1 | version="$Id: brnoeth.lib,v 1.12 2001-08-27 14:47:46 Singular Exp $"; |
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2 | category="Miscellaneous"; |
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3 | info=" |
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4 | LIBRARY: brnoeth.lib Brill-Noether Algorithm, Weierstrass-SG and AG-codes |
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5 | AUTHORS: Jose Ignacio Farran Martin, ignfar@eis.uva.es |
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6 | Christoph Lossen, lossen@mathematik.uni-kl.de |
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7 | |
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8 | OVERVIEW: |
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9 | Implementation of the Brill-Noether algorithm for solving the |
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10 | Riemann-Roch problem and applications in Algebraic Geometry codes. |
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11 | The computation of Weierstrass semigroups is also implemented.@* |
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12 | The procedures are intended only for plane (singular) curves defined over |
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13 | a prime field of positive charateristic.@* |
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14 | For more information about the library see the end of the file brnoeth.lib. |
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15 | |
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16 | MAIN PROCEDURES: |
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17 | Adj_div(f); computes the conductor of a curve |
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18 | NSplaces(h,A); computes non-singular places up to given degree |
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19 | BrillNoether(D,C); computes a vector space basis of the linear system L(D) |
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20 | Weierstrass(P,m,C); computes the Weierstrass semigroup of C at P up to m |
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21 | extcurve(d,C); extends the curve C to an extension of degree d |
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22 | AGcode_L(G,D,E); computes the evaluation AG code with divisors G and D |
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23 | AGcode_Omega(G,D,E); computes the residual AG code with divisors G and D |
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24 | prepSV(G,D,F,E); preprocessing for the basic decoding algorithm |
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25 | decodeSV(y,K); decoding of a word with the basic decoding algorithm |
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26 | |
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27 | AUXILIARY PROCEDURES: |
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28 | closed_points(I); computes the zero-set of a zero-dim. ideal in 2 vars |
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29 | dual_code(C); computes the dual code |
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30 | sys_code(C); computes an equivalent systematic code |
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31 | permute_L(L,P); applies a permutation to a list |
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32 | |
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33 | SEE ALSO: hnoether_lib, triang_lib |
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34 | |
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35 | KEYWORDS: Weierstrass semigroup; Algebraic Geometry codes; |
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36 | Brill-Noether algorithm |
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37 | "; |
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38 | |
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39 | LIB "matrix.lib"; |
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40 | LIB "triang.lib"; |
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41 | LIB "hnoether.lib"; |
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42 | LIB "inout.lib"; |
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43 | |
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44 | /////////////////////////////////////////////////////////////////////////////// |
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45 | |
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46 | // ********************************************************** |
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47 | // * POINTS, PLACES AND DIVISORS OF (SINGULAR) PLANE CURVES * |
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48 | // ********************************************************** |
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49 | |
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50 | proc closed_points (ideal I) |
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51 | "USAGE: closed_points(I); I an ideal |
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52 | RETURN: list of prime ideals (each a Groebner basis), corresponding to |
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53 | the (distinct affine closed) points of V(I) |
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54 | NOTE: The ideal must have dimension 0, the basering must have 2 |
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55 | variables, the ordering must be lp, and the base field must |
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56 | be finite and prime.@* |
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57 | It might be convenient to set the option(redSB) in advance. |
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58 | SEE ALSO: triang_lib |
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59 | EXAMPLE: example closed_points; shows an example |
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60 | " |
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61 | { |
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62 | ideal II=std(I); |
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63 | if (II==1) |
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64 | { |
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65 | return(list()); |
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66 | } |
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67 | list TL=triangMH(II); |
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68 | int s=size(TL); |
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69 | list L=list(); |
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70 | int i,j,k; |
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71 | ideal Facts; |
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72 | poly G2; |
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73 | for (i=1;i<=s;i=i+1) |
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74 | { |
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75 | Facts=factorize(TL[i][1],1); |
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76 | k=size(Facts); |
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77 | G2=TL[i][2]; |
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78 | for (j=1;j<=k;j=j+1) |
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79 | { |
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80 | L=L+pd2(Facts[j],G2); |
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81 | } |
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82 | } |
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83 | // eliminate possible repetitions |
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84 | s=size(L); |
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85 | list LP=list(); |
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86 | LP[1]=std(L[1]); |
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87 | int counter=1; |
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88 | for (i=2;i<=s;i=i+1) |
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89 | { |
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90 | if (isPinL(L[i],LP)==0) |
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91 | { |
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92 | counter=counter+1; |
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93 | LP[counter]=std(L[i]); |
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94 | } |
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95 | } |
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96 | return(LP); |
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97 | } |
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98 | example |
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99 | { |
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100 | "EXAMPLE:"; echo = 2; |
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101 | ring s=2,(x,y),lp; |
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102 | // this is just the affine plane over F_4 : |
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103 | ideal I=x4+x,y4+y; |
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104 | list L=closed_points(I); |
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105 | // and here you have all the points : |
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106 | L; |
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107 | } |
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108 | /////////////////////////////////////////////////////////////////////////////// |
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109 | |
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110 | static proc pd2 (poly g1,poly g2) |
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111 | { |
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112 | // If g1,g2 is a std. resp. lex. in (x,y) then the procedure |
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113 | // factorizes g2 in the "extension given by g1" |
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114 | // (then g1 must be irreducible) and returns a list of |
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115 | // ideals with always g1 as first component and the |
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116 | // distinct factors of g2 as second components |
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117 | list L=list(); |
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118 | ideal J=g1; |
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119 | int i,s; |
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120 | if (deg(g1)==1) |
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121 | { |
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122 | poly A=-subst(g1,var(2),0); |
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123 | poly B=subst(g2,var(2),A); |
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124 | ideal facts=factorize(B,1); |
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125 | s=size(facts); |
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126 | for (i=1;i<=s;i=i+1) |
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127 | { |
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128 | J[2]=facts[i]; |
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129 | L[i]=J; |
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130 | } |
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131 | } |
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132 | else |
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133 | { |
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134 | def BR=basering; |
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135 | poly A=g1; |
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136 | poly B=g2; |
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137 | ring raux1=char(basering),(x,y,a),lp; |
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138 | poly G; |
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139 | ring raux2=(char(basering),a),(x,y),lp; |
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140 | map psi=BR,x,a; |
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141 | minpoly=number(psi(A)); |
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142 | poly f=psi(B); |
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143 | ideal facts=factorize(f,1); |
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144 | s=size(facts); |
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145 | poly g; |
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146 | string sg; |
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147 | for (i=1;i<=s;i=i+1) |
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148 | { |
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149 | g=facts[i]; |
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150 | sg=string(g); |
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151 | setring raux1; |
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152 | execute("G="+sg+";"); |
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153 | G=subst(G,a,y); |
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154 | setring BR; |
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155 | map ppssii=raux1,var(1),var(2),0; |
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156 | J[2]=ppssii(G); |
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157 | L[i]=J; |
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158 | kill(ppssii); |
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159 | setring raux2; |
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160 | } |
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161 | setring BR; |
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162 | kill(raux1,raux2); |
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163 | } |
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164 | return(L); |
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165 | } |
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166 | /////////////////////////////////////////////////////////////////////////////// |
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167 | static proc isPinL (ideal P,list L) |
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168 | { |
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169 | // checks if a (plane) point P is in a list of (plane) points L |
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170 | // by just comparing generators |
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171 | // it works only if all (prime) ideals are given in a "canonical way", |
