1 | // $Id: poly.lib,v 1.2 1997-04-28 19:27:22 obachman Exp $ |
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2 | //system("random",787422842); |
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3 | //(GMG, last modified 22.06.96) |
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4 | /////////////////////////////////////////////////////////////////////////////// |
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5 | |
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6 | LIBRARY: poly.lib PROCEDURES FOR MANIPULATING POLYS, IDEALS, MODULES |
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7 | |
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8 | cyclic(int); ideal of cyclic n-roots |
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9 | freerank(poly/...) rank of coker(input) if coker is free else -1 |
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10 | is_homog(poly/...); int, =1 resp. =0 if input is homogeneous resp. not |
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11 | is_zero(poly/...); int, =1 resp. =0 if coker(input) is 0 resp. not |
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12 | maxcoef(poly/...); maximal length of coefficient occuring in poly/... |
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13 | maxdeg(poly/...); int/intmat = degree/s of terms of maximal order |
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14 | maxdeg1(poly/...); int = [weighted] maximal degree of input |
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15 | mindeg(poly/...); int/intmat = degree/s of terms of minimal order |
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16 | mindeg1(poly/...); int = [weighted] minimal degree of input |
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17 | normalize(poly/...); normalize poly/... such that leading coefficient is 1 |
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18 | (parameters in square brackets [] are optional) |
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19 | |
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20 | LIB "general.lib"; |
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21 | /////////////////////////////////////////////////////////////////////////////// |
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22 | |
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23 | proc cyclic (int n) |
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24 | USAGE: cyclic(n); n integer |
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25 | RETURN: ideal of cyclic n-roots from 1-st n variables of basering |
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26 | EXAMPLE: example cyclic; shows examples |
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27 | { |
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28 | //----------------------------- procedure body -------------------------------- |
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29 | ideal m = maxideal(1); |
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30 | m = m[1..n],m[1..n]; |
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31 | int i,j; |
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32 | ideal s; poly t; |
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33 | for ( j=0; j<=n-2; j=j+1 ) |
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34 | { |
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35 | t=0; |
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36 | for( i=1;i<=n;i=i+1 ) { t=t+product(m,i..i+j); } |
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37 | s=s+t; |
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38 | } |
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39 | s=s,product(m,1..n)-1; |
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40 | return (s); |
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41 | } |
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42 | //-------------------------------- examples ----------------------------------- |
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43 | example |
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44 | { "EXAMPLE:"; echo = 2; |
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45 | ring r=0,(u,v,w,x,y,z),lp; |
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46 | cyclic(nvars(basering)); |
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47 | homog(cyclic(5),z); |
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48 | } |
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49 | /////////////////////////////////////////////////////////////////////////////// |
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50 | |
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51 | proc freerank |
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52 | USAGE: freerank(M[,any]); M=poly/ideal/vector/module/matrix |
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53 | COMPUTE: rank of module presented by M in case it is free. By definition this |
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54 | is vdim(coker(M)/m*coker(M)) if coker(M) is free, where m = maximal |
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55 | ideal of basering and M is considered as matrix (the 0-module is |
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56 | free of rank 0) |
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57 | RETURN: rank of coker(M) if coker(M) is free and -1 else; |
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58 | in case of a second argument return a list: |
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59 | L[1] = rank of coker(M) or -1 |
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60 | L[2] = minbase(M) |
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61 | NOTE: freerank(syz(M)); computes the rank of M if M is free (and -1 else) |
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62 | //* Zur Zeit noch ein Bug, da erste Bettizahl falsch berechnet wird: |
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63 | //betti(0) ist -1 statt 0 |
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64 | EXAMPLE: example freerank; shows examples |
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65 | { |
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66 | int rk; |
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67 | def M = simplify(#[1],10); |
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68 | list mre = mres(M,2); |
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69 | intmat B = betti(mre); |
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70 | if ( ncols(B)>1 ) { rk = -1; } |
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71 | else { rk = sum(B[1..nrows(B),1]); } |
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72 | if (size(#) == 2) { list L=rk,mre[1]; return(L);} |
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73 | return(rk); |
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74 | } |
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75 | example |
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76 | {"EXAMPLE"; echo=2; |
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77 | ring r; |
