1 | // $Id: invar.lib,v 1.4 1998-02-16 12:07:01 pfister Exp $ |
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2 | /////////////////////////////////////////////////////// |
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3 | // invar.lib |
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4 | // algorithm for computing the ring of invariants under |
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5 | // the action of the additive group (C,+) |
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6 | // written by Gerhard Pfister |
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7 | ////////////////////////////////////////////////////// |
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8 | |
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9 | LIBRARY: invar.lib PROCEDURES FOR COMPUTING INVARIANTS OF (C,+)-ACTIONS |
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10 | |
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11 | invariantRing(matrix m,poly p,poly q,int choose) |
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12 | // ring of invariants of the action of the additive group |
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13 | // defined by the vectorfield corresponding to the matrix m |
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14 | // (m[i,1] are the coefficients of d/dx(i)) |
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15 | // the polys p and q are assumed to be variables x(i) and x(j) |
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16 | // such that m[j,1]=0 and m[i,1]=x(j) |
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17 | // if choose=0 the computation stops if generators of the ring |
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18 | // of invariants are computed (to be used only if you know that |
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19 | // the ring of invariants is finitey generated) |
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20 | // if choose<>0 it computes invariants up to degree choose |
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21 | |
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22 | actionIsProper(matrix m) |
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23 | // returns 1 if the action of the additive group defined by the |
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24 | // matrix m as above i proper and 0 if not. |
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25 | |
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26 | /////////////////////////////////////////////////////////////////////////////// |
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27 | |
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28 | LIB "inout.lib"; |
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29 | LIB "general.lib"; |
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30 | |
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31 | /////////////////////////////////////////////////////////////////////////////// |
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32 | |
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33 | |
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34 | proc sortier(ideal id) |
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35 | { |
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36 | if(size(id)==0) |
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37 | { |
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38 | return(id); |
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39 | } |
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40 | intvec i=sortvec(id); |
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41 | int j; |
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42 | ideal m; |
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43 | for (j=1;j<=size(i);j++) |
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44 | { |
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45 | m[j] = id[i[j]]; |
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46 | } |
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47 | return(m); |
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48 | } |
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49 | example |
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50 | { "EXAMPLE:"; echo = 2; |
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51 | ring q=0,(x,y,z,u,v,w),dp; |
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52 | ideal i=w,x,z,y,v; |
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53 | ideal j=sortier(i); |
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54 | j; |
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55 | } |
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56 | |
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57 | |
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58 | /////////////////////////////////////////////////////////////////////////////// |
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59 | |
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60 | |
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61 | proc der (matrix m, poly f) |
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62 | USAGE: der(m,f); m matrix, f poly |
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63 | RETURN: poly= application of the vectorfield m befined by the matrix m |
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64 | (m[i,1] are the coefficients of d/dx(i)) to f |
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65 | NOTE: |
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66 | EXAMPLE: example der; shows an example |
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67 | { |
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68 | matrix mh=matrix(jacob(f))*m; |
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69 | return(mh[1,1]); |
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70 | } |
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71 | example |
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72 | { "EXAMPLE:"; echo = 2; |
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73 | ring q=0,(x,y,z,u,v,w),dp; |
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74 | poly f=2xz-y2; |
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75 | matrix m[6][1]; |
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76 | m[2,1]=x; |
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77 | m[3,1]=y; |
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78 | m[5,1]=u; |
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79 | m[6,1]=v; |
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80 | der(m,f); |
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81 | } |
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82 | |
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83 | /////////////////////////////////////////////////////////////////////////////// |
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84 | |
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85 | |
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86 | proc actionIsProper(matrix m) |
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87 | USAGE: actionIsProper(m); m matrix |
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88 | RETURN: int= 1 if is proper, 0 else |