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172 | // namely: |
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173 | // the first generator is monic and irreducible, |
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174 | // and depends only on the second variable, |
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175 | // and the second one is monic in the first variable |
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176 | // and irreducible over the field extension determined by |
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177 | // the second variable and the first generator as minpoly |
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178 | int s=size(L); |
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179 | int i; |
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180 | for (i=1;i<=s;i=i+1) |
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181 | { |
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182 | if ( P[1]==L[i][1] && P[2]==L[i][2] ) |
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183 | { |
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184 | return(1); |
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185 | } |
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186 | } |
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187 | return(0); |
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188 | } |
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189 | /////////////////////////////////////////////////////////////////////////////// |
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190 | static proc s_locus (poly f) |
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191 | { |
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192 | // computes : ideal of affine singular locus |
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193 | // the equation f must be affine |
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194 | // warning : if there is an error message then the output is "none" |
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195 | // option(redSB) is convenient to be set in advance |
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196 | ideal I=f,jacob(f); |
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197 | I=std(I); |
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198 | if (dim(I)>0) |
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199 | { |
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200 | // dimension check (has to be 0) |
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201 | ERROR("something was wrong; possibly non-reduced curve"); |
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202 | } |
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203 | else |
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204 | { |
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205 | return(I); |
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206 | } |
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207 | } |
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208 | /////////////////////////////////////////////////////////////////////////////// |
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209 | static proc curve (poly f) |
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210 | "USAGE: curve(f), where f is a polynomial (affine or projective) |
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211 | CREATE: poly CHI in both rings aff_r=p,(x,y),lp and Proj_R=p,(x,y,z),lp |
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212 | also ideal (std) Aff_SLocus of affine singular locus in the ring |
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213 | aff_r |
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214 | RETURN: list (size 3) with two rings aff_r,Proj_R and an integer deg(f) |
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215 | NOTE: f must be absolutely irreducible, but this is not checked |
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216 | it is not implemented yet for extensions of prime fields |
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217 | " |
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218 | { |
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219 | def base_r=basering; |
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220 | ring aff_r=char(basering),(x,y),lp; |
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221 | ring Proj_R=char(basering),(x,y,z),lp; |
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222 | setring base_r; |
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223 | int degX=deg(f); |
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224 | if (nvars(basering)==2) |
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225 | { |
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226 | setring aff_r; |
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227 | map embpol=base_r,x,y; |
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228 | poly CHI=embpol(f); |
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229 | export(CHI); |
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230 | kill(embpol); |
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231 | ideal Aff_SLocus=s_locus(CHI); |
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232 | export(Aff_SLocus); |
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233 | setring Proj_R; |
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234 | poly CHI=homog(imap(aff_r,CHI),z); |
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235 | export(CHI); |
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236 | setring base_r; |
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237 | list L=list(); |
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238 | L[1]=aff_r; |
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239 | L[2]=Proj_R; |
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240 | L[3]=degX; |
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241 | kill(aff_r,Proj_R); |
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242 | return(L); |
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243 | } |
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244 | if (nvars(basering)==3) |
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245 | { |
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246 | setring Proj_R; |
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247 | map embpol=base_r,x,y,z; |
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248 | poly CHI=embpol(f); |
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249 | export(CHI); |
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250 | kill(embpol); |
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251 | string s=string(subst(CHI,z,1)); |
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252 | setring aff_r; |
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253 | execute("poly CHI="+s+";"); |
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254 | export(CHI); |
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255 | ideal Aff_SLocus=s_locus(CHI); |
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256 | export(Aff_SLocus); |
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257 | setring base_r; |
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258 | list L=list(); |
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259 | L[1]=aff_r; |
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260 | L[2]=Proj_R; |
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261 | L[3]=degX; |
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262 | kill(aff_r,Proj_R); |
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263 | return(L); |
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264 | } |
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265 | ERROR("basering must have 2 or 3 variables"); |
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266 | } |
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267 | /////////////////////////////////////////////////////////////////////////////// |
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268 | static proc Aff_SL (ideal ISL) |
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269 | { |
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270 | // computes : affine singular (closed) points as a list of lists of |
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271 | // prime ideals and intvec (for storing the places over each point) |
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272 | // the ideal ISL=s_locus(CHI) is assumed to be computed in advance for |
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273 | // a plane curve CHI, and it must be given by a standard basis |
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274 | // for our purpose the function must called with the "global" ideal |
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275 | // "Aff_SLocus" |
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276 | list SL=list(); |
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277 | ideal I=ISL; |
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278 | if ( I != 1 ) |
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279 | { |
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280 | list L=list(); |
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281 | ideal aux; |
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282 | intvec iv; |
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283 | int i,s; |
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284 | L=closed_points(I); |
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285 | s=size(L); |
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286 | for (i=1;i<=s;i=i+1) |
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287 | { |
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288 | aux=std(L[i]); |
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289 | SL[i]=list(aux,iv); |
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290 | } |
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291 | } |
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292 | return(SL); |
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293 | } |
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294 | /////////////////////////////////////////////////////////////////////////////// |
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295 | static proc inf_P (poly f) |
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296 | { |
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297 | // computes : all (closed) points at infinity as homogeneous polynomials |
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298 | // output : two lists with respectively singular and non-singular points |
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299 | intvec iv; |
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300 | def base_r=basering; |
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301 | ring r_auxz=char(basering),(x,y,z),lp; |
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302 | poly f=imap(base_r,f); |
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303 | poly F=homog(f,z); // equation of projective curve |
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304 | poly f_inf=subst(F,z,0); |
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305 | setring base_r; |
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306 | poly f_inf=imap(r_auxz,f_inf); |
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307 | ideal I=factorize(f_inf,1); // points at infinity as homogeneous |
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308 | // polynomials |
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309 | int s=size(I); |
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310 | int i; |
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311 | list IP_S=list(); // for singular points at infinity |
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312 | list IP_NS=list(); // for non-singular points at infinity |
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313 | int counter_S; |
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314 | int counter_NS; |
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315 | poly aux; |
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316 | for (i=1;i<=s;i=i+1) |
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317 | { |
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318 | aux=subst(I[i],y,1); |
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319 | if (aux==1) |
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320 | { |
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321 | // the point is (1:0:0) |
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322 | setring r_auxz; |
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323 | poly f_yz=subst(F,x,1); |
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324 | if ( subst(subst(diff(f_yz,y),y,0),z,0)==0 && |
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325 | subst(subst(diff(f_yz,z),y,0),z,0)==0 ) |
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326 | { |
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327 | // the point is singular |
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328 | counter_S=counter_S+1; |
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329 | kill(f_yz); |
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330 | setring base_r; |
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331 | IP_S[counter_S]=list(I[i],iv); |
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332 | } |
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333 | else |
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334 | { |
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335 | // the point is non-singular |
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336 | counter_NS=counter_NS+1; |
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337 | kill(f_yz); |
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338 | setring base_r; |
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339 | IP_NS[counter_NS]=list(I[i],iv); |
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340 | } |
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341 | } |
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342 | else |
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343 | { |
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344 | // the point is (a:1:0) | a is root of aux |
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345 | if (deg(aux)==1) |
