1 | /////////////////////////////////////////////////////////////////////////////// |
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2 | version="$Id$"; |
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3 | category="real algebra"; |
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4 | info=" |
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5 | LIBRARY: realrad.lib Computation of real radicals |
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6 | AUTHOR : Silke Spang |
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7 | |
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8 | OVERVIEW: |
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9 | Algorithms about the computation of the real |
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10 | radical of an arbitary ideal over the rational numbers |
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11 | and transcendetal extensions thereof |
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12 | |
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13 | PROCEDURES: |
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14 | realpoly(f); Computes the real part of the univariate polynomial f |
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15 | realzero(j); Computes the real radical of the zerodimensional ideal j |
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16 | realrad(j); Computes the real radical of an arbitary ideal over |
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17 | transcendental extension of the rational numbers |
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18 | "; |
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19 | |
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20 | LIB "inout.lib"; |
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21 | LIB "poly.lib"; |
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22 | LIB "matrix.lib"; |
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23 | LIB "general.lib"; |
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24 | LIB "rootsur.lib"; |
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25 | LIB "algebra.lib"; |
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26 | LIB "standard.lib"; |
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27 | LIB "primdec.lib"; |
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28 | LIB "elim.lib"; |
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29 | |
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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 | //// the main procedure ////////////////////////////////////////////////////// |
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35 | ////////////////////////////////////////////////////////////////////////////// |
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36 | proc realrad(ideal id) |
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37 | "USAGE: realrad(id), id an ideal of arbitary dimension |
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38 | RETURN: the real radical of id |
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39 | EXAMPE: example realrad; shows an example" |
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40 | { |
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41 | |
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42 | def r=basering; |
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43 | int n=nvars(basering); |
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44 | // for faster Groebner basis and dimension compuations |
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45 | string neuring ="ring schnell=("+charstr(r)+"),("+varstr(r)+"),dp;"; |
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46 | execute(neuring); |
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47 | def ri=basering; |
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48 | |
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49 | list reddim;//reduct dimension to 0 |
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50 | list lpar,lvar,sub;//for the ringchange |
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51 | string pari,vari; |
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52 | int i,siz,l,j; |
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53 | string less="list lessvar="+varstr(r)+";"; |
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54 | execute(less); |
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55 | ideal id=imap(r,id); |
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56 | l=size(id); |
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57 | for (i=1;i<=l;i++) |
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58 | { |
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59 | id[i]=simplify_gen(id[i]); |
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60 | } |
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61 | id=groebner(id); |
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62 | if (dim(id)<=0) |
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63 | { |
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64 | id=realzero(id); |
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65 | setring r; |
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66 | id=imap(ri,id); |
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67 | return(id); |
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68 | } |
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69 | //sub are the subsets of {x_1,...,x_n} |
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70 | sub=subsets(n); |
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71 | siz=size(sub)-1;//we dont want to localize on all variables |
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72 | |
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73 | //for the empty set |
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74 | reddim[1]=zeroreduct(id); |
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75 | reddim[1]=realzero(reddim[1]); |
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76 | for (i=1;i<=siz;i++) |
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77 | { |
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78 | |
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79 | lvar=lessvar; |
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80 | lpar=list(); |
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81 | l=size(sub[i]); |
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82 | for (j=1;j<=l;j++) |
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83 | { |
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84 | lpar=lpar+list(lvar[sub[i][j]-j+1]); |
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85 | lvar=delete(lvar,sub[i][j]-j+1); |
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86 | } |
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87 | for(j=1;j<=l;j++)//there are l entries in lpar |
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88 | { |
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89 | pari=pari+","+string(lpar[j]); |
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90 | } |
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91 | l=n-l;//there are the remaining n-l entries in lvar |
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92 | for(j=1;j<=l;j++)//there are l entries in lpar |
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93 | { |
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94 | vari=vari+","+string(lvar[j]); |
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95 | } |
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96 | vari=vari[2..size(vari)]; |
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97 | neuring="ring neu=("+charstr(r)+pari+"),("+vari+"),dp;"; |
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98 | execute(neuring); |
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99 | ideal id=imap(r,id); |