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89 | NOTE: |
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90 | EXAMPLE: example actionIsProper; shows an example |
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91 | { |
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92 | int i; |
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93 | ideal id=maxideal(1); |
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94 | def bsr=basering; |
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95 | |
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96 | //changes the basering bsr to bsr[@t] |
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97 | execute "ring s="+charstr(basering)+",("+varstr(basering)+",@t),dp;"; |
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98 | poly inv,delta,tee,j; |
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99 | ideal id=imap(bsr,id); |
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100 | matrix @m[size(id)+1][1]; |
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101 | @m=imap(bsr,m),0; |
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102 | |
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103 | //computes the exp(@t*m)(var(i)) for all i |
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104 | for(i=1;i<=nvars(basering)-1;i++) |
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105 | { |
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106 | inv=var(i); |
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107 | delta=der(@m,inv); |
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108 | j=1; |
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109 | tee=@t; |
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110 | while(delta!=0) |
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111 | { |
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112 | inv=inv+1/j*delta*tee; |
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113 | j=j*(j+1); |
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114 | tee=tee*@t; |
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115 | delta=der(@m,delta); |
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116 | } |
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117 | id=id+ideal(inv); |
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118 | } |
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119 | i=inSubring(@t,id)[1]; |
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120 | setring(bsr); |
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121 | return(i); |
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122 | } |
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123 | example |
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124 | { "EXAMPLE:"; echo = 2; |
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125 | |
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126 | ring rf=0,(x(1..7)),dp; |
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127 | matrix m[7][1]; |
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128 | m[4,1]=x(1)^3; |
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129 | m[5,1]=x(2)^3; |
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130 | m[6,1]=x(3)^3; |
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131 | m[7,1]=(x(1)*x(2)*x(3))^2; |
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132 | actionIsProper(m); |
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133 | |
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134 | ring rd=0,(x(1..5)),dp; |
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135 | matrix m[5][1]; |
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136 | m[3,1]=x(1); |
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137 | m[4,1]=x(2); |
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138 | m[5,1]=1+x(1)*x(4)^2 |
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139 | actionIsProper(m); |
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140 | } |
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141 | /////////////////////////////////////////////////////////////////////////////// |
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142 | |
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143 | |
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144 | proc reduction(poly p, ideal dom, list #) |
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145 | USAGE: reduction(p,dom,q); p poly, dom ideal, q (optional) monomial |
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146 | RETURN: poly= (p-H(f1,...,fr))/q^a, if Lt(p)=H(Lt(f1),...,Lt(fr)) for |
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147 | some polynomial H in r variables over the base field, |
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148 | a maximal such that q^a devides p-H(f1,...,fr), |
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149 | dom =(f1,...,fr) |
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150 | NOTE: |
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151 | EXAMPLE: example reduction; shows an example |
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152 | { |
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153 | int i; |
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154 | int z=size(dom); |
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155 | def bsr=basering; |
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156 | |
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157 | //arranges the monomial v for elimination |
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158 | poly v=var(1); |
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159 | for (i=2;i<=nvars(basering);i=i+1) |
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160 | { |
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161 | v=v*var(i); |
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162 | } |
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163 | |
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164 | //changes the basering bsr to bsr[@(0),...,@(z)] |
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165 | execute "ring s="+charstr(basering)+",("+varstr(basering)+",@(0..z)),dp;"; |
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166 | |
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167 | //costructes the ideal dom=(p-@(0),dom[1]-@(1),...,dom[z]-@(z)) |
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168 | ideal dom=imap(bsr,dom); |
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169 | for (i=1;i<=z;i++) |
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170 | { |
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171 | dom[i]=lead(dom[i])-var(nvars(bsr)+i+1); |
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172 | } |
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173 | dom=lead(imap(bsr,p))-@(0),dom; |
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174 | |
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175 | //eliminates the variables of the basering bsr |
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176 | //i.e. computes dom intersected with K[@(0),...,@(z)] |
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177 | ideal kern=eliminate(dom,imap(bsr,v)); |