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346 | { |
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347 | // the point is rational and no field extension is needed |
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348 | setring r_auxz; |
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349 | poly f_xz=subst(F,y,1); |
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350 | poly aux=imap(base_r,aux); |
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351 | number A=-number(subst(aux,x,0)); |
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352 | map phi=r_auxz,x+A,0,z; |
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353 | poly f_origin=phi(f_xz); |
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354 | if ( subst(subst(diff(f_origin,x),x,0),z,0)==0 && |
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355 | subst(subst(diff(f_origin,z),x,0),z,0)==0 ) |
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356 | { |
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357 | // the point is singular |
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358 | counter_S=counter_S+1; |
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359 | kill(f_xz,aux,A,phi,f_origin); |
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360 | setring base_r; |
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361 | IP_S[counter_S]=list(I[i],iv); |
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362 | } |
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363 | else |
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364 | { |
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365 | // the point is non-singular |
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366 | counter_NS=counter_NS+1; |
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367 | kill(f_xz,aux,A,phi,f_origin); |
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368 | setring base_r; |
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369 | IP_NS[counter_NS]=list(I[i],iv); |
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370 | } |
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371 | } |
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372 | else |
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373 | { |
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374 | // the point is non-rational and a field extension with minpoly=aux |
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375 | // is needed |
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376 | ring r_ext=(char(basering),a),(x,y,z),lp; |
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377 | poly F=imap(r_auxz,F); |
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378 | poly f_xz=subst(F,y,1); |
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379 | poly aux=imap(base_r,aux); |
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380 | minpoly=number(subst(aux,x,a)); |
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381 | map phi=r_ext,x+a,0,z; |
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382 | poly f_origin=phi(f_xz); |
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383 | if ( subst(subst(diff(f_origin,x),x,0),z,0)==0 && |
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384 | subst(subst(diff(f_origin,z),x,0),z,0)==0 ) |
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385 | { |
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386 | // the point is singular |
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387 | counter_S=counter_S+1; |
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388 | setring base_r; |
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389 | kill(r_ext); |
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390 | IP_S[counter_S]=list(I[i],iv); |
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391 | } |
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392 | else |
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393 | { |
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394 | // the point is non-singular |
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395 | counter_NS=counter_NS+1; |
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396 | setring base_r; |
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397 | kill(r_ext); |
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398 | IP_NS[counter_NS]=list(I[i],iv); |
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399 | } |
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400 | } |
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401 | } |
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402 | } |
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403 | kill(r_auxz); |
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404 | return(list(IP_S,IP_NS)); |
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405 | } |
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406 | /////////////////////////////////////////////////////////////////////////////// |
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407 | static proc closed_points_ext (poly f,int d,ideal SL) |
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408 | { |
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409 | // computes : (closed affine non-singular) points over an extension of |
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410 | // degree d |
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411 | // remark(1) : singular points are supposed to be listed appart |
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412 | // remark(2) : std SL=s_locus(f) is supposed to be computed in advance |
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413 | // remark(3) : ideal SL is used to remove those points which are singular |
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414 | // output : list of list of prime ideals with an intvec for storing the |
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415 | // places |
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416 | int Q=char(basering)^d; // cardinality of the extension field |
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417 | ideal I=f,x^Q-x,y^Q-y; // ideal of the searched points |
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418 | I=std(I); |
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419 | if (I==1) |
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420 | { |
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421 | return(list()); |
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422 | } |
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423 | list LP=list(); |
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424 | int m=size(SL); |
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425 | list L=list(); |
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426 | ideal aux; |
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427 | intvec iv; |
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428 | int i,s,j,counter; |
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429 | L=closed_points(I); |
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430 | s=size(L); |
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431 | for (i=1;i<=s;i=i+1) |
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432 | { |
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433 | aux=std(L[i]); |
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434 | for (j=1;j<=m;j=j+1) |
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435 | { |
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436 | // check if singular i.e. if SL is contained in aux |
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437 | if ( NF(SL[j],aux) != 0 ) |
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438 | { |
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439 | counter=counter+1; |
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440 | LP[counter]=list(aux,iv); |
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441 | break; |
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442 | } |
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443 | } |
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444 | } |
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445 | return(LP); |
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446 | } |
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447 | /////////////////////////////////////////////////////////////////////////////// |
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448 | static proc degree_P (list P) |
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449 | "USAGE: degree_P(P), where P is either a polynomial or an ideal |
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450 | RETURN: integer with the degree of the closed point given by P |
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451 | SEE ALSO: closed_points |
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452 | NOTE: If P is a (homogeneous irreducible) polynomial the point is at |
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453 | infinity, and if P is a (prime) ideal the points is affine, and |
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454 | the ideal must be given by 2 generators: the first one irreducible |
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455 | and depending only on y, and the second one irreducible over the |
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456 | extension given by y with the first generator as minimal polynomial |
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457 | " |
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458 | { |
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459 | // computes : the degree of a given point |
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460 | // remark(1) : if the input is (irreducible homogeneous) poly => the point |
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461 | // is at infinity |
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462 | // remark(2) : it the input is (std. resp. lp. prime) ideal => the point is |
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463 | // affine |
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464 | if (typeof(P[1])=="ideal") |
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465 | { |
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466 | if (size(P[1])==2) |
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467 | { |
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468 | int d=deg(P[1][1]); |
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469 | poly aux=subst(P[1][2],y,1); |
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470 | d=d*deg(aux); |
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471 | return(d); |
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472 | } |
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473 | else |
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474 | { |
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475 | // this should not happen in principle |
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476 | ERROR("non-valid parameter"); |
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477 | } |
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478 | } |
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479 | else |
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480 | { |
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481 | if (typeof(P[1])=="poly") |
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482 | { |
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483 | return(deg(P[1])); |
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484 | } |
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485 | else |
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486 | { |
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487 | ERROR("parameter must have a poly or ideal in the first component"); |
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488 | } |
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489 | } |
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490 | } |
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491 | /////////////////////////////////////////////////////////////////////////////// |
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492 | static proc closed_points_deg (poly f,int d,ideal SL) |
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493 | { |
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494 | // computes : (closed affine non-singular) points of degree d |
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495 | // remark(1) : singular points are supposed to be listed appart |
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496 | // remark(2) : std SL=s_locus(f) is supposed to be computed in advance |
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497 | list L=closed_points_ext(f,d,SL); |
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498 | int s=size(L); |
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499 | int i,counter; |
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500 | list LP=list(); |
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501 | for (i=1;i<=s;i=i+1) |
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502 | { |
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503 | if (degree_P(L[i])==d) |
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504 | { |
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505 | counter=counter+1; |
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506 | LP[counter]=L[i]; |
---|
507 | } |
---|
508 | } |
---|
509 | return(LP); |
---|
510 | } |
---|
511 | /////////////////////////////////////////////////////////////////////////////// |
---|
512 | static proc subset (ideal I,ideal J) |
---|
513 | { |
---|
514 | // checks wether I is contained in J and returns a boolean |
---|
515 | // remark : J is assumed to be given by a standard basis |
---|
516 | int s=size(I); |
---|
517 | int i; |
---|
518 | for (i=1;i<=s;i=i+1) |
---|
519 | { |
---|
520 | if ( NF(I[i],std(J)) != 0 ) |
---|
521 | { |
---|
522 | return(0); |
---|
523 | } |
---|
524 | } |
---|
525 | return(1); |
---|
526 | } |
---|
527 | /////////////////////////////////////////////////////////////////////////////// |
---|
528 | static proc belongs (list P,ideal I) |
---|
529 | { |
---|
530 | // checks if affine point P is contained in V(I) and returns a boolean |
---|
531 | // remark : P[1] is assumed to be an ideal given by a standard basis |
---|
532 | if (typeof(P[1])=="ideal") |
---|
533 | { |
---|
534 | return(subset(I,P[1])); |
---|
535 | } |
---|
