1 | ////////////////////////////////////////////////////////////////////////////// |
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2 | version="$Id: sagbi.lib,v 1.11 2007-07-13 12:31:06 Singular Exp $"; |
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3 | category="Commutative Algebra"; |
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4 | info=" |
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5 | LIBRARY: sagbi.lib Compute Subalgebras bases Analogous to Groebner bases for ideals |
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6 | AUTHORS: Gerhard Pfister, pfister@mathematik.uni-kl.de, |
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7 | @* Anen Lakhal, alakhal@mathematik.uni-kl.de |
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8 | |
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9 | PROCEDURES: |
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10 | sagbiRreduction(p,I); perform one step subalgebra reducton (for short S-reduction) of p w.r.t I |
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11 | sagbiSPoly(I); compute the S-polynomials of the Subalgebra defined by the genartors of I |
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12 | sagbiNF(id,I); perform iterated S-reductions in order to compute Subalgebras normal forms |
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13 | sagbi(I); construct SAGBI basis for the Subalgebra defined by I |
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14 | sagbiPart(I); construct partial SAGBI basis for the Subalgebra defined by I |
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15 | "; |
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16 | |
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17 | LIB "algebra.lib"; |
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18 | LIB "elim.lib"; |
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19 | |
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20 | /////////////////////////////////////////////////////////////////////////////// |
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21 | proc sagbiSPoly(id ,list #) |
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22 | "USAGE: sagbiSPoly(id [,n]); id ideal, n positive integer. |
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23 | RETURN: an ideal |
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24 | @format |
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25 | - If (n=0 or default) an ideal, whose generators are the S-polynomials. |
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26 | - If (n=1) a list of size 2: |
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27 | the first element of this list is the ideal of S-polynomials. |
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28 | the second element of this list is the ring in which is defined |
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29 | the ideal of algebraic relations. |
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30 | @end format |
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31 | EXAMPLE: example sagbiSPoly; show an example " |
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32 | { |
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33 | if(size(#)==0) |
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34 | { |
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35 | #[1]=0; |
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36 | } |
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37 | degBound=0; |
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38 | def bsr=basering; |
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39 | ideal vars=maxideal(1); |
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40 | ideal B=ideal(bsr);//when the basering is quotient ring this "type casting" |
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41 | //gives th quotient ideal. |
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42 | int b=size(B); |
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43 | |
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44 | //In quotient rings,SINGULAR does not reduce polynomials w.r.t the |
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45 | //quotient ideal,therefore we should first 'reduce';if it is necessary for |
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46 | //computations to have a uniquely determined representant for each equivalent |
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47 | //class,which is the case of this procedure. |
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48 | |
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49 | if(b!=0) |
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50 | { |
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51 | id =reduce(id,groebner(0)); |
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52 | } |
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53 | int n,m=nvars(bsr),ncols(id); |
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54 | int z; |
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55 | string mp=string(minpoly); |
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56 | ideal P; |
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57 | list L; |
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58 | |
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59 | if(id==0) |
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60 | { |
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61 | if(#[1]==0) |
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62 | { |
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63 | return(P); |
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64 | } |
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65 | else |
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66 | { |
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67 | return(L); |
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68 | } |
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69 | } |
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70 | else |
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71 | { |
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72 | //==================create anew ring with extra variables================ |
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73 | |
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74 | execute("ring R1=("+charstr(bsr)+"),("+varstr(bsr)+",@y(1..m)),(dp(n),dp(m));"); |
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75 | execute("minpoly=number("+mp+");"); |
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76 | ideal id=imap(bsr,id); |
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77 | ideal A; |
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78 | |
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79 | for(z=1;z<=m;z++) |
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80 | { |
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81 | A[z]=lead(id[z])-@y(z); |
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82 | } |
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83 | |
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84 | A=groebner(A); |
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85 | ideal kern=nselect(A,1,n);// "kern" is the kernel of the ring map: |
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86 | // R1----->bsr ;y(z)----> lead(id[z]). |
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87 | //"kern" is the ideal of algebraic relations between |
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88 | // lead(id[z]). |
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89 | export kern,A;// we export: |
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90 | // * the ideal A to avoid useless computations |
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91 | // between 2 steps in sagbi procedure. |
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92 | // * the ideal kern : some times we can get intresting |
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93 | // informations from this ideal. |
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94 | setring bsr; |
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95 | map phi=R1,vars,id; |
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96 | |