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78 | ideal i=x; |
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79 | module M=[x,0,1],[-x,0,-1]; |
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80 | freerank(M); // should be -1, coker(M) is not free |
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81 | // [1] should be 1, coker(syz(M))=M is free of rank 1 |
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82 | freerank(syz (M),""); // [2] should be gen(2)+gen(1) (minimal relation of M) |
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83 | freerank(i); |
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84 | freerank(syz(i)); //* bug, should be 1, coker(syz(i))=i is free of rank 1 |
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85 | } |
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86 | /////////////////////////////////////////////////////////////////////////////// |
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87 | |
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88 | proc is_homog (id) |
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89 | USAGE: is_homog(id); id poly/ideal/vector/module/matrix |
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90 | RETURN: integer which is 1 if input is homogeneous (resp. weighted homogeneous |
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91 | if the monomial ordering consists of one block of type ws,Ws,wp or Wp, |
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92 | assuming that all weights are positive) and 0 otherwise |
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93 | NOTE: A vector is homogeneous, if the components are homogeneous of same |
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94 | degree, a module/matrix is homogeneous if all column vectors are |
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95 | homogeneous |
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96 | //*** ergaenzen, wenn Matrizen-Spalten Gewichte haben |
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97 | EXAMPLE: example is_homog; shows examples |
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98 | { |
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99 | //----------------------------- procedure body -------------------------------- |
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100 | module M = module(matrix(id)); |
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101 | M = simplify(M,2); // remove 0-columns |
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102 | intvec v = ringweights(basering); |
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103 | int i,j=1,1; |
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104 | for (i=1; i<=ncols(M); i=i+1) |
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105 | { |
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106 | if( M[i]!=jet(M[i],deg(lead(M[i])),v)-jet(M[i],deg(lead(M[i]))-1,v)) |
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107 | { return(0); } |
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108 | } |
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109 | return(1); |
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110 | } |
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111 | //-------------------------------- examples ----------------------------------- |
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112 | example |
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113 | { "EXAMPLE:"; echo = 2; |
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114 | ring r = 0,(x,y,z),wp(1,2,3); |
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115 | is_homog(x5-yz+y3); |
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116 | ideal i = x6+y3+z2, x9-z3; |
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117 | is_homog(i); |
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118 | ring s = 0,(a,b,c),ds; |
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119 | vector v = [a2,0,ac+bc]; |
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120 | vector w = [a3,b3,c4]; |
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121 | is_homog(v); |
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122 | is_homog(w); |
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123 | } |
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124 | /////////////////////////////////////////////////////////////////////////////// |
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125 | |
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126 | proc is_zero |
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127 | USAGE: is_zero(M[,any]); M=poly/ideal/vector/module/matrix |
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128 | RETURN: integer, 1 if coker(M)=0 resp. 0 if coker(M)!=0, where M is considered |
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129 | as matrix |
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130 | if a second argument is given, return a list: |
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131 | L[1] = 1 if coker(M)=0 resp. 0 if coker(M)!=0 |
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132 | L[2] = dim(M) |
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133 | EXAMPLE: example is_zero; shows examples |
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134 | { |
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135 | int d=dim(std(#[1])); |
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136 | int a = ( d==-1 ); |
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137 | if( size(#) >1 ) { list L=a,d; return(L); } |
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138 | return(a); |
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139 | } |
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140 | example |
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141 | { "EXAMPLE:"; echo=2; |
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142 | ring r; |
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143 | module m = [x],[y],[1,z]; |
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144 | is_zero(m,1); |
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145 | qring q = std(ideal(x2+y3+z2)); |
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146 | ideal j = x2+y3+z2-37; |
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147 | is_zero(j); |
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148 | } |
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149 | //////////////////////////////////////////////////////////////////////////////// |
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150 | |
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151 | proc maxcoef (f) |
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152 | USAGE: maxcoef(f); f poly/ideal/vector/module/matrix |
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153 | RETURN: maximal length of coefficient of f of type int (by counting the |
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154 | length of the string of each coefficient) |
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155 | EXAMPLE: example maxcoef; shows examples |
