // intvec representation |
---|
1131 | ideal G = p,q,w; |
---|
1132 | // define the ideal containing p,q and w |
---|
1133 | lp2ivId(G); // and return the list of intmats for this ideal |
---|
1134 | } |
---|
1135 | |
---|
1136 | // -----------------main procedures---------------------- |
---|
1137 | |
---|
1138 | proc lpUfGkDim(ideal G) |
---|
1139 | { |
---|
1140 | G = lead(G); |
---|
1141 | |
---|
1142 | int l = 0; |
---|
1143 | for (int i = 1; i <= size(G); i++) { |
---|
1144 | // find the max degree in G |
---|
1145 | int d = deg(G[i]); |
---|
1146 | if (d > l) { |
---|
1147 | l = d; |
---|
1148 | } |
---|
1149 | |
---|
1150 | // also if G is the whole ring return minus infinity |
---|
1151 | if (leadmonom(G[i]) == 1) { |
---|
1152 | return(-2); // minus infinity |
---|
1153 | } |
---|
1154 | } |
---|
1155 | // if longet word has length 1 we handle it as a special case |
---|
1156 | if (l == 1) { |
---|
1157 | int n = attrib(basering, "lV"); // variable count |
---|
1158 | int k = size(G); |
---|
1159 | if (k == n) { |
---|
1160 | return(0); |
---|
1161 | } |
---|
1162 | if (k == n-1) { |
---|
1163 | return(1); |
---|
1164 | } |
---|
1165 | if (k <= n-2) { |
---|
1166 | return(-1); |
---|
1167 | } |
---|
1168 | } |
---|
1169 | |
---|
1170 | // otherwise count cycles in the Ufnarovskij Graph |
---|
1171 | return(UfGraphGrowth(lpUfGraph(G))); |
---|
1172 | } |
---|
1173 | |
---|
1174 | proc UfGraphGrowth(intmat UG) |
---|
1175 | { |
---|
1176 | int n = ncols(UG); // number of vertices |
---|
1177 | // iterate through all vertices |
---|
1178 | |
---|
1179 | intvec visited; |
---|
1180 | visited[n] = 0; |
---|
1181 | |
---|
1182 | intvec cyclic; |
---|
1183 | cyclic[n] = 0; |
---|
1184 | |
---|
1185 | int maxCycleCount = 0; |
---|
1186 | for (int v = 1; v <= n; v++) { |
---|
1187 | int cycleCount = Cycles(UG, v, visited, cyclic, 0); |
---|
1188 | if (cycleCount == -1) { |
---|
1189 | return(-1); |
---|
1190 | } |
---|
1191 | if (cycleCount > maxCycleCount) { |
---|
1192 | maxCycleCount = cycleCount; |
---|
1193 | } |
---|
1194 | } |
---|
1195 | return(maxCycleCount); |
---|
1196 | } |
---|
1197 | |
---|
1198 | proc Cycles(intmat G, int v, intvec visited, intvec cyclic, intvec path) |
---|
1199 | { |
---|
1200 | // Mark the current vertex as visited |
---|
1201 | visited[v] = 1; |
---|
1202 | |
---|
1203 | // Store the current vertex in path |
---|
1204 | if (path[1] == 0) { |
---|
1205 | path[1] = v; |
---|
1206 | } else { |
---|
1207 | path[size(path) + 1] = v; |
---|
1208 | } |
---|
1209 | |
---|
1210 | int cycles = 0; |
---|
1211 | for (int w = 1; w <= ncols(G); w++) { |
---|
1212 | if (G[v,w] == 1) { |
---|
1213 | if (visited[w] == 1) { // neuer zykel gefunden |
---|
1214 | // 1. alle Knoten in path bis w ÃŒberprÃŒfen ob in cyclic |
---|
1215 | for (int j = size(path); j >= 1; j--) { |
---|
1216 | if(cyclic[path[j]] == 1) { |
---|
1217 | // 1.1 falls ja return -1 |
---|
1218 | return (-1); |
---|
1219 | } |
---|
1220 | if (path[j] == w) { |
---|
1221 | break; |
---|
1222 | } |
---|
1223 | } |
---|
1224 | |
---|
1225 | // 2. ansonsten cycles++ |
---|
1226 | for (int j = size(path); j >= 1; j--) { |
---|
1227 | // 2.2 Kanten in diesem Zykel entfernen; Knoten cyclic |
---|
1228 | if (j == size(path)) { // Sonderfall bei der ersten Iteration |
---|
1229 | cyclic[v] = 1; |
---|
1230 | G[v, w] = 0; |
---|
1231 | } else { |
---|
1232 | cyclic[path[j]] = 1; |
---|
1233 | G[path[j], path[j+1]] = 0; |
---|
1234 | } |
---|
1235 | if (path[j] == w) { |
---|
1236 | break; |
---|
1237 | } |
---|
1238 | } |
---|
1239 | |
---|
1240 | // 3. auf jedem dieser Knoten Cycles() aufrufen |
---|
1241 | int maxCycleCount = 0; |
---|
1242 | for (int j = size(path); j >= 1; j--) { |
---|
1243 | int cycleCount = Cycles(G, path[j], visited, cyclic, path); |
---|
1244 | if(cycleCount == -1) { |
---|
1245 | return (-1); |
---|
1246 | } |
---|
1247 | if (cycleCount > maxCycleCount) { |
---|
1248 | maxCycleCount = cycleCount; |
---|
1249 | } |
---|
1250 | if (path[j] == w) { |
---|
1251 | break; |
---|
1252 | } |
---|
1253 | } |
---|
1254 | if (maxCycleCount >= cycles) { |
---|
1255 | cycles = maxCycleCount + 1; |
---|
1256 | } |
---|
1257 | } else { |
---|
1258 | int cycleCount = Cycles(G, w, visited, cyclic, path); |
---|
1259 | if (cycleCount == -1) { |
---|
1260 | return(-1); |
---|
1261 | } |
---|
1262 | if (cycleCount > cycles) { |
---|
1263 | cycles = cycleCount; |
---|
1264 | } |
---|
1265 | } |
---|
1266 | } |
---|
1267 | } |
---|
1268 | //printf("Path: %s Cycles: %s", path, cycles); |
---|
1269 | return(cycles); |
---|
1270 | } |
---|
1271 | |
---|
1272 | proc lpUfGraph(ideal G) |
---|
1273 | "USUAGE: lpUfGraph(G); G a set of monomials in a letterplace ring |
---|
1274 | RETURN: intmat |
---|
1275 | PURPOSE: Constructs the Ufnarovskij graph induced by G |
---|
1276 | @* the adjacency matrix of the Ufnarovskij graph induced by G |
---|
1277 | ASSUME: - basering is a Letterplace ring |
---|
1278 | - G are the leading monomials of a Groebner basis |
---|
1279 | " |
---|
1280 | { |
---|
1281 | int l = 0; // length of longest word (monomial) in G |
---|
1282 | for (int i = 1; i <= size(G); i++) { // find the max degree in G |
---|
1283 | int d = deg(G[i]); |
---|
1284 | if (d > l) { |
---|
1285 | l = d; |
---|
1286 | } |
---|
1287 | } |
---|
1288 | ideal V = lpStandardWords(G, l - 1); // vertices |
---|
1289 | int n = size(V); |
---|
1290 | intmat UG[n][n]; // Ufnarovskij graph |
---|
1291 | for (int i = 1; i <= n; i++) { |
---|
1292 | for (int j = 1; j <= n; j++) { |
---|
1293 | // V[i] = v, V[j] = w [Studzinski page 76] |
---|
1294 | intvec v = lp2iv(V[i]); |
---|
1295 | intvec w = lp2iv(V[j]); |
---|
1296 | intvec v_overlap; |
---|
1297 | intvec w_overlap; |
---|
1298 | //TODO how should the graph look like when l - 1 = 0 ? |
---|
1299 | if (l - 1 > 1) { |
---|
1300 | v_overlap = v[2 .. l-1]; |
---|
1301 | w_overlap = w[1 .. l-2]; |
---|
1302 | } |
---|
1303 | intvec vw = v; |
---|
1304 | vw[l] = w[l-1]; |
---|
1305 | if (v_overlap == w_overlap && !divides(G, vw)) { |
---|
1306 | UG[i,j] = 1; |
---|
1307 | } |
---|
1308 | } |
---|
1309 | } |
---|
1310 | return (UG); |
---|
1311 | } |
---|
1312 | |
---|
1313 | proc lpStandardWords(ideal G, int length) { |
---|
1314 | if (length == 0) { |
---|
1315 | return (1); |
---|
1316 | } |
---|
1317 | int lV = attrib(basering, "lV"); // variable count |
---|
1318 | // recursion could cause problems |
---|
1319 | ideal prevWords = lpStandardWords(G, length - 1); |
---|
1320 | ideal words; |
---|
1321 | for (int i = 1; i <= lV; i++) { |
---|
1322 | for (int j = 1; j <= size(prevWords); j++) { |
---|
1323 | int placeShift = (length - 1) * lV; // letterplace |
---|
1324 | // multiply every previous word with every variable |
---|
1325 | poly word = prevWords[j] * var(i + placeShift); |
---|
1326 | // assumes that G is simplified! |
---|
1327 | if (!divides(G, lp2iv(word))) { |
---|
1328 | words = words, word; |
---|
1329 | } |
---|
1330 | } |
---|
1331 | } |
---|
1332 | words = simplify(words, 2); // remove zeroes |
---|
1333 | return (words); |
---|
1334 | } |
---|
1335 | |
---|
1336 | proc divides(ideal G, intvec iv) { |
---|
1337 | for (int k = 1; k <= size(G); k++) { |
---|
1338 | if (isIF(lp2iv(G[k]), iv)) { |
---|
1339 | return (1); |
---|
1340 | } else { |
---|
1341 | if (k == size(G)) { |
---|
1342 | return (0); |
---|
1343 | } |
---|
1344 | } |
---|
1345 | } |
---|
1346 | } |
---|
1347 | |
---|
1348 | proc lpGraphOfNormalWords(ideal G) |
---|
1349 | "USUAGE: lpGraphOfNormalWords(G); G a set of monomials in a letterplace ring |
---|
1350 | RETURN: intmat |
---|
1351 | PURPOSE: Constructs the graph of normal words induced by G |
---|
1352 | @* the adjacency matrix of the graph of normal words induced by G |
---|
1353 | ASSUME: - basering is a Letterplace ring |
---|
1354 | - G are the leading monomials of a Groebner basis |
---|
1355 | " |
---|
1356 | { |
---|
1357 | // construct the Graph of normal words [Studzinski page 78] |
---|
1358 | // construct set of vertices |
---|
1359 | int v = attrib(basering,"lV"); int d = attrib(basering,"uptodeg"); |
---|
1360 | ideal V; poly p,q,w; |
---|
1361 | ideal LG = lead(G); |
---|
1362 | int i,j,k,b; intvec E,Et; |
---|
1363 | for (i = 1; i <= v; i++){V = V, var(i);} |
---|
1364 | for (i = 1; i <= size(LG); i++) |
---|
1365 | { |
---|
1366 | E = leadexp(LG[i]); |
---|
1367 | if (E == intvec(0)) {V = V,monomial(intvec(0));} |
---|
1368 | else |
---|
1369 | { |
---|
1370 | for (j = 1; j < d; j++) |
---|
1371 | { |
---|
1372 | Et = E[(j*v+1)..(d*v)]; |
---|
1373 | if (Et == intvec(0)) {break;} |
---|
1374 | else {V = V, monomial(Et);} |
---|
1375 | } |
---|
1376 | } |
---|
1377 | } |
---|
1378 | V = simplify(V,2+4); |
---|
1379 | printf("V = %p", V); |
---|
1380 | |
---|
1381 | |
---|
1382 | // construct incidence matrix |
---|
1383 | |
---|
1384 | list LV = lpId2ivLi(V); |
---|
1385 | intvec Ip,Iw; |
---|
1386 | int n = size(V); |
---|
1387 | intmat T[n+1][n]; |
---|
1388 | for (i = 1; i <= n; i++) |
---|
1389 | { |
---|
1390 | // printf("for1 (i=%p, n=%p)", i, n); |
---|
1391 | p = V[i]; Ip = lp2iv(p); |
---|
1392 | for (j = 1; j <= n; j++) |
---|
1393 | { |
---|
1394 | // printf("for2 (j=%p, n=%p)", j, n); |
---|
1395 | k = 1; b = 1; |
---|
1396 | q = V[j]; |
---|
1397 | w = lpNF(lpMult(p,q),LG); |
---|
1398 | if (w <> 0) |
---|
1399 | { |
---|
1400 | Iw = lp2iv(w); |
---|
1401 | while (k <= n) |
---|
1402 | { |
---|
1403 | // printf("while (k=%p, n=%p)", k, n); |
---|
1404 | if (isPF(LV[k],Iw) > 0) |
---|
1405 | {if (isPF(LV[k],Ip) == 0) {b = 0; k = n+1;} else {k++;} |
---|
1406 | } |
---|
1407 | else {k++;} |
---|
1408 | } |
---|
1409 | T[i,j] = b; |
---|
1410 | // print("Incidence Matrix:"); |
---|
1411 | // print(T); |
---|
1412 | } |
---|
1413 | } |
---|
1414 | } |
---|
1415 | return(T); |
---|
1416 | } |
---|
1417 | |
---|
1418 | proc lpGkDim(ideal G) |
---|
1419 | "USUAGE: lpGkDim(G); G an ideal in a letterplace ring |
---|
1420 | RETURN: int |
---|