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178 | |
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179 | // test wether @(0)-h(@(1),...,@(z)) is in ker for some poly h |
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180 | poly h; |
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181 | z=size(kern); |
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182 | for (i=1;i<=z;i++) |
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183 | { |
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184 | h=kern[i]/@(0); |
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185 | if (deg(h)==0) |
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186 | { |
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187 | h=(1/h)*kern[i]; |
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188 | // defines the map psi : s ---> bsr defined by @(i) ---> p,dom[i] |
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189 | setring bsr; |
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190 | map psi=s,maxideal(1),p,dom; |
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191 | poly re=psi(h); |
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192 | |
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193 | // devides by the maximal power of #[1] |
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194 | if (size(#)>0) |
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195 | { |
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196 | while ((re!=0) && (re!=#[1]) &&(subst(re,#[1],0)==0)) |
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197 | { |
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198 | re=re/#[1]; |
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199 | } |
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200 | } |
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201 | |
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202 | return(re); |
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203 | } |
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204 | } |
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205 | setring bsr; |
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206 | return(p); |
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207 | } |
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208 | example |
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209 | { "EXAMPLE:"; echo = 2; |
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210 | ring q=0,(x,y,z,u,v,w),dp; |
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211 | poly p=x2yz-x2v; |
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212 | ideal dom =x-w,u2w+1,yz-v; |
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213 | reduction(p,dom); |
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214 | reduction(p,dom,w); |
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215 | } |
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216 | |
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217 | /////////////////////////////////////////////////////////////////////////////// |
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218 | |
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219 | proc completeReduction(poly p, ideal dom, list #) |
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220 | USAGE: completeReduction(p,dom,q); p poly, dom ideal, |
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221 | q (optional) monomial |
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222 | RETURN: poly= the polynomial p reduced with dom via the procedure |
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223 | reduction as long as possible |
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224 | NOTE: |
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225 | EXAMPLE: example completeReduction; shows an example |
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226 | { |
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227 | poly p1=p; |
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228 | poly p2=reduction(p,dom,#); |
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229 | while (p1!=p2) |
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230 | { |
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231 | p1=p2; |
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232 | p2=reduction(p1,dom,#); |
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233 | } |
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234 | return(p2); |
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235 | } |
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236 | example |
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237 | { "EXAMPLE:"; echo = 2; |
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238 | ring q=0,(x,y,z,u,v,w),dp; |
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239 | poly p=x2yz-x2v; |
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240 | ideal dom =x-w,u2w+1,yz-v; |
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241 | completeReduction(p,dom); |
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242 | completeReduction(p,dom,w); |
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243 | } |
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244 | /////////////////////////////////////////////////////////////////////////////// |
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245 | |
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246 | proc inSubring(poly p, ideal dom) |
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247 | USAGE: inSubring(p,dom); p poly, dom ideal |
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248 | RETURN: list= 1,string(@(0)-h(@(1),...,@(size(dom)))) :if p = h(dom[1],...,dom[size(dom)]) |
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249 | 0,string(h(@(0),...,@(size(dom)))) :if there is only a nonlinear relation |
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250 | h(p,dom[1],...,dom[size(dom)])=0. |
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251 | NOTE: |
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252 | EXAMPLE: example inSubring; shows an example |
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253 | { |
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254 | int z=size(dom); |
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255 | int i; |
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256 | def gnir=basering; |
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257 | list l; |
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258 | poly mile=var(1); |
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259 | for (i=2;i<=nvars(basering);i++) |
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260 | { |
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261 | mile=mile*var(i); |
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262 | } |
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263 | string eli=string(mile); |
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264 | // the intersection of ideal nett=(p-@(0),dom[1]-@(1),...) |
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265 | // with the ring k[@(0),...,@(n)] is computed, the result is ker |
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266 | execute "ring r1="+charstr(basering)+",("+varstr(basering)+",@(0..z)),dp;"; |