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156 | { |
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157 | //----------------------------- procedure body -------------------------------- |
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158 | int max,s,ii,jj; string t; |
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159 | ideal i = ideal(matrix(f)); |
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160 | i = simplify(i,6); // delete 0's and keep first of equal elements |
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161 | poly m = var(1); matrix C; |
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162 | for (ii=2;ii<=nvars(basering);ii=ii+1) { m = m*var(ii); } |
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163 | for (ii=1; ii<=size(i); ii=ii+1) |
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164 | { |
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165 | C = coef(i[ii],m); |
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166 | for (jj=1; jj<=ncols(C); jj=jj+1) |
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167 | { |
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168 | t = string(C[2,jj]); s = size(t); |
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169 | if ( t[1] == "-" ) { s = s - 1; } |
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170 | if ( s > max ) { max = s; } |
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171 | } |
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172 | } |
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173 | return(max); |
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174 | } |
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175 | //-------------------------------- examples ----------------------------------- |
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176 | example |
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177 | { "EXAMPLE:"; echo = 2; |
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178 | ring r= 0,(x,y,z),ds; |
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179 | poly g = 345x2-1234567890y+7/4z; |
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180 | maxcoef(g); |
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181 | ideal i = g,10/1234567890; |
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182 | maxcoef(i); |
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183 | // since i[2]=1/123456789 |
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184 | } |
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185 | /////////////////////////////////////////////////////////////////////////////// |
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186 | |
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187 | proc maxdeg (id) |
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188 | USAGE: maxdeg(id); id poly/ideal/vector/module/matrix |
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189 | RETURN: int/intmat, each component equals maximal degree of monomials in the |
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190 | corresponding component of id, independent of ring ordering |
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191 | (maxdeg of each var is 1) |
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192 | of type int if id is of type poly, of type intmat else |
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193 | NOTE: proc maxdeg1 returns 1 integer, the absolut maximum; moreover, it has |
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194 | an option for computing weighted degrees |
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195 | EXAMPLE: example maxdeg; shows examples |
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196 | { |
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197 | //-------- subprocedure to find maximal degree of given component ---------- |
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198 | proc findmaxdeg |
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199 | { |
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200 | poly c = #[1]; |
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201 | if (c==0) { return(-1); } |
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202 | //--- guess upper 'o' and lower 'u' bound, in case of negative weights ----- |
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203 | int d = (deg(c)>=0)*deg(c)-(deg(c)<0)*deg(c); |
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204 | int i = d; |
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205 | while ( c-jet(c,i) != 0 ) { i = 2*(i+1); } |
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206 | int o = i-1; |
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207 | int u = (d != i)*((i/ 2)-1); |
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208 | //----------------------- "quick search" for maxdeg ------------------------ |
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209 | while ( (c-jet(c,i)==0)*(c-jet(c,i-1)!=0) == 0) |
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210 | { |
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211 | i = (o+1+u)/ 2; |
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212 | if (c-jet(c,i)!=0) { u = i+1; } |
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213 | else { o = i-1; } |
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214 | } |
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215 | return(i); |
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216 | } |
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217 | //------------------------------ main program --------------------------------- |
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218 | matrix M = matrix(id); |
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219 | int r,c = nrows(M), ncols(M); int i,j; |
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220 | intmat m[r][c]; |
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221 | for (i=r; i>0; i=i-1) |
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222 | { |
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223 | for (j=c; j>0; j=j-1) { m[i,j] = findmaxdeg(M[i,j]); } |
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224 | } |
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225 | if (typeof(id)=="poly") { return(m[1,1]); } |
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226 | return(m); |
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227 | } |
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228 | //-------------------------------- examples ----------------------------------- |
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229 | example |
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230 | { "EXAMPLE:"; echo = 2; |
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231 | ring r = 0,(x,y,z),wp(-1,-2,-3); |
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232 | poly f = x+y2+z3; |
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233 | deg(f); //deg; returns weighted degree (in case of 1 block)! |
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234 | maxdeg(f); |