1421 | PURPOSE: Determines the Gelfand Kirillov dimension of A/<G> |
---|
1422 | @*: -1 means it is infinite |
---|
1423 | ASSUME: - basering is a Letterplace ring |
---|
1424 | - G is a Groebner basis |
---|
1425 | " |
---|
1426 | { |
---|
1427 | return(growthAlg(lpGraphOfNormalWords(G))); |
---|
1428 | } |
---|
1429 | example |
---|
1430 | { |
---|
1431 | "EXAMPLE:"; echo = 2; |
---|
1432 | ring r = 0,(x,y,z),dp; |
---|
1433 | def R = makeLetterplaceRing(5); // constructs a Letterplace ring |
---|
1434 | R; |
---|
1435 | setring R; // sets basering to Letterplace ring |
---|
1436 | ideal I = z(1);//an example of infinite GK dimension |
---|
1437 | lpGkDim(I); |
---|
1438 | I = x(1),y(1),z(1); // gkDim = 0 |
---|
1439 | lpGkDim(I); |
---|
1440 | I = x(1)*y(2), x(1)*z(2), z(1)*y(2), z(1)*z(2);//gkDim = 2 |
---|
1441 | lpGkDim(I); |
---|
1442 | } |
---|
1443 | |
---|
1444 | |
---|
1445 | proc ivDHilbert(list L, int n, list #) |
---|
1446 | "USAGE: ivDHilbert(L,n[,degbound]); L a list of intmats, n an integer, |
---|
1447 | @* degbound an optional integer |
---|
1448 | RETURN: list |
---|
1449 | PURPOSE:Computing the K-dimension and the Hilbert series |
---|
1450 | ASSUME: - basering is a Letterplace ring |
---|
1451 | @* - all rows of each intmat correspond to a Letterplace monomial |
---|
1452 | @* - if you specify a different degree bound degbound, |
---|
1453 | @* degbound <= attrib(basering,uptodeg) holds |
---|
1454 | NOTE: - If L is the list returned, then L[1] is an integer corresponding to the |
---|
1455 | @* dimension, L[2] is an intvec which contains the coefficients of the |
---|
1456 | @* Hilbert series |
---|
1457 | @* - If degbound is set, there will be a degree bound added. By default there |
---|
1458 | @* is no degree bound |
---|
1459 | @* - n is the number of variables |
---|
1460 | @* - If I = L[2] is the intvec returned, then I[k] is the (k-1)-th coefficient of |
---|
1461 | @* the Hilbert series. |
---|
1462 | @* - If the K-dimension is known to be infinite, a degree bound is needed |
---|
1463 | EXAMPLE: example ivDHilbert; shows examples |
---|
1464 | " |
---|
1465 | {int degbound = 0; |
---|
1466 | if (size(#) > 0){if (typeof(#[1])=="int"){if (#[1] > 0){degbound = #[1];}}} |
---|
1467 | checkAssumptions(degbound,L); |
---|
1468 | intvec H; int i,dimen; |
---|
1469 | H = ivHilbert(L,n,degbound); |
---|
1470 | for (i = 1; i <= size(H); i++){dimen = dimen + H[i];} |
---|
1471 | L = dimen,H; |
---|
1472 | return(L); |
---|
1473 | } |
---|
1474 | example |
---|
1475 | { |
---|
1476 | "EXAMPLE:"; echo = 2; |
---|
1477 | ring r = 0,(x,y),dp; |
---|
1478 | def R = makeLetterplaceRing(5); // constructs a Letterplace ring |
---|
1479 | R; |
---|
1480 | setring R; // sets basering to Letterplace ring |
---|
1481 | //some intmats, which contain monomials in intvec representation as rows |
---|
1482 | intmat I1 [2][2] = 1,1,2,2; intmat I2 [1][3] = 1,2,1; |
---|
1483 | intmat J1 [1][2] = 1,1; intmat J2 [2][3] = 2,1,2,1,2,1; |
---|
1484 | print(I1); |
---|
1485 | print(I2); |
---|
1486 | print(J1); |
---|
1487 | print(J2); |
---|
1488 | list G = I1,I2; // ideal, which is already a Groebner basis |
---|
1489 | list I = J1,J2; // ideal, which is already a Groebner basis |
---|
1490 | //the procedure without a degree bound |
---|
1491 | ivDHilbert(G,2); |
---|
1492 | // the procedure with degree bound 5 |
---|
1493 | ivDHilbert(I,2,5); |
---|
1494 | } |
---|
1495 | |
---|
1496 | proc ivDHilbertSickle(list L, int n, list #) |
---|
1497 | "USAGE: ivDHilbertSickle(L,n[,degbound]); L a list of intmats, n an integer, |
---|
1498 | @* degbound an optional integer |
---|
1499 | RETURN: list |
---|
1500 | PURPOSE:Computing K-dimension, Hilbert series and mistletoes |
---|
1501 | ASSUME: - basering is a Letterplace ring. |
---|
1502 | @* - All rows of each intmat correspond to a Letterplace monomial. |
---|
1503 | @* - If you specify a different degree bound degbound, |
---|
1504 | @* degbound <= attrib(basering,uptodeg) holds. |
---|
1505 | NOTE: - If L is the list returned, then L[1] is an integer, L[2] is an intvec |
---|
1506 | @* which contains the coefficients of the Hilbert series and L[3] |
---|
1507 | @* is a list, containing the mistletoes as intvecs. |
---|
1508 | @* - If degbound is set, a degree bound will be added. By default there |
---|
1509 | @* is no degree bound. |
---|
1510 | @* - n is the number of variables. |
---|
1511 | @* - If I = L[2] is the intvec returned, then I[k] is the (k-1)-th |
---|
1512 | @* coefficient of the Hilbert series. |
---|
1513 | @* - If the K-dimension is known to be infinite, a degree bound is needed |
---|
1514 | EXAMPLE: example ivDHilbertSickle; shows examples |
---|
1515 | " |
---|
1516 | {int degbound = 0; |
---|
1517 | if (size(#) > 0){if (typeof(#[1])=="int"){if (#[1] > 0){degbound = #[1];}}} |
---|
1518 | checkAssumptions(degbound,L); |
---|
1519 | int i,dimen; list R; |
---|
1520 | R = ivSickleHil(L,n,degbound); |
---|
1521 | for (i = 1; i <= size(R[1]); i++){dimen = dimen + R[1][i];} |
---|
1522 | R[3] = R[2]; R[2] = R[1]; R[1] = dimen; |
---|
1523 | return(R); |
---|
1524 | } |
---|
1525 | example |
---|
1526 | { |
---|
1527 | "EXAMPLE:"; echo = 2; |
---|
1528 | ring r = 0,(x,y),dp; |
---|
1529 | def R = makeLetterplaceRing(5); // constructs a Letterplace ring |
---|
1530 | R; |
---|
1531 | setring R; // sets basering to Letterplace ring |
---|
1532 | //some intmats, which contain monomials in intvec representation as rows |
---|
1533 | intmat I1 [2][2] = 1,1,2,2; intmat I2 [1][3] = 1,2,1; |
---|
1534 | intmat J1 [1][2] = 1,1; intmat J2 [2][3] = 2,1,2,1,2,1; |
---|
1535 | print(I1); |
---|
1536 | print(I2); |
---|
1537 | print(J1); |
---|
1538 | print(J2); |
---|
1539 | list G = I1,I2;// ideal, which is already a Groebner basis |
---|
1540 | list I = J1,J2; // ideal, which is already a Groebner basis |
---|
1541 | ivDHilbertSickle(G,2); // invokes the procedure without a degree bound |
---|
1542 | ivDHilbertSickle(I,2,3); // invokes the procedure with degree bound 3 |
---|
1543 | } |
---|
1544 | |
---|
1545 | proc ivDimCheck(list L, int n) |
---|
1546 | "USAGE: ivDimCheck(L,n); L a list of intmats, n an integer |
---|
1547 | RETURN: int, 0 if the dimension is finite, or 1 otherwise |
---|
1548 | PURPOSE:Decides, whether the K-dimension is finite or not |
---|
1549 | ASSUME: - basering is a Letterplace ring. |
---|
1550 | @* - All rows of each intmat correspond to a Letterplace monomial. |
---|
1551 | NOTE: - n is the number of variables. |
---|
1552 | EXAMPLE: example ivDimCheck; shows examples |
---|
1553 | " |
---|
1554 | {checkAssumptions(0,L); |
---|
1555 | int i,r; |
---|
1556 | intvec P,H; |
---|
1557 | for (i = 1; i <= size(L); i++) |
---|
1558 | {P[i] = ncols(L[i]); |
---|
1559 | if (P[i] == 1) {if (isInMat(H,L[i]) > 0) {ERROR("Quotient algebra is trivial");}} |
---|
1560 | } |
---|
1561 | if (size(L) == 0) {ERROR("GB is empty, quotient algebra corresponds to free algebra");} |
---|
1562 | kill H; |
---|
1563 | intmat S; int sd,ld; intvec V; |
---|
1564 | sd = P[1]; ld = P[1]; |
---|
1565 | for (i = 2; i <= size(P); i++) |
---|
1566 | {if (P[i] < sd) {sd = P[i];} |
---|
1567 | if (P[i] > ld) {ld = P[i];} |
---|
1568 | } |
---|
1569 | sd = (sd - 1); ld = ld - 1; |
---|
1570 | if (ld == 0) { return(allVars(L,P,n));} |
---|
1571 | if (sd == 0) { for (i = 1; i <= size(L); i++){if (ncols(L[i]) == 1){S = createStartMat1(n,L[i]); break;}}} |
---|
1572 | else {S = createStartMat(sd,n);} |
---|
1573 | module M; |
---|
1574 | for (i = 1; i <= nrows(S); i++) |
---|
1575 | {V = S[i,1..ncols(S)]; |
---|
1576 | if (findCycle(V,L,P,n,ld,M)) {r = 1; break;} |
---|
1577 | } |
---|
1578 | return(r); |
---|
1579 | } |
---|
1580 | example |
---|
1581 | { |
---|
1582 | "EXAMPLE:"; echo = 2; |
---|
1583 | ring r = 0,(x,y),dp; |
---|
1584 | def R = makeLetterplaceRing(5); // constructs a Letterplace ring |
---|
1585 | R; |
---|
1586 | setring R; // sets basering to Letterplace ring |
---|
1587 | //some intmats, which contain monomials in intvec representation as rows |
---|
1588 | intmat I1 [2][2] = 1,1,2,2; intmat I2 [1][3] = 1,2,1; |
---|
1589 | intmat J1 [1][2] = 1,1; intmat J2 [2][3] = 2,1,2,1,2,1; |
---|
1590 | print(I1); |
---|
1591 | print(I2); |
---|
1592 | print(J1); |
---|
1593 | print(J2); |
---|
1594 | list G = I1,I2;// ideal, which is already a Groebner basis |
---|
1595 | list I = J1,J2; // ideal, which is already a Groebner basis and which |
---|
1596 | ivDimCheck(G,2); // invokes the procedure, factor is of finite K-dimension |
---|
1597 | ivDimCheck(I,2); // invokes the procedure, factor is not of finite K-dimension |
---|
1598 | } |
---|
1599 | |
---|
1600 | proc ivHilbert(list L, int n, list #) |
---|
1601 | "USAGE: ivHilbert(L,n[,degbound]); L a list of intmats, n an integer, |
---|
1602 | @* degbound an optional integer |
---|
1603 | RETURN: intvec, containing the coefficients of the Hilbert series |
---|
1604 | PURPOSE:Computing the Hilbert series |
---|
1605 | ASSUME: - basering is a Letterplace ring. |
---|
1606 | @* - all rows of each intmat correspond to a Letterplace monomial |
---|
1607 | @* - if you specify a different degree bound degbound, |
---|
1608 | @* degbound <= attrib(basering,uptodeg) holds. |
---|
1609 | NOTE: - If degbound is set, a degree bound will be added. By default there |
---|
1610 | @* is no degree bound. |
---|
1611 | @* - n is the number of variables. |
---|
1612 | @* - If I is returned, then I[k] is the (k-1)-th coefficient of the Hilbert |
---|
1613 | @* series. |
---|
1614 | @* - If the K-dimension is known to be infinite, a degree bound is needed |
---|
1615 | EXAMPLE: example ivHilbert; shows examples |
---|
1616 | " |
---|
1617 | {int degbound = 0; |
---|
1618 | if (size(#) > 0) {if (typeof(#[1])=="int"){if (#[1] > 0) {degbound = #[1];}}} |
---|
1619 | intvec P,H; int i; |
---|
1620 | for (i = 1; i <= size(L); i++) |
---|
1621 | {P[i] = ncols(L[i]); |
---|
1622 | if (P[i] == 1) {if ( isInMat(H,L[i]) > 0) {ERROR("Quotient algebra is trivial");}} |
---|
1623 | } |
---|