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267 | ideal nett=imap(gnir,dom); |
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268 | poly p; |
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269 | for (i=1;i<=z;i++) |
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270 | { |
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271 | execute "p=@("+string(i)+");"; |
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272 | nett[i]=nett[i]-p; |
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273 | } |
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274 | nett=imap(gnir,p)-@(0),nett; |
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275 | execute "ideal ker=eliminate(nett,"+eli+");"; |
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276 | // test wether @(0)-h(@(1),...,@(z)) is in ker |
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277 | l[1]=0; |
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278 | l[2]=""; |
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279 | for (i=1;i<=size(ker);i++) |
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280 | { |
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281 | if (deg(ker[i]/@(0))==0) |
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282 | { |
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283 | string str=string(ker[i]); |
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284 | setring gnir; |
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285 | l[1]=1; |
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286 | l[2]=str; |
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287 | return(l); |
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288 | } |
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289 | if (deg(ker[i]/@(0))>0) |
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290 | { |
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291 | l[2]=l[2]+string(ker[i]); |
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292 | } |
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293 | } |
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294 | setring gnir; |
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295 | return(l); |
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296 | } |
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297 | example |
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298 | { "EXAMPLE:"; echo = 2; |
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299 | ring q=0,(x,y,z,u,v,w),dp; |
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300 | poly p=xyzu2w-1yzu2w2+u4w2-1xu2vw+u2vw2+xyz-1yzw+2u2w-1xv+vw+2; |
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301 | ideal dom =x-w,u2w+1,yz-v; |
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302 | inSubring(p,dom); |
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303 | } |
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304 | |
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305 | /////////////////////////////////////////////////////////////////////////////// |
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306 | |
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307 | proc localInvar(matrix m, poly p, poly q, poly h) |
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308 | USAGE: localInvar(m,p,q,h); m matrix, p,q,h poly |
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309 | RETURN: poly= the invariant of the vectorfield m=Sum m[i,1]*d/dx(i) with respect |
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310 | to p,q,h, i.e. |
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311 | Sum (-1)^v*(1/v!)*m^v(p)*(q/m(q))^v)*m(q)^N, m^N(q)=0, m^(N-1)(q)<>0 |
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312 | it is assumed that m(q) and h are invariant |
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313 | the sum above is divided by h as much as possible |
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314 | NOTE: |
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315 | EXAMPLE: example localInvar; shows an example |
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316 | { |
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317 | if ((der(m,h) !=0) || (der(m,der(m,q)) !=0)) |
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318 | { |
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319 | "the last variable defines not an invariant function "; |
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320 | return(q); |
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321 | } |
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322 | poly inv=p; |
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323 | poly dif= der(m,inv); |
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324 | poly a=der(m,q); |
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325 | poly sgn=-1; |
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326 | poly coeff=sgn*q; |
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327 | int k=1; |
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328 | if (dif==0) |
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329 | { |
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330 | return(inv); |
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331 | } |
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332 | while (dif!=0) |
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333 | { |
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334 | inv=(a*inv)+(coeff*dif); |
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335 | dif=der(m,dif); |
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336 | k=k+1; |
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337 | coeff=q*coeff*sgn/k; |
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338 | } |
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339 | while ((inv!=0) && (inv!=h) &&(subst(inv,h,0)==0)) |
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340 | { |
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341 | inv=inv/h; |
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342 | } |
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343 | return(inv); |
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344 | } |
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345 | example |
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346 | { "EXAMPLE:"; echo = 2; |
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347 | ring q=0,(x,y,z),dp; |
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348 | matrix m[3][1]; |
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349 | m[2,1]=x; |
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350 | m[3,1]=y; |
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351 | poly in=localInvar(m,z,y,x); |
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352 | in; |
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353 | } |
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354 | /////////////////////////////////////////////////////////////////////////////// |
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355 | |
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356 | |
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357 | |
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358 | proc furtherInvar(matrix m, ideal id, ideal karl, poly q) |