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235 | matrix m[2][2]=f+x10,1,0,f^2; |
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236 | maxdeg(m); |
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237 | } |
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238 | /////////////////////////////////////////////////////////////////////////////// |
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239 | |
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240 | proc maxdeg1 (id,list #) |
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241 | USAGE: maxdeg1(id[,v]); id=poly/ideal/vector/module/matrix, v=intvec |
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242 | RETURN: integer, maximal [weighted] degree of monomials of id independent of |
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243 | ring ordering, maxdeg1 of i-th variable is v[i] (default: v=1..1). |
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244 | NOTE: This proc returns one integer while maxdeg returns, in general, |
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245 | a matrix of integers. For one polynomial and if no intvec v is given |
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246 | maxdeg is faster |
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247 | EXAMPLE: example maxdeg1; shows examples |
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248 | { |
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249 | //-------- subprocedure to find maximal degree of given component ---------- |
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250 | proc findmaxdeg |
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251 | { |
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252 | poly c = #[1]; |
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253 | if (c==0) { return(-1); } |
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254 | intvec v = #[2]; |
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255 | //--- guess upper 'o' and lower 'u' bound, in case of negative weights ----- |
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256 | int d = (deg(c)>=0)*deg(c)-(deg(c)<0)*deg(c); |
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257 | int i = d; |
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258 | if ( c == jet(c,-1,v)) //case: maxdeg is negative |
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259 | { |
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260 | i = -d; |
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261 | while ( c == jet(c,i,v) ) { i = 2*(i-1); } |
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262 | int o = (d != -i)*((i/ 2)+2) - 1; |
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263 | int u = i+1; |
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264 | int e = -1; |
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265 | } |
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266 | else //case: maxdeg is nonnegative |
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267 | { |
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268 | while ( c != jet(c,i,v) ) { i = 2*(i+1); } |
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269 | int o = i-1; |
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270 | int u = (d != i)*((i/ 2)-1); |
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271 | int e = 1; |
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272 | } |
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273 | //----------------------- "quick search" for maxdeg ------------------------ |
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274 | while ( ( c==jet(c,i,v) )*( c!=jet(c,i-1,v) ) == 0 ) |
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275 | { |
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276 | i = (o+e+u)/ 2; |
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277 | if ( c!=jet(c,i,v) ) { u = i+1; } |
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278 | else { o = i-1; } |
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279 | } |
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280 | return(i); |
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281 | } |
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282 | //------------------------------ main program --------------------------------- |
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283 | ideal M = simplify(ideal(matrix(id)),8); //delete scalar multiples from id |
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284 | int c = ncols(M); |
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285 | int i,n; |
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286 | if( size(#)==0 ) |
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287 | { |
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288 | int m = maxdeg(M[c]); |
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289 | for (i=c-1; i>0; i=i-1) |
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290 | { |
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291 | n = maxdeg(M[i]); |
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292 | m = (m>=n)*m + (m<n)*n; //let m be the maximum of m and n |
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293 | } |
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294 | } |
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295 | else |
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296 | { |
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297 | intvec v=#[1]; //weight vector for the variables |
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298 | int m = findmaxdeg(M[c],v); |
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299 | for (i=c-1; i>0; i--) |
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300 | { |
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301 | n = findmaxdeg(M[i],v); |
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302 | if( n>m ) { m=n; } |
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303 | } |
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304 | } |
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305 | return(m); |
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306 | } |
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307 | //-------------------------------- examples ----------------------------------- |
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308 | example |
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309 | { "EXAMPLE:"; echo = 2; |
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310 | ring r = 0,(x,y,z),wp(-1,-2,-3); |
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311 | poly f = x+y2+z3; |
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312 | deg(f); //deg returns weighted degree (in case of 1 block)! |
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313 | maxdeg1(f); |
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314 | intvec v = ringweights(r); |
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315 | maxdeg1(f,v); //weighted maximal degree |
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316 | matrix m[2][2]=f+x10,1,0,f^2; |