1624 | if (size(L) == 0) {ERROR("GB is empty, quotient algebra corresponds to free algebra");} |
---|
1625 | H[1] = 1; |
---|
1626 | checkAssumptions(degbound,L); |
---|
1627 | if (degbound == 0) |
---|
1628 | {int sd; |
---|
1629 | intmat S; |
---|
1630 | sd = P[1]; |
---|
1631 | for (i = 2; i <= size(P); i++) {if (P[i] < sd) {sd = P[i];}} |
---|
1632 | sd = (sd - 1); |
---|
1633 | if (sd == 0) { for (i = 1; i <= size(L); i++){if (ncols(L[i]) == 1){S = createStartMat1(n,L[i]); break;}}} |
---|
1634 | else {S = createStartMat(sd,n);} |
---|
1635 | if (intvec(S) == 0) {return(H);} |
---|
1636 | for (i = 1; i <= sd; i++) {H = H,(n^i);} |
---|
1637 | for (i = 1; i <= nrows(S); i++) |
---|
1638 | {intvec St = S[i,1..ncols(S)]; |
---|
1639 | H = findHCoeff(St,n,L,P,H); |
---|
1640 | kill St; |
---|
1641 | } |
---|
1642 | return(H); |
---|
1643 | } |
---|
1644 | else |
---|
1645 | {for (i = 1; i <= size(P); i++) |
---|
1646 | {if (P[i] > degbound) {ERROR("degreebound is too small, GB contains elements of higher degree");}} |
---|
1647 | int sd; |
---|
1648 | intmat S; |
---|
1649 | sd = P[1]; |
---|
1650 | for (i = 2; i <= size(P); i++) {if (P[i] < sd) {sd = P[i];}} |
---|
1651 | sd = (sd - 1); |
---|
1652 | if (sd == 0) { for (i = 1; i <= size(L); i++){if (ncols(L[i]) == 1){S = createStartMat1(n,L[i]); break;}}} |
---|
1653 | else {S = createStartMat(sd,n);} |
---|
1654 | if (intvec(S) == 0) {return(H);} |
---|
1655 | for (i = 1; i <= sd; i++) {H = H,(n^i);} |
---|
1656 | for (i = 1; i <= nrows(S); i++) |
---|
1657 | {intvec St = S[i,1..ncols(S)]; |
---|
1658 | H = findHCoeff(St,n,L,P,H,degbound); |
---|
1659 | kill St; |
---|
1660 | } |
---|
1661 | return(H); |
---|
1662 | } |
---|
1663 | } |
---|
1664 | example |
---|
1665 | { |
---|
1666 | "EXAMPLE:"; echo = 2; |
---|
1667 | ring r = 0,(x,y),dp; |
---|
1668 | def R = makeLetterplaceRing(5); // constructs a Letterplace ring |
---|
1669 | R; |
---|
1670 | setring R; // sets basering to Letterplace ring |
---|
1671 | //some intmats, which contain monomials in intvec representation as rows |
---|
1672 | intmat I1 [2][2] = 1,1,2,2; intmat I2 [1][3] = 1,2,1; |
---|
1673 | intmat J1 [1][2] = 1,1; intmat J2 [2][3] = 2,1,2,1,2,1; |
---|
1674 | print(I1); |
---|
1675 | print(I2); |
---|
1676 | print(J1); |
---|
1677 | print(J2); |
---|
1678 | list G = I1,I2; // ideal, which is already a Groebner basis |
---|
1679 | list I = J1,J2; // ideal, which is already a Groebner basis |
---|
1680 | ivHilbert(G,2); // invokes the procedure without any degree bound |
---|
1681 | ivHilbert(I,2,5); // invokes the procedure with degree bound 5 |
---|
1682 | } |
---|
1683 | |
---|
1684 | |
---|
1685 | proc ivKDim(list L, int n, list #) |
---|
1686 | "USAGE: ivKDim(L,n[,degbound]); L a list of intmats, |
---|
1687 | @* n an integer, degbound an optional integer |
---|
1688 | RETURN: int, the K-dimension of A/<L> |
---|
1689 | PURPOSE:Computing the K-dimension of A/<L> |
---|
1690 | ASSUME: - basering is a Letterplace ring. |
---|
1691 | @* - all rows of each intmat correspond to a Letterplace monomial |
---|
1692 | @* - if you specify a different degree bound degbound, |
---|
1693 | @* degbound <= attrib(basering,uptodeg) holds. |
---|
1694 | NOTE: - If degbound is set, a degree bound will be added. By default there |
---|
1695 | @* is no degree bound. |
---|
1696 | @* - n is the number of variables. |
---|
1697 | @* - If the K-dimension is known to be infinite, a degree bound is needed |
---|
1698 | EXAMPLE: example ivKDim; shows examples |
---|
1699 | " |
---|
1700 | {int degbound = 0; |
---|
1701 | if (size(#) > 0) {if (typeof(#[1])=="int"){if (#[1] > 0) {degbound = #[1];}}} |
---|
1702 | intvec P,H; int i; |
---|
1703 | for (i = 1; i <= size(L); i++) |
---|
1704 | {P[i] = ncols(L[i]); |
---|
1705 | if (P[i] == 1) {if ( isInMat(H,L[i]) > 0) {ERROR("Quotient algebra is trivial");}} |
---|
1706 | } |
---|
1707 | if (size(L) == 0) {ERROR("GB is empty, quotient algebra corresponds to free algebra");} |
---|
1708 | kill H; |
---|
1709 | checkAssumptions(degbound,L); |
---|
1710 | if (degbound == 0) |
---|
1711 | {int sd; int dimen = 1; |
---|
1712 | intmat S; |
---|
1713 | sd = P[1]; |
---|
1714 | for (i = 2; i <= size(P); i++) {if (P[i] < sd) {sd = P[i];}} |
---|
1715 | sd = (sd - 1); |
---|
1716 | if (sd == 0) { for (i = 1; i <= size(L); i++){if (ncols(L[i]) == 1){S = createStartMat1(n,L[i]); break;}}} |
---|
1717 | else {S = createStartMat(sd,n);} |
---|
1718 | if (intvec(S) == 0) {return(dimen);} |
---|
1719 | for (i = 1; i <= sd; i++) {dimen = dimen +(n^i);} |
---|
1720 | for (i = 1; i <= nrows(S); i++) |
---|
1721 | {intvec St = S[i,1..ncols(S)]; |
---|
1722 | dimen = dimen + findDimen(St,n,L,P); |
---|
1723 | kill St; |
---|
1724 | } |
---|
1725 | return(dimen); |
---|
1726 | } |
---|
1727 | else |
---|
1728 | {for (i = 1; i <= size(P); i++) |
---|
1729 | {if (P[i] > degbound) {ERROR("degreebound is too small, GB contains elements of higher degree");}} |
---|
1730 | int sd; int dimen = 1; |
---|
1731 | intmat S; |
---|
1732 | sd = P[1]; |
---|
1733 | for (i = 2; i <= size(P); i++) {if (P[i] < sd) {sd = P[i];}} |
---|
1734 | sd = (sd - 1); |
---|
1735 | if (sd == 0) { for (i = 1; i <= size(L); i++){if (ncols(L[i]) == 1){S = createStartMat1(n,L[i]); break;}}} |
---|
1736 | else {S = createStartMat(sd,n);} |
---|
1737 | if (intvec(S) == 0) {return(dimen);} |
---|
1738 | for (i = 1; i <= sd; i++) {dimen = dimen +(n^i);} |
---|
1739 | for (i = 1; i <= nrows(S); i++) |
---|
1740 | {intvec St = S[i,1..ncols(S)]; |
---|
1741 | dimen = dimen + findDimen(St,n,L,P, degbound); |
---|
1742 | kill St; |
---|
1743 | } |
---|
1744 | return(dimen); |
---|
1745 | } |
---|
1746 | } |
---|
1747 | example |
---|
1748 | { |
---|
1749 | "EXAMPLE:"; echo = 2; |
---|
1750 | ring r = 0,(x,y),dp; |
---|
1751 | def R = makeLetterplaceRing(5); // constructs a Letterplace ring |
---|
1752 | R; |
---|
1753 | setring R; // sets basering to Letterplace ring |
---|
1754 | //some intmats, which contain monomials in intvec representation as rows |
---|
1755 | intmat I1 [2][2] = 1,1,2,2; intmat I2 [1][3] = 1,2,1; |
---|
1756 | intmat J1 [1][2] = 1,1; intmat J2 [2][3] = 2,1,2,1,2,1; |
---|
1757 | print(I1); |
---|
1758 | print(I2); |
---|
1759 | print(J1); |
---|
1760 | print(J2); |
---|
1761 | list G = I1,I2; // ideal, which is already a Groebner basis |
---|
1762 | list I = J1,J2; // ideal, which is already a Groebner basis |
---|
1763 | ivKDim(G,2); // invokes the procedure without any degree bound |
---|
1764 | ivKDim(I,2,5); // invokes the procedure with degree bound 5 |
---|
1765 | } |
---|
1766 | |
---|
1767 | proc ivMis2Dim(list M) |
---|
1768 | "USAGE: ivMis2Dim(M); M a list of intvecs |
---|
1769 | RETURN: int, the K-dimension of the given algebra |
---|
1770 | PURPOSE:Computing the K-dimension out of given mistletoes |
---|
1771 | ASSUME: - The mistletoes have to be ordered lexicographically -> OrdMisLex. |
---|
1772 | @* Otherwise the returned value may differ from the K-dimension. |
---|
1773 | @* - basering is a Letterplace ring. |
---|
1774 | EXAMPLE: example ivMis2Dim; shows examples |
---|
1775 | " |
---|
1776 | {checkAssumptions(0,M); |
---|
1777 | intvec L; |
---|
1778 | if (size(M) == 0){ERROR("There are no mistletoes, so it appears your dimension is infinite!");} |
---|
1779 | if (isInList(L,M) > 0) {print("1 is a mistletoe, therefore dim = 1"); return(1);} |
---|
1780 | int i,j,d,s; |
---|
1781 | j = 1; |
---|
1782 | d = 1 + size(M[1]); |
---|
1783 | for (i = 1; i < size(M); i++) |
---|
1784 | {s = size(M[i]); if (s > size(M[i+1])){s = size(M[i+1]);} |
---|
1785 | while ((M[i][j] == M[i+1][j]) && (j <= s)){j = j + 1;} |
---|
1786 | d = d + size(M[i+1])- j + 1; |
---|
1787 | } |
---|
1788 | return(d); |
---|
1789 | } |
---|
1790 | example |
---|
1791 | { |
---|
1792 | "EXAMPLE:"; echo = 2; |
---|
1793 | ring r = 0,(x,y),dp; |
---|
1794 | def R = makeLetterplaceRing(5); // constructs a Letterplace ring |
---|
1795 | R; |
---|
1796 | setring R; // sets basering to Letterplace ring |
---|
1797 | intvec i1 = 1,2; intvec i2 = 2,1,2; |
---|
1798 | // the mistletoes are xy and yxy, which are already ordered lexicographically |
---|
1799 | list L = i1,i2; |
---|
1800 | ivMis2Dim(L); // returns the dimension of the factor algebra |
---|
1801 | } |
---|
1802 | |
---|
1803 | proc ivOrdMisLex(list M) |
---|
1804 | "USAGE: ivOrdMisLex(M); M a list of intvecs |
---|
1805 | RETURN: list, containing the ordered intvecs of M |
---|
1806 | PURPOSE:Orders a given set of mistletoes lexicographically |
---|
1807 | ASSUME: - basering is a Letterplace ring. |
---|
1808 | - intvecs correspond to monomials |
---|
1809 | NOTE: - This is preprocessing, it's not needed if the mistletoes are returned |
---|
1810 | @* from the sickle algorithm. |
---|
1811 | @* - Each entry of the list returned is an intvec. |
---|
1812 | EXAMPLE: example ivOrdMisLex; shows examples |
---|
1813 | " |
---|
1814 | {checkAssumptions(0,M); |
---|
1815 | return(sort(M)[1]); |
---|
1816 | } |
---|
1817 | example |
---|
1818 | { |
---|
1819 | "EXAMPLE:"; echo = 2; |
---|
1820 | ring r = 0,(x,y),dp; |
---|
1821 | def R = makeLetterplaceRing(5); // constructs a Letterplace ring |
---|
1822 | setring R; // sets basering to Letterplace ring |
---|
1823 | intvec i1 = 1,2,1; intvec i2 = 2,2,1; intvec i3 = 1,1; intvec i4 = 2,1,1,1; |
---|
1824 | // the corresponding monomials are xyx,y^2x,x^2,yx^3 |
---|
1825 | list M = i1,i2,i3,i4; |
---|
1826 | M; |
---|
1827 | ivOrdMisLex(M);// orders the list of monomials |
---|
1828 | } |
---|
1829 | |
---|
1830 | proc ivSickle(list L, int n, list #) |
---|
1831 | "USAGE: ivSickle(L,n,[degbound]); L a list of intmats, n an int, degbound an |
---|
1832 | @* optional integer |
---|
1833 | RETURN: list, containing intvecs, the mistletoes of A/<L> |
---|
1834 | PURPOSE:Computing the mistletoes for a given Groebner basis L |
---|
1835 | ASSUME: - basering is a Letterplace ring. |
---|
1836 | @* - all rows of each intmat correspond to a Letterplace monomial |
---|