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100 | ideal buffer=zeroreduct(id); |
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101 | buffer=realzero(buffer); |
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102 | setring ri; |
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103 | reddim[i+1]=imap(neu,buffer); |
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104 | kill neu; |
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105 | //compute the intersection of buffer with r |
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106 | reddim[i+1]=contnonloc(reddim[i+1],pari,vari); |
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107 | vari=""; |
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108 | pari=""; |
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109 | } |
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110 | id=intersect(reddim[1..(siz+1)]); |
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111 | //id=timeStd(id,301);//simplify the output |
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112 | id=interred(id); // timeStd does not work yet |
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113 | setring r; |
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114 | id=imap(ri,id); |
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115 | return(id); |
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116 | |
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117 | } |
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118 | example |
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119 | { "EXAMPLE:"; echo = 2; |
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120 | ring r1=0,(x,y,z),lp; |
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121 | //dimension 0 |
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122 | ideal i0=(x2+1)*(x3-2),(y3-2)*(y2+y+1),z3+2; |
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123 | //dimension 1 |
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124 | ideal i1=(y3+3y2+y+1)*(y2+4y+4)*(x2+1),(x2+y)*(x2-y2)*(x2+2xy+y2)*(y2+y+1); |
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125 | ideal i=intersect(i0,i1); |
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126 | realrad(i); |
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127 | } |
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128 | |
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129 | |
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130 | /*static*/ proc zeroreduct(ideal i) |
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131 | "USAGE:zeroreduct(i), i an arbitary ideal |
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132 | RETURN: an ideal j of dimension <=0 s.th. i is contained in |
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133 | j and j is contained in i_{Iso} which is the zariski closure |
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134 | of all real isolated points of i |
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135 | " |
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136 | { |
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137 | list equi; |
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138 | int d,n,di; |
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139 | n=nvars(basering); |
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140 | def r=basering; |
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141 | |
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142 | //chance ring to get faster groebner bases computation for dimensions |
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143 | |
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144 | string rneu="ring neu=("+charstr(r)+"),("+varstr(r)+"),dp;"; |
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145 | execute(rneu); |
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146 | ideal i=imap(r,i); |
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147 | |
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148 | i=groebner(i); |
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149 | while (dim(i)> 0) |
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150 | { |
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151 | equi=equidim(i); |
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152 | d=size(equi); |
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153 | equi[d]=radical(equi[d]); |
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154 | di=dim(std(equi[d])); |
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155 | equi[d]=equi[d],minor(jacob(equi[d]),n-di); |
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156 | equi[d]=radical(equi[d]); |
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157 | i=intersect(equi[1..d]); |
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158 | i=groebner(i); |
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159 | } |
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160 | |
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161 | setring r; |
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162 | i=imap(neu,i); |
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163 | //i=timeStd(i,301); |
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164 | i=interred(i); // timeStd does not work yet |
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165 | return(i); |
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166 | } |
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167 | ////////////////////////////////////////////////////////////////////////////// |
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168 | ///////the zero-dimensional case ///////////////////////////////////////////// |
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169 | ////////////////////////////////////////////////////////////////////////////// |
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170 | proc realzero(ideal j) |
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171 | "USAGE: realzero(j); a zero-dimensional ideal j |
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172 | RETURN: j: a zero dimensional ideal, which is the real radical |
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173 | of i, if dim(i)=0 |
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174 | 0: otherwise |
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175 | this acts via |
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176 | primary decomposition (i=1) |
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177 | listdecomp (i=2) or facstd (i=3) |
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178 | EXAMPLE: example realzero; shows an example" |
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179 | |
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180 | |
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181 | { |
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182 | list prim,prepared,nonshape,realu; |
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183 | int r;//counter |
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184 | int l;//number of first polynomial with degree >1 or even |
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185 | l=size(j); |
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186 | for (r=1;r<=l;r++) |
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187 | { |
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188 | j[r]=simplify_gen(j[r]); |
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189 | if (j[r]==1) |
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190 | { |
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191 | return(ideal(1)); |
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192 | } |
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193 | } |
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194 | option(redSB); |