536 | else |
---|
537 | { |
---|
538 | ERROR("first argument must be an affine point"); |
---|
539 | } |
---|
540 | } |
---|
541 | /////////////////////////////////////////////////////////////////////////////// |
---|
542 | static proc equals (ideal I,ideal J) |
---|
543 | { |
---|
544 | // checks if I is equal to J and returns a boolean |
---|
545 | // remark : I and J are assumed to be given by a standard basis |
---|
546 | int answer=0; |
---|
547 | if (subset(I,J)==1) |
---|
548 | { |
---|
549 | if (subset(J,I)==1) |
---|
550 | { |
---|
551 | answer=1; |
---|
552 | } |
---|
553 | } |
---|
554 | return(answer); |
---|
555 | } |
---|
556 | /////////////////////////////////////////////////////////////////////////////// |
---|
557 | static proc isInLP (ideal P,list LP) |
---|
558 | { |
---|
559 | // checks if affine point P is a list LP and returns either its position or |
---|
560 | // zero |
---|
561 | // remark : all points in LP and P itself are assumed to be given by a |
---|
562 | // standard basis |
---|
563 | // warning : the procedure does not check whether the points are affine or |
---|
564 | // not |
---|
565 | int s=size(LP); |
---|
566 | if (s==0) |
---|
567 | { |
---|
568 | return(0); |
---|
569 | } |
---|
570 | int i; |
---|
571 | for (i=1;i<=s;i=i+1) |
---|
572 | { |
---|
573 | if (equals(P,LP[i][1])==1) |
---|
574 | { |
---|
575 | return(i); |
---|
576 | } |
---|
577 | } |
---|
578 | return(0); |
---|
579 | } |
---|
580 | /////////////////////////////////////////////////////////////////////////////// |
---|
581 | static proc res_deg () |
---|
582 | { |
---|
583 | // computes the residual degree of the basering with respect to its prime |
---|
584 | // field |
---|
585 | // warning : minpoly must depend on a parameter called "a" |
---|
586 | int ext; |
---|
587 | string s_m=string(minpoly); |
---|
588 | if (s_m=="0") |
---|
589 | { |
---|
590 | ext=1; |
---|
591 | } |
---|
592 | else |
---|
593 | { |
---|
594 | ring auxr=char(basering),a,lp; |
---|
595 | execute("poly minp="+s_m+";"); |
---|
596 | ext=deg(minp); |
---|
597 | } |
---|
598 | return(ext); |
---|
599 | } |
---|
600 | /////////////////////////////////////////////////////////////////////////////// |
---|
601 | static proc Frobenius (etwas,int r) |
---|
602 | { |
---|
603 | // applies the Frobenius map over F_{p^r} to an object defined over an |
---|
604 | // extension of such field |
---|
605 | // usually it is called with r=1, i.e. the Frobenius map over the prime |
---|
606 | // field F_p |
---|
607 | // returns always an object of the same type, and works correctly on |
---|
608 | // numbers, polynomials, ideals, matrices or lists of the above types |
---|
609 | // maybe : types vector and module should be added in the future, but they |
---|
610 | // are not needed now |
---|
611 | int q=char(basering)^r; |
---|
612 | if (typeof(etwas)=="number") |
---|
613 | { |
---|
614 | return(etwas^q); |
---|
615 | } |
---|
616 | if (typeof(etwas)=="poly") |
---|
617 | { |
---|
618 | int s=size(etwas); |
---|
619 | poly f; |
---|
620 | int i; |
---|
621 | for (i=1;i<=s;i=i+1) |
---|
622 | { |
---|
623 | f=f+(leadcoef(etwas[i])^q)*leadmonom(etwas[i]); |
---|
624 | } |
---|
625 | return(f); |
---|
626 | } |
---|
627 | if (typeof(etwas)=="ideal") |
---|
628 | { |
---|
629 | int s=ncols(etwas); |
---|
630 | ideal I; |
---|
631 | int i; |
---|
632 | for (i=1;i<=s;i=i+1) |
---|
633 | { |
---|
634 | I[i]=Frobenius(etwas[i],r); |
---|
635 | } |
---|
636 | return(I); |
---|
637 | } |
---|
638 | if (typeof(etwas)=="matrix") |
---|
639 | { |
---|
640 | int m=nrows(etwas); |
---|
641 | int n=ncols(etwas); |
---|
642 | matrix A[m][n]; |
---|
643 | int i,j; |
---|
644 | for (i=1;i<=m;i=i+1) |
---|
645 | { |
---|
646 | for (j=1;j<=n;j=j+1) |
---|
647 | { |
---|
648 | A[i,j]=Frobenius(etwas[i,j],r); |
---|
649 | } |
---|
650 | } |
---|
651 | return(A); |
---|
652 | } |
---|
653 | if (typeof(etwas)=="list") |
---|
654 | { |
---|
655 | int s=size(etwas); |
---|
656 | list L=list(); |
---|
657 | int i; |
---|
658 | for (i=1;i<=s;i=i+1) |
---|
659 | { |
---|
660 | if (typeof(etwas[i])<>"none") |
---|
661 | { |
---|
662 | L[i]=Frobenius(etwas[i],r); |
---|
663 | } |
---|
664 | } |
---|
665 | return(L); |
---|
666 | } |
---|
667 | return(etwas); |
---|
668 | } |
---|
669 | /////////////////////////////////////////////////////////////////////////////// |
---|
670 | static proc conj_b (list L,int r) |
---|
671 | { |
---|
672 | // applies the Frobenius map over F_{p^r} to a list of type HNE defined over |
---|
673 | // a larger extension |
---|
674 | // when r=1 it turns to be the Frobenius map over the prime field F_{p} |
---|
675 | // returns : a list of type HNE which is either conjugate of the input or |
---|
676 | // the same list in case of L being actually defined over the base field |
---|
677 | // F_{p^r} |
---|
678 | int p=char(basering); |
---|
679 | int Q=p^r; |
---|
680 | list LL=list(); |
---|
681 | int m=nrows(L[1]); |
---|
682 | int n=ncols(L[1]); |
---|
683 | matrix A[m][n]; |
---|
684 | poly f; |
---|
685 | poly aux; |
---|
686 | int i,j; |
---|
687 | for (i=1;i<=m;i=i+1) |
---|
688 | { |
---|
689 | for (j=1;j<=n;j=j+1) |
---|
690 | { |
---|
691 | aux=L[1][i,j]; |
---|
692 | if (aux<>x) |
---|
693 | { |
---|
694 | A[i,j]=aux^Q; |
---|
695 | } |
---|
696 | else |
---|
697 | { |
---|
698 | A[i,j]=aux; |
---|
699 | break; |
---|
700 | } |
---|
701 | } |
---|
702 | } |
---|
703 | m=size(L[4]); |
---|
704 | for (i=1;i<=m;i=i+1) |
---|
705 | { |
---|
706 | f=f+(leadcoef(L[4][i])^Q)*leadmonom(L[4][i]); |
---|
707 | } |
---|
708 | LL[1]=A; |
---|
709 | LL[2]=L[2]; |
---|
710 | LL[3]=L[3]; |
---|
711 | LL[4]=f; |
---|
712 | return(LL); |
---|
713 | } |
---|
714 | /////////////////////////////////////////////////////////////////////////////// |
---|
715 | static proc grad_b (list L,int r) |
---|
716 | { |
---|
717 | // computes the degree of a list of type HNE which is actually defined over |
---|
718 | // F_{p^r} eventhough it is given in an extension of such field |
---|
719 | int gr=1; |
---|
720 | int rd=res_deg() div r; |
---|
721 | list LL=L; |
---|
722 | int i; |
---|
723 | for (i=1;i<=rd;i=i+1) |
---|
724 | { |
---|
725 | LL=conj_b(LL,r); |
---|
726 | if ( LL[1]==L[1] && LL[4]==L[4] ) |
---|
727 | { |
---|
728 | break; |
---|
729 | } |
---|
730 | else |
---|
731 | { |
---|
732 | gr=gr+1; |
---|
733 | } |
---|
734 | } |
---|
735 | return(gr); |
---|
736 | } |
---|
737 | /////////////////////////////////////////////////////////////////////////////// |
---|
738 | static proc conj_bs (list L,int r) |
---|
739 | { |
---|
740 | // computes all the conjugates over F_{p^r} of a list of type HNE defined |
---|
741 | // over an extension |
---|
742 | // returns : a list of lists of type HNE, where the first one is the input |
---|
743 | // list |
---|
744 | // remark : notice that the degree of the branch is then the size of the |
---|
745 | // output |
---|
746 | list branches=list(); |
---|
747 | int gr=1; |
---|
748 | branches[1]=L; |
---|
749 | int rd=res_deg() div r; |
---|
750 | list LL=L; |
---|
751 | int i; |
---|
752 | for (i=1;i<=rd;i=i+1) |
---|
753 | { |
---|
754 | LL=conj_b(LL,r); |
---|
755 | if ( LL[1]==L[1] && LL[4]==L[4] ) |
---|
756 | { |
---|
757 | break; |
---|
758 | } |
---|
759 | else |
---|
760 | { |
---|
761 | gr=gr+1; |
---|
762 | branches[gr]=LL; |
---|
763 | } |
---|
764 | } |
---|
765 | return(branches); |
---|
766 | } |
---|
767 | /////////////////////////////////////////////////////////////////////////////// |
---|
768 | static proc subfield (sf) |
---|
769 | { |
---|
770 | // writes the generator "a" of a subfield of the coefficients field of |
---|
771 | // basering in terms of the the current generator (also called "a") as a |
---|
772 | // string sf is an existing ring whose coefficient field is such a subfield |
---|
773 | // warning : in basering there must be a variable called "x" and subfield |
---|
774 | // must not be prime |
---|
775 | def base_r=basering; |
---|
776 | string new_m=string(minpoly); |
---|
777 | setring sf; |
---|
778 | string old_m=string(minpoly); |
---|
779 | if (old_m==new_m) |
---|
780 | { |
---|
781 | setring base_r; |
---|
782 | return("a"); |
---|
783 | } |
---|
784 | else |
---|
785 | { |
---|
786 | if (old_m<>string(0)) |
---|
787 | { |
---|
788 | ring auxring=char(basering),(a,x),lp; |
---|
789 | execute("poly mpol="+old_m+";"); |
---|
790 | mpol=subst(mpol,a,x); |
---|
791 | setring base_r; |
---|
792 | poly mpol=imap(auxring,mpol); |
---|
793 | kill(auxring); |
---|
794 | string answer="? error : non-primitive element"; |
---|
795 | int r=res_deg(); |
---|
796 | int q=char(basering)^r; |
---|
797 | int i; |
---|
798 | number b; |
---|
799 | for (i=1;i<=q-2;i=i+1) |
---|
800 | { |
---|
801 | b=a^i; |
---|
802 | if (subst(mpol,x,b)==0) |
---|
803 | { |
---|
804 | answer=string(b); |
---|
805 | break; |
---|
806 | } |
---|
807 | } |
---|
808 | if (answer<>"? error : non-primitive element") |
---|
809 | { |
---|
810 | return(answer); |
---|
811 | } |
---|
812 | else |
---|
813 | { |
---|
814 | // list all the elements of the finite field F_q |
---|
815 | int p=char(basering); |
---|
816 | list FF1,FF2; |
---|
817 | for (i=0;i<p;i=i+1) |
---|
818 | { |
---|
819 | FF1[i+1]=number(i); |
---|
820 | } |
---|
821 | int s,j,k; |
---|
822 | for (i=1;i<r;i=i+1) |
---|
823 | { |
---|
824 | s=size(FF1); |
---|
825 | for (j=1;j<=s;j=j+1) |
---|
826 | { |
---|
827 | FF1[j]=FF1[j]*a; |
---|
828 | } |
---|
829 | FF2=FF1; |
---|
830 | for (k=1;k<p;k=k+1) |
---|
831 | { |
---|
832 | for (j=1;j<=s;j=j+1) |
---|
833 | { |
---|
834 | FF1[j]=FF1[j]+number(1); |
---|
835 | } |
---|
836 | FF2=FF2+FF1; |
---|
837 | } |
---|
838 | FF1=FF2; |
---|
839 | FF2=list(); |
---|
840 | } |
---|
841 | kill(FF2); |
---|
842 | // list Fp; |
---|
843 | // ideal facs=factorize(x^(q-1)-1,1); |
---|
844 | // for (i=1;i<=q-1;i=i+1) |
---|
845 | // { |
---|
846 | // Fq[i]=number(subst(facs[i],x,0)); |
---|
847 | // } |
---|
848 | for (i=1;i<=q-1;i=i+1) |
---|
849 | { |
---|
850 | // b=Fq[i]; |
---|
851 | b=FF1[i]; |
---|
852 | if (subst(mpol,x,b)==0) |
---|
853 | { |
---|
854 | answer=string(b); |
---|
855 | break; |
---|
856 | } |
---|
857 | } |
---|
858 | } |
---|
859 | kill(FF1); |
---|
860 | return(answer); |
---|
861 | } |
---|
862 | else |
---|
863 | { |
---|
864 | dbprint(printlevel+1,"warning : minpoly=0 in the subfield; |
---|
865 | you should check that nothing is wrong"); |
---|
866 | return(string(1)); |
---|
867 | } |
---|
868 | } |
---|
869 | } |
---|
870 | /////////////////////////////////////////////////////////////////////////////// |
---|
871 | static proc importdatum (sf,string datum,string rel) |
---|
872 | { |
---|
873 | // fetchs a poly with name "datum" to the current basering from the ring sf |
---|
874 | // such that the generator is given by string "rel" |
---|
875 | // warning : ring sf must have only variables (x,y) and basering must have |
---|
876 | // at least (x,y) |
---|
877 | // warning : the case of minpoly=0 is not regarded; there you can use "imap" |
---|
878 | // instead |
---|
879 | def base_r=basering; |
---|
880 | if (rel=="a") |
---|
881 | { |
---|
882 | setring sf; |
---|
883 | execute("poly pdatum="+datum+";"); |
---|
884 | setring base_r; |
---|
885 | poly pdatum=imap(sf,pdatum); |
---|
886 | return(pdatum); |
---|
887 | } |
---|
888 | else |
---|
889 | { |
---|
890 | setring sf; |
---|
891 | execute("string sdatum=string("+datum+");"); |
---|
892 | ring auxring=char(basering),(a,x,y),lp; |
---|
893 | execute("poly pdatum="+sdatum+";"); |
---|
894 | execute("map phi=basering,"+rel+",x,y;"); |
---|
895 | pdatum=phi(pdatum); |
---|
896 | string snewdatum=string(pdatum); |
---|
897 | setring base_r; |
---|
898 | execute("poly pdatum="+snewdatum+";"); |
---|
899 | kill(auxring); |
---|
900 | return(pdatum); |
---|
901 | } |
---|
902 | } |
---|
903 | /////////////////////////////////////////////////////////////////////////////// |
---|
904 | static proc rationalize (lf,string datum,string rel) |
---|
905 | { |
---|
906 | // fetchs a poly with name "datum" to the current basering from the ring lf |
---|
907 | // and larger coefficients field, where the generator of current ring is |
---|
908 | // given by string "rel" and "datum" is actually defined over the small |
---|
909 | // field |
---|
910 | // warning : "ring lf must have only variables (x,y) and basering must have |
---|
911 | // at least (x,y) |
---|
912 | // warning : the case of minpoly=0 is supposed unnecessary, since then |
---|
913 | // "datum" should be |
---|
914 | // already written only int the right way, i.e. in terms of the prime field |
---|
915 | def base_r=basering; |
---|
916 | if (rel=="a") |
---|
917 | { |
---|
918 | setring lf; |
---|
919 | execute("poly pdatum="+datum+";"); |
---|
920 | setring base_r; |
---|
921 | poly pdatum=imap(lf,pdatum); |
---|
922 | return(pdatum); |
---|
923 | } |
---|
924 | else |
---|
925 | { |
---|
926 | setring lf; |
---|
927 | execute("string sdatum=string("+datum+");"); |
---|
928 | ring auxring=char(basering),(a,b,x,y,t),lp; |
---|
929 | execute("poly pdatum="+sdatum+";"); |
---|
930 | execute("poly prel=b-"+rel+";"); |
---|
931 | ideal I=pdatum,prel; |
---|
932 | I=eliminate(I,a); |
---|
933 | poly newdatum=I[1]; // hopefully it was done correctly and size(I)=1 !!!! |
---|
934 | newdatum=subst(newdatum,b,a); |
---|
935 | string snewdatum=string(newdatum); |
---|
936 | setring base_r; |
---|
937 | execute("poly newdatum="+snewdatum+";"); |
---|
938 | kill(auxring); |
---|
939 | return(newdatum); |
---|
940 | } |
---|
941 | } |
---|
942 | /////////////////////////////////////////////////////////////////////////////// |
---|
943 | static proc place (intvec Pp,int sing,list CURVE) |
---|
944 | { |
---|
945 | // computes the "rational" places which are defined over a (closed) point |
---|
946 | // Pp points to an appropriate point of the given curve |
---|
947 | // creates : local rings (if they do not exist yet) and then add the |
---|
948 | // places to a list |
---|
949 | // each place is given basically by the coordinates of the point and a |
---|
950 | // list HNdevelop |
---|
951 | // returns : list with all updated data of the curve |
---|
952 | // if the places already exist they are not computed again |
---|
953 | // if sing==1 the point is assumed singular and computes the local conductor |
---|
954 | // for all places using the local invariants of the branches |
---|
955 | // if sing==2 the point is assumed singular and computes the local conductor |
---|
956 | // for all places using the Dedekind formula and local parametrizations |
---|
957 | // of the branches |
---|
958 | // if sing<>1&2 the point is assumed non-singular and the local conductor |