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359 | USAGE: furtherInvar(m,id,karl,q); m matrix, id,karl ideal,q poly |
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360 | RETURN: ideal= further invariants of the vectorfield m=Sum m[i,1]*d/dx(i) with respect |
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361 | to id,p,q, i.e. |
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362 | the ideal id contains invariants of m and we are looking for elements |
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363 | in the subring generated by id which are divisible by q |
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364 | it is assumed that m(p) and q are invariant |
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365 | the elements mentioned above are computed and divided by q |
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366 | as much as possible |
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367 | the ideal karl contains all invariants computed yet |
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368 | NOTE: |
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369 | EXAMPLE: example furtherInvar; shows an example |
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370 | { |
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371 | int i; |
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372 | ideal null; |
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373 | int z=size(id); |
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374 | intvec v; |
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375 | def @r=basering; |
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376 | ideal su; |
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377 | for (i=1;i<=z;i++) |
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378 | { |
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379 | su[i]=subst(id[i],q,0); |
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380 | } |
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381 | // defines the map phi : r1 ---> @r defined by |
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382 | // y(i) ---> id[i](q=0) |
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383 | execute "ring r1="+charstr(basering)+",(y(1..z)),dp"; |
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384 | setring @r; |
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385 | map phi=r1,su; |
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386 | setring r1; |
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387 | // computes the kernel of phi |
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388 | execute "ideal ker=preimage(@r,phi,null)"; |
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389 | // defines the map psi : r1 ---> @r defined by y(i) ---> id[i] |
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390 | setring @r; |
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391 | map psi=r1,id; |
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392 | // computes psi(ker(phi)) |
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393 | ideal rel=psi(ker); |
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394 | // devides by the maximal power of q |
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395 | // and tests wether we really obtain invariants |
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396 | for (i=1;i<=size(rel);i++) |
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397 | { |
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398 | while ((rel[i]!=0) && (rel[i]!=q) &&(subst(rel[i],q,0)==0)) |
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399 | { |
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400 | rel[i]=rel[i]/q; |
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401 | if (der(m,rel[i])!=0) |
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402 | { |
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403 | "error in furtherInvar, function not invariant"; |
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404 | rel[i]; |
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405 | } |
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406 | } |
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407 | rel[i]=simplify(rel[i],1); |
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408 | } |
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409 | // test whether some variables occur linearly |
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410 | // and deletes the corresponding invariant function |
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411 | setring r1; |
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412 | int j; |
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413 | for (i=1;i<=size(ker);i=i+1) |
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414 | { |
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415 | for (j=1;j<=z;j++) |
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416 | { |
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417 | if (deg(ker[i]/y(j))==0) |
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418 | { |
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419 | setring @r; |
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420 | rel[i]= completeReduction(rel[i],karl,q); |
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421 | if(rel[i]!=0) |
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422 | { |
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423 | karl[j+1]=rel[i]; |
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424 | rel[i]=0; |
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425 | } |
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426 | setring r1; |
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427 | } |
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428 | } |
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429 | |
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430 | } |
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431 | setring @r; |
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432 | list l=rel+null,karl; |
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433 | return(l); |
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434 | } |
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435 | example |
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436 | { "EXAMPLE:"; echo = 2; |
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437 | ring r=0,(x,y,z,u),dp; |
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438 | matrix m[4][1]; |
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439 | m[2,1]=x; |
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440 | m[3,1]=y; |
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441 | m[4,1]=z; |
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442 | ideal id=localInvar(m,z,y,x),localInvar(m,u,y,x); |
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443 | ideal karl=id,x; |
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444 | list in=furtherInvar(m,id,karl,x); |
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445 | in; |
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446 | } |
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447 | /////////////////////////////////////////////////////////////////////////////// |