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195 | //j=groebner(j); |
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196 | //special case |
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197 | //if (j==1) |
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198 | //{ |
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199 | // return(j); |
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200 | //} |
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201 | if (nvars(basering)==1) |
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202 | { |
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203 | j=groebner(j); |
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204 | j=realpoly(j[1]); |
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205 | return(j); |
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206 | } |
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207 | |
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208 | |
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209 | //if (dim(j)>0) {return(0);} |
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210 | |
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211 | def r_alt=basering; |
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212 | //store the ring |
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213 | //for a ring chance to the ordering lp; |
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214 | execute("ring r_neu =("+charstr(basering)+"),("+varstr(basering)+"),lp;"); |
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215 | setring r_neu; |
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216 | ideal boeser,max; |
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217 | prepared[1]=ideal(1); |
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218 | ideal j=imap(r_alt,j); |
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219 | //ideal j=fglm(r_alt,j); |
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220 | prim=primdecGTZ(j); |
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221 | for (r=1;r<=size(prim);r++) |
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222 | { |
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223 | max=prim[r][2]; |
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224 | max=groebner(max); |
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225 | realu=prepare_max(max); |
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226 | max=realu[1]; |
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227 | if (max!=1) |
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228 | { |
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229 | if (realu[2]==1) |
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230 | { |
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231 | prepared=insert(prepared,max); |
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232 | } |
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233 | else |
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234 | { |
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235 | nonshape=insert(nonshape,max); |
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236 | } |
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237 | } |
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238 | } |
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239 | j=intersect(prepared[1..size(prepared)]); |
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240 | |
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241 | //use a variable change into general position to obtain |
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242 | //the shape via radzero |
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243 | if (size(nonshape)>0) |
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244 | { |
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245 | boeser=GeneralPos(nonshape); |
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246 | j=intersect(j,boeser); |
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247 | } |
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248 | //j=timeStd(j,301); |
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249 | j=interred(j); // timeStd does not work yet |
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250 | setring r_alt; |
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251 | j=fetch(r_neu,j); |
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252 | return(j); |
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253 | } |
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254 | example |
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255 | { "EXAMPLE:"; echo = 2; |
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256 | //in non parametric fields |
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257 | ring r=0,(x,y),dp; |
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258 | ideal i=(y3+3y2+y+1)*(y2+4y+4)*(x2+1),(x2+y)*(x2-y2)*(x2+2xy+y2)*(y2+y+1); |
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259 | realzero(i); |
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260 | ideal j=(y3+3y2+y+1)*(y2-2y+1),(x2+y)*(x2-y2); |
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261 | realzero(j); |
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262 | |
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263 | //to get every path |
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264 | ring r1=(0,t),(x,y),lp; |
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265 | ideal m1=x2+1-t,y3+t2; |
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266 | ideal m2=x2+t2+1,y2+t; |
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267 | ideal m3=x2+1-t,y2-t; |
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268 | ideal m4=x^2+1+t,y2-t; |
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269 | ideal i=intersect(m1,m2,m3,m4); |
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270 | realzero(i); |
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271 | |
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272 | } |
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273 | |
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274 | static proc GeneralPos(list buffer) |
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275 | "USAGE: GeneralPos(buffer); |
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276 | buffer a list of maximal ideals which failed the prepare_max-test |
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277 | RETURN: j: the intersection of their realradicals |
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278 | EXAMPLE: example radzero; shows no example" |
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279 | { |
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280 | def r=basering; |
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281 | int n,ll; |
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282 | //for the mapping in general position |
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283 | map phi,psi; |
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284 | ideal j; |
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285 | ideal jmap=randomLast(20); |
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286 | string ri; |
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287 | intvec @hilb; |
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288 | ideal trans,transprep;// the transformation ideals |
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289 | int nva=nvars(r); |
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290 | int zz,k,l;//counter |
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291 | poly randp; |
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292 | for (zz=1;zz<nva;zz++) |
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293 | { |
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294 | if (npars(basering)>0) |