1837 | @* - if you specify a different degree bound degbound, |
---|
1838 | @* degbound <= attrib(basering,uptodeg) holds. |
---|
1839 | NOTE: - If degbound is set, a degree bound will be added. By default there |
---|
1840 | @* is no degree bound. |
---|
1841 | @* - n is the number of variables. |
---|
1842 | @* - If the K-dimension is known to be infinite, a degree bound is needed |
---|
1843 | EXAMPLE: example ivSickle; shows examples |
---|
1844 | " |
---|
1845 | {list M; |
---|
1846 | int degbound = 0; |
---|
1847 | if (size(#) > 0){if (typeof(#[1])=="int"){if (#[1] > 0){degbound = #[1];}}} |
---|
1848 | int i; |
---|
1849 | intvec P,H; |
---|
1850 | for (i = 1; i <= size(L); i++) |
---|
1851 | {P[i] = ncols(L[i]); |
---|
1852 | if (P[i] == 1) {if (isInMat(H,L[i]) > 0) {ERROR("Quotient algebra is trivial");}} |
---|
1853 | } |
---|
1854 | if (size(L) == 0) {ERROR("GB is empty, quotient algebra corresponds to free algebra");} |
---|
1855 | kill H; |
---|
1856 | checkAssumptions(degbound,L); |
---|
1857 | if (degbound == 0) |
---|
1858 | {intmat S; int sd; |
---|
1859 | sd = P[1]; |
---|
1860 | for (i = 2; i <= size(P); i++) {if (P[i] < sd) {sd = P[i];}} |
---|
1861 | sd = (sd - 1); |
---|
1862 | if (sd == 0) { for (i = 1; i <= size(L); i++){if (ncols(L[i]) == 1){S = createStartMat1(n,L[i]); break;}}} |
---|
1863 | else {S = createStartMat(sd,n);} |
---|
1864 | if (intvec(S) == 0) {return(list (intvec(0)));} |
---|
1865 | for (i = 1; i <= nrows(S); i++) |
---|
1866 | {intvec St = S[i,1..ncols(S)]; |
---|
1867 | M = M + findmistletoes(St,n,L,P); |
---|
1868 | kill St; |
---|
1869 | } |
---|
1870 | return(M); |
---|
1871 | } |
---|
1872 | else |
---|
1873 | {for (i = 1; i <= size(P); i++) |
---|
1874 | {if (P[i] > degbound) {ERROR("degreebound is too small, GB contains elements of higher degree");}} |
---|
1875 | intmat S; int sd; |
---|
1876 | sd = P[1]; |
---|
1877 | for (i = 2; i <= size(P); i++) {if (P[i] < sd) {sd = P[i];}} |
---|
1878 | sd = (sd - 1); |
---|
1879 | if (sd == 0) { for (i = 1; i <= size(L); i++){if (ncols(L[i]) == 1){S = createStartMat1(n,L[i]); break;}}} |
---|
1880 | else {S = createStartMat(sd,n);} |
---|
1881 | if (intvec(S) == 0) {return(list (intvec(0)));} |
---|
1882 | for (i = 1; i <= nrows(S); i++) |
---|
1883 | {intvec St = S[i,1..ncols(S)]; |
---|
1884 | M = M + findmistletoes(St,n,L,P,degbound); |
---|
1885 | kill St; |
---|
1886 | } |
---|
1887 | return(M); |
---|
1888 | } |
---|
1889 | } |
---|
1890 | example |
---|
1891 | { |
---|
1892 | "EXAMPLE:"; echo = 2; |
---|
1893 | ring r = 0,(x,y),dp; |
---|
1894 | def R = makeLetterplaceRing(5); // constructs a Letterplace ring |
---|
1895 | setring R; // sets basering to Letterplace ring |
---|
1896 | //some intmats, which contain monomials in intvec representation as rows |
---|
1897 | intmat I1 [2][2] = 1,1,2,2; intmat I2 [1][3] = 1,2,1; |
---|
1898 | intmat J1 [1][2] = 1,1; intmat J2 [2][3] = 2,1,2,1,2,1; |
---|
1899 | print(I1); |
---|
1900 | print(I2); |
---|
1901 | print(J1); |
---|
1902 | print(J2); |
---|
1903 | list G = I1,I2; // ideal, which is already a Groebner basis |
---|
1904 | list I = J1,J2; // ideal, which is already a Groebner basis |
---|
1905 | ivSickle(G,2); // invokes the procedure without any degree bound |
---|
1906 | ivSickle(I,2,5); // invokes the procedure with degree bound 5 |
---|
1907 | } |
---|
1908 | |
---|
1909 | proc ivSickleDim(list L, int n, list #) |
---|
1910 | "USAGE: ivSickleDim(L,n[,degbound]); L a list of intmats, n an integer, degbound |
---|
1911 | @* an optional integer |
---|
1912 | RETURN: list |
---|
1913 | PURPOSE:Computing mistletoes and the K-dimension |
---|
1914 | ASSUME: - basering is a Letterplace ring. |
---|
1915 | @* - all rows of each intmat correspond to a Letterplace monomial |
---|
1916 | @* - if you specify a different degree bound degbound, |
---|
1917 | @* degbound <= attrib(basering,uptodeg) holds. |
---|
1918 | NOTE: - If L is the list returned, then L[1] is an integer, L[2] is a list, |
---|
1919 | @* containing the mistletoes as intvecs. |
---|
1920 | @* - If degbound is set, a degree bound will be added. By default there |
---|
1921 | @* is no degree bound. |
---|
1922 | @* - n is the number of variables. |
---|
1923 | @* - If the K-dimension is known to be infinite, a degree bound is needed |
---|
1924 | EXAMPLE: example ivSickleDim; shows examples |
---|
1925 | " |
---|
1926 | {list M; |
---|
1927 | int degbound = 0; |
---|
1928 | if (size(#) > 0){if (typeof(#[1])=="int"){if (#[1] > 0){degbound = #[1];}}} |
---|
1929 | int i,dimen; list R; |
---|
1930 | intvec P,H; |
---|
1931 | for (i = 1; i <= size(L); i++) |
---|
1932 | {P[i] = ncols(L[i]); |
---|
1933 | if (P[i] == 1) {if (isInMat(H,L[i]) > 0) {ERROR("Quotient algebra is trivial, dimension equals zero");}} |
---|
1934 | } |
---|
1935 | if (size(L) == 0) {ERROR("GB is empty, quotient algebra corresponds to free algebra");} |
---|
1936 | kill H; |
---|
1937 | checkAssumptions(degbound,L); |
---|
1938 | if (degbound == 0) |
---|
1939 | {int sd; dimen = 1; |
---|
1940 | intmat S; |
---|
1941 | sd = P[1]; |
---|
1942 | for (i = 2; i <= size(P); i++) {if (P[i] < sd) {sd = P[i];}} |
---|
1943 | sd = (sd - 1); |
---|
1944 | if (sd == 0) { for (i = 1; i <= size(L); i++){if (ncols(L[i]) == 1){S = createStartMat1(n,L[i]); break;}}} |
---|
1945 | else {S = createStartMat(sd,n);} |
---|
1946 | if (intvec(S) == 0) {return(list(dimen,list(intvec(0))));} |
---|
1947 | for (i = 1; i <= sd; i++) {dimen = dimen +(n^i);} |
---|
1948 | R[1] = dimen; |
---|
1949 | for (i = 1; i <= nrows(S); i++) |
---|
1950 | {intvec St = S[i,1..ncols(S)]; |
---|
1951 | R = findMisDim(St,n,L,P,R); |
---|
1952 | kill St; |
---|
1953 | } |
---|
1954 | return(R); |
---|
1955 | } |
---|
1956 | else |
---|
1957 | {for (i = 1; i <= size(P); i++) |
---|
1958 | {if (P[i] > degbound) {ERROR("degreebound is too small, GB contains elements of higher degree");}} |
---|
1959 | int sd; dimen = 1; |
---|
1960 | intmat S; |
---|
1961 | sd = P[1]; |
---|
1962 | for (i = 2; i <= size(P); i++) {if (P[i] < sd) {sd = P[i];}} |
---|
1963 | sd = (sd - 1); |
---|
1964 | if (sd == 0) { for (i = 1; i <= size(L); i++){if (ncols(L[i]) == 1){S = createStartMat1(n,L[i]); break;}}} |
---|
1965 | else {S = createStartMat(sd,n);} |
---|
1966 | if (intvec(S) == 0) {return(list(dimen,list(intvec(0))));} |
---|
1967 | for (i = 1; i <= sd; i++) {dimen = dimen +(n^i);} |
---|
1968 | R[1] = dimen; |
---|
1969 | for (i = 1; i <= nrows(S); i++) |
---|
1970 | {intvec St = S[i,1..ncols(S)]; |
---|
1971 | R = findMisDim(St,n,L,P,R,degbound); |
---|
1972 | kill St; |
---|
1973 | } |
---|
1974 | return(R); |
---|
1975 | } |
---|
1976 | } |
---|
1977 | example |
---|
1978 | { |
---|
1979 | "EXAMPLE:"; echo = 2; |
---|
1980 | ring r = 0,(x,y),dp; |
---|
1981 | def R = makeLetterplaceRing(5); // constructs a Letterplace ring |
---|
1982 | setring R; // sets basering to Letterplace ring |
---|
1983 | //some intmats, which contain monomials in intvec representation as rows |
---|
1984 | intmat I1 [2][2] = 1,1,2,2; intmat I2 [1][3] = 1,2,1; |
---|
1985 | intmat J1 [1][2] = 1,1; intmat J2 [2][3] = 2,1,2,1,2,1; |
---|
1986 | print(I1); |
---|
1987 | print(I2); |
---|
1988 | print(J1); |
---|
1989 | print(J2); |
---|
1990 | list G = I1,I2;// ideal, which is already a Groebner basis |
---|
1991 | list I = J1,J2; // ideal, which is already a Groebner basis |
---|
1992 | ivSickleDim(G,2); // invokes the procedure without any degree bound |
---|
1993 | ivSickleDim(I,2,5); // invokes the procedure with degree bound 5 |
---|
1994 | } |
---|
1995 | |
---|
1996 | proc ivSickleHil(list L, int n, list #) |
---|
1997 | "USAGE:ivSickleHil(L,n[,degbound]); L a list of intmats, n an integer, |
---|
1998 | @* degbound an optional integer |
---|
1999 | RETURN: list |
---|
2000 | PURPOSE:Computing the mistletoes and the Hilbert series |
---|
2001 | ASSUME: - basering is a Letterplace ring. |
---|
2002 | @* - all rows of each intmat correspond to a Letterplace monomial |
---|
2003 | @* - if you specify a different degree bound degbound, |
---|
2004 | @* degbound <= attrib(basering,uptodeg) holds. |
---|
2005 | NOTE: - If L is the list returned, then L[1] is an intvec, L[2] is a list, |
---|
2006 | @* containing the mistletoes as intvecs. |
---|
2007 | @* - If degbound is set, a degree bound will be added. By default there |
---|
2008 | @* is no degree bound. |
---|
2009 | @* - n is the number of variables. |
---|
2010 | @* - If I = L[1] is the intvec returned, then I[k] is the (k-1)-th |
---|
2011 | @* coefficient of the Hilbert series. |
---|
2012 | @* - If the K-dimension is known to be infinite, a degree bound is needed |
---|
2013 | EXAMPLE: example ivSickleHil; shows examples |
---|
2014 | " |
---|
2015 | {int degbound = 0; |
---|
2016 | if (size(#) > 0) {if (typeof(#[1])=="int"){if (#[1] > 0) {degbound = #[1];}}} |
---|
2017 | intvec P,H; int i; list R; |
---|
2018 | for (i = 1; i <= size(L); i++) |
---|
2019 | {P[i] = ncols(L[i]); |
---|
2020 | if (P[i] == 1) {if ( isInMat(H,L[i]) > 0) {ERROR("Quotient algebra is trivial");}} |
---|
2021 | } |
---|
2022 | if (size(L) == 0) {ERROR("GB is empty, quotient algebra corresponds to free algebra");} |
---|
2023 | H[1] = 1; |
---|
2024 | checkAssumptions(degbound,L); |
---|
2025 | if (degbound == 0) |
---|
2026 | {int sd; |
---|
2027 | intmat S; |
---|
2028 | sd = P[1]; |
---|
2029 | for (i = 2; i <= size(P); i++) {if (P[i] < sd) {sd = P[i];}} |
---|
2030 | sd = (sd - 1); |
---|
2031 | if (sd == 0) { for (i = 1; i <= size(L); i++){if (ncols(L[i]) == 1){S = createStartMat1(n,L[i]); break;}}} |
---|
2032 | else {S = createStartMat(sd,n);} |
---|
2033 | if (intvec(S) == 0) {return(list(H,list(intvec (0))));} |
---|
2034 | for (i = 1; i <= sd; i++) {H = H,(n^i);} |
---|
2035 | R[1] = H; kill H; |
---|
2036 | for (i = 1; i <= nrows(S); i++) |
---|
2037 | {intvec St = S[i,1..ncols(S)]; |
---|
2038 | R = findHCoeffMis(St,n,L,P,R); |
---|
2039 | kill St; |
---|
2040 | } |
---|
2041 | return(R); |
---|