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97 | // the sagbiSPolynomials are the image by phi of the generators of kern |
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98 | |
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99 | P=simplify(phi(kern),1); |
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100 | if(#[1]==0) |
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101 | { |
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102 | return(P); |
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103 | } |
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104 | else |
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105 | { |
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106 | L=P,R1; |
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107 | kill phi,vars; |
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108 | |
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109 | dbprint(printlevel-voice+3," |
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110 | // 'sagbiSPoly' created a ring as 2nd element of the list. |
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111 | // The ring contains the ideal 'kern' of algebraic relations between the |
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112 | //leading terms of the generators of I. |
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113 | // To access to this ring and see 'kern' you should give the ring a name, |
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114 | // e.g.: |
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115 | def S = L[2]; setring S; kern; |
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116 | "); |
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117 | return(L); |
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118 | } |
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119 | } |
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120 | } |
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121 | example |
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122 | { "EXAMPLE:"; echo = 2; |
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123 | ring r=0, (x,y),dp; |
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124 | poly f1,f2,f3,f4=x2,y2,xy+y,2xy2; |
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125 | ideal I=f1,f2,f3,f4; |
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126 | sagbiSPoly(I); |
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127 | list L=sagbiSPoly(I,1); |
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128 | L[1]; |
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129 | def S= L[2]; setring S; kern; |
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130 | } |
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131 | |
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132 | /////////////////////////////////////////////////////////////////////////////// |
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133 | static proc std1(ideal J,ideal I,list #) |
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134 | // I is contained in J, and it is assumed to be a standard bases! |
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135 | //This procedure computes a Standard basis for J from I one's |
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136 | //This procedure is essential for Spoly1 procedure. |
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137 | { |
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138 | def br=basering; |
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139 | int tt; |
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140 | ideal Res,@result; |
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141 | |
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142 | if(size(#)>0) {tt=#[1];} |
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143 | |
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144 | if(size(I)==0) {@result=groebner(J);} |
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145 | |
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146 | if((size(I)!=0) && (size(J)-size(I)>=1)) |
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147 | { |
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148 | qring Q=I; |
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149 | ideal J=fetch(br,J); |
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150 | J=groebner(J); |
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151 | setring br; |
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152 | Res=fetch(Q,J);// Res contains the generators that we add to I |
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153 | // to get the generators of std(J); |
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154 | @result=Res+I; |
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155 | } |
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156 | |
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157 | if(tt==0) { return(@result);} |
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158 | else { return(Res);} |
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159 | } |
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160 | |
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161 | /////////////////////////////////////////////////////////////////////////////// |
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162 | |
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163 | static proc Spoly1(list l,ideal I,ideal J,int a) |
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164 | //an implementation of SAGBI construction Algorithm using Spoly |
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165 | //procedure leads to useless computations and affect the efficiency |
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166 | //of SAGBI bases computations. This procedure is a variant of Spoly |
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167 | //in order to avoid these useless compuations. |
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168 | { |
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169 | degBound=0; |
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170 | def br=basering; |
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171 | ideal vars=maxideal(1); |
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172 | ideal B=ideal(br); |
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173 | int b=size(B); |
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174 | |
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175 | if(b!=0) |
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176 | { |
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177 | I=reduce(I,groebner(0)); |
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178 | J=reduce(J,groebner(0)); |
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179 | } |
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180 | int n,ii,jj=nvars(br),ncols(I),ncols(J); |
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181 | int z; |
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182 | list @L; |
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183 | string mp =string(minpoly); |
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184 | |
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185 | if(size(J)==0) |
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186 | { |
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187 | @L =sagbiSPoly(I,1); |
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188 | } |
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189 | else |
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190 | { |
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191 | ideal @sum=I+J; |
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192 | ideal P1; |
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193 | ideal P=l[1];//P is the ideal of spolynomials of I; |
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194 | def R=l[2];setring R;int kk=nvars(R); |
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195 | ideal J=fetch(br,J); |
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196 | |
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197 | //================create a new ring with extra variables============== |