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295 | { |
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296 | randp=randp+(random(0,5)*par(1)+random(0,5)*par(1)^2+random(0,5))*var(zz); |
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297 | } |
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298 | else |
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299 | { |
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300 | randp=randp+random(0,5)*var(zz); |
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301 | } |
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302 | } |
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303 | randp=randp+var(nva); |
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304 | |
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305 | //now they are all irreducible in the non univariate case and |
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306 | //real in the univariate case |
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307 | |
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308 | int m=size(buffer); |
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309 | for (l=1;l<=m;l++) |
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310 | { |
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311 | //searching first non univariate polynomial with an even degree |
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312 | //for odd degree we could use the fundamental theorem of algebra and |
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313 | //get real zeros |
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314 | |
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315 | //this will act via a coordinate chance into general position |
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316 | //denote that this random chance doesn't work allways |
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317 | //the ideas for the transformation into general position are |
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318 | //used from the primdec.lib |
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319 | transprep=buffer[l]; |
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320 | if (voice>=10) |
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321 | { |
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322 | jmap[size(jmap)]=randp; |
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323 | } |
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324 | |
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325 | |
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326 | for (k=2;k<=n;k++) |
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327 | { |
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328 | if (ord(buffer[l][k])==1) |
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329 | { |
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330 | for (zz=1;zz<=nva;zz++) |
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331 | { |
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332 | if (lead(buffer[l][k])/var(zz)!=0) |
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333 | { |
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334 | transprep[k]=var(zz); |
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335 | } |
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336 | } |
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337 | jmap[nva]=subst(jmap[nva],lead(buffer[l][k]),0); |
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338 | } |
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339 | } |
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340 | phi =r,jmap; |
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341 | for (k=1;k<=nva;k++) |
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342 | { |
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343 | jmap[k]=-(jmap[k]-2*var(k)); |
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344 | } |
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345 | psi =r,jmap; |
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346 | |
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347 | //coordinate chance |
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348 | trans=phi(transprep); |
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349 | |
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350 | //acting with the chanced ideal |
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351 | |
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352 | trans=groebner(trans); |
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353 | trans[1]=realpoly(trans[1]); |
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354 | |
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355 | //special case |
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356 | if (trans==1) |
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357 | { |
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358 | buffer[l]=trans; |
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359 | } |
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360 | else |
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361 | { |
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362 | ri="ring rhelp=("+charstr(r)+ "),(" +varstr(r)+ ",@t),dp;"; |
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363 | execute(ri); |
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364 | ideal trans=homog(imap(r,trans),@t); |
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365 | |
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366 | ideal trans1=std(trans); |
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367 | @hilb=hilb(trans1,1); |
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368 | ri= "ring rhelp1=(" |
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369 | +charstr(r)+ "),(" +varstr(rhelp)+ "),lp;"; |
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370 | execute(ri); |
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371 | ideal trans=homog(imap(r,trans),@t); |
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372 | kill rhelp; |
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373 | trans=std(trans,@hilb); |
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374 | trans=subst(trans,@t,1);//dehomogenising |
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375 | setring r; |
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376 | trans=imap(rhelp1,trans); |
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377 | kill rhelp1; |
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378 | trans=std(trans); |
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379 | attrib(trans,"isSB",1); |
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380 | |
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381 | trans=realzero(trans); |
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382 | |
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383 | //going back |
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384 | buffer[l]=psi(trans); |
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385 | //buffer[l]=timeStd(buffer[l],301);//timelimit for std computation |
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386 | buffer[l]=interred(buffer[l]);//timeStd does not work yet |
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387 | } |
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388 | } |
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389 | //option(returnSB); |
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390 | j=intersect(buffer[1..m]); |
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391 | return(j); |
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392 | |
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393 | } |
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394 | |
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395 | /*proc minAssReal(ideal i, int erg) |