2042 | } |
---|
2043 | else |
---|
2044 | {for (i = 1; i <= size(P); i++) |
---|
2045 | {if (P[i] > degbound) {ERROR("degreebound is too small, GB contains elements of higher degree");}} |
---|
2046 | int sd; |
---|
2047 | intmat S; |
---|
2048 | sd = P[1]; |
---|
2049 | for (i = 2; i <= size(P); i++) {if (P[i] < sd) {sd = P[i];}} |
---|
2050 | sd = (sd - 1); |
---|
2051 | if (sd == 0) { for (i = 1; i <= size(L); i++){if (ncols(L[i]) == 1){S = createStartMat1(n,L[i]); break;}}} |
---|
2052 | else {S = createStartMat(sd,n);} |
---|
2053 | if (intvec(S) == 0) {return(list(H,list(intvec(0))));} |
---|
2054 | for (i = 1; i <= sd; i++) {H = H,(n^i);} |
---|
2055 | R[1] = H; kill H; |
---|
2056 | for (i = 1; i <= nrows(S); i++) |
---|
2057 | {intvec St = S[i,1..ncols(S)]; |
---|
2058 | R = findHCoeffMis(St,n,L,P,R,degbound); |
---|
2059 | kill St; |
---|
2060 | } |
---|
2061 | return(R); |
---|
2062 | } |
---|
2063 | } |
---|
2064 | example |
---|
2065 | { |
---|
2066 | "EXAMPLE:"; echo = 2; |
---|
2067 | ring r = 0,(x,y),dp; |
---|
2068 | def R = makeLetterplaceRing(5); // constructs a Letterplace ring |
---|
2069 | setring R; // sets basering to Letterplace ring |
---|
2070 | //some intmats, which contain monomials in intvec representation as rows |
---|
2071 | intmat I1[2][2] = 1,1,2,2; intmat I2[1][3] = 1,2,1; |
---|
2072 | intmat J1[1][2] = 1,1; intmat J2[2][3] = 2,1,2,1,2,1; |
---|
2073 | print(I1); |
---|
2074 | print(I2); |
---|
2075 | print(J1); |
---|
2076 | print(J2); |
---|
2077 | list G = I1,I2;// ideal, which is already a Groebner basis |
---|
2078 | list I = J1,J2; // ideal, which is already a Groebner basis |
---|
2079 | ivSickleHil(G,2); // invokes the procedure without any degree bound |
---|
2080 | ivSickleHil(I,2,5); // invokes the procedure with degree bound 5 |
---|
2081 | } |
---|
2082 | |
---|
2083 | proc lpDHilbert(ideal G, list #) |
---|
2084 | "USAGE: lpDHilbert(G[,degbound,n]); G an ideal, degbound, n optional integers |
---|
2085 | RETURN: list |
---|
2086 | PURPOSE:Computing K-dimension and Hilbert series, starting with a lp-ideal |
---|
2087 | ASSUME: - basering is a Letterplace ring. |
---|
2088 | @* - if you specify a different degree bound degbound, |
---|
2089 | @* degbound <= attrib(basering,uptodeg) holds. |
---|
2090 | NOTE: - If L is the list returned, then L[1] is an integer corresponding to the |
---|
2091 | @* dimension, L[2] is an intvec which contains the coefficients of the |
---|
2092 | @* Hilbert series |
---|
2093 | @* - If degbound is set, there will be a degree bound added. 0 means no |
---|
2094 | @* degree bound. Default: attrib(basering,uptodeg). |
---|
2095 | @* - n can be set to a different number of variables. |
---|
2096 | @* Default: n = attrib(basering, lV). |
---|
2097 | @* - If I = L[2] is the intvec returned, then I[k] is the (k-1)-th |
---|
2098 | @* coefficient of the Hilbert series. |
---|
2099 | @* - If the K-dimension is known to be infinite, a degree bound is needed |
---|
2100 | EXAMPLE: example lpDHilbert; shows examples |
---|
2101 | " |
---|
2102 | {int degbound = attrib(basering,"uptodeg");int n = attrib(basering, "lV"); |
---|
2103 | if (size(#) > 0){if (typeof(#[1])=="int"){if (#[1] >= 0){degbound = #[1];}}} |
---|
2104 | if (size(#) > 1){if (typeof(#[1])=="int"){if (#[2] > 0){n = #[2];}}} |
---|
2105 | list L; |
---|
2106 | L = lp2ivId(normalize(lead(G))); |
---|
2107 | return(ivDHilbert(L,n,degbound)); |
---|
2108 | } |
---|
2109 | example |
---|
2110 | { |
---|
2111 | "EXAMPLE:"; echo = 2; |
---|
2112 | ring r = 0,(x,y),dp; |
---|
2113 | def R = makeLetterplaceRing(5); // constructs a Letterplace ring |
---|
2114 | setring R; // sets basering to Letterplace ring |
---|
2115 | ideal G = x(1)*x(2), y(1)*y(2),x(1)*y(2)*x(3); // ideal G contains a |
---|
2116 | //Groebner basis |
---|
2117 | lpDHilbert(G,5,2); // invokes procedure with degree bound 5 and 2 variables |
---|
2118 | // note that the optional parameters are not necessary, due to the finiteness |
---|
2119 | // of the K-dimension of the factor algebra |
---|
2120 | lpDHilbert(G); // procedure with ring parameters |
---|
2121 | lpDHilbert(G,0); // procedure without degreebound |
---|
2122 | } |
---|
2123 | |
---|
2124 | proc lpDHilbertSickle(ideal G, list #) |
---|
2125 | "USAGE: lpDHilbertSickle(G[,degbound,n]); G an ideal, degbound, n optional |
---|
2126 | @* integers |
---|
2127 | RETURN: list |
---|
2128 | PURPOSE:Computing K-dimension, Hilbert series and mistletoes at once |
---|
2129 | ASSUME: - basering is a Letterplace ring. |
---|
2130 | @* - if you specify a different degree bound degbound, |
---|
2131 | @* degbound <= attrib(basering,uptodeg) holds. |
---|
2132 | NOTE: - If L is the list returned, then L[1] is an integer, the K-dimension, |
---|
2133 | @* L[2] is an intvec, the Hilbert series and L[3] is an ideal, |
---|
2134 | @* the mistletoes |
---|
2135 | @* - If degbound is set, there will be a degree bound added. 0 means no |
---|
2136 | @* degree bound. Default: attrib(basering,uptodeg). |
---|
2137 | @* - n can be set to a different number of variables. |
---|
2138 | @* Default: n = attrib(basering, lV). |
---|
2139 | @* - If I = L[1] is the intvec returned, then I[k] is the (k-1)-th |
---|
2140 | @* coefficient of the Hilbert series. |
---|
2141 | @* - If the K-dimension is known to be infinite, a degree bound is needed |
---|
2142 | EXAMPLE: example lpDHilbertSickle; shows examples |
---|
2143 | " |
---|
2144 | {int degbound = attrib(basering,"uptodeg");int n = attrib(basering, "lV"); |
---|
2145 | if (size(#) > 0){if (typeof(#[1])=="int"){if (#[1] >= 0){degbound = #[1];}}} |
---|
2146 | if (size(#) > 1){if (typeof(#[1])=="int"){if (#[2] > 0){n = #[2];}}} |
---|
2147 | list L; |
---|
2148 | L = lp2ivId(normalize(lead(G))); |
---|
2149 | L = ivDHilbertSickle(L,n,degbound); |
---|
2150 | L[3] = ivL2lpI(L[3]); |
---|
2151 | return(L); |
---|
2152 | } |
---|
2153 | example |
---|
2154 | { |
---|
2155 | "EXAMPLE:"; echo = 2; |
---|
2156 | ring r = 0,(x,y),dp; |
---|
2157 | def R = makeLetterplaceRing(5); // constructs a Letterplace ring |
---|
2158 | setring R; // sets basering to Letterplace ring |
---|
2159 | ideal G = x(1)*x(2), y(1)*y(2),x(1)*y(2)*x(3); // ideal G contains a |
---|
2160 | //Groebner basis |
---|
2161 | lpDHilbertSickle(G,5,2); //invokes procedure with degree bound 5 and 2 variables |
---|
2162 | // note that the optional parameters are not necessary, due to the finiteness |
---|
2163 | // of the K-dimension of the factor algebra |
---|
2164 | lpDHilbertSickle(G); // procedure with ring parameters |
---|
2165 | lpDHilbertSickle(G,0); // procedure without degreebound |
---|
2166 | } |
---|
2167 | |
---|
2168 | proc lpGlDimBound (ideal I) |
---|
2169 | "USAGE: lpGlDimBound(I); I an ideal |
---|
2170 | RETURN: int, an upper bound for the global dimension, -1 means infinity |
---|
2171 | PURPOSE: computing an upper bound for the global dimension |
---|
2172 | ASSUME: - basering is a Letterplace ring. |
---|
2173 | NOTE: -1 is also returned when degree bound is reached |
---|
2174 | EXAMPLE: example lpGlDimBound; shows example |
---|
2175 | " |
---|
2176 | { |
---|
2177 | ideal G = lead(I); |
---|
2178 | list L = lpId2ivLi(G); list RL; |
---|
2179 | int n = attrib(basering,"lV"); |
---|
2180 | int d = attrib(basering,"uptodeg"); |
---|
2181 | int i,j,r; list V; intvec g,v,s,s1,vs; |
---|
2182 | for (i = 1; i <= n; i++) {L[i] = i;} |
---|
2183 | for (j = 1; j <= size(L); j++) |
---|
2184 | { |
---|
2185 | i=1; |
---|
2186 | while (i <= size(V)) |
---|
2187 | { |
---|
2188 | v = V[i], g = L[i]; s = GLDimSuffix(v,g); |
---|
2189 | if (size(s)>1) {s1 = s[1..(size(s)-1)];} |
---|
2190 | else {s1 = s[1];} |
---|
2191 | vs = v,s1; |
---|
2192 | if (isNormal(vs,L)) |
---|
2193 | { |
---|
2194 | if (ContainedIn(s1,V)==0) |
---|
2195 | { |
---|
2196 | V = insert(V,s1,size(V)); |
---|
2197 | } |
---|
2198 | } |
---|
2199 | } |
---|
2200 | } |
---|
2201 | r=findDChain(L); |
---|
2202 | if (r == d) {return(-1);} |
---|
2203 | else { return(r);} |
---|
2204 | } |
---|
2205 | example |
---|
2206 | { |
---|
2207 | "EXAMPLE:"; echo = 2; |
---|
2208 | ring r = 0,(x,y),dp; |
---|
2209 | def R = makeLetterplaceRing(5); // constructs a Letterplace ring |
---|
2210 | setring R; // sets basering to Letterplace ring |
---|
2211 | ideal G = x(1)*x(2), y(1)*y(2),x(1)*y(2)*x(3); // ideal G contains a |
---|
2212 | //Groebner basis |
---|
2213 | lpGlDimBound(G); // invokes procedure with Groebner basis G |
---|
2214 | } |
---|
2215 | |
---|
2216 | proc lpHilbert(ideal G, list #) |
---|
2217 | "USAGE: lpHilbert(G[,degbound,n]); G an ideal, degbound, n optional integers |
---|
2218 | RETURN: intvec, containing the coefficients of the Hilbert series |
---|
2219 | PURPOSE:Computing the Hilbert series |
---|
2220 | ASSUME: - basering is a Letterplace ring. |
---|
2221 | @* - if you specify a different degree bound degbound, |
---|
2222 | @* degbound <= attrib(basering,uptodeg) holds. |
---|
2223 | NOTE: - If degbound is set, there will be a degree bound added. 0 means no |
---|
2224 | @* degree bound. Default: attrib(basering,uptodeg). |
---|
2225 | @* - n is the number of variables, which can be set to a different number. |
---|
2226 | @* Default: attrib(basering, lV). |
---|
2227 | @* - If I is returned, then I[k] is the (k-1)-th coefficient of the Hilbert |
---|
2228 | @* series. |
---|
2229 | @* - If the K-dimension is known to be infinite, a degree bound is needed |
---|
2230 | EXAMPLE: example lpHilbert; shows examples |
---|
2231 | " |
---|
2232 | {int degbound = attrib(basering,"uptodeg");int n = attrib(basering, "lV"); |
---|
2233 | if (size(#) > 0){if (typeof(#[1])=="int"){if (#[1] >= 0){degbound = #[1];}}} |
---|
2234 | if (size(#) > 1){if (typeof(#[1])=="int"){if (#[2] > 0){n = #[2];}}} |
---|
2235 | list L; |
---|
2236 | L = lp2ivId(normalize(lead(G))); |
---|
2237 | return(ivHilbert(L,n,degbound)); |
---|
2238 | } |
---|
2239 | example |
---|
2240 | { |
---|
2241 | "EXAMPLE:"; echo = 2; |
---|