---|
959 | // should be zero |
---|
960 | list PP=list(); |
---|
961 | if (Pp[1]==0) |
---|
962 | { |
---|
963 | if (Pp[2]==0) |
---|
964 | { |
---|
965 | PP=Aff_SPoints[Pp[3]]; |
---|
966 | } |
---|
967 | if (Pp[2]==1) |
---|
968 | { |
---|
969 | PP=Inf_Points[1][Pp[3]]; |
---|
970 | } |
---|
971 | if (Pp[2]==2) |
---|
972 | { |
---|
973 | PP=Inf_Points[2][Pp[3]]; |
---|
974 | } |
---|
975 | } |
---|
976 | else |
---|
977 | { |
---|
978 | PP=Aff_Points(Pp[2])[Pp[3]]; |
---|
979 | } |
---|
980 | if (PP[2]<>0) |
---|
981 | { |
---|
982 | return(CURVE); |
---|
983 | } |
---|
984 | intvec PtoPl; |
---|
985 | def base_r=basering; |
---|
986 | int ext1; |
---|
987 | list Places=CURVE[3]; |
---|
988 | intvec Conductor=CURVE[4]; |
---|
989 | list update_CURVE=CURVE; |
---|
990 | if (typeof(PP[1])=="ideal") |
---|
991 | { |
---|
992 | ideal P=PP[1]; |
---|
993 | if (size(P)==2) |
---|
994 | { |
---|
995 | int d=deg(P[1]); |
---|
996 | poly aux=subst(P[2],y,1); |
---|
997 | d=d*deg(aux); |
---|
998 | ext1=d; |
---|
999 | // the point is (A:B:1) but one must distinguish several cases |
---|
1000 | // P is assumed to be a std. resp. "(x,y),lp" and thus P[1] depends |
---|
1001 | // only on "y" |
---|
1002 | if (d==1) |
---|
1003 | { |
---|
1004 | // the point is rational |
---|
1005 | number B=-number(subst(P[1],y,0)); |
---|
1006 | poly aux2=subst(P[2],y,B); |
---|
1007 | number A=-number(subst(aux2,x,0)); |
---|
1008 | // the point is (A:B:1) |
---|
1009 | ring local_aux=char(basering),(x,y),ls; |
---|
1010 | number coord@1=imap(base_r,A); |
---|
1011 | number coord@2=imap(base_r,B); |
---|
1012 | number coord@3=number(1); |
---|
1013 | map phi=base_r,x+coord@1,y+coord@2; |
---|
1014 | poly CHI=phi(CHI); |
---|
1015 | } |
---|
1016 | else |
---|
1017 | { |
---|
1018 | if (deg(P[1])==1) |
---|
1019 | { |
---|
1020 | // the point is non-rational but the second component needs no |
---|
1021 | // field extension |
---|
1022 | number B=-number(subst(P[1],y,0)); |
---|
1023 | poly aux2=subst(P[2],y,B); |
---|
1024 | // the point has degree d>1 |
---|
1025 | // careful : the parameter will be called "a" anyway |
---|
1026 | ring local_aux=(char(basering),a),(x,y),ls; |
---|
1027 | map psi=base_r,a,0; |
---|
1028 | minpoly=number(psi(aux2)); |
---|
1029 | number coord@1=a; |
---|
1030 | number coord@2=imap(base_r,B); |
---|
1031 | number coord@3=number(1); |
---|
1032 | // the point is (a:B:1) |
---|
1033 | map phi=base_r,x+a,y+coord@2; |
---|
1034 | poly CHI=phi(CHI); |
---|
1035 | } |
---|
1036 | else |
---|
1037 | { |
---|
1038 | if (deg(subst(P[2],y,1))==1) |
---|
1039 | { |
---|
1040 | // the point is non-rational but the needed minpoly is just P[1] |
---|
1041 | // careful : the parameter will be called "a" anyway |
---|
1042 | poly P1=P[1]; |
---|
1043 | poly P2=P[2]; |
---|
1044 | ring local_aux=(char(basering),a),(x,y),ls; |
---|
1045 | map psi=base_r,0,a; |
---|
1046 | minpoly=number(psi(P1)); |
---|
1047 | // the point looks like (A:a:1) |
---|
1048 | // A is computed by substituting y=a in P[2] |
---|
1049 | poly aux1=imap(base_r,P2); |
---|
1050 | poly aux2=subst(aux1,y,a); |
---|
1051 | number coord@1=-number(subst(aux2,x,0)); |
---|
1052 | number coord@2=a; |
---|
1053 | number coord@3=number(1); |
---|
1054 | map phi=base_r,x+coord@1,y+a; |
---|
1055 | poly CHI=phi(CHI); |
---|
1056 | } |
---|
1057 | else |
---|
1058 | { |
---|
1059 | // this is the most complicated case of non-rational point |
---|
1060 | // firstly : construct an extension of degree d and guess the |
---|
1061 | // minpoly |
---|
1062 | poly P1=P[1]; |
---|
1063 | poly P2=P[2]; |
---|
1064 | int p=char(basering); |
---|
1065 | int Q=p^d; |
---|
1066 | ring aux_r=(Q,a),(x,y,t),ls; |
---|
1067 | string minpoly_string=string(minpoly); |
---|
1068 | ring local_aux=(char(basering),a),(x,y),ls; |
---|
1069 | execute("minpoly="+minpoly_string+";"); |
---|
1070 | // secondly : compute one root of P[1] |
---|
1071 | poly P_1=imap(base_r,P1); |
---|
1072 | poly P_2=imap(base_r,P2); |
---|
1073 | ideal factors1=factorize(P_1,1); // hopefully this works !!!! |
---|
1074 | number coord@2=-number(subst(factors1[1],y,0)); |
---|
1075 | // thirdly : compute one of the first components for the above root |
---|
1076 | poly P_0=subst(P_2,y,coord@2); |
---|
1077 | ideal factors2=factorize(P_0,1); // hopefully this works !!!! |
---|
1078 | number coord@1=-number(subst(factors2[1],x,0)); |
---|
1079 | number coord@3=number(1); |
---|
1080 | map phi=base_r,x+coord@1,y+coord@2; |
---|
1081 | poly CHI=phi(CHI); |
---|
1082 | kill(aux_r); |
---|
1083 | } |
---|
1084 | } |
---|
1085 | } |
---|
1086 | } |
---|
1087 | else |
---|
1088 | { |
---|
1089 | // this should not happen in principle |
---|
1090 | ERROR("non-valid parameter"); |
---|
1091 | } |
---|
1092 | } |
---|
1093 | else |
---|
1094 | { |
---|
1095 | if (typeof(PP[1])=="poly") |
---|
1096 | { |
---|
1097 | poly P=PP[1]; |
---|
1098 | ring r_auxz=char(basering),(x,y,z),lp; |
---|
1099 | poly CHI=imap(base_r,CHI); |
---|
1100 | CHI=homog(CHI,z); |
---|
1101 | setring base_r; |
---|
1102 | poly aux=subst(P,y,1); |
---|
1103 | if (aux==1) |
---|
1104 | { |
---|
1105 | // the point is (1:0:0) |
---|
1106 | ring local_aux=char(basering),(x,y),ls; |
---|
1107 | number coord@1=number(1); |
---|
1108 | number coord@2=number(0); |
---|
1109 | number coord@3=number(0); |
---|
1110 | map Phi=r_auxz,1,x,y; |
---|
1111 | poly CHI=Phi(CHI); |
---|
1112 | ext1=1; |
---|
1113 | } |
---|
1114 | else |
---|
1115 | { |
---|
1116 | // the point is (A:1:0) where A is a root of aux |
---|
1117 | int d=deg(aux); |
---|
1118 | ext1=d; |
---|
1119 | if (d==1) |
---|
1120 | { |
---|
1121 | // the point is rational |
---|
1122 | number A=-number(subst(aux,x,0)); |
---|
1123 | ring local_aux=char(basering),(x,y),ls; |
---|
1124 | number coord@1=imap(base_r,A); |
---|
1125 | number coord@2=number(1); |
---|
1126 | number coord@3=number(0); |
---|
1127 | map Phi=r_auxz,x+coord@1,1,y; |
---|
1128 | poly CHI=Phi(CHI); |
---|
1129 | } |
---|
1130 | else |
---|
1131 | { |
---|
1132 | // the point has degree d>1 |
---|
1133 | // careful : the parameter will be called "a" anyway |
---|
1134 | ring local_aux=(char(basering),a),(x,y),ls; |
---|
1135 | map psi=base_r,a,1; |
---|
1136 | minpoly=number(psi(P)); |
---|
1137 | number coord@1=a; |
---|
1138 | number coord@2=number(1); |
---|
1139 | number coord@3=number(0); |
---|
1140 | map Phi=r_auxz,x+a,1,y; |
---|
1141 | poly CHI=Phi(CHI); |
---|
1142 | } |
---|
1143 | } |
---|
1144 | kill(r_auxz); |
---|
1145 | } |
---|
1146 | else |
---|
1147 | { |
---|
1148 | ERROR("a point must have a poly or ideal in the first component"); |
---|
1149 | } |
---|
1150 | } |
---|
1151 | export(coord@1); |
---|
1152 | export(coord@2); |
---|
1153 | export(coord@3); |
---|
1154 | export(CHI); |
---|
1155 | int i,j,k; |
---|
1156 | int m,n; |
---|
1157 | list L@HNE=essdevelop(CHI); |
---|
1158 | export(L@HNE); |
---|
1159 | int n_branches=size(L@HNE); |
---|
1160 | list Li_aux=list(); |
---|
1161 | int N_branches; |
---|
1162 | int N=size(Places); |
---|
1163 | if (sing==1) |
---|
1164 | { |
---|
1165 | list delta2=list(); |
---|
1166 | for (i=1;i<=n_branches;i=i+1) |
---|
1167 | { |
---|
1168 | delta2[i]=invariants(L@HNE[i])[5]; |
---|
1169 | } |
---|
1170 | int dq; |
---|
1171 | } |
---|
1172 | int ext2=res_deg(); |
---|
1173 | list dgs=list(); |
---|
1174 | int ext_0; |
---|
1175 | int check; |
---|
1176 | string sss,olda,newa; |
---|
1177 | if (defined(Q)==0) |
---|
1178 | { |
---|
1179 | int Q; |
---|
1180 | } |
---|
1181 | if (ext1==1) |
---|
1182 | { |
---|
1183 | if (ext2==1) |
---|
1184 | { |
---|
1185 | if (sing==1) |
---|
1186 | { |
---|
1187 | intmat I_mult[n_branches][n_branches]; |
---|
1188 | if (n_branches>1) |
---|
1189 | { |
---|
1190 | for (i=1;i<=n_branches-1;i=i+1) |
---|
1191 | { |
---|
1192 | for (j=i+1;j<=n_branches;j=j+1) |
---|
1193 | { |
---|
1194 | I_mult[i,j]=intersection(L@HNE[i],L@HNE[j]); |
---|
1195 | I_mult[j,i]=I_mult[i,j]; |
---|
1196 | } |
---|
1197 | } |
---|
1198 | } |
---|
1199 | } |
---|
1200 | if (size(update_CURVE[5])>0) |
---|
1201 | { |
---|
1202 | if (typeof(update_CURVE[5][1])=="list") |
---|
1203 | { |
---|
1204 | check=1; |
---|
1205 | } |
---|
1206 | } |
---|
1207 | if (check==0) |
---|
1208 | { |
---|
1209 | intvec dgs_points(1); |
---|
1210 | ring S(1)=char(basering),(x,y,t),ls; |
---|
1211 | list BRANCHES=list(); |
---|
1212 | list POINTS=list(); |
---|
1213 | list LOC_EQS=list(); |
---|
1214 | list PARAMETRIZATIONS=list(); |
---|
1215 | export(BRANCHES); |
---|
1216 | export(POINTS); |
---|
1217 | export(LOC_EQS); |
---|
1218 | export(PARAMETRIZATIONS); |
---|
1219 | } |
---|
1220 | else |
---|
1221 | { |
---|
1222 | intvec dgs_points(1)=update_CURVE[5][1][2]; |
---|
1223 | def S1=update_CURVE[5][1][1]; |
---|
1224 | execute("ring S(1)="+string(update_CURVE[5][1][1])+";"); |
---|
1225 | fetchall(S1); |
---|
1226 | kill(S1); |
---|
1227 | } |
---|
1228 | N_branches=size(BRANCHES); |
---|
1229 | for (i=1;i<=n_branches;i=i+1) |
---|
1230 | { |
---|
1231 | dgs_points(1)[N_branches+i]=1; |
---|
1232 | POINTS[N_branches+i]=list(); |
---|
1233 | POINTS[N_branches+i][1]=imap(local_aux,coord@1); |
---|
1234 | POINTS[N_branches+i][2]=imap(local_aux,coord@2); |
---|
1235 | POINTS[N_branches+i][3]=imap(local_aux,coord@3); |
---|
1236 | LOC_EQS[N_branches+i]=imap(local_aux,CHI); |
---|
1237 | setring HNEring; |
---|
1238 | Li_aux=L@HNE[i]; |
---|
1239 | setring S(1); |
---|
1240 | BRANCHES=insert(BRANCHES,imap(HNEring,Li_aux),N_branches+i-1); |
---|
1241 | PARAMETRIZATIONS[N_branches+i]=param(BRANCHES[N_branches+i],0); |
---|
1242 | N=N+1; |
---|
1243 | intvec iw=1,N_branches+i; |
---|
1244 | Places[N]=iw; |
---|
1245 | if (sing==1) |
---|
1246 | { |
---|
1247 | dq=delta2[i]; |
---|
1248 | for (j=1;j<=n_branches;j=j+1) |
---|
1249 | { |
---|
1250 | dq=dq+I_mult[i,j]; |
---|
1251 | } |
---|
1252 | Conductor[N]=dq; |
---|
1253 | } |
---|
1254 | if (sing==2) |
---|
1255 | { |
---|
1256 | Conductor[N]=local_conductor(iw[2],S(1)); |
---|
1257 | } |
---|
1258 | kill(iw); |
---|
1259 | PtoPl[i]=N; |
---|
1260 | } |
---|
1261 | setring base_r; |
---|
1262 | update_CURVE[5][1]=list(); |
---|
1263 | update_CURVE[5][1][1]=S(1); |
---|
1264 | update_CURVE[5][1][2]=dgs_points(1); |
---|
1265 | } |
---|
1266 | else |
---|
1267 | { |
---|
1268 | // we start with a rational point but we get non-rational branches |
---|
1269 | // they may have different degrees and then we may need reduce the |
---|
1270 | // field extensions for each one, and finally check if the minpoly |
---|
1271 | // fetchs with S(i) or not |
---|
1272 | // if one of the branches is rational, we may trust that is is written |
---|
1273 | // correctly |
---|
1274 | if (sing==1) |
---|
1275 | { |
---|
1276 | int n_geobrs; |
---|
1277 | int counter_c; |
---|
1278 | list auxgb=list(); |
---|
1279 | list geobrs=list(); |
---|
1280 | for (i=1;i<=n_branches;i=i+1) |
---|
1281 | { |
---|
1282 | auxgb=conj_bs(L@HNE[i],1); |
---|
1283 | dgs[i]=size(auxgb); |
---|
1284 | n_geobrs=n_geobrs+dgs[i]; |
---|
1285 | for (j=1;j<=dgs[i];j=j+1) |
---|
1286 | { |
---|
1287 | counter_c=counter_c+1; |
---|
1288 | geobrs[counter_c]=auxgb[j]; |
---|
1289 | } |
---|
1290 | } |
---|
1291 | intmat I_mult[n_geobrs][n_geobrs]; |
---|
1292 | for (i=1;i<n_geobrs;i=i+1) |
---|
1293 | { |
---|
1294 | for (j=i+1;j<=n_geobrs;j=j+1) |
---|
1295 | { |
---|
1296 | I_mult[i,j]=intersection(geobrs[i],geobrs[j]); |
---|
1297 | I_mult[j,i]=I_mult[i,j]; |
---|
1298 | } |
---|
1299 | } |
---|
1300 | kill(auxgb,geobrs); |
---|
1301 | } |
---|
1302 | else |
---|
1303 | { |
---|
1304 | for (i=1;i<=n_branches;i=i+1) |
---|
1305 | { |
---|
1306 | dgs[i]=grad_b(L[i],1); |
---|
1307 | } |
---|
1308 | } |
---|
1309 | // the actual degree of each branch is computed and now check if the |
---|
1310 | // local ring exists |
---|
1311 | for (i=1;i<=n_branches;i=i+1) |
---|
1312 | { |
---|
1313 | ext_0=dgs[i]; |
---|
1314 | if (size(update_CURVE[5])>=ext_0) |
---|
1315 | { |
---|
1316 | if (typeof(update_CURVE[5][ext_0])=="list") |
---|
1317 | { |
---|
1318 | check=1; |
---|
1319 | } |
---|
1320 | } |
---|
1321 | if (check==0) |
---|
1322 | { |
---|
1323 | if (ext_0>1) |
---|
1324 | { |
---|
1325 | if (ext_0==ext2) |
---|
1326 | { |
---|
1327 | sss=string(minpoly); |
---|
1328 | } |
---|
1329 | else |
---|
1330 | { |
---|
1331 | Q=char(basering)^ext_0; |
---|
1332 | ring auxxx=(Q,a),z,lp; |
---|
1333 | sss=string(minpoly); |
---|
1334 | setring base_r; |
---|
1335 | kill(auxxx); |
---|
1336 | } |
---|
1337 | ring S(ext_0)=(char(basering),a),(x,y,t),ls; |
---|
1338 | execute("minpoly="+sss+";"); |
---|
1339 | } |
---|
1340 | else |
---|
1341 | { |
---|
1342 | ring S(ext_0)=char(basering),(x,y,t),ls; |
---|
1343 | } |
---|
1344 | intvec dgs_points(ext_0); |
---|
1345 | list BRANCHES=list(); |
---|
1346 | list POINTS=list(); |
---|
1347 | list LOC_EQS=list(); |
---|
1348 | list PARAMETRIZATIONS=list(); |
---|
1349 | export(BRANCHES); |
---|
1350 | export(POINTS); |
---|
1351 | export(LOC_EQS); |
---|
1352 | export(PARAMETRIZATIONS); |
---|
1353 | } |
---|
1354 | else |
---|
1355 | { |
---|
1356 | intvec dgs_points(ext_0)=update_CURVE[5][ext_0][2]; |
---|
1357 | def Sext_0=update_CURVE[5][ext_0][1]; |
---|
1358 | setring Sext_0; |
---|
1359 | string SM=string(minpoly); |
---|
1360 | string SR=string(update_CURVE[5][ext_0][1]); |
---|
1361 | execute("ring S("+string(ext_0)+")="+SR+";"); |
---|
1362 | execute("minpoly="+SM+";"); |
---|
1363 | kill(SM,SR); |
---|
1364 | fetchall(Sext_0); |
---|
1365 | kill(Sext_0); |
---|
1366 | } |
---|
1367 | N_branches=size(BRANCHES); |
---|
1368 | dgs_points(ext_0)[N_branches+1]=1; |
---|
1369 | POINTS[N_branches+1]=list(); |
---|
1370 | POINTS[N_branches+1][1]=imap(local_aux,coord@1); |
---|
1371 | POINTS[N_branches+1][2]=imap(local_aux,coord@2); |
---|
1372 | POINTS[N_branches+1][3]=imap(local_aux,coord@3); |
---|
1373 | LOC_EQS[N_branches+1]=imap(local_aux,CHI); |
---|
1374 | // now fetch the branches into the new local ring |
---|
1375 | if (ext_0==1) |
---|
1376 | { |
---|
1377 | setring HNEring; |
---|
1378 | Li_aux=L@HNE[i]; |
---|
1379 | setring S(1); |
---|
1380 | BRANCHES=insert(BRANCHES,imap(HNEring,Li_aux),N_branches); |
---|
1381 | } |
---|
1382 | else |
---|
1383 | { |
---|
1384 | // rationalize branche |
---|
1385 | setring HNEring; |
---|
1386 | newa=subfield(S(ext_0)); |
---|
1387 | m=nrows(L@HNE[i][1]); |
---|
1388 | n=ncols(L@HNE[i][1]); |
---|
1389 | setring S(ext_0); |
---|
1390 | list Laux=list(); |
---|
1391 | poly paux=rationalize(HNEring,"L@HNE["+string(i)+"][4]",newa); |
---|
1392 | matrix Maux[m][n]; |
---|
1393 | for (j=1;j<=m;j=j+1) |
---|
1394 | { |