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396 | { |
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397 | int l,m,d,e,r,fac; |
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398 | ideal buffer,factor; |
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399 | list minreal; |
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400 | l=size(i); |
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401 | for (r=1;r<=l;r++) |
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402 | { |
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403 | i[r]=simplify_gen(i[r]); |
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404 | |
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405 | } |
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406 | |
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407 | list pr=primdecGTZ(i); |
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408 | m=size(pr); |
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409 | for (l=1;l<=m;l++) |
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410 | { |
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411 | d=dim(std(pr[l][2])); |
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412 | buffer=realrad(pr[l][2]); |
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413 | buffer=std(buffer); |
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414 | e=dim(buffer); |
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415 | if (d==e) |
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416 | { |
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417 | minreal=minreal+list(pr[l]); |
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418 | } |
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419 | } |
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420 | if (erg==0) |
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421 | { |
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422 | return(minreal); |
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423 | } |
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424 | else |
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425 | { |
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426 | pr=list(); |
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427 | m=size(minreal); |
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428 | for (l=1;l<=m;l++) |
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429 | { |
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430 | pr=insert(pr,minreal[l][2]); |
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431 | } |
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432 | i=intersect(pr[1..m]); |
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433 | //i=timeStd(i,301); |
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434 | i=interred(i);//timeStd does not work yet |
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435 | list realmin=minreal+list(i); |
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436 | return(realmin); |
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437 | } |
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438 | }*/ |
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439 | ////////////////////////////////////////////////////////////////////////////// |
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440 | ///////the univariate case /////////////////////////////////////////////////// |
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441 | ////////////////////////////////////////////////////////////////////////////// |
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442 | proc realpoly(poly f) |
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443 | "USAGE: realpoly(f); a univariate polynomial f; |
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444 | RETURN: poly f, where f is the real part of the input f |
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445 | EXAMPLE: example realpoly; shows an example" |
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446 | { |
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447 | def r=basering; |
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448 | int tester; |
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449 | if (size(parstr(r))!=0) |
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450 | { |
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451 | string changering="ring rneu=0,("+parstr(r)+","+varstr(r)+"),lp"; |
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452 | execute(changering); |
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453 | poly f=imap(r,f); |
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454 | tester=1; |
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455 | } |
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456 | f=simplify(f,1);//wlog f is monic |
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457 | if (f==1) |
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458 | { |
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459 | setring r; |
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460 | return(f); |
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461 | } |
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462 | ideal j=factorize(f,1);//for getting the squarefree factorization |
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463 | poly erg=1; |
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464 | for (int i=1;i<=size(j);i=i+1) |
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465 | { |
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466 | if (is_real(j[i])==1) {erg=erg*j[i];} |
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467 | //we only need real primes |
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468 | } |
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469 | if (tester==1) |
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470 | { |
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471 | setring(r); |
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472 | poly erg=imap(rneu,erg); |
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473 | } |
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474 | return(erg); |
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475 | } |
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476 | example |
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477 | { "EXAMPLE:"; echo = 2; |
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478 | ring r1 = 0,x,dp; |
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479 | poly f=x5+16x2+x+1; |
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480 | realpoly(f); |
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481 | realpoly(f*(x4+2)); |
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482 | ring r2=0,(x,y),dp; |
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483 | poly f=x6-3x4y2 + y6 + x2y2 -6y+5; |
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484 | realpoly(f); |
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485 | ring r3=0,(x,y,z),dp; |
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486 | poly f=x4y4-2x5y3z2+x6y2z4+2x2y3z-4x3y2z3+2x4yz5+z2y2-2z4yx+z6x2; |
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487 | realpoly(f); |
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488 | realpoly(f*(x2+y2+1)); |
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489 | } |
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490 | |
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491 | |
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492 | |
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493 | |
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494 | /////////////////////////////////////////////////////////////////////////////// |
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495 | //// for semi-definiteness///////////////////////////////////////////////////// |