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317 | maxdeg1(m,v); //absolut weighted maximal degree |
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318 | } |
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319 | /////////////////////////////////////////////////////////////////////////////// |
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320 | |
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321 | proc mindeg (id) |
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322 | USAGE: mindeg(id); id poly/ideal/vector/module/matrix |
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323 | RETURN: minimal degree/s of monomials of id, independent of ring ordering |
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324 | (mindeg of each variable is 1) of type int if id of type poly, else |
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325 | of type intmat. |
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326 | NOTE: proc mindeg1 returns one integer, the absolut minimum; moreover it |
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327 | has an option for computing weighted degrees. |
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328 | EXAMPLE: example mindeg; shows examples |
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329 | { |
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330 | //--------- subprocedure to find minimal degree of given component --------- |
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331 | proc findmindeg |
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332 | { |
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333 | poly c = #[1]; |
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334 | if (c==0) { return(-1); } |
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335 | //--- guess upper 'o' and lower 'u' bound, in case of negative weights ----- |
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336 | int d = (ord(c)>=0)*ord(c)-(ord(c)<0)*ord(c); |
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337 | int i = d; |
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338 | while ( jet(c,i) == 0 ) { i = 2*(i+1); } |
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339 | int o = i-1; |
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340 | int u = (d != i)*((i/ 2)-1); |
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341 | //----------------------- "quick search" for mindeg ------------------------ |
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342 | while ( (jet(c,u)==0)*(jet(c,o)!=0) ) |
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343 | { |
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344 | i = (o+u)/ 2; |
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345 | if (jet(c,i)==0) { u = i+1; } |
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346 | else { o = i-1; } |
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347 | } |
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348 | if (jet(c,u)!=0) { return(u); } |
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349 | else { return(o+1); } |
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350 | } |
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351 | //------------------------------ main program --------------------------------- |
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352 | matrix M = matrix(id); |
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353 | int r,c = nrows(M), ncols(M); int i,j; |
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354 | intmat m[r][c]; |
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355 | for (i=r; i>0; i=i-1) |
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356 | { |
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357 | for (j=c; j>0; j=j-1) { m[i,j] = findmindeg(M[i,j]); } |
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358 | } |
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359 | if (typeof(id)=="poly") { return(m[1,1]); } |
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360 | return(m); |
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361 | } |
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362 | //-------------------------------- examples ----------------------------------- |
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363 | example |
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364 | { "EXAMPLE:"; echo = 2; |
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365 | ring r = 0,(x,y,z),ls; |
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366 | poly f = x5+y2+z3; |
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367 | ord(f); // ord returns weighted order of leading term! |
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368 | mindeg(f); // computes minimal degree |
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369 | matrix m[2][2]=x10,1,0,f^2; |
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370 | mindeg(m); // computes matrix of minimum degrees |
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371 | } |
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372 | /////////////////////////////////////////////////////////////////////////////// |
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373 | |
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374 | proc mindeg1 (id, list #) |
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375 | USAGE: mindeg1(id[,v]); id=poly/ideal/vector/module/matrix, v=intvec |
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376 | RETURN: integer, minimal [weighted] degree of monomials of id independent of |
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377 | ring ordering, mindeg1 of i-th variable is v[i] (default v=1..1). |
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378 | NOTE: This proc returns one integer while mindeg returns, in general, |
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379 | a matrix of integers. For one polynomial and if no intvec v is given |
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380 | mindeg is faster. |
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381 | EXAMPLE: example mindeg1; shows examples |
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382 | { |
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383 | //--------- subprocedure to find minimal degree of given component --------- |
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384 | proc findmindeg |
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385 | { |
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386 | poly c = #[1]; |
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387 | intvec v = #[2]; |
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388 | if (c==0) { return(-1); } |
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389 | //--- guess upper 'o' and lower 'u' bound, in case of negative weights ----- |
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390 | int d = (ord(c)>=0)*ord(c)-(ord(c)<0)*ord(c); |
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391 | int i = d; |
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392 | if ( jet(c,-1,v) !=0 ) //case: mindeg is negative |