---|
1395 | for (k=1;k<=n;k=k+1) |
---|
1396 | { |
---|
1397 | Maux[j,k]=rationalize(HNEring,"L@HNE["+string(i)+"][1]["+ |
---|
1398 | string(j)+","+string(k)+"]",newa); |
---|
1399 | } |
---|
1400 | } |
---|
1401 | setring HNEring; |
---|
1402 | intvec Li2=L@HNE[i][2]; |
---|
1403 | int Li3=L@HNE[i][3]; |
---|
1404 | setring S(ext_0); |
---|
1405 | Laux[1]=Maux; |
---|
1406 | Laux[2]=Li2; |
---|
1407 | Laux[3]=Li3; |
---|
1408 | Laux[4]=paux; |
---|
1409 | BRANCHES=insert(BRANCHES,Laux,N_branches); |
---|
1410 | kill(Laux,Maux,paux,Li2,Li3); |
---|
1411 | } |
---|
1412 | PARAMETRIZATIONS[N_branches+1]=param(BRANCHES[N_branches+1],0); |
---|
1413 | N=N+1; |
---|
1414 | intvec iw=ext_0,N_branches+1; |
---|
1415 | Places[N]=iw; |
---|
1416 | if (sing==2) |
---|
1417 | { |
---|
1418 | Conductor[N]=local_conductor(iw[2],S(ext_0)); |
---|
1419 | } |
---|
1420 | kill(iw); |
---|
1421 | PtoPl[i]=N; |
---|
1422 | setring HNEring; |
---|
1423 | update_CURVE[5][ext_0]=list(); |
---|
1424 | update_CURVE[5][ext_0][1]=S(ext_0); |
---|
1425 | update_CURVE[5][ext_0][2]=dgs_points(ext_0); |
---|
1426 | } |
---|
1427 | if (sing==1) |
---|
1428 | { |
---|
1429 | int N_ini=N-n_branches; |
---|
1430 | counter_c=1; |
---|
1431 | for (i=1;i<=n_branches;i=i+1) |
---|
1432 | { |
---|
1433 | dq=delta2[i]; |
---|
1434 | for (j=1;j<=n_geobrs;j=j+1) |
---|
1435 | { |
---|
1436 | dq=dq+I_mult[counter_c,j]; |
---|
1437 | } |
---|
1438 | Conductor[N_ini+i]=dq; |
---|
1439 | counter_c=counter_c+dgs[i]; |
---|
1440 | } |
---|
1441 | } |
---|
1442 | setring base_r; |
---|
1443 | } |
---|
1444 | } |
---|
1445 | else |
---|
1446 | { |
---|
1447 | if (ext1==ext2) |
---|
1448 | { |
---|
1449 | // the degree of the point equals to the degree of all branches |
---|
1450 | // one must just fetch the minpoly's of local_aux, HNEring and S(ext2) |
---|
1451 | if (sing==1) |
---|
1452 | { |
---|
1453 | intmat I_mult[n_branches][n_branches]; |
---|
1454 | if (n_branches>1) |
---|
1455 | { |
---|
1456 | for (i=1;i<=n_branches-1;i=i+1) |
---|
1457 | { |
---|
1458 | for (j=i+1;j<=n_branches;j=j+1) |
---|
1459 | { |
---|
1460 | I_mult[i,j]=intersection(L@HNE[i],L@HNE[j]); |
---|
1461 | I_mult[j,i]=I_mult[i,j]; |
---|
1462 | } |
---|
1463 | } |
---|
1464 | } |
---|
1465 | } |
---|
1466 | if (size(update_CURVE[5])>=ext2) |
---|
1467 | { |
---|
1468 | if (typeof(update_CURVE[5][ext2])=="list") |
---|
1469 | { |
---|
1470 | check=1; |
---|
1471 | } |
---|
1472 | } |
---|
1473 | if (check==0) |
---|
1474 | { |
---|
1475 | sss=string(minpoly); |
---|
1476 | ring S(ext2)=(char(basering),a),(x,y,t),ls; |
---|
1477 | execute("minpoly="+sss+";"); |
---|
1478 | intvec dgs_points(ext2); |
---|
1479 | list BRANCHES=list(); |
---|
1480 | list POINTS=list(); |
---|
1481 | list LOC_EQS=list(); |
---|
1482 | list PARAMETRIZATIONS=list(); |
---|
1483 | export(BRANCHES); |
---|
1484 | export(POINTS); |
---|
1485 | export(LOC_EQS); |
---|
1486 | export(PARAMETRIZATIONS); |
---|
1487 | } |
---|
1488 | else |
---|
1489 | { |
---|
1490 | intvec dgs_points(ext2)=update_CURVE[5][ext2][2]; |
---|
1491 | def Sext2=update_CURVE[5][ext2][1]; |
---|
1492 | setring Sext2; |
---|
1493 | string SM=string(minpoly); |
---|
1494 | string SR=string(update_CURVE[5][ext2][1]); |
---|
1495 | execute("ring S("+string(ext2)+")="+SR+";"); |
---|
1496 | execute("minpoly="+SM+";"); |
---|
1497 | kill(SM,SR); |
---|
1498 | fetchall(Sext2); |
---|
1499 | kill(Sext2); |
---|
1500 | } |
---|
1501 | N_branches=size(BRANCHES); |
---|
1502 | for (i=1;i<=n_branches;i=i+1) |
---|
1503 | { |
---|
1504 | // fetch all the data into the new local ring |
---|
1505 | olda=subfield(local_aux); |
---|
1506 | dgs_points(ext2)[N_branches+i]=ext1; |
---|
1507 | POINTS[N_branches+i]=list(); |
---|
1508 | POINTS[N_branches+i][1]=number(importdatum(local_aux,"coord@1",olda)); |
---|
1509 | POINTS[N_branches+i][2]=number(importdatum(local_aux,"coord@2",olda)); |
---|
1510 | POINTS[N_branches+i][3]=number(importdatum(local_aux,"coord@3",olda)); |
---|
1511 | LOC_EQS[N_branches+i]=importdatum(local_aux,"CHI",olda); |
---|
1512 | newa=subfield(HNEring); |
---|
1513 | setring HNEring; |
---|
1514 | m=nrows(L@HNE[i][1]); |
---|
1515 | n=ncols(L@HNE[i][1]); |
---|
1516 | setring S(ext2); |
---|
1517 | list Laux=list(); |
---|
1518 | poly paux=importdatum(HNEring,"L@HNE["+string(i)+"][4]",newa); |
---|
1519 | matrix Maux[m][n]; |
---|
1520 | for (j=1;j<=m;j=j+1) |
---|
1521 | { |
---|
1522 | for (k=1;k<=n;k=k+1) |
---|
1523 | { |
---|
1524 | Maux[j,k]=importdatum(HNEring,"L@HNE["+string(i)+"][1]["+ |
---|
1525 | string(j)+","+string(k)+"]",newa); |
---|
1526 | } |
---|
1527 | } |
---|
1528 | setring HNEring; |
---|
1529 | intvec Li2=L@HNE[i][2]; |
---|
1530 | int Li3=L@HNE[i][3]; |
---|
1531 | setring S(ext2); |
---|
1532 | Laux[1]=Maux; |
---|
1533 | Laux[2]=Li2; |
---|
1534 | Laux[3]=Li3; |
---|
1535 | Laux[4]=paux; |
---|
1536 | BRANCHES=insert(BRANCHES,Laux,N_branches+i-1); |
---|
1537 | kill(Laux,Maux,paux,Li2,Li3); |
---|
1538 | PARAMETRIZATIONS[N_branches+i]=param(BRANCHES[N_branches+i],0); |
---|
1539 | N=N+1; |
---|
1540 | intvec iw=ext2,N_branches+i; |
---|
1541 | Places[N]=iw; |
---|
1542 | if (sing==1) |
---|
1543 | { |
---|
1544 | dq=delta2[i]; |
---|
1545 | for (j=1;j<=n_branches;j=j+1) |
---|
1546 | { |
---|
1547 | dq=dq+I_mult[i,j]; |
---|
1548 | } |
---|
1549 | Conductor[N]=dq; |
---|
1550 | } |
---|
1551 | if (sing==2) |
---|
1552 | { |
---|
1553 | Conductor[N]=local_conductor(iw[2],S(ext2)); |
---|
1554 | } |
---|
1555 | kill(iw); |
---|
1556 | PtoPl[i]=N; |
---|
1557 | } |
---|
1558 | setring base_r; |
---|
1559 | update_CURVE[5][ext2]=list(); |
---|
1560 | update_CURVE[5][ext2][1]=S(ext2); |
---|
1561 | update_CURVE[5][ext2][2]=dgs_points(ext2); |
---|
1562 | } |
---|
1563 | else |
---|
1564 | { |
---|
1565 | // this is the most complicated case |
---|
1566 | if (sing==1) |
---|
1567 | { |
---|
1568 | int n_geobrs; |
---|
1569 | int counter_c; |
---|
1570 | list auxgb=list(); |
---|
1571 | list geobrs=list(); |
---|
1572 | for (i=1;i<=n_branches;i=i+1) |
---|
1573 | { |
---|
1574 | auxgb=conj_bs(L@HNE[i],ext1); |
---|
1575 | dgs[i]=size(auxgb); |
---|
1576 | n_geobrs=n_geobrs+dgs[i]; |
---|
1577 | for (j=1;j<=dgs[i];j=j+1) |
---|
1578 | { |
---|
1579 | counter_c=counter_c+1; |
---|
1580 | geobrs[counter_c]=auxgb[j]; |
---|
1581 | } |
---|
1582 | } |
---|
1583 | intmat I_mult[n_geobrs][n_geobrs]; |
---|
1584 | for (i=1;i<n_geobrs;i=i+1) |
---|
1585 | { |
---|
1586 | for (j=i+1;j<=n_geobrs;j=j+1) |
---|
1587 | { |
---|
1588 | I_mult[i,j]=intersection(geobrs[i],geobrs[j]); |
---|
1589 | I_mult[j,i]=I_mult[i,j]; |
---|
1590 | } |
---|
1591 | } |
---|
1592 | kill(auxgb,geobrs); |
---|
1593 | } |
---|
1594 | else |
---|
1595 | { |
---|
1596 | for (i=1;i<=n_branches;i=i+1) |
---|
1597 | { |
---|
1598 | dgs[i]=grad_b(L@HNE[i],ext1); |
---|
1599 | } |
---|
1600 | } |
---|
1601 | for (i=1;i<=n_branches;i=i+1) |
---|
1602 | { |
---|
1603 | // first compute the actual degree of each branch and check if the |
---|
1604 | // local ring exists |
---|
1605 | ext_0=ext1*dgs[i]; |
---|
1606 | if (size(update_CURVE[5])>=ext_0) |
---|
1607 | { |
---|
1608 | if (typeof(update_CURVE[5][ext_0])=="list") |
---|
1609 | { |
---|
1610 | check=1; |
---|
1611 | } |
---|
1612 | } |
---|
1613 | if (check==0) |
---|
1614 | { |
---|
1615 | if (ext_0>ext1) |
---|
1616 | { |
---|
1617 | if (ext_0==ext2) |
---|
1618 | { |
---|
1619 | sss=string(minpoly); |
---|
1620 | } |
---|
1621 | else |
---|
1622 | { |
---|
1623 | Q=char(basering)^ext_0; |
---|
1624 | ring auxxx=(Q,a),z,lp; |
---|
1625 | sss=string(minpoly); |
---|
1626 | setring base_r; |
---|
1627 | kill(auxxx); |
---|
1628 | } |
---|
1629 | } |
---|
1630 | else |
---|
1631 | { |
---|
1632 | setring local_aux; |
---|
1633 | sss=string(minpoly); |
---|
1634 | } |
---|
1635 | ring S(ext_0)=(char(basering),a),(x,y,t),ls; |
---|
1636 | execute("minpoly="+sss+";"); |
---|
1637 | intvec dgs_points(ext_0); |
---|
1638 | list BRANCHES=list(); |
---|
1639 | list POINTS=list(); |
---|
1640 | list LOC_EQS=list(); |
---|
1641 | list PARAMETRIZATIONS=list(); |
---|
1642 | export(BRANCHES); |
---|
1643 | export(POINTS); |
---|
1644 | export(LOC_EQS); |
---|
1645 | export(PARAMETRIZATIONS); |
---|
1646 | } |
---|
1647 | else |
---|
1648 | { |
---|
1649 | intvec dgs_points(ext_0)=update_CURVE[5][ext_0][2]; |
---|
1650 | def Sext_0=update_CURVE[5][ext_0][1]; |
---|
1651 | setring Sext_0; |
---|
1652 | string SM=string(minpoly); |
---|
1653 | string SR=string(update_CURVE[5][ext_0][1]); |
---|
1654 | execute("ring S("+string(ext_0)+")="+SR+";"); |
---|
1655 | execute("minpoly="+SM+";"); |
---|
1656 | kill(SM,SR); |
---|
1657 | fetchall(badring); |
---|
1658 | kill(badring); |
---|
1659 | } |
---|
1660 | N_branches=size(BRANCHES); |
---|
1661 | // now fetch all the data into the new local ring |
---|
1662 | olda=subfield(local_aux); |
---|
1663 | dgs_points(ext_0)[N_branches+1]=ext1; |
---|
1664 | POINTS[N_branches+1]=list(); |
---|
1665 | POINTS[N_branches+1][1]=number(importdatum(local_aux,"coord@1",olda)); |
---|
1666 | POINTS[N_branches+1][2]=number(importdatum(local_aux,"coord@2",olda)); |
---|
1667 | POINTS[N_branches+1][3]=number(importdatum(local_aux,"coord@3",olda)); |
---|
1668 | LOC_EQS[N_branches+1]=importdatum(local_aux,"CHI",olda); |
---|
1669 | setring HNEring; |
---|
1670 | newa=subfield(S(ext_0)); |
---|
1671 | m=nrows(L@HNE[i][1]); |
---|
1672 | n=ncols(L@HNE[i][1]); |
---|
1673 | setring S(ext_0); |
---|
1674 | list Laux=list(); |
---|
1675 | poly paux=rationalize(HNEring,"L@HNE["+string(i)+"][4]",newa); |
---|
1676 | matrix Maux[m][n]; |
---|
1677 | for (j=1;j<=m;j=j+1) |
---|
1678 | { |
---|
1679 | for (k=1;k<=n;k=k+1) |
---|
1680 | { |
---|
1681 | Maux[j,k]=rationalize(HNEring,"L@HNE["+string(i)+"][1]["+ |
---|
1682 | string(j)+","+string(k)+"]",newa); |
---|
1683 | } |
---|
1684 | } |
---|
1685 | setring HNEring; |
---|
1686 | intvec Li2=L@HNE[i][2]; |
---|
1687 | int Li3=L@HNE[i][3]; |
---|
1688 | setring S(ext_0); |
---|
1689 | Laux[1]=Maux; |
---|
1690 | Laux[2]=Li2; |
---|
1691 | Laux[3]=Li3; |
---|
1692 | Laux[4]=paux; |
---|
1693 | BRANCHES=insert(BRANCHES,Laux,N_branches); |
---|
1694 | kill(Laux,Maux,paux,Li2,Li3); |
---|
1695 | PARAMETRIZATIONS[N_branches+1]=param(BRANCHES[N_branches+1],0); |
---|
1696 | N=N+1; |
---|
1697 | intvec iw=ext_0,N_branches+1; |
---|
1698 | Places[N]=iw; |
---|
1699 | if (sing==2) |
---|
1700 | { |
---|
1701 | Conductor[N]=local_conductor(iw[2],S(ext_0)); |
---|
1702 | } |
---|
1703 | kill(iw); |
---|
1704 | PtoPl[i]=N; |
---|
1705 | setring HNEring; |
---|
1706 | update_CURVE[5][ext_0]=list(); |
---|
1707 | update_CURVE[5][ext_0][1]=S(ext_0); |
---|
1708 | update_CURVE[5][ext_0][2]=dgs_points(ext_0); |
---|
1709 | } |
---|
1710 | if (sing==1) |
---|
1711 | { |
---|
1712 | int N_ini=N-n_branches; |
---|
1713 | counter_c=1; |
---|
1714 | for (i=1;i<=n_branches;i=i+1) |
---|
1715 | { |
---|
1716 | dq=delta2[i]; |
---|
1717 | for (j=1;j<=n_geobrs;j=j+1) |
---|
1718 | { |
---|
1719 | dq=dq+I_mult[counter_c,j]; |
---|
1720 | } |
---|
1721 | Conductor[N_ini+i]=dq; |
---|
1722 | counter_c=counter_c+dgs[i]; |
---|
1723 | } |
---|
1724 | } |
---|
1725 | setring base_r; |
---|
1726 | } |
---|
1727 | } |
---|
1728 | update_CURVE[3]=Places; |
---|
1729 | update_CURVE[4]=Conductor; |
---|
1730 | PP[2]=PtoPl; |
---|
1731 | if (Pp[1]==0) |
---|
1732 | { |
---|
1733 | if (Pp[2]==0) |
---|
1734 | { |
---|
1735 | Aff_SPoints[Pp[3]]=PP; |
---|
1736 | } |
---|
1737 | if (Pp[2]==1) |
---|
1738 | { |
---|
1739 | Inf_Points[1][Pp[3]]=PP; |
---|
1740 | } |
---|
1741 | if (Pp[2]==2) |
---|
1742 | { |
---|
1743 | Inf_Points[2][Pp[3]]=PP; |
---|
1744 | } |
---|
1745 | } |
---|
1746 | else |
---|
1747 | { |
---|
1748 | Aff_Points(Pp[2])[Pp[3]]=PP; |
---|
1749 | } |
---|
1750 | update_CURVE[1][1]=base_r; |
---|
1751 | kill(HNEring); |
---|
1752 | kill(local_aux); |
---|
1753 | return(update_CURVE); |
---|
1754 | } |
---|
1755 | /////////////////////////////////////////////////////////////////////////////// |
---|
1756 | static proc local_conductor (int k,SS) |
---|
1757 | { |
---|
1758 | // computes the degree of the local conductor at a place of a plane curve |
---|
1759 | // if the point is non-singular the result will be zero |
---|
1760 | // the computation is carried out with the "Dedekind formula" via |
---|
1761 | // parametrizations |
---|
1762 | int a,b,Cq; |
---|
1763 | def b_ring=basering; |
---|
1764 | setring SS; |
---|
1765 | poly fx=diff(LOC_EQS[k],x); |
---|
1766 | poly fy=diff(LOC_EQS[k],y); |
---|
1767 | int nr=ncols(BRANCHES[k][1]); |
---|
1768 | poly xt=PARAMETRIZATIONS[k][1][1]; |
---|
1769 | poly yt=PARAMETRIZATIONS[k][1][2]; |
---|
1770 | int ordx=PARAMETRIZATIONS[k][2][1]; |
---|
1771 | int ordy=PARAMETRIZATIONS[k][2][2]; |
---|
1772 | map phi_t=basering,xt,yt,1; |
---|
1773 | poly derf; |
---|
1774 | if (fx<>0) |
---|
1775 | { |
---|
1776 | derf=fx; |
---|
1777 | poly tt=diff(yt,t); |
---|
1778 | b=mindeg(tt); |
---|
1779 | if (ordy>-1) |
---|
1780 | { |
---|
1781 | while (b>=ordy) |
---|
1782 | { |
---|
1783 | BRANCHES[k]=extdevelop(BRANCHES[k],2*nr); |
---|
1784 | nr=ncols(BRANCHES[k][1]); |
---|
1785 | PARAMETRIZATIONS[k]=param(BRANCHES[k],0); |
---|
1786 | ordy=PARAMETRIZATIONS[k][2][2]; |
---|
1787 | yt=PARAMETRIZATIONS[k][1][2]; |
---|
1788 | tt=diff(yt,t); |
---|
1789 | b=mindeg(tt); |
---|
1790 | } |
---|
1791 | xt=PARAMETRIZATIONS[k][1][1]; |
---|
1792 | ordx=PARAMETRIZATIONS[k][2][1]; |
---|
1793 | } |
---|
1794 | poly ft=phi_t(derf); |
---|
1795 | } |
---|
1796 | else |
---|
1797 | { |
---|
1798 | derf=fy; |
---|
1799 | poly tt=diff(xt,t); |
---|
1800 | b=mindeg(tt); |
---|
1801 | if (ordx>-1) |
---|
1802 | { |
---|
1803 | while (b>=ordx) |
---|
1804 | { |
---|
1805 | BRANCHES[k]=extdevelop(BRANCHES[k],2*nr); |
---|
1806 | nr=ncols(BRANCHES[k][1]); |
---|
1807 | PARAMETRIZATIONS[k]=param(BRANCHES[k],0); |
---|
1808 | ordx=PARAMETRIZATIONS[k][2][1]; |
---|
1809 | xt=PARAMETRIZATIONS[k][1][1]; |
---|
1810 | tt=diff(xt,t); |
---|
1811 | b=mindeg(tt); |
---|
1812 | } |
---|
1813 | yt=PARAMETRIZATIONS[k][1][2]; |
---|
1814 | ordy=PARAMETRIZATIONS[k][2][2]; |
---|
1815 | } |
---|
1816 | poly ft=phi_t(derf); |
---|
1817 | } |
---|
1818 | a=mindeg(ft); |
---|
1819 | if ( ordx>-1 || ordy>-1 ) |
---|
1820 | { |
---|
1821 | if (ordy==-1) |
---|
1822 | { |
---|
1823 | while (a>ordx) |
---|
1824 | { |
---|
1825 | BRANCHES[k]=extdevelop(BRANCHES[k],2*nr); |
---|
1826 | nr=ncols(BRANCHES[k][1]); |
---|
1827 | PARAMETRIZATIONS[k]=param(BRANCHES[k],0); |
---|
1828 | ordx=PARAMETRIZATIONS[k][2][1]; |
---|
1829 | xt=PARAMETRIZATIONS[k][1][1]; |
---|
1830 | ft=phi_t(derf); |
---|
1831 | a=mindeg(ft); |
---|
1832 | } |