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496 | /////////////////////////////////////////////////////////////////////////////// |
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497 | proc decision(poly f) |
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498 | " USAGE: decission(f); a multivariate polynomial f in Q[x_1,..,x_n] and lc f=0 |
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499 | RETURN: assume that the basering has a lexicographical ordering, |
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500 | 1 if f is positive semidefinite 0 if f is indefinite |
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501 | EXAMPLE: decision shows an example |
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502 | { |
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503 | string ri,lessvar,parvar,perm; |
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504 | ideal jac; |
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505 | list varlist,buffer,isol,@s,lhelp,lhelp1,lfac,worklist; |
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506 | poly p,g; |
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507 | def rbuffer; |
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508 | def r=basering; |
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509 | //diverse zaehler |
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510 | int @z,zz,count,tester; |
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511 | int n=nvars(r); |
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512 | //specialcases |
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513 | |
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514 | if (leadcoef(f)<0) |
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515 | { |
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516 | return(0); |
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517 | } |
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518 | lfac=factorize(f,2); |
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519 | ideal factor=lfac[1]; |
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520 | intvec @ex=lfac[2]; |
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521 | factor=factor[1]; |
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522 | zz=size(factor); |
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523 | f=1; |
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524 | for (@z=1;@z<=zz;@z++) |
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525 | { |
---|
526 | if ((@ex[@z] mod 2)==1) |
---|
527 | { |
---|
528 | f=f*factor[@z]; |
---|
529 | } |
---|
530 | } |
---|
531 | if (deg(f)<=0) |
---|
532 | { |
---|
533 | if (leadcoef(f)>=0) |
---|
534 | { |
---|
535 | return(1); |
---|
536 | } |
---|
537 | return(0); |
---|
538 | } |
---|
539 | //for recursion |
---|
540 | if (n==1) |
---|
541 | { |
---|
542 | if (sturm(f,-length(f),length(f))==0) |
---|
543 | { |
---|
544 | return(1); |
---|
545 | } |
---|
546 | return(0); |
---|
547 | } |
---|
548 | //search for a p in Q[x_n] such that f is pos. sem. definite |
---|
549 | //if and only if for every isolating setting S={a_1,...,a_r} holds that |
---|
550 | //every f(x_1,..,x_n-1, a_i) is positiv semidefinite |
---|
551 | //recursion of variables |
---|
552 | /////////////////////////////////////////////////////////////////////////// |
---|
553 | /////////////////////////////////////////////////////////////////////////// |
---|
554 | ideal II = maxideal(1); |
---|
555 | varlist = II[1..n-1]; |
---|
556 | lessvar=string(varlist); |
---|
557 | |
---|
558 | parvar=string(var(n)); |
---|
559 | ri="ring r_neu="+charstr(r)+",(@t,"+parvar+","+lessvar+"),dp;"; |
---|
560 | execute(ri); |
---|
561 | poly f=imap(r,f); |
---|
562 | list varlist=imap(r,varlist); |
---|
563 | ideal jac=jacob(@t+f); |
---|
564 | jac=jac[3..(n+1)]; |
---|
565 | ideal eins=std(jac); |
---|
566 | ideal i=@t+f,jac; |
---|
567 | //use Wu method |
---|
568 | if (eins==1) |
---|
569 | { |
---|
570 | zz=0; |
---|
571 | } |
---|
572 | else |
---|
573 | { |
---|
574 | matrix m=char_series(i); |
---|
575 | zz=nrows(m);//number of rows |
---|
576 | } |
---|
577 | poly p=1; |
---|
578 | for (@z=1;@z<=zz;@z++) |
---|
579 | { |
---|
580 | p=p*m[@z,1]; |
---|
581 | } |
---|
582 | //trailing coefficient of p |
---|
583 | p=subst(p,@t,0); |
---|
584 | p=realpoly(p); |
---|
585 | @s=subsets(n-1); |
---|
586 | ideal jacs; |
---|
587 | for (@z=1;@z<=size(@s);@z++) |
---|
588 | { |
---|
589 | perm=""; |
---|
590 | lhelp=list(); |
---|
591 | |
---|
592 | worklist=varlist; |
---|
593 | buffer=jac[1..(n-1)]; |
---|
594 | //vorbereitungen fuer den Ringwechsel |
---|
595 | //setze worklist=x_1,..,x_(n-1) |
---|
596 | |
---|
597 | for (zz=1;zz<=size(@s[@z]);zz++) |
---|
598 | { |
---|
599 | buffer =delete(buffer ,@s[@z][zz]-zz+1); |
---|
600 | worklist=delete(worklist,@s[@z][zz]-zz+1); |
---|
601 | lhelp=lhelp+list(string(var(@s[@z][zz]+2))); |
---|
602 | lhelp1=insert(lhelp,string(var(@s[@z][zz]+2))); |
---|
603 | } |
---|
604 | //worklist=(x_1,...,x_n-1)\(x_i1,...,x_ik) |
---|
605 | //lhelp =(x_i1,...,x_ik) |
---|
606 | //buffer=diff(f,x_i) i not in (i1,..,ik); |
---|
607 | |
---|
608 | worklist=list("@t",string(var(2)))+lhelp+worklist; |
---|
609 | for (zz=1;zz<=n+1;zz++) |
---|
610 | { |
---|
611 | perm=perm+","+string(worklist[zz]); |
---|
612 | } |
---|
613 | perm=perm[2..size(perm)]; |
---|
614 | if (size(buffer)!=0) |
---|
615 | { |
---|
616 | jacs=buffer[1..size(buffer)]; |
---|
617 | jacs=@t+f,jacs; |
---|
618 | } |
---|
619 | else |
---|
620 | { |
---|
621 | jacs=@t+f; |
---|
622 | } |
---|
623 | rbuffer=basering; |
---|
624 | //perm=@t,x_n,x_1,..,x_ik,x\(x_i1,..,x_ik) |
---|
625 | ri="ring rh=0,("+perm+"),dp;"; |
---|
626 | execute(ri); |
---|
627 | ideal jacs=imap(rbuffer,jacs); |
---|
628 | poly p=imap(rbuffer,p); |
---|
629 | matrix m=char_series(jacs); |
---|
630 | poly e=1; |
---|
631 | for (count=1;count<=nrows(m);count++) |
---|
632 | { |
---|
633 | e=e*m[count,1]; |
---|
634 | } |
---|
635 | //search for the leading coefficient of e in |
---|
636 | //Q(@t,x_n)[x_@s[@z][1],..,x_@s[@z][size(@s[@z])] |
---|
637 | intmat l[n-1][n-1]; |
---|
638 | for (zz=1;zz<n;zz++) |
---|
639 | { |
---|
640 | l[zz,n-zz]=1; |
---|
641 | } |
---|
642 | ri="ring rcoef="+"(0,@t,"+parvar+"), |
---|
643 | ("+lessvar+"),M(l);"; |
---|
644 | execute(ri); |
---|
645 | kill l; |
---|
646 | poly e=imap(rh,e); |
---|
647 | e=leadcoef(e); |
---|
648 | setring rh; |
---|
649 | e=imap(rcoef,e); |
---|
650 | e=subst(e,@t,0); |
---|
651 | e=realpoly(e); |
---|
652 | p=p*e; |
---|
653 | setring r_neu; |
---|
654 | p=imap(rh,p); |
---|
655 | kill rh,rcoef; |
---|
656 | } |
---|
657 | setring r; |
---|
658 | p=imap(r_neu,p); |
---|
659 | /////////////////////////////////////////////////////////////////////////// |
---|
660 | ///////////found polynomial p ///////////////////////////////////////////// |
---|
661 | /////////////////////////////////////////////////////////////////////////// |
---|
662 | //Compute an isolating set for p |
---|
663 | ri="ring iso="+charstr(r)+","+parvar+",lp;"; |
---|
664 | execute(ri); |
---|
665 | poly p=imap(r,p); |
---|
666 | isol=isolset(p); |
---|
667 | setring r; |
---|
668 | list isol=imap(iso,isol); |
---|
669 | tester=1; |
---|
670 | for (@z=1;@z<=size(isol);@z++) |
---|
671 | { |
---|
672 | ri="ring rless="+charstr(r)+",("+lessvar+"),lp;"; |
---|
673 | g=subst(f,var(n),isol[@z]); |
---|
674 | execute(ri); |
---|
675 | poly g=imap(r,g); |
---|
676 | tester=tester*decision(g); |
---|
677 | setring r; |
---|
678 | kill rless; |
---|
679 | } |
---|
680 | return(tester); |
---|
681 | } |
---|
682 | |
---|
683 | |
---|
684 | proc isolset(poly f) |
---|
685 | "USAGE: isolset(f); f a univariate polynomial over the rational numbers |
---|
686 | RETURN: An isolating set of f |
---|
687 | NOTE: algorithm can be found in M-F. Roy,R: Pollack, S. Basu page 373 |