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448 | |
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449 | |
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450 | |
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451 | proc invariantRing(matrix m, poly p, poly q,list #) |
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452 | USAGE: invariantRing(m,p,q); m matrix, p,q poly |
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453 | RETURN: ideal= the invariants of the vectorfield m=Sum m[i,1]*d/dx(i) |
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454 | p,q variables with m(p)=q invariant |
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455 | NOTE: |
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456 | EXAMPLE: example furtherInvar; shows an example |
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457 | { |
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458 | ideal j; |
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459 | int i,it; |
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460 | int bou=-1; |
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461 | if(size(#)>0) |
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462 | { |
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463 | bou=#[1]; |
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464 | } |
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465 | int z; |
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466 | ideal karl; |
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467 | ideal k1=1; |
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468 | list k2; |
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469 | // computation of local invariants |
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470 | for (i=1;i<=nvars(basering);i++) |
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471 | { |
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472 | karl=karl+localInvar(m,var(i),p,q); |
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473 | } |
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474 | if(bou==0) |
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475 | { |
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476 | " "; |
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477 | "the local invariants:"; |
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478 | " "; |
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479 | karl; |
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480 | // pause; |
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481 | " "; |
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482 | } |
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483 | // computation of further invariants |
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484 | it=0; |
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485 | while (size(k1)!=0) |
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486 | { |
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487 | // test if the new invariants are already in the ring generated |
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488 | // by the invariants we constructed already |
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489 | it++; |
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490 | karl=sortier(karl); |
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491 | j=q; |
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492 | for (i=1;i<=size(karl);i++) |
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493 | { |
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494 | j=j+ simplify(completeReduction(karl[i],j,q),1); |
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495 | } |
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496 | karl=j; |
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497 | j[1]=0; |
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498 | j=simplify(j,2); |
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499 | k2=furtherInvar(m,j,karl,q); |
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500 | k1=k2[1]; |
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501 | karl=k2[2]; |
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502 | k1=sortier(k1); |
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503 | z=size(k1); |
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504 | for (i=1;i<=z;i++) |
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505 | { |
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506 | k1[i]= completeReduction(k1[i],karl,q); |
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507 | if (k1[i]!=0) |
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508 | { |
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509 | karl=karl+simplify(k1[i],1); |
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510 | } |
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511 | } |
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512 | if(bou==0) |
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513 | { |
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514 | " "; |
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515 | "the invariants after the iteration"; |
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516 | it; |
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517 | " "; |
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518 | karl; |
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519 | // pause; |
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520 | " "; |
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521 | } |
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522 | if((bou>0) && (size(k1)>0)) |
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523 | { |
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524 | if(deg(k1[size(k1)])>bou) |
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525 | { |
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526 | return(karl); |
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527 | } |
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528 | } |
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529 | } |
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530 | return(karl); |
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531 | } |
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532 | example |
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533 | { "EXAMPLE:"; echo = 2; |
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534 | |
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535 | //Winkelmann: free action but Spec k[x(1),...,x(5)]---> Spec In- |
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536 | //variantring is not surjective |
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537 | |
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538 | ring rw=0,(x(1..5)),dp; |
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539 | matrix m[5][1]; |
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540 | m[3,1]=x(1); |
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541 | m[4,1]=x(2); |
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542 | m[5,1]=1+x(1)*x(4)+x(2)*x(3); |
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543 | ideal in=invariantRing(m,x(3),x(1),0); |