---|
1833 | } |
---|
1834 | else |
---|
1835 | { |
---|
1836 | if (ordx==-1) |
---|
1837 | { |
---|
1838 | while (a>ordy) |
---|
1839 | { |
---|
1840 | BRANCHES[k]=extdevelop(BRANCHES[k],2*nr); |
---|
1841 | nr=ncols(BRANCHES[k][1]); |
---|
1842 | PARAMETRIZATIONS[k]=param(BRANCHES[k],0); |
---|
1843 | ordy=PARAMETRIZATIONS[k][2][2]; |
---|
1844 | yt=PARAMETRIZATIONS[k][1][2]; |
---|
1845 | ft=phi_t(derf); |
---|
1846 | a=mindeg(ft); |
---|
1847 | } |
---|
1848 | } |
---|
1849 | else |
---|
1850 | { |
---|
1851 | int ordf=ordx; |
---|
1852 | if (ordx>ordy) |
---|
1853 | { |
---|
1854 | ordf=ordy; |
---|
1855 | } |
---|
1856 | while (a>ordf) |
---|
1857 | { |
---|
1858 | BRANCHES[k]=extdevelop(BRANCHES[k],2*nr); |
---|
1859 | nr=ncols(BRANCHES[k][1]); |
---|
1860 | PARAMETRIZATIONS[k]=param(BRANCHES[k],0); |
---|
1861 | ordx=PARAMETRIZATIONS[k][2][1]; |
---|
1862 | ordy=PARAMETRIZATIONS[k][2][2]; |
---|
1863 | ordf=ordx; |
---|
1864 | if (ordx>ordy) |
---|
1865 | { |
---|
1866 | ordf=ordy; |
---|
1867 | } |
---|
1868 | xt=PARAMETRIZATIONS[k][1][1]; |
---|
1869 | yt=PARAMETRIZATIONS[k][1][2]; |
---|
1870 | ft=phi_t(derf); |
---|
1871 | a=mindeg(ft); |
---|
1872 | } |
---|
1873 | } |
---|
1874 | } |
---|
1875 | } |
---|
1876 | Cq=a-b; |
---|
1877 | setring b_ring; |
---|
1878 | return(Cq); |
---|
1879 | } |
---|
1880 | /////////////////////////////////////////////////////////////////////////////// |
---|
1881 | static proc max_D (intvec D1,intvec D2) |
---|
1882 | { |
---|
1883 | // computes the maximum of two divisors (intvec) |
---|
1884 | int s1=size(D1); |
---|
1885 | int s2=size(D2); |
---|
1886 | int i; |
---|
1887 | if (s1>s2) |
---|
1888 | { |
---|
1889 | for (i=1;i<=s2;i=i+1) |
---|
1890 | { |
---|
1891 | if (D2[i]>D1[i]) |
---|
1892 | { |
---|
1893 | D1[i]=D2[i]; |
---|
1894 | } |
---|
1895 | } |
---|
1896 | for (i=s2+1;i<=s1;i=i+1) |
---|
1897 | { |
---|
1898 | if (D1[i]<0) |
---|
1899 | { |
---|
1900 | D1[i]=0; |
---|
1901 | } |
---|
1902 | } |
---|
1903 | return(D1); |
---|
1904 | } |
---|
1905 | else |
---|
1906 | { |
---|
1907 | for (i=1;i<=s1;i=i+1) |
---|
1908 | { |
---|
1909 | if (D1[i]>D2[i]) |
---|
1910 | { |
---|
1911 | D2[i]=D1[i]; |
---|
1912 | } |
---|
1913 | } |
---|
1914 | for (i=s1+1;i<=s2;i=i+1) |
---|
1915 | { |
---|
1916 | if (D2[i]<0) |
---|
1917 | { |
---|
1918 | D2[i]=0; |
---|
1919 | } |
---|
1920 | } |
---|
1921 | return(D2); |
---|
1922 | } |
---|
1923 | } |
---|
1924 | /////////////////////////////////////////////////////////////////////////////// |
---|
1925 | static proc deg_D (intvec D,list PP) |
---|
1926 | { |
---|
1927 | // computes the degree of a divisor (intvec or list of integers) |
---|
1928 | int i; |
---|
1929 | int d=0; |
---|
1930 | int s=size(D); |
---|
1931 | for (i=1;i<=s;i=i+1) |
---|
1932 | { |
---|
1933 | d=d+D[i]*PP[i][1]; |
---|
1934 | } |
---|
1935 | return(d); |
---|
1936 | } |
---|
1937 | /////////////////////////////////////////////////////////////////////////////// |
---|
1938 | |
---|
1939 | // ============================================================================ |
---|
1940 | // ******* MAIN PROCEDURES for the "preprocessing" of Brill-Noether ******** |
---|
1941 | // ============================================================================ |
---|
1942 | |
---|
1943 | proc Adj_div (poly f,list #) |
---|
1944 | "USAGE: Adj_div( f [,l] ); f a poly, [l a list] |
---|
1945 | RETURN: list L with the computed data: |
---|
1946 | @format |
---|
1947 | L[1] a list of rings: L[1][1]=aff_r (affine), L[1][2]=Proj_R (projective), |
---|
1948 | L[2] an intvec with 2 entries (degree, genus), |
---|
1949 | L[3] a list of intvec (closed places), |
---|
1950 | L[4] an intvec (conductor), |
---|
1951 | L[5] a list of lists: |
---|
1952 | L[5][d][1] a (local) ring over an extension of degree d, |
---|
1953 | L[5][d][2] an intvec (degrees of base points of places of degree d) |
---|
1954 | @end format |
---|
1955 | NOTE: @code{Adj_div(f);} computes and stores the fundamental data of the |
---|
1956 | plane curve defined by f as needed for AG codes. |
---|
1957 | In the affine ring you can find the following data: |
---|
1958 | @format |
---|
1959 | poly CHI: affine equation of the curve, |
---|
1960 | ideal Aff_SLocus: affine singular locus (std), |
---|
1961 | list Inf_Points: points at infinity |
---|
1962 | Inf_Points[1]: singular points |
---|
1963 | Inf_Points[2]: non-singular points, |
---|
1964 | list Aff_SPoints: affine singular points (if not empty). |
---|
1965 | @end format |
---|
1966 | In the projective ring you can find the projective equation |
---|
1967 | CHI of the curve (poly). |
---|
1968 | In the local rings L[5][d][1] you find: |
---|
1969 | @format |
---|
1970 | list POINTS: base points of the places of degree d, |
---|
1971 | list LOC_EQS: local equations of the curve at the base points, |
---|
1972 | list BRANCHES: Hamburger-Noether developments of the places, |
---|
1973 | list PARAMETRIZATIONS: local parametrizations of the places, |
---|
1974 | @end format |
---|
1975 | Each entry of the list L[3] corresponds to one closed place (i.e., |
---|
1976 | a place and all its conjugates) which is represented by an intvec |
---|
1977 | of size two, the first entry is the degree of the place (in |
---|
1978 | particular, it tells the local ring where to find the data |
---|
1979 | describing one representative of the closed place), and the |
---|
1980 | second one is the position of those data in the lists POINTS, etc., |
---|
1981 | inside this local ring.@* |
---|
1982 | In the intvec L[4] (conductor) the i-th entry corresponds to the |
---|
1983 | i-th entry in the list of places L[3]. |
---|
1984 | |
---|
1985 | With no optional arguments, the conductor is computed by |
---|
1986 | local invariants of the singularities; otherwise it is computed |
---|
1987 | by the Dedekind formula. @* |
---|
1988 | An affine point is represented by a list P where P[1] is std |
---|
1989 | of a prime ideal and P[2] is an intvec containing the position |
---|
1990 | of the places above P in the list of closed places L[3]. @* |
---|
1991 | If the point is at infinity, P[1] is a homogeneous irreducible |
---|
1992 | polynomial in two variables. |
---|
1993 | |
---|
1994 | If @code{printlevel>=0} additional comments are displayed (default: |
---|
1995 | @code{printlevel=0}). |
---|
1996 | KEYWORDS: Hamburger-Noether expansions; adjunction divisor |
---|
1997 | SEE ALSO: closed_points, NSplaces |
---|
1998 | EXAMPLE: example Adj_div; shows an example |
---|
1999 | " |
---|
2000 | { |
---|
2001 | // computes the adjunction divisor and the genus of a (singular) plane curve |
---|
2002 | // as a byproduct, it computes all the singular points with the corresponding |
---|
2003 | // places and the genus of the curve |
---|
2004 | // the adjunction divisor is stored in an intvec |
---|
2005 | // also the non-singular places at infinity are computed |
---|
2006 | // returns a list with all the computed data |
---|
2007 | if (char(basering)==0) |
---|
2008 | { |
---|
2009 | ERROR("Base field not implemented"); |
---|
2010 | } |
---|
2011 | if (npars(basering)>0) |
---|
2012 | { |
---|
2013 | ERROR("Base field not implemented"); |
---|
2014 | } |
---|
2015 | intvec opgt=option(get); |
---|
2016 | option(redSB); |
---|
2017 | def Base_R=basering; |
---|
2018 | list CURVE=curve(f); |
---|
2019 | def aff_r=CURVE[1]; |
---|
2020 | def Proj_R=CURVE[2]; |
---|
2021 | int degX=CURVE[3]; |
---|
2022 | int genusX=(degX-1)*(degX-2); |
---|
2023 | genusX = genusX div 2; |
---|
2024 | intvec iivv=degX,genusX; |
---|
2025 | intvec Conductor; |
---|
2026 | setring aff_r; |
---|
2027 | dbprint(printlevel+1,"Computing affine singular points ... "); |
---|
2028 | list Aff_SPoints=Aff_SL(Aff_SLocus); |
---|
2029 | int s=size(Aff_SPoints); |
---|
2030 | if (s>0) |
---|
2031 | { |
---|
2032 | export(Aff_SPoints); |
---|
2033 | } |
---|
2034 | dbprint(printlevel+1,"Computing all points at infinity ... "); |
---|
2035 | list Inf_Points=inf_P(CHI); |
---|
2036 | export(Inf_Points); |
---|
2037 | list update_CURVE=list(); |
---|
2038 | update_CURVE[1]=list(); |
---|
2039 | update_CURVE[1][1]=aff_r; |
---|
2040 | update_CURVE[1][2]=Proj_R; |
---|
2041 | update_CURVE[2]=iivv; |
---|
2042 | update_CURVE[3]=list(); |
---|
2043 | update_CURVE[4]=Conductor; |
---|
2044 | update_CURVE[5]=list(); |
---|
2045 | int i; |
---|
2046 | intvec pP=0,0,0; |
---|
2047 | if (size(#)==0) |
---|
2048 | { |
---|
2049 | dbprint(printlevel+1,"Computing affine singular places ... "); |
---|
2050 | if (s>0) |
---|
2051 | { |
---|
2052 | for (i=1;i<=s;i=i+1) |
---|
2053 | { |
---|
2054 | pP[3]=i; |
---|
2055 | update_CURVE=place(pP,1,update_CURVE); |
---|
2056 | } |
---|
2057 | } |
---|
2058 | dbprint(printlevel+1,"Computing singular places at infinity ... "); |
---|
2059 | s=size(Inf_Points[1]); |
---|
2060 | if (s>0) |
---|
2061 | { |
---|
2062 | pP[2]=1; |
---|
2063 | for (i=1;i<=s;i=i+1) |
---|
2064 | { |
---|
2065 | pP[3]=i; |
---|
2066 | update_CURVE=place(pP,1,update_CURVE); |
---|
2067 | } |
---|
2068 | } |
---|
2069 | } |
---|
2070 | else |
---|
2071 | { |
---|
2072 | dbprint(printlevel+1,"Computing affine singular places ... "); |
---|
2073 | s=size(Aff_SPoints); |
---|
2074 | if (s>0) |
---|
2075 | { |
---|
2076 | for (i=1;i<=s;i=i+1) |
---|
2077 | { |
---|
2078 | pP[3]=i; |
---|
2079 | update_CURVE=place(pP,2,update_CURVE); |
---|
2080 | } |
---|
2081 | } |
---|
2082 | dbprint(printlevel+1,"Computing singular places at infinity ... "); |
---|
2083 | s=size(Inf_Points[1]); |
---|
2084 | if (s>0) |
---|
2085 | { |
---|
2086 | pP[2]=1; |
---|
2087 | for (i=1;i<=s;i=i+1) |
---|
2088 | { |
---|
2089 | pP[3]=i; |
---|
2090 | update_CURVE=place(pP,2,update_CURVE); |
---|
2091 | } |
---|
2092 | } |
---|
2093 | } |
---|
2094 | dbprint(printlevel+1,"Computing non-singular places at infinity ... "); |
---|
2095 | s=size(Inf_Points[2]); |
---|
2096 | if (s>0) |
---|
2097 | { |
---|
2098 | pP[2]=2; |
---|
2099 | for (i=1;i<=s;i=i+1) |
---|
2100 | { |
---|
2101 | pP[3]=i; |
---|
2102 | update_CURVE=place(pP,0,update_CURVE); |
---|
2103 | } |
---|
2104 | } |
---|
2105 | dbprint(printlevel+1,"Adjunction divisor computed successfully"); |
---|
2106 | list Places=update_CURVE[3]; |
---|
2107 | Conductor=update_CURVE[4]; |
---|
2108 | genusX = genusX - (deg_D(Conductor,Places) div 2); |
---|
2109 | update_CURVE[2][2]=genusX; |
---|
2110 | setring Base_R; |
---|
2111 | dbprint(printlevel+1," "); |
---|
2112 | dbprint(printlevel+2,"The genus of the curve is "+string(genusX)); |
---|
2113 | option(set,opgt); |
---|
2114 | return(update_CURVE); |
---|
2115 | } |
---|
2116 | example |
---|
2117 | { |
---|
2118 | "EXAMPLE:"; echo = 2; |
---|
2119 | int plevel=printlevel; |
---|
2120 | printlevel=-1; |
---|
2121 | ring s=2,(x,y),lp; |
---|
2122 | list C=Adj_div(y9+y8+xy6+x2y3+y2+x3); |
---|
2123 | def aff_R=C[1][1]; // the affine ring |
---|
2124 | setring aff_R; |
---|
2125 | listvar(aff_R); // data in the affine ring |
---|
2126 | CHI; // affine equation of the curve |
---|
2127 | Aff_SLocus; // ideal of the affine singular locus |
---|
2128 | Aff_SPoints[1]; // 1st affine singular point: (1:1:1), no.1 |
---|
2129 | Inf_Points[1]; // singular point(s) at infinity: (1:0:0), no.4 |
---|
2130 | Inf_Points[2]; // list of non-singular points at infinity |
---|
2131 | // |
---|
2132 | pause("press RETURN"); |
---|
2133 | def proj_R=C[1][2]; // the projective ring |
---|
2134 | setring proj_R; |
---|
2135 | CHI; // projective equation of the curve |
---|
2136 | C[2][1]; // degree of the curve |
---|
2137 | C[2][2]; // genus of the curve |
---|
2138 | C[3]; // list of computed places |
---|
2139 | C[4]; // adjunction divisor (all points are singular!) |
---|
2140 | // |
---|
2141 | pause("press RETURN"); |
---|
2142 | // we look at the place(s) of degree 2 by changing to the ring |
---|
2143 | C[5][2][1]; |
---|
2144 | def S(2)=C[5][2][1]; |
---|
2145 | setring S(2); |
---|
2146 | POINTS; // base point(s) of place(s) of degree 2: (1:a:1) |
---|
2147 | LOC_EQS; // local equation(s) |
---|
2148 | PARAMETRIZATIONS; // parametrization(s) and exactness |
---|
2149 | BRANCHES; // Hamburger-Noether development |
---|
2150 | printlevel=plevel; |
---|
2151 | } |
---|
2152 | /////////////////////////////////////////////////////////////////////////////// |
---|
2153 | |
---|
2154 | proc NSplaces (int h,list CURVE) |
---|
2155 | "USAGE: NSplaces( h, CURVE ), where h is an integer and CURVE is a list |
---|
2156 | RETURN: list L with updated data of CURVE after computing all |
---|
2157 | points up to degree H+h (H the maximum degree of the previously |
---|
2158 | computed places): @* |
---|
2159 | @format |
---|
2160 | in L[1][1]: (affine ring) lists Aff_Points(d) with affine non-singular |
---|
2161 | points of degree d (if non-empty) |
---|
2162 | in L[3]: the newly computed closed places are added, |
---|
2163 | in L[5]: local rings created/updated to store (repres. of) new places. |
---|
2164 | @end format |
---|
2165 | See @ref{Adj_div} for a description of the entries in L. |
---|
2166 | NOTE: The list_expression should be the output of the procedure Adj_div.@* |
---|
2167 | If @code{printlevel>=0} additional comments are displayed (default: |
---|
2168 | @code{printlevel=0}). |
---|
2169 | SEE ALSO: closed_points, Adj_div |
---|
2170 | EXAMPLE: example NSplaces; shows an example |
---|
2171 | " |
---|
2172 | { |
---|
2173 | // computes all the non-singular closed places with degree up to a certain |
---|
2174 | // bound; this bound is the maximum degree of an existing singular place or |
---|
2175 | // non-singular place at infinity plus an increment h>=0 which is given as |
---|
2176 | // input |
---|
2177 | // creates lists of points and the corresponding places |
---|
2178 | // list CURVE must be the output of the procedure "Adj_div" |
---|
2179 | // warning : if h<0 then it will be replaced by h=0 |
---|
2180 | intvec opgt=option(get); |
---|
2181 | option(redSB); |
---|
2182 | def Base_R=basering; |
---|
2183 | def aff_r=CURVE[1][1]; |
---|
2184 | int M=size(CURVE[5]); |
---|
2185 | if (h>0) |
---|
2186 | { |
---|
2187 | M=M+h; |
---|
2188 | } |
---|
2189 | list update_CURVE=CURVE; |
---|
2190 | int i,j,s; |
---|
2191 | setring aff_r; |
---|
2192 | intvec pP=1,0,0; |
---|
2193 | for (i=1;i<=M;i=i+1) |
---|
2194 | { |
---|
2195 | dbprint(printlevel+1,"Computing non-singular affine places of degree " |
---|
2196 | +string(i)+" ... "); |
---|
2197 | if (defined(Aff_Points(i))==0) |
---|
2198 | { |
---|
2199 | list Aff_Points(i)=closed_points_deg(CHI,i,Aff_SLocus); |
---|
2200 | s=size(Aff_Points(i)); |
---|
2201 | if (s>0) |
---|
2202 | { |
---|
2203 | export(Aff_Points(i)); |
---|