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198 | execute("ring R1=("+charstr(R)+"),("+varstr(R)+",@y((ii+1)..(ii+jj))),(dp(n),dp(kk+jj-n));"); |
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199 | // *levandov: would it not be easier and better to use |
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200 | // ring @Y = char(R),(@y((ii+1)..(ii+jj))),dp; |
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201 | // def R1 = R + @Y; |
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202 | // setring R1; |
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203 | // -> thus |
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204 | ideal kern1; |
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205 | ideal A=fetch(R,A); |
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206 | attrib(A,"isSB",1); |
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207 | ideal J=fetch(R,J); |
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208 | ideal kern=fetch(R,kern); |
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209 | ideal A1; |
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210 | for(z=1;z<=jj;z++) |
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211 | { |
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212 | A1[z]=lead(J[z])-var(z+kk); |
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213 | } |
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214 | A1=A+A1; |
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215 | ideal @Res=std1(A1,A,1);// the generators of @Res are whose we have to add |
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216 | // to A to get std(A1). |
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217 | A=A+@Res; |
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218 | kern1=nselect(@Res,1,n); |
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219 | kern=kern+kern1; |
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220 | export kern,kern1,A; |
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221 | setring br; |
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222 | map phi=R1,vars,@sum; |
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223 | P1=simplify(phi(kern1),1);//P1 is th ideal we add to P to get the ideal |
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224 | //of Spolynomials of @sum. |
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225 | P=P+P1; |
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226 | |
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227 | if (a==1) |
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228 | { |
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229 | @L=P,R1; |
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230 | kill phi,vars; |
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231 | dbprint(printlevel-voice+3," |
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232 | // 'Spoly1' created a ring as 2nd element of the list. |
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233 | // The ring contains the ideal 'kern' of algebraic relations between the |
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234 | //generators of I+J. |
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235 | // To access to this ring and see 'kern' you should give the ring a name, |
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236 | // e.g.: |
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237 | def @ring = L[2]; setring @ring ; kern; |
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238 | "); |
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239 | } |
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240 | if(a==2) |
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241 | { |
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242 | @L=P1,R1; |
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243 | kill phi,vars; |
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244 | } |
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245 | } |
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246 | return(@L); |
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247 | } |
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248 | /////////////////////////////////////////////////////////////////////////////// |
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249 | |
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250 | proc sagbiReduction(poly p,ideal dom,list #) |
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251 | "USAGE: sagbiReduction(p,dom[,n]); p poly , dom ideal |
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252 | RETURN: a polynomial, after one step subalgebra reduction |
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253 | @format |
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254 | Three algorithm variants are used to perform subalgebra reduction. |
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255 | The positive interger n determines which variant should be used. |
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256 | n may take the values 0 (default), 1 or 2. |
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257 | @end format |
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258 | EXAMPLE: sagbiReduction; shows an example" |
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259 | { |
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260 | def bsr=basering; |
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261 | ideal B=ideal(bsr);//When the basering is quotient ring this type casting |
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262 | // gives the quotient ideal. |
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263 | int b=size(B); |
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264 | int n=nvars(bsr); |
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265 | |
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266 | //In quotient rings, SINGULAR, usually does not reduce polynomials w.r.t the |
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267 | //quotient ideal,therefore we should first reduce ,when it is necessary for computations, |
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268 | // to have a uniquely determined representant for each equivalent |
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269 | //class,which is the case of this algorithm. |
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270 | |
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271 | if(b !=0) //means that the basering is a quotient ring |
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272 | { |
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273 | p=reduce(p,std(0)); |
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274 | dom=reduce(dom,std(0)); |
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275 | } |
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276 | |
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277 | int i,choose; |
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278 | int z=ncols(dom); |
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279 | |
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280 | if((size(#)>0) && (typeof(#[1])=="int")) |
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281 | { |
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282 | choose = #[1]; |
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283 | } |
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284 | if (size(#)>1) |
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285 | { |
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286 | choose =#[2]; |
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287 | } |
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288 | |
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289 | //=======================first algorithm(default)========================= |
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290 | if ( choose == 0 ) |
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291 | { |
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292 | list L = algebra_containment(lead(p),lead(dom),1); |
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293 | if( L[1]==1 ) |
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294 | { |
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295 | // the ring L[2] = char(bsr),(x(1..nvars(bsr)),y(1..z)),(dp(n),dp(m)), |