---|
688 | EXAMPLE: example isolset; shows an example" |
---|
689 | { |
---|
690 | int i,case; |
---|
691 | number m; |
---|
692 | list buffer; |
---|
693 | //only real roots count |
---|
694 | f=realpoly(f); |
---|
695 | poly seppart=f; |
---|
696 | seppart=simplify(seppart,1); |
---|
697 | //int N=binlog(length(seppart)); |
---|
698 | //number zweihochN=exp(2,N+1); |
---|
699 | number zweihochN=length(f); |
---|
700 | //a special case |
---|
701 | if (deg(seppart)==0) |
---|
702 | { |
---|
703 | return(list(number(0))); |
---|
704 | } |
---|
705 | if (sturm(seppart,-zweihochN,zweihochN)==1) |
---|
706 | { |
---|
707 | return(list(-zweihochN,zweihochN)); |
---|
708 | } |
---|
709 | //getting bernstein coeffs |
---|
710 | ideal id=isuni(f)-zweihochN; |
---|
711 | map jmap=basering,id; |
---|
712 | seppart=jmap(seppart); |
---|
713 | |
---|
714 | id=2*zweihochN*var(1); |
---|
715 | jmap=basering,id; |
---|
716 | seppart=jmap(seppart); |
---|
717 | |
---|
718 | matrix c=coeffs(seppart,var(1)); |
---|
719 | int s=size(c); |
---|
720 | poly recproc; |
---|
721 | //Reciprocal polynomial |
---|
722 | for (i=1;i<=s;i++) |
---|
723 | { |
---|
724 | recproc=recproc+c[s+1-i,1]*(var(1)^(i-1)); |
---|
725 | } |
---|
726 | jmap=basering,var(1)+1; |
---|
727 | seppart=jmap(recproc); |
---|
728 | list bernsteincoeffs,bern; |
---|
729 | c=coeffs(seppart,var(1)); |
---|
730 | for (i=1;i<=s;i++) |
---|
731 | { |
---|
732 | bern[i]=number(c[s+1-i,1])/binomial(s-1,i-1); |
---|
733 | } |
---|
734 | bernsteincoeffs=bern,list(-zweihochN,zweihochN); |
---|
735 | list POS; |
---|
736 | POS[1]=bernsteincoeffs; |
---|
737 | list L; |
---|
738 | while (size(POS)!=0) |
---|
739 | { |
---|
740 | if (varsigns(POS[1][1])<2) |
---|
741 | { |
---|
742 | case=varsigns(POS[1][1]); |
---|
743 | } |
---|
744 | else |
---|
745 | { |
---|
746 | case=2; |
---|
747 | } |
---|
748 | //case Anweisung |
---|
749 | buffer=POS[1]; |
---|
750 | POS=delete(POS,1); |
---|
751 | while(1) |
---|
752 | { |
---|
753 | if (case==1) |
---|
754 | { |
---|
755 | L=L+buffer[2]; |
---|
756 | break; |
---|
757 | } |
---|
758 | |
---|
759 | if (case==2) |
---|
760 | { |
---|
761 | m=number(buffer[2][1]+buffer[2][2])/2; |
---|
762 | bern=BernsteinCoefficients(buffer[1],buffer[2],m); |
---|
763 | POS=bern+POS; |
---|
764 | if (leadcoef(sign(leadcoef(subst(f,isuni(f),m))))==0) |
---|
765 | { |
---|
766 | number epsilon=1/10; |
---|
767 | while (sturm(f,m-epsilon,m+epsilon)!=1) |
---|
768 | { |
---|
769 | epsilon=epsilon/10; |
---|
770 | } |
---|
771 | L=L+list(m-epsilon,m+epsilon); |
---|
772 | } |
---|
773 | break; |
---|
774 | } |
---|
775 | break; |
---|
776 | } |
---|
777 | } |
---|
778 | i=1; |
---|
779 | while (i<size(L)) |
---|
780 | { |
---|
781 | if (L[i]==L[i+1]) |
---|
782 | { |
---|
783 | L=delete(L,i); |
---|
784 | } |
---|
785 | else |
---|
786 | { |
---|
787 | i=i+1; |
---|
788 | } |
---|
789 | } |
---|
790 | return(L); |
---|
791 | } |
---|
792 | |
---|
793 | static proc BernsteinCoefficients(list bern,list lr,number m) |
---|
794 | "USAGE :BernsteinCoefficients(bern,lr,m); |
---|
795 | a list bern=b_0,...,b_p representing a polynomial P of degree <=p |
---|
796 | in the Bernstein basis pf lr=(l,r) an a number m in Q |
---|
797 | RETURN:a list erg=erg1,erg2 s.th. erg1=erg1[1],erg[2] and erg1[1] are |
---|
798 | the bernstein coefficients of P w.r.t. to erg1[2]=(l,m) and erg2[1] |
---|
799 | is one for erg2[2]=(m,r) |
---|
800 | EXAMPLE: Bernsteincoefficients shows no example |
---|
801 | " |
---|
802 | { |
---|
803 | //Zaehler |
---|
804 | int i,j; |
---|
805 | list erg,erg1,erg2; |
---|
806 | number a=(lr[2]-m)/(lr[2]-lr[1]); |
---|
807 | number b=(m-lr[1])/(lr[2]-lr[1]); |
---|
808 | int p=size(bern); |
---|
809 | list berns,buffer,buffer2; |
---|
810 | berns[1]=bern; |
---|
811 | for (i=2;i<=p;i++) |
---|
812 | { |
---|
813 | for (j=1;j<=p+1-i;j++) |
---|
814 | { |
---|
815 | buffer[j]=a*berns[i-1][j]+b*berns[i-1][j+1]; |
---|
816 | } |
---|
817 | berns[i]=buffer; |
---|
818 | buffer=list(); |
---|
819 | } |
---|
820 | |
---|
821 | for (i=1;i<=p;i++) |
---|
822 | { |
---|
823 | buffer[i]=berns[i][1]; |
---|
824 | buffer2[i]=berns[p+1-i][i]; |
---|
825 | } |
---|
826 | erg1=buffer,list(lr[1],m); |
---|
827 | erg2=buffer2,list(m,lr[2]); |
---|
828 | erg=erg1,erg2; |
---|
829 | return(erg); |
---|
830 | } |
---|
831 | |
---|
832 | static proc binlog(number i) |
---|
833 | { |
---|
834 | int erg; |
---|
835 | if (i<2) {return(0);} |
---|
836 | else |
---|
837 | { |
---|
838 | erg=1+binlog(i/2); |
---|
839 | return(erg); |
---|
840 | } |
---|
841 | } |
---|
842 | |
---|
843 | ////////////////////////////////////////////////////////////////////////////// |
---|
844 | ///////diverse Hilfsprozeduren /////////////////////////////////////////////// |
---|
845 | ////////////////////////////////////////////////////////////////////////////// |
---|
846 | |
---|
847 | ///////////////////////////////////////////////////////////////////////////// |
---|
848 | /////wichtig fuers Verstaendnis////////////////////////////////////////////// |
---|
849 | ///////////////////////////////////////////////////////////////////////////// |
---|
850 | static proc is_real(poly f) |
---|
851 | "USAGE: is_real(f);a univariate irreducible polynomial f; |
---|
852 | RETURN: 1: if f is real |
---|
853 | 0: is f is not real |
---|
854 | EXAMPLE: example is_real; shows an example" |
---|
855 | |
---|
856 | { |
---|
857 | int d,anz,i; |
---|
858 | def r=basering; |
---|
859 | |
---|
860 | if (f==1) {return(1);} |
---|
861 | if (isuniv(f)==0) |
---|
862 | { |
---|
863 | for (i=1;i<=nvars(r);i++) |
---|
864 | { |
---|
865 | d=size(coeffs(f,var(i)))+1; |
---|
866 | if ((d mod 2)==1) |
---|
867 | { |
---|
868 | return(1); |
---|
869 | } |
---|
870 | } |
---|
871 | d=1-decision(f); |
---|
872 | return(d); |
---|
873 | } |
---|
874 | d=deg(f) mod 2; |
---|
875 | if (d==1) |
---|
876 | { |
---|
877 | return(1);//because of fundamental theorem of algebra |
---|
878 | } |
---|
879 | else |
---|
880 | { |
---|
881 | f=simplify(f,1);//wlog we can assume that f is monic |
---|
882 | number a=leadcoef(sign(leadcoef(subst(f,isuni(f),-length(f))))); |
---|
883 | number b=leadcoef(sign(leadcoef(subst(f,isuni(f),length(f))))); |
---|
884 | if |
---|
885 | (a*b!=1) |
---|
886 | //polynomials are contineous so the image is an interval |
---|
887 | //referres to analysis |
---|
888 | { |
---|
889 | return(1); |
---|
890 | } |
---|
891 | else |
---|
892 | { |
---|
893 | anz=sturm(f,-length(f),length(f)); |
---|
894 | if (anz==0) {return(0);} |
---|
895 | else {return(1);} |
---|
896 | } |
---|
897 | } |
---|
898 | } |
---|
899 | example |
---|
900 | { "EXAMPLE:"; echo = 2; |
---|
901 | ring r1 = 0,x,dp; |
---|
902 | poly f=x2+1; |
---|
903 | is_real(f); |
---|
904 | |
---|
905 | } |
---|
906 | |
---|
907 | |
---|
908 | static proc prepare_max(ideal m) |
---|
909 | "USAGE: prepare_max(m); m a maximal ideal in Q(y_1,...,y_m)[x_1,...,x_n] |
---|
910 | RETURN: a list erg=(id,j); where id is the real radical of m if j=1 (i.e. m |
---|
911 | satisfies the shape lemma in one variable x_i) else id=m and j=0; |
---|
912 | EXAMPLE: is_in_shape shows an exmaple; |
---|
913 | " |
---|
914 | |
---|
915 | { |
---|
916 | int j,k,i,l,fakul; |
---|
917 | def r=basering; |
---|
918 | int n=nvars(r); |
---|
919 | list erg,varlist,perm; |
---|
920 | string wechsler,vari; |
---|
921 | //option(redSB); |
---|
922 | |
---|
923 | for (i=1;i<=n;i++) |
---|
924 | { |
---|
925 | varlist=varlist+list(var(i)); |
---|
926 | } |
---|
927 | perm=permutation(varlist); |
---|
928 | fakul=size(perm); |
---|
929 | for (i=1;i<=fakul;i++) |
---|
930 | { |
---|
931 | for (j=1;j<=n;j++) |
---|
932 | { |
---|
933 | vari=vari+","+string(perm[i][j]); |
---|
934 | } |
---|
935 | vari=vari[2..size(vari)]; |
---|
936 | wechsler="ring r_neu=("+charstr(r)+"),("+vari+"),lp;"; |
---|
937 | execute(wechsler); |
---|
938 | ideal id=imap(r,m); |
---|
939 | id=groebner(id); |