2242 | ring r = 0,(x,y),dp; |
---|
2243 | def R = makeLetterplaceRing(5); // constructs a Letterplace ring |
---|
2244 | setring R; // sets basering to Letterplace ring |
---|
2245 | ideal G = x(1)*x(2), y(1)*y(2),x(1)*y(2)*x(3); // ideal G contains a |
---|
2246 | //Groebner basis |
---|
2247 | lpHilbert(G,5,2); // invokes procedure with degree bound 5 and 2 variables |
---|
2248 | // note that the optional parameters are not necessary, due to the finiteness |
---|
2249 | // of the K-dimension of the factor algebra |
---|
2250 | lpDHilbert(G); // procedure with ring parameters |
---|
2251 | lpDHilbert(G,0); // procedure without degreebound |
---|
2252 | } |
---|
2253 | |
---|
2254 | proc lpDimCheck(ideal G) |
---|
2255 | "USAGE: lpDimCheck(G); |
---|
2256 | RETURN: int, 1 if K-dimension of the factor algebra is infinite, 0 otherwise |
---|
2257 | PURPOSE:Checking a factor algebra for finiteness of the K-dimension |
---|
2258 | ASSUME: - basering is a Letterplace ring. |
---|
2259 | EXAMPLE: example lpDimCheck; shows examples |
---|
2260 | " |
---|
2261 | {int n = attrib(basering,"lV"); |
---|
2262 | list L; |
---|
2263 | ideal R; |
---|
2264 | R = normalize(lead(G)); |
---|
2265 | L = lp2ivId(R); |
---|
2266 | return(ivDimCheck(L,n)); |
---|
2267 | } |
---|
2268 | example |
---|
2269 | { |
---|
2270 | "EXAMPLE:"; echo = 2; |
---|
2271 | ring r = 0,(x,y),dp; |
---|
2272 | def R = makeLetterplaceRing(5); // constructs a Letterplace ring |
---|
2273 | setring R; // sets basering to Letterplace ring |
---|
2274 | ideal G = x(1)*x(2), y(1)*y(2),x(1)*y(2)*x(3); |
---|
2275 | // Groebner basis |
---|
2276 | ideal I = x(1)*x(2), y(1)*x(2)*y(3), x(1)*y(2)*x(3); |
---|
2277 | // Groebner basis |
---|
2278 | lpDimCheck(G); // invokes procedure, factor algebra is of finite K-dimension |
---|
2279 | lpDimCheck(I); // invokes procedure, factor algebra is of infinite Kdimension |
---|
2280 | } |
---|
2281 | |
---|
2282 | proc lpKDim(ideal G, list #) |
---|
2283 | "USAGE: lpKDim(G[,degbound, n]); G an ideal, degbound, n optional integers |
---|
2284 | RETURN: int, the K-dimension of the factor algebra |
---|
2285 | PURPOSE:Computing the K-dimension of a factor algebra, given via an ideal |
---|
2286 | ASSUME: - basering is a Letterplace ring |
---|
2287 | @* - if you specify a different degree bound degbound, |
---|
2288 | @* degbound <= attrib(basering,uptodeg) holds. |
---|
2289 | NOTE: - If degbound is set, there will be a degree bound added. 0 means no |
---|
2290 | @* degree bound. Default: attrib(basering, uptodeg). |
---|
2291 | @* - n is the number of variables, which can be set to a different number. |
---|
2292 | @* Default: attrib(basering, lV). |
---|
2293 | @* - If the K-dimension is known to be infinite, a degree bound is needed |
---|
2294 | EXAMPLE: example lpKDim; shows examples |
---|
2295 | " |
---|
2296 | {int degbound = attrib(basering, "uptodeg");int n = attrib(basering, "lV"); |
---|
2297 | if (size(#) > 0){if (typeof(#[1])=="int"){if (#[1] >= 0){degbound = #[1];}}} |
---|
2298 | if (size(#) > 1){if (typeof(#[1])=="int"){if (#[2] > 0){n = #[2];}}} |
---|
2299 | list L; |
---|
2300 | L = lp2ivId(normalize(lead(G))); |
---|
2301 | return(ivKDim(L,n,degbound)); |
---|
2302 | } |
---|
2303 | example |
---|
2304 | { |
---|
2305 | "EXAMPLE:"; echo = 2; |
---|
2306 | ring r = 0,(x,y),dp; |
---|
2307 | def R = makeLetterplaceRing(5); // constructs a Letterplace ring |
---|
2308 | setring R; // sets basering to Letterplace ring |
---|
2309 | ideal G = x(1)*x(2), y(1)*y(2),x(1)*y(2)*x(3); |
---|
2310 | // ideal G contains a Groebner basis |
---|
2311 | lpKDim(G); //procedure invoked with ring parameters |
---|
2312 | // the factor algebra is finite, so the degree bound given by the Letterplace |
---|
2313 | // ring is not necessary |
---|
2314 | lpKDim(G,0); // procedure without any degree bound |
---|
2315 | } |
---|
2316 | |
---|
2317 | proc lpMis2Dim(ideal M) |
---|
2318 | "USAGE: lpMis2Dim(M); M an ideal |
---|
2319 | RETURN: int, the K-dimension of the factor algebra |
---|
2320 | PURPOSE:Computing the K-dimension out of given mistletoes |
---|
2321 | ASSUME: - basering is a Letterplace ring. |
---|
2322 | @* - M contains only monomials |
---|
2323 | NOTE: - The mistletoes have to be ordered lexicographically -> OrdMisLex. |
---|
2324 | EXAMPLE: example lpMis2Dim; shows examples |
---|
2325 | " |
---|
2326 | {list L; |
---|
2327 | L = lpId2ivLi(M); |
---|
2328 | return(ivMis2Dim(L)); |
---|
2329 | } |
---|
2330 | example |
---|
2331 | { |
---|
2332 | "EXAMPLE:"; echo = 2; |
---|
2333 | ring r = 0,(x,y),dp; |
---|
2334 | def R = makeLetterplaceRing(5); // constructs a Letterplace ring |
---|
2335 | setring R; // sets basering to Letterplace ring |
---|
2336 | ideal L = x(1)*y(2),y(1)*x(2)*y(3); |
---|
2337 | // ideal containing the mistletoes |
---|
2338 | lpMis2Dim(L); // returns the K-dimension of the factor algebra |
---|
2339 | } |
---|
2340 | |
---|
2341 | proc lpOrdMisLex(ideal M) |
---|
2342 | "USAGE: lpOrdMisLex(M); M an ideal of mistletoes |
---|
2343 | RETURN: ideal, containing the mistletoes, ordered lexicographically |
---|
2344 | PURPOSE:A given set of mistletoes is ordered lexicographically |
---|
2345 | ASSUME: - basering is a Letterplace ring. |
---|
2346 | NOTE: This is preprocessing, it is not needed if the mistletoes are returned |
---|
2347 | @* from the sickle algorithm. |
---|
2348 | EXAMPLE: example lpOrdMisLex; shows examples |
---|
2349 | " |
---|
2350 | {return(ivL2lpI(sort(lpId2ivLi(M))[1]));} |
---|
2351 | example |
---|
2352 | { |
---|
2353 | "EXAMPLE:"; echo = 2; |
---|
2354 | ring r = 0,(x,y),dp; |
---|
2355 | def R = makeLetterplaceRing(5); // constructs a Letterplace ring |
---|
2356 | setring R; // sets basering to Letterplace ring |
---|
2357 | ideal M = x(1)*y(2)*x(3), y(1)*y(2)*x(3), x(1)*x(2), y(1)*x(2)*x(3)*x(4); |
---|
2358 | // some monomials |
---|
2359 | lpOrdMisLex(M); // orders the monomials lexicographically |
---|
2360 | } |
---|
2361 | |
---|
2362 | proc lpSickle(ideal G, list #) |
---|
2363 | "USAGE: lpSickle(G[,degbound,n]); G an ideal, degbound, n optional integers |
---|
2364 | RETURN: ideal |
---|
2365 | PURPOSE:Computing the mistletoes of K[X]/<G> |
---|
2366 | ASSUME: - basering is a Letterplace ring. |
---|
2367 | @* - if you specify a different degree bound degbound, |
---|
2368 | @* degbound <= attrib(basering,uptodeg) holds. |
---|
2369 | NOTE: - If degbound is set, there will be a degree bound added. 0 means no |
---|
2370 | @* degree bound. Default: attrib(basering,uptodeg). |
---|
2371 | @* - n is the number of variables, which can be set to a different number. |
---|
2372 | @* Default: attrib(basering, lV). |
---|
2373 | @* - If the K-dimension is known to be infinite, a degree bound is needed |
---|
2374 | EXAMPLE: example lpSickle; shows examples |
---|
2375 | " |
---|
2376 | {int degbound = attrib(basering,"uptodeg"); int n = attrib(basering, "lV"); |
---|
2377 | if (size(#) > 0){if (typeof(#[1])=="int"){if (#[1] >= 0){degbound = #[1];}}} |
---|
2378 | if (size(#) > 1){if (typeof(#[1])=="int"){if (#[2] > 0){n = #[2];}}} |
---|
2379 | list L; ideal R; |
---|
2380 | R = normalize(lead(G)); |
---|
2381 | L = lp2ivId(R); |
---|
2382 | L = ivSickle(L,n,degbound); |
---|
2383 | R = ivL2lpI(L); |
---|
2384 | return(R); |
---|
2385 | } |
---|
2386 | example |
---|
2387 | { |
---|
2388 | "EXAMPLE:"; echo = 2; |
---|
2389 | ring r = 0,(x,y),dp; |
---|
2390 | def R = makeLetterplaceRing(5); // constructs a Letterplace ring |
---|
2391 | setring R; // sets basering to Letterplace ring |
---|
2392 | ideal G = x(1)*x(2), y(1)*y(2),x(1)*y(2)*x(3); // ideal G contains a |
---|
2393 | //Groebner basis |
---|
2394 | lpSickle(G); //invokes the procedure with ring parameters |
---|
2395 | // the factor algebra is finite, so the degree bound given by the Letterplace |
---|
2396 | // ring is not necessary |
---|
2397 | lpSickle(G,0); // procedure without any degree bound |
---|
2398 | } |
---|
2399 | |
---|
2400 | proc lpSickleDim(ideal G, list #) |
---|
2401 | "USAGE: lpSickleDim(G[,degbound,n]); G an ideal, degbound, n optional integers |
---|
2402 | RETURN: list |
---|
2403 | PURPOSE:Computing the K-dimension and the mistletoes |
---|
2404 | ASSUME: - basering is a Letterplace ring. |
---|
2405 | @* - if you specify a different degree bound degbound, |
---|
2406 | @* degbound <= attrib(basering,uptodeg) holds. |
---|
2407 | NOTE: - If L is the list returned, then L[1] is an integer, the K-dimension, |
---|
2408 | @* L[2] is an ideal, the mistletoes. |
---|
2409 | @* - If degbound is set, there will be a degree bound added. 0 means no |
---|
2410 | @* degree bound. Default: attrib(basering,uptodeg). |
---|
2411 | @* - n is the number of variables, which can be set to a different number. |
---|
2412 | @* Default: attrib(basering, lV). |
---|
2413 | @* - If the K-dimension is known to be infinite, a degree bound is needed |
---|
2414 | EXAMPLE: example lpSickleDim; shows examples |
---|
2415 | " |
---|
2416 | {int degbound = attrib(basering,"uptodeg");int n = attrib(basering, "lV"); |
---|
2417 | if (size(#) > 0){if (typeof(#[1])=="int"){if (#[1] >= 0){degbound = #[1];}}} |
---|
2418 | if (size(#) > 1){if (typeof(#[1])=="int"){if (#[2] > 0){n = #[2];}}} |
---|
2419 | list L; |
---|
2420 | L = lp2ivId(normalize(lead(G))); |
---|
2421 | L = ivSickleDim(L,n,degbound); |
---|
2422 | L[2] = ivL2lpI(L[2]); |
---|
2423 | return(L); |
---|
2424 | } |
---|
2425 | example |
---|
2426 | { |
---|
2427 | "EXAMPLE:"; echo = 2; |
---|
2428 | ring r = 0,(x,y),dp; |
---|
2429 | def R = makeLetterplaceRing(5); // constructs a Letterplace ring |
---|
2430 | setring R; // sets basering to Letterplace ring |
---|
2431 | ideal G = x(1)*x(2), y(1)*y(2),x(1)*y(2)*x(3); // ideal G contains a |
---|
2432 | //Groebner basis |
---|
2433 | lpSickleDim(G); // invokes the procedure with ring parameters |
---|
2434 | // the factor algebra is finite, so the degree bound given by the Letterplace |
---|
2435 | // ring is not necessary |
---|