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296 | // contains poly check s.t. LT(p) is of the form check(LT(f1),...,LT(fr)) |
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297 | def s1 = L[2]; |
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298 | map psi = s1,maxideal(1),dom; |
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299 | poly re = p - psi(check); |
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300 | // divide by the maximal power of #[1] |
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301 | if ( (size(#)>0) && (typeof(#[1])=="poly") ) |
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302 | { |
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303 | while ((re!=0) && (re!=#[1]) &&(subst(re,#[1],0)==0)) |
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304 | { |
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305 | re=re/#[1]; |
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306 | } |
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307 | } |
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308 | return(re); |
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309 | } |
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310 | return(p); |
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311 | } |
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312 | //======================2end variant of algorithm========================= |
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313 | //It uses two different commands for elimaination. |
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314 | //if(choose==1):"elimainate"command. |
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315 | //if (choose==2):"nselect" command. |
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316 | else |
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317 | { |
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318 | poly v=product(maxideal(1)); |
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319 | |
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320 | //------------- change the basering bsr to bsr[@(0),...,@(z)] ---------- |
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321 | execute("ring s=("+charstr(basering)+"),("+varstr(basering)+",@(0..z)),dp;"); |
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322 | // Ev hier die Reihenfolge der Vars aendern. Dazu muss unten aber entsprechend |
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323 | // geaendert werden: |
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324 | // execute("ring s="+charstr(basering)+",(@(0..z),"+varstr(basering)+"),dp;"); |
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325 | |
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326 | //constructs the leading ideal of dom=(p-@(0),dom[1]-@(1),...,dom[z]-@(z)) |
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327 | ideal dom=imap(bsr,dom); |
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328 | for (i=1;i<=z;i++) |
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329 | { |
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330 | dom[i]=lead(dom[i])-var(nvars(bsr)+i+1); |
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331 | } |
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332 | dom=lead(imap(bsr,p))-@(0),dom; |
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333 | |
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334 | //---------- eliminate the variables of the basering bsr -------------- |
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335 | //i.e. computes dom intersected with K[@(0),...,@(z)]. |
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336 | |
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337 | if(choose==1) |
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338 | { |
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339 | ideal kern=eliminate(dom,imap(bsr,v));//eliminate does not need a |
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340 | //standard basis as input. |
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341 | } |
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342 | if(choose==2) |
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343 | { |
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344 | ideal kern= nselect(groebner(dom),1,n);//"nselect" is combinatorial command |
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345 | //which uses the internal command |
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346 | // "simplify" |
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347 | } |
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348 | |
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349 | //--------- test wether @(0)-h(@(1),...,@(z)) is in ker --------------- |
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350 | // for some poly h and divide by maximal power of q=#[1] |
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351 | poly h; |
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352 | z=size(kern); |
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353 | for (i=1;i<=z;i++) |
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354 | { |
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355 | h=kern[i]/@(0); |
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356 | if (deg(h)==0) |
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357 | { |
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358 | h=(1/h)*kern[i]; |
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359 | // define the map psi : s ---> bsr defined by @(i) ---> p,dom[i] |
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360 | setring bsr; |
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361 | map psi=s,maxideal(1),p,dom; |
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362 | poly re=psi(h); |
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363 | // divide by the maximal power of #[1] |
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364 | if ((size(#)>0) && (typeof(#[1])== "poly") ) |
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365 | { |
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366 | while ((re!=0) && (re!=#[1]) &&(subst(re,#[1],0)==0)) |
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367 | { |
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368 | re=re/#[1]; |
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369 | } |
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370 | } |
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371 | return(re); |
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372 | } |
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373 | } |
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374 | setring bsr; |
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375 | return(p); |
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376 | } |
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377 | } |
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378 | example |
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379 | {"EXAMPLE:"; echo = 2; |
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380 | ring r= 0,(x,y),dp; |
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381 | ideal dom =x2,y2,xy-y; |
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382 | poly p=x4+x3y+xy2-y2; |
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383 | sagbiReduction(p,dom); |
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384 | sagbiReduction(p,dom,1); |
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385 | sagbiReduction(p,dom,2); |
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386 | } |
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387 | |
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388 | /////////////////////////////////////////////////////////////////////////////// |
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389 | static proc completeReduction(poly p,ideal dom,list#)//reduction |
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390 | { |
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391 | poly p1=p; |
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392 | poly p2=sagbiReduction(p,dom,#); |
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393 | while (p1!=p2) |
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394 | { |