---|
940 | k=search_first(id,2,2); |
---|
941 | setring r; |
---|
942 | m=imap(r_neu,id); |
---|
943 | m[1]=realpoly(m[1]); |
---|
944 | if (m[1]==1) |
---|
945 | { |
---|
946 | erg[1]=ideal(1); |
---|
947 | erg[2]=1; |
---|
948 | return(erg); |
---|
949 | } |
---|
950 | if (k>n) |
---|
951 | { |
---|
952 | erg[1]=m; |
---|
953 | erg[2]=1; |
---|
954 | return(erg); |
---|
955 | } |
---|
956 | else |
---|
957 | { |
---|
958 | for (l=k;l<=n;l++) |
---|
959 | { |
---|
960 | if (realpoly(m[l])==1) |
---|
961 | { |
---|
962 | erg[1]=ideal(1); |
---|
963 | erg[2]=1; |
---|
964 | return(erg); |
---|
965 | } |
---|
966 | } |
---|
967 | } |
---|
968 | vari=""; |
---|
969 | kill r_neu; |
---|
970 | } |
---|
971 | if (size(parstr(r))==0) |
---|
972 | { |
---|
973 | erg[1]=m; |
---|
974 | j=1; |
---|
975 | for (i=1;i<=n;i++) |
---|
976 | { |
---|
977 | j=j*isuniv(m[i]); |
---|
978 | } |
---|
979 | erg[2]=j; |
---|
980 | return(erg); |
---|
981 | } |
---|
982 | erg[1]=m; |
---|
983 | erg[2]=0; |
---|
984 | return(erg); |
---|
985 | } |
---|
986 | |
---|
987 | static proc length(poly f) |
---|
988 | "USAGE: length(f); poly f; |
---|
989 | RETURN: sum of the absolute Value of all coeffients of an irreducible |
---|
990 | poly nomial f |
---|
991 | EXAMPLE: example length; shows an example" |
---|
992 | |
---|
993 | { |
---|
994 | number erg,buffer; |
---|
995 | f=simplify(f,1);//wlog f is monic |
---|
996 | int n=size(f); |
---|
997 | for (int i=1;i<=n;i=i+1) |
---|
998 | { |
---|
999 | buffer= leadcoef(f[i]); |
---|
1000 | erg=erg + absValue(buffer); |
---|
1001 | } |
---|
1002 | |
---|
1003 | return(erg); |
---|
1004 | } |
---|
1005 | example |
---|
1006 | { "EXAMPLE:"; echo = 2; |
---|
1007 | ring r1 = 0,x,dp; |
---|
1008 | poly f=x4-6x3+x2+1; |
---|
1009 | norm(f); |
---|
1010 | |
---|
1011 | ring r2=0,(x,y),dp; |
---|
1012 | poly g=x2-y3; |
---|
1013 | length(g); |
---|
1014 | |
---|
1015 | } |
---|
1016 | ////////////////////////////////////////////////////////////////////////////// |
---|
1017 | //////////////weniger wichtig fuers Verstaendnis////////////////////////////// |
---|
1018 | ////////////////////////////////////////////////////////////////////////////// |
---|
1019 | static proc isuniv(poly f) |
---|
1020 | { |
---|
1021 | int erg; |
---|
1022 | if (f==0) |
---|
1023 | { |
---|
1024 | erg=1; |
---|
1025 | } |
---|
1026 | else |
---|
1027 | { |
---|
1028 | erg=(isuni(f)!=0); |
---|
1029 | } |
---|
1030 | return(erg); |
---|
1031 | } |
---|
1032 | static proc search_first(ideal j,int start, int i) |
---|
1033 | "USAGE: searchfirst(j, start, i); |
---|
1034 | id a reduced groebner basis w.r.t. lex |
---|
1035 | RETURN: if i=1 then turns the number of the first non univariate entry |
---|
1036 | with order >1 in its leading term after start |
---|
1037 | else the first non univariate of even order |
---|
1038 | EXAMPLE: example norm; shows no example" |
---|
1039 | { |
---|
1040 | int n=size(j); |
---|
1041 | int k=start;//counter |
---|
1042 | j=j,0; |
---|
1043 | if (i==1) |
---|
1044 | { |
---|
1045 | while |
---|
1046 | ((k<=n)&&(ord(j[k])==1)) |
---|
1047 | { |
---|
1048 | k=k+1; |
---|
1049 | } |
---|
1050 | } |
---|
1051 | else |
---|
1052 | { |
---|
1053 | while |
---|
1054 | ((k<=n)&&(ord(j[k]) mod 2==1)) |
---|
1055 | { |
---|
1056 | k=k+1; |
---|
1057 | } |
---|
1058 | |
---|
1059 | } |
---|
1060 | return(k); |
---|
1061 | } |
---|
1062 | |
---|
1063 | static proc subsets(int n) |
---|
1064 | "USAGE :subsets(n); n>=0 in Z |
---|
1065 | RETURN :l a list of all non-empty subsets of {1,..,n} |
---|
1066 | EXAMPLE:subsets(n) shows an example; |
---|
1067 | " |
---|
1068 | { |
---|
1069 | list l,buffer; |
---|
1070 | int i,j,binzahl; |
---|
1071 | if (n<=0) |
---|
1072 | { |
---|
1073 | return(l); |
---|
1074 | } |
---|
1075 | int grenze=2**n-1; |
---|
1076 | for (i=1;i<=grenze;i++) |
---|
1077 | { |
---|
1078 | binzahl=i; |
---|
1079 | for (j=1;j<=n;j++) |
---|
1080 | { |
---|
1081 | if ((binzahl mod 2)==1) |
---|
1082 | { |
---|
1083 | buffer=buffer+list(j); |
---|
1084 | } |
---|
1085 | binzahl=binzahl div 2; |
---|
1086 | } |
---|
1087 | l[i]=buffer; |
---|
1088 | buffer=list(); |
---|
1089 | } |
---|
1090 | return(l); |
---|
1091 | } |
---|
1092 | example |
---|
1093 | { "EXAMPLE:"; echo = 2; |
---|
1094 | subsets(3); |
---|
1095 | subsets(4); |
---|
1096 | } |
---|
1097 | |
---|
1098 | proc permutation(list L) |
---|
1099 | " USAGE: permutation(L); L a list |
---|
1100 | OUTPUT: a list of all permutation lists of L |
---|
1101 | EXAMPLE: permutation(L) gives an example" |
---|
1102 | { |
---|
1103 | list erg,buffer,permi,einfueger; |
---|
1104 | int i,j,l; |
---|
1105 | int n=size(L); |
---|
1106 | if (n==0) |
---|
1107 | { |
---|
1108 | return(erg); |
---|
1109 | } |
---|
1110 | if (n==1) |
---|
1111 | { |
---|
1112 | erg=list(L); |
---|
1113 | return(erg); |
---|
1114 | } |
---|
1115 | for (i=1;i<=n;i++) |
---|
1116 | { |
---|
1117 | buffer=delete(L,i); |
---|
1118 | einfueger=permutation(buffer); |
---|
1119 | l=size(einfueger); |
---|
1120 | for (j=1;j<=l;j++) |
---|
1121 | { |
---|
1122 | permi=list(L[i])+einfueger[j]; |
---|
1123 | erg=insert(erg,permi); |
---|
1124 | } |
---|
1125 | } |
---|
1126 | return(erg); |
---|
1127 | } |
---|
1128 | example |
---|
1129 | { "EXAMPLE:"; echo = 2; |
---|
1130 | list L1="Just","an","example"; |
---|
1131 | permutation(L1); |
---|
1132 | list L2=1,2,3,4; |
---|
1133 | permutation(L2); |
---|
1134 | } |
---|
1135 | static proc simplify_gen(poly f) |
---|
1136 | "USAGE : simplify_gen(f); f a polymimial in Q(y_1,..,y_m)[x_1,..,x_n] |
---|
1137 | RETURN : a polynomial g such that g is the square-free part of f and |
---|
1138 | every real univariate factor of f is cancelled out |
---|
1139 | EXAMPLE:simplify_gen gives no example" |
---|
1140 | { |
---|
1141 | int i,l; |
---|
1142 | ideal factor; |
---|
1143 | poly g=1; |
---|
1144 | factor=factorize(f,2)[1]; |
---|
1145 | l=size(factor); |
---|
1146 | for (i=1;i<=l;i++) |
---|
1147 | { |
---|
1148 | if (isuniv(factor[i])) |
---|
1149 | { |
---|
1150 | g=g*realpoly(factor[i]); |
---|
1151 | } |
---|
1152 | else |
---|
1153 | { |
---|
1154 | g=g*factor[i]; |
---|
1155 | } |
---|
1156 | } |
---|
1157 | return(g); |
---|
1158 | } |
---|
1159 | static proc contnonloc(ideal id,string pari, string vari) |
---|
1160 | "INPUT : a radical ideal id in in F[pari+vari] which is radical in |
---|
1161 | F(pari)[vari), pari and vari strings of variables |
---|
1162 | OUTPUT : the contraction ideal of id, i.e. idF(pari)[vari]\cap F[pari+vari] |
---|
1163 | EXAMPLE: contnonloc shows an example |
---|
1164 | " |
---|
1165 | { |
---|
1166 | list pr; |
---|
1167 | list contractpr; |
---|
1168 | int i,l,tester; |
---|
1169 | ideal primcomp; |
---|
1170 | def r=basering; |
---|
1171 | string neu="ring r_neu=("+charstr(r)+pari+"),("+vari+"),dp;"; |
---|
1172 | execute(neu); |
---|
1173 | def r1=basering; |
---|
1174 | ideal buffer; |
---|
1175 | setring r; |
---|
1176 | pr=primdecGTZ(id); |
---|
1177 | l=size(pr); |
---|
1178 | contractpr[1]=ideal(1); |
---|
1179 | for (i=1;i<=l;i++) |
---|
1180 | { |
---|
1181 | primcomp=pr[i][2]; |
---|
1182 | setring r1; |
---|
1183 | buffer=imap(r,primcomp); |
---|
1184 | buffer=groebner(buffer); |
---|
1185 | if (buffer==1) |
---|
1186 | { |
---|
1187 | tester=0; |
---|
1188 | } |
---|
1189 | else |
---|
1190 | { |
---|
1191 | tester=1; |
---|
1192 | } |
---|
1193 | setring r; |
---|
1194 | |
---|
1195 | //id only consits of non units in F(pari) |
---|
1196 | if (tester==1) |
---|
1197 | { |
---|
1198 | contractpr=insert(contractpr,primcomp); |
---|
1199 | } |
---|
1200 | } |
---|
1201 | l=size(contractpr); |
---|
1202 | id=intersect(contractpr[1..l]); |
---|
1203 | return(id); |
---|
1204 | } |
---|
1205 | example |
---|
1206 | { "EXAMPLE:"; echo = 2; |
---|
1207 | ring r = 0,(a,b,c),lp; |
---|
1208 | ideal i=b3+c5,ab2+c3; |
---|
1209 | ideal j=contnonloc(i,",b","a,c"); |
---|
1210 | j; |
---|
1211 | } |
---|