2436 | lpSickleDim(G,0); // procedure without any degree bound |
---|
2437 | } |
---|
2438 | |
---|
2439 | proc lpSickleHil(ideal G, list #) |
---|
2440 | "USAGE: lpSickleHil(G); |
---|
2441 | RETURN: list |
---|
2442 | PURPOSE:Computing the Hilbert series and the mistletoes |
---|
2443 | ASSUME: - basering is a Letterplace ring. |
---|
2444 | @* - if you specify a different degree bound degbound, |
---|
2445 | @* degbound <= attrib(basering,uptodeg) holds. |
---|
2446 | NOTE: - If L is the list returned, then L[1] is an intvec, corresponding to the |
---|
2447 | @* Hilbert series, L[2] is an ideal, the mistletoes. |
---|
2448 | @* - If degbound is set, there will be a degree bound added. 0 means no |
---|
2449 | @* degree bound. Default: attrib(basering,uptodeg). |
---|
2450 | @* - n is the number of variables, which can be set to a different number. |
---|
2451 | @* Default: attrib(basering, lV). |
---|
2452 | @* - If I = L[1] is the intvec returned, then I[k] is the (k-1)-th |
---|
2453 | @* coefficient of the Hilbert series. |
---|
2454 | @* - If the K-dimension is known to be infinite, a degree bound is needed |
---|
2455 | EXAMPLE: example lpSickleHil; shows examples |
---|
2456 | " |
---|
2457 | {int degbound = attrib(basering,"uptodeg");int n = attrib(basering, "lV"); |
---|
2458 | if (size(#) > 0){if (typeof(#[1])=="int"){if (#[1] >= 0){degbound = #[1];}}} |
---|
2459 | if (size(#) > 1){if (typeof(#[1])=="int"){if (#[2] > 0){n = #[2];}}} |
---|
2460 | list L; |
---|
2461 | L = lp2ivId(normalize(lead(G))); |
---|
2462 | L = ivSickleHil(L,n,degbound); |
---|
2463 | L[2] = ivL2lpI(L[2]); |
---|
2464 | return(L); |
---|
2465 | } |
---|
2466 | example |
---|
2467 | { |
---|
2468 | "EXAMPLE:"; echo = 2; |
---|
2469 | ring r = 0,(x,y),dp; |
---|
2470 | def R = makeLetterplaceRing(5); // constructs a Letterplace ring |
---|
2471 | setring R; // sets basering to Letterplace ring |
---|
2472 | ideal G = x(1)*x(2), y(1)*y(2),x(1)*y(2)*x(3); // ideal G contains a |
---|
2473 | //Groebner basis |
---|
2474 | lpSickleHil(G); // invokes the procedure with ring parameters |
---|
2475 | // the factor algebra is finite, so the degree bound given by the Letterplace |
---|
2476 | // ring is not necessary |
---|
2477 | lpSickleHil(G,0); // procedure without any degree bound |
---|
2478 | } |
---|
2479 | |
---|
2480 | proc sickle(ideal G, list #) |
---|
2481 | "USAGE: sickle(G[,m, d, h, degbound]); G an ideal; m,d,h,degbound optional |
---|
2482 | @* integers |
---|
2483 | RETURN: list |
---|
2484 | PURPOSE:Allowing the user to access all procs with one command |
---|
2485 | ASSUME: - basering is a Letterplace ring. |
---|
2486 | @* - if you specify a different degree bound degbound, |
---|
2487 | @* degbound <= attrib(basering,uptodeg) holds. |
---|
2488 | NOTE: The returned object will always be a list, but the entries of the |
---|
2489 | @* returned list may be very different |
---|
2490 | @* case m=1,d=1,h=1: see lpDHilbertSickle |
---|
2491 | @* case m=1,d=1,h=0: see lpSickleDim |
---|
2492 | @* case m=1,d=0,h=1: see lpSickleHil |
---|
2493 | @* case m=1,d=0,h=0: see lpSickle (this is the default case) |
---|
2494 | @* case m=0,d=1,h=1: see lpDHilbert |
---|
2495 | @* case m=0,d=1,h=0: see lpKDim |
---|
2496 | @* case m=0,d=0,h=1: see lpHilbert |
---|
2497 | @* case m=0,d=0,h=0: returns an error |
---|
2498 | @* - If degbound is set, there will be a degree bound added. 0 means no |
---|
2499 | @* degree bound. Default: attrib(basering,uptodeg). |
---|
2500 | @* - If the K-dimension is known to be infinite, a degree bound is needed |
---|
2501 | EXAMPLE: example sickle; shows examples |
---|
2502 | " |
---|
2503 | {int m,d,h,degbound; |
---|
2504 | m = 1; d = 0; h = 0; degbound = attrib(basering,"uptodeg"); |
---|
2505 | if (size(#) > 0) {if (typeof(#[1])=="int"){if (#[1] < 1) {m = 0;}}} |
---|
2506 | if (size(#) > 1) {if (typeof(#[1])=="int"){if (#[2] > 0) {d = 1;}}} |
---|
2507 | if (size(#) > 2) {if (typeof(#[1])=="int"){if (#[3] > 0) {h = 1;}}} |
---|
2508 | if (size(#) > 3) {if (typeof(#[1])=="int"){if (#[4] >= 0) {degbound = #[4];}}} |
---|
2509 | if (m == 1) |
---|
2510 | {if (d == 0) |
---|
2511 | {if (h == 0) {return(lpSickle(G,degbound,attrib(basering,"lV")));} |
---|
2512 | else {return(lpSickleHil(G,degbound,attrib(basering,"lV")));} |
---|
2513 | } |
---|
2514 | else |
---|
2515 | {if (h == 0) {return(lpSickleDim(G,degbound,attrib(basering,"lV")));} |
---|
2516 | else {return(lpDHilbertSickle(G,degbound,attrib(basering,"lV")));} |
---|
2517 | } |
---|
2518 | } |
---|
2519 | else |
---|
2520 | {if (d == 0) |
---|
2521 | {if (h == 0) {ERROR("You request to do nothing, so relax and do so");} |
---|
2522 | else {return(lpHilbert(G,degbound,attrib(basering,"lV")));} |
---|
2523 | } |
---|
2524 | else |
---|
2525 | {if (h == 0) {return(lpKDim(G,degbound,attrib(basering,"lV")));} |
---|
2526 | else {return(lpDHilbert(G,degbound,attrib(basering,"lV")));} |
---|
2527 | } |
---|
2528 | } |
---|
2529 | } |
---|
2530 | example |
---|
2531 | { |
---|
2532 | "EXAMPLE:"; echo = 2; |
---|
2533 | ring r = 0,(x,y),dp; |
---|
2534 | def R = makeLetterplaceRing(5); // constructs a Letterplace ring |
---|
2535 | setring R; // sets basering to Letterplace ring |
---|
2536 | ideal G = x(1)*x(2), y(1)*y(2),x(1)*y(2)*x(3); |
---|
2537 | // G contains a Groebner basis |
---|
2538 | sickle(G,1,1,1); // computes mistletoes, K-dimension and the Hilbert series |
---|
2539 | sickle(G,1,0,0); // computes mistletoes only |
---|
2540 | sickle(G,0,1,0); // computes K-dimension only |
---|
2541 | sickle(G,0,0,1); // computes Hilbert series only |
---|
2542 | } |
---|
2543 | |
---|
2544 | /////////////////////////////////////////////////////////////////////////////// |
---|
2545 | /* vl: new stuff for conversion to Magma and to SD |
---|
2546 | todo: doc, example |
---|
2547 | */ |
---|
2548 | proc extractVars(r) |
---|
2549 | { |
---|
2550 | int i = 1; |
---|
2551 | int j = 1; |
---|
2552 | string candidate; |
---|
2553 | list result = list(); |
---|
2554 | for (i = 1; i<=nvars(r);i++) |
---|
2555 | { |
---|
2556 | candidate = string(var(i))[1,find(string(var(i)),"(")-1]; |
---|
2557 | if (!inList(result, candidate)) |
---|
2558 | { |
---|
2559 | result = insert(result,candidate,size(result)); |
---|
2560 | } |
---|
2561 | } |
---|
2562 | return(result); |
---|
2563 | } |
---|
2564 | |
---|
2565 | proc letterPlacePoly2MagmaString(poly h) |
---|
2566 | { |
---|
2567 | int pos; |
---|
2568 | string s = string(h); |
---|
2569 | while(find(s,"(")) |
---|
2570 | { |
---|
2571 | pos = find(s,"("); |
---|
2572 | while(s[pos]!=")") |
---|
2573 | { |
---|
2574 | s = s[1,pos-1]+s[pos+1,size(s)-pos]; |
---|
2575 | } |
---|
2576 | if (size(s)!=pos) |
---|
2577 | { |
---|
2578 | s = s[1,pos-1]+s[pos+1,size(s)-pos]; // The last (")") |
---|
2579 | } |
---|
2580 | else |
---|
2581 | { |
---|
2582 | s = s[1,pos-1]; |
---|
2583 | } |
---|
2584 | } |
---|
2585 | return(s); |
---|
2586 | } |
---|
2587 | |
---|
2588 | proc letterPlaceIdeal2SD(ideal I, int upToDeg) |
---|
2589 | { |
---|
2590 | int i; |
---|
2591 | print("Don't forget to fill in the formal Data in the file"); |
---|
2592 | string result = "<?xml version=\"1.0\"?>"+newline+"<FREEALGEBRA createdAt=\"\" createdBy=\"Singular\" id=\"FREEALGEBRA/\">"+newline; |
---|
2593 | result = result + "<vars>"+string(extractVars(basering))+"</vars>"+newline; |
---|
2594 | result = result + "<basis>"+newline; |
---|
2595 | for (i = 1;i<=size(I);i++) |
---|
2596 | { |
---|
2597 | result = result + "<poly>"+letterPlacePoly2MagmaString(I[i])+"</poly>"+newline; |
---|
2598 | } |
---|
2599 | result = result + "</basis>"+newline; |
---|
2600 | result = result + "<uptoDeg>"+ string(upToDeg)+"</uptoDeg>"+newline; |
---|
2601 | result = result + "<Comment></Comment>"+newline; |
---|
2602 | result = result + "<Version></Version>"+newline; |
---|
2603 | result = result + "</FREEALGEBRA>"; |
---|
2604 | return(result); |
---|
2605 | } |
---|
2606 | |
---|
2607 | |
---|
2608 | /////////////////////////////////////////////////////////////////////////////// |
---|
2609 | |
---|
2610 | |
---|
2611 | proc tst_fpadim() |
---|
2612 | { |
---|
2613 | example ivDHilbert; |
---|
2614 | example ivDHilbertSickle; |
---|
2615 | example ivDimCheck; |
---|
2616 | example ivHilbert; |
---|
2617 | example ivKDim; |
---|
2618 | example ivMis2Dim; |
---|
2619 | example ivOrdMisLex; |
---|
2620 | example ivSickle; |
---|
2621 | example ivSickleHil; |
---|
2622 | example ivSickleDim; |
---|
2623 | example lpDHilbert; |
---|
2624 | example lpDHilbertSickle; |
---|
2625 | example lpHilbert; |
---|
2626 | example lpDimCheck; |
---|
2627 | example lpKDim; |
---|
2628 | example lpMis2Dim; |
---|
2629 | example lpOrdMisLex; |
---|
2630 | example lpSickle; |
---|
2631 | example lpSickleHil; |
---|
2632 | example lpSickleDim; |
---|
2633 | example sickle; |
---|
2634 | example ivL2lpI; |
---|
2635 | example iv2lp; |
---|
2636 | example iv2lpList; |
---|
2637 | example iv2lpMat; |
---|
2638 | example lp2iv; |
---|
2639 | example lp2ivId; |
---|
2640 | example lpId2ivLi; |
---|
2641 | } |
---|
2642 | |
---|
2643 | |
---|
2644 | |
---|
2645 | |
---|
2646 | |
---|
2647 | /* |
---|
2648 | Here are some examples one may try. Just copy them into your console. |
---|
2649 | These are relations for braid groups, up to degree d: |
---|
2650 | |
---|
2651 | |
---|
2652 | LIB "fpadim.lib"; |
---|
2653 | ring r = 0,(x,y,z),dp; |
---|
2654 | int d =10; // degree |
---|
2655 | def R = makeLetterplaceRing(d); |
---|
2656 | setring R; |
---|
2657 | ideal I = y(1)*x(2)*y(3) - z(1)*y(2)*z(3), x(1)*y(2)*x(3) - z(1)*x(2)*y(3), |
---|
2658 | z(1)*x(2)*z(3) - y(1)*z(2)*x(3), x(1)*x(2)*x(3) + y(1)*y(2)*y(3) + |
---|
2659 | z(1)*z(2)*z(3) + x(1)*y(2)*z(3); |
---|
2660 | option(prot); |
---|
2661 | option(redSB);option(redTail);option(mem); |
---|
2662 | ideal J = system("freegb",I,d,3); |
---|
2663 | lpDimCheck(J); |
---|
2664 | sickle(J,1,1,1,d);//Computes mistletoes, K-dimension and the Hilbert series |
---|