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395 | p1=p2; |
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396 | p2=sagbiReduction(p1,dom,#); |
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397 | } |
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398 | return(p2); |
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399 | } |
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400 | /////////////////////////////////////////////////////////////////////////////// |
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401 | |
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402 | static proc completeReduction1(poly p,ideal dom,list #) //tail reduction |
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403 | { |
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404 | poly p1,p2,re; |
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405 | p1=p; |
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406 | while(p1!=0) |
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407 | { |
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408 | p2=sagbiReduction(p1,dom,#); |
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409 | if(p2!=p1) |
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410 | { |
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411 | p1=p2; |
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412 | } |
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413 | else |
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414 | { |
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415 | re=re+lead(p2); |
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416 | p1=p2-lead(p2); |
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417 | } |
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418 | } |
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419 | return(re); |
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420 | } |
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421 | |
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422 | |
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423 | |
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424 | /////////////////////////////////////////////////////////////////////////////// |
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425 | |
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426 | proc sagbiNF(id,ideal dom,int k,list#) |
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427 | "USAGE: sagbiNF(id,dom,k[,n]); id either poly or ideal,dom ideal, k and n positive intergers. |
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428 | RETURN: depends On the type of id; ideal or polynomial. |
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429 | @format |
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430 | The integer k determines what kind of s-reduction is performed: |
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431 | - if (k=0) no tail s-reduction is performed. |
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432 | - if (k=1) tail s-reduction is performed. |
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433 | Three Algorthim variants are used to perform Subalgebra reduction. |
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434 | The positive integer n determine which variant should be used. |
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435 | n may take the values (0 or default),1 or 2. |
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436 | @end format |
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437 | NOTE: computation of Subalgebras normal forms may be performed either |
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438 | in polynomial rings or quotient polynomial rings |
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439 | EXAMPLE: example sagbiNF; show example " |
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440 | { |
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441 | int z; |
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442 | ideal Red; |
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443 | poly re; |
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444 | if(typeof(id)=="ideal") |
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445 | { |
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446 | int i=ncols(id); |
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447 | for(z=1;z<=i;z++) |
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448 | { |
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449 | if(k==0) |
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450 | { |
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451 | id[z]=completeReduction(id[z],dom,#); |
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452 | } |
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453 | else |
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454 | { |
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455 | id[z]=completeReduction1(id[z],dom,#);//tail reduction. |
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456 | } |
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457 | } |
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458 | Red=simplify(id,7); |
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459 | return(Red); |
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460 | } |
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461 | if(typeof(id)=="poly") |
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462 | { |
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463 | if(k==0) |
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464 | { |
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465 | re=completeReduction(id,dom,#); |
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466 | } |
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467 | else |
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468 | { |
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469 | re=completeReduction1(id,dom,#); |
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470 | } |
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471 | return(re); |
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472 | } |
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473 | } |
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474 | example |
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475 | {"EXAMPLE:"; echo = 2; |
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476 | ring r=0,(x,y),dp; |
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477 | ideal I= x2-xy; |
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478 | qring Q=std(I); |
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479 | ideal dom =x2,x2y+y,x3y2; |
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480 | poly p=x4+x2y+y; |
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481 | sagbiNF(p,dom,0); |
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482 | sagbiNF(p,dom,1);// tail subalgebra reduction is performed |
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483 | } |
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484 | |
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485 | |
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486 | /////////////////////////////////////////////////////////////////////////////// |
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487 | |
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488 | static proc intRed(id,int k, list #) |
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489 | { |
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490 | int i,z; |
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491 | ideal Rest,intRed; |
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492 | z=ncols(id); |
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493 | for(i=1;i<=z;i++) |
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494 | { |
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495 | Rest=id; |
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496 | Rest[i]=0; |
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497 | Rest=simplify(Rest,2); |
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498 | if(k==0) |
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499 | { |