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393 | { |
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394 | i = -d; |
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395 | while ( jet(c,i,v) != 0 ) { i = 2*(i-1); } |
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396 | int o = (d != -i)*((i/ 2)+2) - 1; |
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397 | int u = i+1; |
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398 | int e = -1; i=u; |
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399 | } |
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400 | else //case: inded is nonnegative |
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401 | { |
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402 | while ( jet(c,i,v) == 0 ) { i = 2*(i+1); } |
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403 | int o = i-1; |
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404 | int u = (d != i)*((i/ 2)-1); |
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405 | int e = 1; i=u; |
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406 | } |
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407 | //----------------------- "quick search" for mindeg ------------------------ |
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408 | while ( (jet(c,i-1,v)==0)*(jet(c,i,v)!=0) == 0 ) |
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409 | { |
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410 | i = (o+e+u)/ 2; |
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411 | if (jet(c,i,v)==0) { u = i+1; } |
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412 | else { o = i-1; } |
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413 | } |
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414 | return(i); |
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415 | } |
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416 | //------------------------------ main program --------------------------------- |
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417 | ideal M = simplify(ideal(matrix(id)),8); //delete scalar multiples from id |
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418 | int c = ncols(M); |
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419 | int i,n; |
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420 | if( size(#)==0 ) |
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421 | { |
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422 | int m = mindeg(M[c]); |
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423 | for (i=c-1; i>0; i=i-1) |
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424 | { |
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425 | n = mindeg(M[i]); |
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426 | m = (m<=n)*m + (m>n)*n; //let m be the maximum of m and n |
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427 | } |
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428 | } |
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429 | else |
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430 | { |
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431 | intvec v=#[1]; //weight vector for the variables |
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432 | int m = findmindeg(M[c],v); |
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433 | for (i=c-1; i>0; i=i-1) |
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434 | { |
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435 | n = findmindeg(M[i],v); |
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436 | m = (m<=n)*m + (m>n)*n; //let m be the maximum of m and n |
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437 | } |
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438 | } |
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439 | return(m); |
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440 | } |
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441 | //-------------------------------- examples ----------------------------------- |
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442 | example |
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443 | { "EXAMPLE:"; echo = 2; |
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444 | ring r = 0,(x,y,z),ls; |
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445 | poly f = x5+y2+z3; |
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446 | ord(f); // ord returns weighted order of leading term! |
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447 | intvec v = 1,-3,2; |
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448 | mindeg1(f,v); // computes minimal weighted degree |
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449 | matrix m[2][2]=x10,1,0,f^2; |
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450 | mindeg1(m,1..3); // computes absolut minimum of weighted degrees |
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451 | } |
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452 | /////////////////////////////////////////////////////////////////////////////// |
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453 | |
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454 | proc normalize (id) |
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455 | USAGE: normalize(id); id=poly/vector/ideal/module |
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456 | RETURN: object of same type with leading coefficient equal to 1 |
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457 | EXAMPLE: example normalize; shows an example |
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458 | { |
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459 | return(simplify(id,1)); |
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460 | } |
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461 | //-------------------------------- examples ----------------------------------- |
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462 | example |
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463 | { "EXAMPLE:"; echo = 2; |
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464 | ring r = 0,(x,y,z),ls; |
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465 | poly f = 2x5+3y2+4z3; |
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466 | normalize(f); |
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467 | module m=[9xy,0,3z3],[4z,6y,2x]; |
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468 | normalize(m); |
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469 | ring s = 0,(x,y,z),(c,ls); |
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470 | module m=[9xy,0,3z3],[4z,6y,2x]; |
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471 | normalize(m); |
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472 | normalize(matrix(m)); // by automatic type conversion to module! |
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473 | } |
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474 | /////////////////////////////////////////////////////////////////////////////// |
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475 | |
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