2665 | |
---|
2666 | |
---|
2667 | |
---|
2668 | LIB "fpadim.lib"; |
---|
2669 | ring r = 0,(x,y,z),dp; |
---|
2670 | int d =11; // degree |
---|
2671 | def R = makeLetterplaceRing(d); |
---|
2672 | setring R; |
---|
2673 | ideal I = y(1)*x(2)*y(3) - z(1)*y(2)*z(3), x(1)*y(2)*z(3) - z(1)*x(2)*y(3), |
---|
2674 | z(1)*x(2)*z(3) - y(1)*z(2)*x(3), x(1)*x(2)*x(3) + y(1)*y(2)*y(3) + |
---|
2675 | z(1)*z(2)*z(3) + x(1)*y(2)*z(3); |
---|
2676 | option(prot); |
---|
2677 | option(redSB);option(redTail);option(mem); |
---|
2678 | ideal J = system("freegb",I,d,3); |
---|
2679 | lpDimCheck(J); |
---|
2680 | sickle(J,1,1,1,d); |
---|
2681 | |
---|
2682 | |
---|
2683 | |
---|
2684 | LIB "fpadim.lib"; |
---|
2685 | ring r = 0,(x,y,z),dp; |
---|
2686 | int d = 6; // degree |
---|
2687 | def R = makeLetterplaceRing(d); |
---|
2688 | setring R; |
---|
2689 | ideal I = y(1)*x(2)*y(3) - z(1)*y(2)*z(3), x(1)*y(2)*x(3) - z(1)*x(2)*y(3), |
---|
2690 | z(1)*x(2)*z(3) - y(1)*z(2)*x(3), x(1)*x(2)*x(3) -2*y(1)*y(2)*y(3) + 3*z(1)*z(2)*z(3) -4*x(1)*y(2)*z(3) + 5*x(1)*z(2)*z(3)- 6*x(1)*y(2)*y(3) +7*x(1)*x(2)*z(3) - 8*x(1)*x(2)*y(3); |
---|
2691 | option(prot); |
---|
2692 | option(redSB);option(redTail);option(mem); |
---|
2693 | ideal J = system("freegb",I,d,3); |
---|
2694 | lpDimCheck(J); |
---|
2695 | sickle(J,1,1,1,d); |
---|
2696 | */ |
---|
2697 | |
---|
2698 | /* |
---|
2699 | Here are some examples, which can also be found in [studzins]: |
---|
2700 | |
---|
2701 | // takes up to 880Mb of memory |
---|
2702 | LIB "fpadim.lib"; |
---|
2703 | ring r = 0,(x,y,z),dp; |
---|
2704 | int d =10; // degree |
---|
2705 | def R = makeLetterplaceRing(d); |
---|
2706 | setring R; |
---|
2707 | ideal I = |
---|
2708 | z(1)*z(2)*z(3)*z(4) + y(1)*x(2)*y(3)*x(4) - x(1)*y(2)*y(3)*x(4) - 3*z(1)*y(2)*x(3)*z(4), x(1)*x(2)*x(3) + y(1)*x(2)*y(3) - x(1)*y(2)*x(3), z(1)*y(2)*x(3)-x(1)*y(2)*z(3) + z(1)*x(2)*z(3); |
---|
2709 | option(prot); |
---|
2710 | option(redSB);option(redTail);option(mem); |
---|
2711 | ideal J = system("freegb",I,d,nvars(r)); |
---|
2712 | lpDimCheck(J); |
---|
2713 | sickle(J,1,1,1,d); // dimension is 24872 |
---|
2714 | |
---|
2715 | |
---|
2716 | LIB "fpadim.lib"; |
---|
2717 | ring r = 0,(x,y,z),dp; |
---|
2718 | int d =10; // degree |
---|
2719 | def R = makeLetterplaceRing(d); |
---|
2720 | setring R; |
---|
2721 | ideal I = x(1)*y(2) + y(1)*z(2), x(1)*x(2) + x(1)*y(2) - y(1)*x(2) - y(1)*y(2); |
---|
2722 | option(prot); |
---|
2723 | option(redSB);option(redTail);option(mem); |
---|
2724 | ideal J = system("freegb",I,d,3); |
---|
2725 | lpDimCheck(J); |
---|
2726 | sickle(J,1,1,1,d); |
---|
2727 | */ |
---|
2728 | |
---|
2729 | |
---|
2730 | /* |
---|
2731 | Example for computing GK dimension: |
---|
2732 | returns a ring which contains an ideal I |
---|
2733 | run gkDim(I) inside this ring and it should return 2n (the GK dimension |
---|
2734 | of n-th Weyl algebra including evaluation operators). |
---|
2735 | |
---|
2736 | proc createWeylEx(int n, int d) |
---|
2737 | " |
---|
2738 | " |
---|
2739 | { |
---|
2740 | int baseringdef; |
---|
2741 | if (defined(basering)) // if a basering is defined, it should be saved for later use |
---|
2742 | { |
---|
2743 | def save = basering; |
---|
2744 | baseringdef = 1; |
---|
2745 | } |
---|
2746 | ring r = 0,(d(1..n),x(1..n),e(1..n)),dp; |
---|
2747 | def R = makeLetterplaceRing(d); |
---|
2748 | setring R; |
---|
2749 | ideal I; int i,j; |
---|
2750 | |
---|
2751 | for (i = 1; i <= n; i++) |
---|
2752 | { |
---|
2753 | for (j = i+1; j<= n; j++) |
---|
2754 | { |
---|
2755 | I[size(I)+1] = lpMult(var(i),var(j)); |
---|
2756 | } |
---|
2757 | } |
---|
2758 | |
---|
2759 | for (i = 1; i <= n; i++) |
---|
2760 | { |
---|
2761 | for (j = i+1; j<= n; j++) |
---|
2762 | { |
---|
2763 | I[size(I)+1] = lpMult(var(n+i),var(n+j)); |
---|
2764 | } |
---|
2765 | } |
---|
2766 | for (i = 1; i <= n; i++) |
---|
2767 | { |
---|
2768 | for (j = 1; j<= n; j++) |
---|
2769 | { |
---|
2770 | I[size(I)+1] = lpMult(var(i),var(n+j)); |
---|
2771 | } |
---|
2772 | } |
---|
2773 | for (i = 1; i <= n; i++) |
---|
2774 | { |
---|
2775 | for (j = 1; j<= n; j++) |
---|
2776 | { |
---|
2777 | I[size(I)+1] = lpMult(var(i),var(2*n+j)); |
---|
2778 | } |
---|
2779 | } |
---|
2780 | for (i = 1; i <= n; i++) |
---|
2781 | { |
---|
2782 | for (j = 1; j<= n; j++) |
---|
2783 | { |
---|
2784 | I[size(I)+1] = lpMult(var(2*n+i),var(n+j)); |
---|
2785 | } |
---|
2786 | } |
---|
2787 | for (i = 1; i <= n; i++) |
---|
2788 | { |
---|
2789 | for (j = 1; j<= n; j++) |
---|
2790 | { |
---|
2791 | I[size(I)+1] = lpMult(var(2*n+i),var(2*n+j)); |
---|
2792 | } |
---|
2793 | } |
---|
2794 | I = simplify(I,2+4); |
---|
2795 | I = letplaceGBasis(I); |
---|
2796 | export(I); |
---|
2797 | if (baseringdef == 1) {setring save;} |
---|
2798 | return(R); |
---|
2799 | } |
---|
2800 | |
---|
2801 | proc TestGKAuslander3() |
---|
2802 | { |
---|
2803 | ring r = (0,q),(z,x,y),(dp(1),dp(2)); |
---|
2804 | def R = makeLetterplaceRing(5); // constructs a Letterplace ring |
---|
2805 | R; setring R; // sets basering to Letterplace ring |
---|
2806 | ideal I; |
---|
2807 | I = q*x(1)*y(2) - y(1)*x(2), z(1)*y(2) - y(1)*z(2), z(1)*x(2) - x(1)*z(2); |
---|
2808 | I = letplaceGBasis(I); |
---|
2809 | lpGkDim(I); // must be 3 |
---|
2810 | I = x(1)*y(2)*z(3) - y(1)*x(2), z(1)*y(2) - y(1)*z(2), z(1)*x(2) - x(1)*z(2);//gkDim = 2 |
---|
2811 | I = letplaceGBasis(I); // not finite BUT contains a poly in x,y only |
---|
2812 | lpGkDim(I); // must be 4 |
---|
2813 | |
---|
2814 | ring r = 0,(y,x,z),dp; |
---|
2815 | def R = makeLetterplaceRing(10); // constructs a Letterplace ring |
---|
2816 | R; setring R; // sets basering to Letterplace ring |
---|
2817 | ideal I; |
---|
2818 | I = x(1)*y(2)*z(3) - y(1)*x(2), z(1)*y(2) - y(1)*z(2), z(1)*x(2) - x(1)*z(2);//gkDim = 2 |
---|
2819 | I = letplaceGBasis(I); // computed as it would be homogenized; infinite |
---|
2820 | poly p = x(1)*y(2)*y(3)*x(4)-y(1)*x(2)*x(3)*y(4); |
---|
2821 | lpNF(p, I); // 0 as expected |
---|
2822 | |
---|
2823 | // with inverse of z |
---|
2824 | ring r = 0,(iz,z,x,y),dp; |
---|
2825 | def R = makeLetterplaceRing(11); // constructs a Letterplace ring |
---|
2826 | R; setring R; // sets basering to Letterplace ring |
---|
2827 | ideal I; |
---|
2828 | I = x(1)*y(2)*z(3) - y(1)*x(2), z(1)*y(2) - y(1)*z(2), z(1)*x(2) - x(1)*z(2), |
---|
2829 | iz(1)*y(2) - y(1)*iz(2), iz(1)*x(2) - x(1)*iz(2), iz(1)*z(2)-1, z(1)*iz(2) -1; |
---|
2830 | I = letplaceGBasis(I); // |
---|
2831 | setring r; |
---|
2832 | def R2 = makeLetterplaceRing(23); // constructs a Letterplace ring |
---|
2833 | setring R2; // sets basering to Letterplace ring |
---|
2834 | ideal I = imap(R,I); |
---|
2835 | lpGkDim(I); |
---|
2836 | |
---|
2837 | |
---|
2838 | ring r = 0,(t,z,x,y),(dp(2),dp(2)); |
---|
2839 | def R = makeLetterplaceRing(20); // constructs a Letterplace ring |
---|
2840 | R; setring R; // sets basering to Letterplace ring |
---|
2841 | ideal I; |
---|
2842 | I = x(1)*y(2)*z(3) - y(1)*x(2)*t(3), z(1)*y(2) - y(1)*z(2), z(1)*x(2) - x(1)*z(2), |
---|
2843 | t(1)*y(2) - y(1)*t(2), t(1)*x(2) - x(1)*t(2), t(1)*z(2) - z(1)*t(2);//gkDim = 2 |
---|
2844 | I = letplaceGBasis(I); // computed as it would be homogenized; infinite |
---|
2845 | LIB "elim.lib"; |
---|
2846 | ideal Inoz = nselect(I,intvec(2,6,10,14,18,22,26,30)); |
---|
2847 | for(int i=1; i<=20; i++) |
---|
2848 | { |
---|
2849 | Inoz=subst(Inoz,t(i),1); |
---|
2850 | } |
---|
2851 | ideal J = x(1)*y(2)*y(3)*x(4)-y(1)*x(2)*x(3)*y(4); |
---|
2852 | J = letplaceGBasis(J); |
---|
2853 | |
---|
2854 | poly p = x(1)*y(2)*y(3)*x(4)-y(1)*x(2)*x(3)*y(4); |
---|
2855 | lpNF(p, I); // 0 as expected |
---|
2856 | |
---|
2857 | ring r2 = 0,(x,y),dp; |
---|
2858 | def R2 = makeLetterplaceRing(50); // constructs a Letterplace ring |
---|
2859 | setring R2; |
---|
2860 | ideal J = x(1)*y(2)*y(3)*x(4)-y(1)*x(2)*x(3)*y(4); |
---|
2861 | J = letplaceGBasis(J); |
---|
2862 | } |
---|
2863 | |
---|
2864 | */ |
---|
2865 | |
---|
2866 | |
---|
2867 | /* actual work: |
---|
2868 | // downup algebra A |
---|
2869 | LIB "fpadim.lib"; |
---|
2870 | ring r = (0,a,b,g),(x,y),Dp; |
---|
2871 | def R = makeLetterplaceRing(6); // constructs a Letterplace ring |
---|
2872 | setring R; |
---|
2873 | poly F1 = g*x(1); |
---|
2874 | poly F2 = g*y(1); |
---|
2875 | ideal J = x(1)*x(2)*y(3)-a*x(1)*y(2)*x(3) - b*y(1)*x(2)*x(3) - F1, |
---|
2876 | x(1)*y(2)*y(3)-a*y(1)*x(2)*y(3) - b*y(1)*y(2)*x(3) - F2; |
---|
2877 | J = letplaceGBasis(J); |
---|
2878 | lpGkDim(J); // 3 == correct |
---|
2879 | |
---|
2880 | // downup algebra B |
---|
2881 | LIB "fpadim.lib"; |
---|
2882 | ring r = (0,a,b,g, p(1..7),q(1..7)),(x,y),Dp; |
---|
2883 | def R = makeLetterplaceRing(6); // constructs a Letterplace ring |
---|
2884 | setring R; |
---|
2885 | ideal imn = 1, y(1)*y(2)*y(3), x(1)*y(2), y(1)*x(2), x(1)*x(2), y(1)*y(2), x(1), y(1); |
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2886 | int i; |
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2887 | poly F1, F2; |
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2888 | for(i=1;i<=7;i++) |
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2889 | { |
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2890 | F1 = F1 + p(i)*imn[i]; |
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2891 | F2 = F2 + q(i)*imn[i]; |
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2892 | } |
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2893 | ideal J = x(1)*x(2)*y(3)-a*x(1)*y(2)*x(3) - b*y(1)*x(2)*x(3) - F1, |
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2894 | x(1)*y(2)*y(3)-a*y(1)*x(2)*y(3) - b*y(1)*y(2)*x(3) - F2; |
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2895 | J = letplaceGBasis(J); |
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2896 | lpGkDim(J); // 3 == correct |
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2897 | |
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2898 | */ |
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