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500 | intRed[i]=completeReduction(id[i],Rest,#); |
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501 | } |
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502 | else |
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503 | { |
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504 | intRed[i]=completeReduction1(id[i],Rest,#); |
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505 | } |
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506 | } |
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507 | intRed=simplify(intRed,7);//1+2+4 in simplify command |
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508 | return(intRed); |
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509 | } |
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510 | ////////////////////////////////////////////////////////////////////////////// |
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511 | |
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512 | proc sagbi(id,int k,list#) |
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513 | "USAGE: sagbi(id,k[,n]); id ideal, k and n positive integers. |
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514 | RETURN: A SAGBI basis for the subalgebra defined by the generators of id. |
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515 | @format |
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516 | k determine what kind of s-reduction is performed: |
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517 | - if (k=0) no tail s-reduction is performed. |
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518 | - if (k=1) tail s-reduction is performed, and S-interreduced SAGBI basis |
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519 | is returned. |
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520 | Three Algorithm variants are used to perform Subalgebra reduction. |
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521 | The positive interger n determine which variant should be used. |
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522 | n may take the values (0 or default),1 or 2. |
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523 | @end format |
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524 | NOTE: SAGBI bases computations may be performed either |
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525 | in polynomial rings or quotient polynomial rings. |
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526 | EXAMPLE: example sagbi; show example " |
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527 | { |
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528 | degBound=0; |
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529 | ideal S,oldS,Red; |
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530 | list L; |
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531 | S=intRed(id,k,#); |
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532 | while(size(S)!=size(oldS)) |
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533 | { |
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534 | L=Spoly1(L,S,Red,2); |
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535 | Red=L[1]; |
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536 | Red=sagbiNF(Red,S,k,#); |
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537 | oldS=S; |
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538 | S=S+Red; |
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539 | } |
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540 | return(S); |
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541 | } |
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542 | example |
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543 | { "EXAMPLE:"; echo = 2; |
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544 | ring r= 0,(x,y),dp; |
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545 | ideal I=x2,y2,xy+y; |
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546 | sagbi(I,1,1); |
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547 | } |
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548 | /////////////////////////////////////////////////////////////////////////////// |
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549 | proc sagbiPart(id,int k,int c,list #) |
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550 | "USAGE: sagbi(id,k,c[,n]); id ideal, k, c and n positive integer. |
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551 | RETURN: A partial SAGBI basis for the subalgebra defined by the generators of id. |
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552 | @format |
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553 | should stop. k determine what kind of s-reduction is performed: |
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554 | - if (k=0) no tail s-reduction is performed. |
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555 | - if (k=1) tail s-reduction is performed, and S-intereduced SAGBI basis |
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556 | is returned. |
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557 | c determines, after which turn Sagbi basis computations should stop |
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558 | Three Algorithm variants are used to perform Subalgebra reduction. |
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559 | The positive integer n determines which variant should be used. |
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560 | n may take the values (0 or default),1 or 2. |
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561 | @end format |
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562 | NOTE:- SAGBI bases computations may be performed either |
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563 | in polynomial rings or quotient polynomial rings. |
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564 | - This version of sagbi procedure is interesting in the case of an Subalgebras |
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565 | with infinte SAGBI basis. In this case, by means of this procedure, |
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566 | we may check for example, if the elements of this basis have a particular form. |
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567 | EXAMPLE: example sagbiPart; show example " |
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568 | { |
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569 | degBound=0; |
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570 | ideal S,oldS,Red; |
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571 | int counter; |
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572 | list L; |
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573 | S=intRed(id,k,#); |
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574 | while((size(S)!=size(oldS))&&(counter<=c)) |
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575 | { |
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576 | L=Spoly1(L,S,Red,2); |
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577 | Red=L[1]; |
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578 | Red=sagbiNF(Red,S,k,#); |
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579 | oldS=S; |
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580 | S=S+Red; |
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581 | counter=counter+1; |
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582 | } |
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583 | return(S); |
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584 | } |
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585 | example |
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586 | { "EXAMPLE:"; echo = 2; |
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587 | ring r= 0,(x,y),dp; |
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588 | ideal I=x,xy-y2,xy2;//the corresponding Subalgebra has an infinte SAGBI basis |
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589 | sagbiPart(I,1,3);// computations should stop after 3 turns. |
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590 | } |
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591 | ////////////////////////////////////////////////////////////////////////////// |
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