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544 | in; |
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545 | |
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546 | //Deveney/Finston: The ring of invariants is not finitely generated |
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547 | |
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548 | ring rf=0,(x(1..7)),dp; |
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549 | matrix m[7][1]; |
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550 | m[4,1]=x(1)^3; |
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551 | m[5,1]=x(2)^3; |
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552 | m[6,1]=x(3)^3; |
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553 | m[7,1]=(x(1)*x(2)*x(3))^2; |
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554 | ideal in=invariantRing(m,x(4),x(1),6); |
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555 | in; |
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556 | |
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557 | |
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558 | //Deveney/Finston:Proper Ga-action which is not locally trivial |
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559 | //r[x(1),...,x(5)] is not flat over the ring of invariants |
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560 | |
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561 | ring rd=0,(x(1..5)),dp; |
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562 | matrix m[5][1]; |
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563 | m[3,1]=x(1); |
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564 | m[4,1]=x(2); |
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565 | m[5,1]=1+x(1)*x(4)^2; |
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566 | ideal in=invariantRing(m,x(3),x(1)); |
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567 | in; |
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568 | |
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569 | actionIsProper(m); |
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570 | |
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571 | //computes the relations between the invariants |
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572 | int z=size(in); |
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573 | ideal null; |
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574 | ring r1=0,(y(1..z)),dp; |
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575 | setring rd; |
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576 | map phi=r1,in; |
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577 | setring r1; |
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578 | ideal ker=preimage(rd,phi,null); |
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579 | ker; |
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580 | |
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581 | //the discriminant |
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582 | |
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583 | ring r=0,(x(1..2),y(1..2),z,t),dp; |
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584 | poly p=z+(1+x(1)*y(2)^2)*t+x(1)*y(1)*y(2)*t^2+(1/3)*x(1)*y(1)^2*t^3; |
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585 | |
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586 | matrix m[5][5]; |
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587 | m[1,1]=z; |
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588 | m[1,2]=x(1)*y(2)^2+1; |
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589 | m[1,3]=x(1)*y(1)*y(2); |
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590 | m[1,4]=1/3*x(1)*y(1)^2; |
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591 | m[1,5]=0; |
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592 | m[2,1]=0; |
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593 | m[2,2]=z; |
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594 | m[2,3]=x(1)*y(2)^2+1; |
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595 | m[2,4]=x(1)*y(1)*y(2); |
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596 | m[2,5]=1/3*x(1)*y(1)^2; |
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597 | m[3,1]=x(1)*y(2)^2+1; |
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598 | m[3,2]=2*x(1)*y(1)*y(2); |
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599 | m[3,3]=x(1)*y(1)^2; |
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600 | m[3,4]=0; |
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601 | m[3,5]=0; |
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602 | m[4,1]=0; |
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603 | m[4,2]=x(1)*y(2)^2+1; |
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604 | m[4,3]=2*x(1)*y(1)*y(2); |
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605 | m[4,4]=x(1)*y(1)^2; |
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606 | m[4,5]=0; |
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607 | m[5,1]=0; |
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608 | m[5,2]=0; |
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609 | m[5,3]=x(1)*y(2)^2+1; |
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610 | m[5,4]=2*x(1)*y(1)*y(2); |
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611 | m[5,5]=x(1)*y(1)^2; |
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612 | |
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613 | poly disc=9*det(m)/(x(1)^2*y(1)^4); |
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614 | |
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615 | LIB "invar.lib"; |
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616 | matrix n[6][1]; |
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617 | n[2,1]=x(1); |
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618 | n[4,1]=y(1); |
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619 | n[5,1]=1+x(1)*y(2)^2; |
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620 | |
---|
621 | der(n,disc); |
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622 | |
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623 | //x(1)^3*y(2)^6-6*x(1)^2*y(1)*y(2)^3*z+6*x(1)^2*y(2)^4+9*x(1)*y(1)^2*z^2-18*x(1)*y(1)*y(2)*z+9*x(1)*y(2)^2+4 |
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624 | |
---|
625 | |
---|
626 | //constructive approach to Weizenbcks theorem |
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627 | |
---|
628 | int n=5; |
---|
629 | |
---|
630 | ring w=0,(x(1..n)),wp(1..n); |
---|
631 | |
---|
632 | // definition of the vectorfield m=sum m[i]*d/dx(i) |
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633 | matrix m[n][1]; |
---|
634 | int i; |
---|
635 | for (i=1;i<=n-1;i=i+1) |
---|
636 | { |
---|
637 | m[i+1,1]=x(i); |
---|
638 | } |
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639 | ideal in=invariantRing(m,x(2),x(1),0); |
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640 | in; |
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641 | |
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642 | |
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643 | |
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644 | } |
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