2204 | pP[2]=i; |
---|
2205 | for (j=1;j<=s;j=j+1) |
---|
2206 | { |
---|
2207 | pP[3]=j; |
---|
2208 | update_CURVE=place(pP,0,update_CURVE); |
---|
2209 | } |
---|
2210 | } |
---|
2211 | } |
---|
2212 | } |
---|
2213 | setring Base_R; |
---|
2214 | option(set,opgt); |
---|
2215 | return(update_CURVE); |
---|
2216 | } |
---|
2217 | example |
---|
2218 | { |
---|
2219 | "EXAMPLE:"; echo = 2; |
---|
2220 | int plevel=printlevel; |
---|
2221 | printlevel=-1; |
---|
2222 | ring s=2,(x,y),lp; |
---|
2223 | list C=Adj_div(x3y+y3+x); |
---|
2224 | // The list of computed places: |
---|
2225 | C[3]; |
---|
2226 | // create places up to degree 1+3 |
---|
2227 | list L=NSplaces(3,C); |
---|
2228 | // The list of computed places is now: |
---|
2229 | L[3]; |
---|
2230 | // e.g., affine non-singular points of degree 4 : |
---|
2231 | def aff_r=L[1][1]; |
---|
2232 | setring aff_r; |
---|
2233 | Aff_Points(4); |
---|
2234 | // e.g., base point of the 1st place of degree 4 : |
---|
2235 | def S(4)=L[5][4][1]; |
---|
2236 | setring S(4); |
---|
2237 | POINTS[1]; |
---|
2238 | printlevel=plevel; |
---|
2239 | } |
---|
2240 | /////////////////////////////////////////////////////////////////////////////// |
---|
2241 | |
---|
2242 | // ** SPECIAL PROCEDURES FOR LINEAR ALGEBRA ** |
---|
2243 | |
---|
2244 | static proc Ker (matrix A) |
---|
2245 | { |
---|
2246 | // warning : "lp" ordering is necessary |
---|
2247 | intvec opgt=option(get); |
---|
2248 | option(redSB); |
---|
2249 | matrix M=transpose(syz(A)); |
---|
2250 | option(set,opgt); |
---|
2251 | return(M); |
---|
2252 | } |
---|
2253 | /////////////////////////////////////////////////////////////////////////////// |
---|
2254 | static proc get_NZsol (matrix A) |
---|
2255 | { |
---|
2256 | matrix sol=Ker(A); |
---|
2257 | return(submat(sol,1..1,1..ncols(sol))); |
---|
2258 | } |
---|
2259 | /////////////////////////////////////////////////////////////////////////////// |
---|
2260 | static proc supplement (matrix W,matrix V) |
---|
2261 | "USAGE: supplement(W,V), where W,V are matrices of numbers such that the |
---|
2262 | vector space generated by the rows of W is contained in that |
---|
2263 | generated by the rows of V |
---|
2264 | RETURN: matrix whose rows generate a supplementary vector space of W in V, |
---|
2265 | or a zero row-matrix if <W>==<V> |
---|
2266 | NOTE: W,V must be given with maximal rank |
---|
2267 | " |
---|
2268 | { |
---|
2269 | // W and V represent independent sets of vectors and <W> is assumed to be |
---|
2270 | // contained in <V> |
---|
2271 | // computes matrix S whose rows are l.i. vectors s.t. <W> union <S> is a |
---|
2272 | // basis of <V> |
---|
2273 | // careful : the size of all vectors is assumed to be the same but it is |
---|
2274 | // not checked and neither the linear independence of the vectors is checked |
---|
2275 | // the trivial case W=0 is not covered by this procedure (and thus V<>0) |
---|
2276 | // if <W>=<V> then a zero row-matrix is returned |
---|
2277 | // warning : option(redSB) must be set in advance |
---|
2278 | int n1=nrows(W); |
---|
2279 | int n2=nrows(V); |
---|
2280 | int s=n2-n1; |
---|
2281 | if (s==0) |
---|
2282 | { |
---|
2283 | int n=ncols(W); |
---|
2284 | matrix HH[1][n]; |
---|
2285 | return(HH); |
---|
2286 | } |
---|
2287 | matrix H=transpose(lift(transpose(V),transpose(W))); |
---|
2288 | H=supplem(H); |
---|
2289 | return(H*V); |
---|
2290 | } |
---|
2291 | /////////////////////////////////////////////////////////////////////////////// |
---|
2292 | static proc supplem (matrix M) |
---|
2293 | "USAGE: suplement(M), where M is a matrix of numbers with maximal rank |
---|
2294 | RETURN: matrix whose rows generate a supplementary vector space of <M> in |
---|
2295 | k^n, where k is the base field and n is the number of columns |
---|
2296 | SEE ALSO: supplement |
---|
2297 | NOTE: The rank r is assumed to be 1<r<n. |
---|
2298 | " |
---|
2299 | { |
---|
2300 | // warning : the linear independence of the rows is not checked |
---|
2301 | int r=nrows(M); |
---|
2302 | int n=ncols(M); |
---|
2303 | int s=n-r; |
---|
2304 | matrix A=M; |
---|
2305 | matrix supl[s][n]; |
---|
2306 | int counter=0; |
---|
2307 | int h=r+1; |
---|
2308 | int i; |
---|
2309 | for (i=1;i<=n;i=i+1) |
---|
2310 | { |
---|
2311 | matrix TT[1][n]; |
---|
2312 | TT[1,i]=1; |
---|
2313 | A=transpose(concat(transpose(A),transpose(TT))); |
---|
2314 | r=mat_rank(A); |
---|
2315 | if (r==h) |
---|
2316 | { |
---|
2317 | h=h+1; |
---|
2318 | counter=counter+1; |
---|
2319 | supl=transpose(concat(transpose(supl),transpose(TT))); |
---|
2320 | if (counter==s) |
---|
2321 | { |
---|
2322 | break; |
---|
2323 | } |
---|
2324 | } |
---|
2325 | kill(TT); |
---|
2326 | } |
---|
2327 | supl=transpose(compress(transpose(supl))); |
---|
2328 | return(supl); |
---|
2329 | } |
---|
2330 | /////////////////////////////////////////////////////////////////////////////// |
---|
2331 | static proc mat_rank (matrix A) |
---|
2332 | { |
---|
2333 | // warning : "lp" ordering is necessary |
---|
2334 | intvec opgt=option(get); |
---|
2335 | option(redSB); |
---|
2336 | int r=size(std(module(transpose(A)))); |
---|
2337 | option(set,opgt); |
---|
2338 | return(r); |
---|
2339 | } |
---|
2340 | /////////////////////////////////////////////////////////////////////////////// |
---|
2341 | |
---|
2342 | // *************************************************************** |
---|
2343 | // * PROCEDURES FOR INTERPOLATION, INTERSECTION AND EXTRA PLACES * |
---|
2344 | // *************************************************************** |
---|
2345 | |
---|
2346 | static proc estim_n (intvec Dplus,int dgX,list PL) |
---|
2347 | { |
---|
2348 | // computes an estimate for the degree n in the Brill-Noether algorithm |
---|
2349 | int estim=2*deg_D(Dplus,PL)+dgX*(dgX-3); |
---|
2350 | estim=estim div (2*dgX); |
---|
2351 | estim=estim+1; |
---|
2352 | if (estim<dgX) |
---|
2353 | { |
---|
2354 | estim=dgX; |
---|
2355 | } |
---|
2356 | return(estim); |
---|
2357 | } |
---|
2358 | /////////////////////////////////////////////////////////////////////////////// |
---|
2359 | static proc nforms (int n) |
---|
2360 | { |
---|
2361 | // computes the list of all homogeneous monomials of degree n>=0 |
---|
2362 | // exports ideal nFORMS(n) whose generators are ranged with lp order |
---|
2363 | // in Proj_R and returns size(nFORMS(n)) |
---|
2364 | // warning : it is supposed to be called inside Proj_R |
---|
2365 | // if n<=0 then nFORMS(0) is "computed fast" |
---|
2366 | ideal nFORMS(n); |
---|
2367 | int N; |
---|
2368 | if (n>0) |
---|
2369 | { |
---|
2370 | N=(n+1)*(n+2); |
---|
2371 | N=N div 2; |
---|
2372 | N=N+1; |
---|
2373 | int i,j,k; |
---|
2374 | for (i=0;i<=n;i=i+1) |
---|
2375 | { |
---|
2376 | for (j=0;j<=n-i;j=j+1) |
---|
2377 | { |
---|
2378 | k=k+1; |
---|
2379 | nFORMS(n)[N-k]=x^i*y^j*z^(n-i-j); |
---|
2380 | } |
---|
2381 | } |
---|
2382 | export(nFORMS(n)); |
---|
2383 | } |
---|
2384 | else |
---|
2385 | { |
---|
2386 | N=2; |
---|
2387 | nFORMS(0)=1; |
---|
2388 | export(nFORMS(0)); |
---|
2389 | } |
---|
2390 | return(N-1); |
---|
2391 | } |
---|
2392 | /////////////////////////////////////////////////////////////////////////////// |
---|
2393 | static proc nmultiples (int n,int dgX,poly f) |
---|
2394 | { |
---|
2395 | // computes a basis of the space of forms of degree n which are multiple of |
---|
2396 | // CHI |
---|
2397 | // returns a matrix whose rows are the coordinates (related to nFORMS(n)) |
---|
2398 | // of such a basis |
---|
2399 | // warning : it is supposed to be called inside Proj_R |
---|
2400 | // warning : nFORMS(n) is created in the way, together with nFORMS(n-degX) |
---|
2401 | // warning : n must be greater or equal than the degree of the curve |
---|
2402 | if (defined(nFORMS(n))==0) |
---|
2403 | { |
---|
2404 | dbprint(printlevel+1,string(nforms(n))); |
---|
2405 | } |
---|
2406 | int m=n-dgX; |
---|
2407 | if (defined(nFORMS(m))==0) |
---|
2408 | { |
---|
2409 | int k=nforms(m); |
---|
2410 | } |
---|
2411 | else |
---|
2412 | { |
---|
2413 | int k=size(nFORMS(m)); |
---|
2414 | } |
---|
2415 | ideal nmults; |
---|
2416 | int i; |
---|
2417 | for (i=1;i<=k;i=i+1) |
---|
2418 | { |
---|
2419 | nmults[i]=f*nFORMS(m)[i]; |
---|
2420 | } |
---|
2421 | return(transpose(lift(nFORMS(n),nmults))); |
---|
2422 | } |
---|
2423 | /////////////////////////////////////////////////////////////////////////////// |
---|
2424 | static proc interpolating_forms (intvec D,int n,list CURVE) |
---|
2425 | { |
---|
2426 | // computes a vector basis of the space of forms of degree n whose |
---|
2427 | // intersection divisor with the curve is greater or equal than D>=0 |
---|
2428 | // the procedure is supposed to be called inside the ring Proj_R and |
---|
2429 | // assumes that the forms nFORMS(n) are previously computed |
---|
2430 | // the output is a matrix whose rows are the coordinates in nFORMS(n) of |
---|
2431 | // such a basis |
---|
2432 | // remark : the support of D may contain "extra" places |
---|
2433 | def BR=basering; |
---|
2434 | def aff_r=CURVE[1][1]; |
---|
2435 | int N=size(nFORMS(n)); |
---|
2436 | matrix totalM[1][N]; |
---|
2437 | int s=size(D); |
---|
2438 | list Places=CURVE[3]; |
---|
2439 | int NPls=size(Places); |
---|
2440 | int i,j,k,kk,id,ip,RR,ordx,ordy,nr,NR; |
---|
2441 | if (s<=NPls) |
---|
2442 | { |
---|
2443 | for (i=1;i<=s;i=i+1) |
---|
2444 | { |
---|
2445 | if (D[i]>0) |
---|
2446 | { |
---|
2447 | id=Places[i][1]; |
---|
2448 | ip=Places[i][2]; |
---|
2449 | RR=D[i]; |
---|
2450 | def SS=CURVE[5][id][1]; |
---|
2451 | setring SS; |
---|
2452 | poly xt=PARAMETRIZATIONS[ip][1][1]; |
---|
2453 | poly yt=PARAMETRIZATIONS[ip][1][2]; |
---|
2454 | ordx=PARAMETRIZATIONS[ip][2][1]; |
---|
2455 | ordy=PARAMETRIZATIONS[ip][2][2]; |
---|
2456 | nr=ncols(BRANCHES[ip][1]); |
---|
2457 | if ( ordx>-1 || ordy>-1 ) |
---|
2458 | { |
---|
2459 | while ( ( RR>ordx && ordx>-1 ) || ( RR>ordy && ordy>-1 ) ) |
---|
2460 | { |
---|
2461 | BRANCHES[ip]=extdevelop(BRANCHES[ip],2*nr); |
---|
2462 | nr=ncols(BRANCHES[ip][1]); |
---|
2463 | PARAMETRIZATIONS[ip]=param(BRANCHES[ip],0); |
---|
2464 | xt=PARAMETRIZATIONS[ip][1][1]; |
---|
2465 | yt=PARAMETRIZATIONS[ip][1][2]; |
---|
2466 | ordx=PARAMETRIZATIONS[ip][2][1]; |
---|
2467 | ordy=PARAMETRIZATIONS[ip][2][2]; |
---|
2468 | } |
---|
2469 | } |
---|
2470 | if (POINTS[ip][3]==number(1)) |
---|
2471 | { |
---|
2472 | number A=POINTS[ip][1]; |
---|
2473 | number B=POINTS[ip][2]; |
---|
2474 | map Mt=BR,A+xt,B+yt,1; |
---|
2475 | kill(A,B); |
---|
2476 | } |
---|
2477 | else |
---|
2478 | { |
---|
2479 | if (POINTS[ip][2]==number(1)) |
---|
2480 | { |
---|
2481 | number A=POINTS[ip][1]; |
---|
2482 | map Mt=BR,A+xt,1,yt; |
---|
2483 | kill(A); |
---|
2484 | } |
---|
2485 | else |
---|
2486 | { |
---|
2487 | map Mt=BR,1,xt,yt; |
---|
2488 | } |
---|
2489 | } |
---|
2490 | ideal nFORMS(n)=Mt(nFORMS(n)); |
---|
2491 | // rewrite properly the above conditions to obtain the local equations |
---|
2492 | matrix partM[RR][N]; |
---|
2493 | matrix auxMC=coeffs(nFORMS(n),t); |
---|
2494 | NR=nrows(auxMC); |
---|
2495 | if (RR<=NR) |
---|
2496 | { |
---|
2497 | for (j=1;j<=RR;j=j+1) |
---|
2498 | { |
---|
2499 | for (k=1;k<=N;k=k+1) |
---|
2500 | { |
---|
2501 | partM[j,k]=number(auxMC[j,k]); |
---|
2502 | } |
---|
2503 | } |
---|
2504 | } |
---|
2505 | else |
---|
2506 | { |
---|
2507 | for (j=1;j<=NR;j=j+1) |
---|
2508 | { |
---|
2509 | for (k=1;k<=N;k=k+1) |
---|
2510 | { |
---|
2511 | partM[j,k]=number(auxMC[j,k]); |
---|
2512 | } |
---|
2513 | } |
---|
2514 | for (j=NR+1;j<=RR;j=j+1) |
---|
2515 | { |
---|
2516 | for (k=1;k<=N;k=k+1) |
---|
2517 | { |
---|
2518 | partM[j,k]=number(0); |
---|
2519 | } |
---|
2520 | } |
---|
2521 | } |
---|
2522 | matrix localM=partM; |
---|
2523 | matrix conjM=partM; |
---|
2524 | for (j=2;j<=id;j=j+1) |
---|
2525 | { |
---|
2526 | conjM=Frobenius(conjM,1); |
---|
2527 | localM=transpose(concat(transpose(localM),transpose(conjM))); |
---|
2528 | } |
---|
2529 | localM=gauss_row(localM); |
---|
2530 | setring BR; |
---|
2531 | totalM=transpose(concat(transpose(totalM),transpose(imap(SS,localM)))); |
---|
2532 | totalM=transpose(compress(transpose(totalM))); |
---|
2533 | setring SS; |
---|
2534 | kill(xt,yt,Mt,nFORMS(n),partM,auxMC,conjM,localM); |
---|
2535 | setring BR; |
---|
2536 | kill(SS); |
---|
2537 | } |
---|
2538 | } |
---|
2539 | } |
---|
2540 | else |
---|
2541 | { |
---|
2542 | // distinguish between "standard" places and "extra" places |
---|
2543 | for (i=1;i<=NPls;i=i+1) |
---|
2544 | { |
---|
2545 | if (D[i]>0) |
---|
2546 | { |
---|
2547 | id=Places[i][1]; |
---|
2548 | ip=Places[i][2]; |
---|
2549 | RR=D[i]; |
---|
2550 | def SS=CURVE[5][id][1]; |
---|
2551 | setring SS; |
---|
2552 | poly xt=PARAMETRIZATIONS[ip][1][1]; |
---|
2553 | poly yt=PARAMETRIZATIONS[ip][1][2]; |
---|
2554 | ordx=PARAMETRIZATIONS[ip][2][1]; |
---|
2555 | ordy=PARAMETRIZATIONS[ip][2][2]; |
---|
2556 | nr=ncols(BRANCHES[ip][1]); |
---|
2557 | if ( ordx>-1 || ordy>-1 ) |
---|
2558 | { |
---|
2559 | while ( ( RR>ordx && ordx>-1 ) || ( RR>ordy && ordy>-1 ) ) |
---|
2560 | { |
---|
2561 | BRANCHES[ip]=extdevelop(BRANCHES[ip],2*nr); |
---|
2562 | nr=ncols(BRANCHES[ip][1]); |
---|
2563 | PARAMETRIZATIONS[ip]=param(BRANCHES[ip],0); |
---|
2564 | xt=PARAMETRIZATIONS[ip][1][1]; |
---|
2565 | yt=PARAMETRIZATIONS[ip][1][2]; |
---|
2566 | ordx=PARAMETRIZATIONS[ip][2][1]; |
---|
2567 | ordy=PARAMETRIZATIONS[ip][2][2]; |
---|
2568 | } |
---|
2569 | } |
---|
2570 | if (POINTS[ip][3]==number(1)) |
---|
2571 | { |
---|
2572 | number A=POINTS[ip][1]; |
---|
2573 | number B=POINTS[ip][2]; |
---|
2574 | map Mt=BR,A+xt,B+yt,1; |
---|
2575 | kill(A,B); |
---|
2576 | } |
---|
2577 | else |
---|
2578 | { |
---|
2579 | if (POINTS[ip][2]==number(1)) |
---|
2580 | { |
---|
2581 | number A=POINTS[ip][1]; |
---|
2582 | map Mt=BR,A+xt,1,yt; |
---|
2583 | kill(A); |
---|
2584 | } |
---|
2585 | else |
---|
2586 | { |
---|
2587 | map Mt=BR,1,xt,yt; |
---|
2588 | } |
---|
2589 | } |
---|
2590 | ideal nFORMS(n)=Mt(nFORMS(n)); |
---|
2591 | // rewrite properly the above conditions to obtain the local equations |
---|
2592 | matrix partM[RR][N]; |
---|
2593 | matrix auxMC=coeffs(nFORMS(n),t); |
---|
2594 | NR=nrows(auxMC); |
---|
2595 | if (RR<=NR) |
---|
2596 | { |
---|
2597 | for (j=1;j<=RR;j=j+1) |
---|
2598 | { |
---|
2599 | for (k=1;k<=N;k=k+1) |
---|
2600 | { |
---|
2601 | partM[j,k]=number(auxMC[j,k]); |
---|
2602 | } |
---|
2603 | } |
---|
2604 | } |
---|
2605 | else |
---|
2606 | { |
---|
2607 | for (j=1;j<=NR;j=j+1) |
---|
2608 | { |
---|
2609 | for (k=1;k<=N;k=k+1) |
---|
2610 | { |
---|
2611 | partM[j,k]=number(auxMC[j,k]); |
---|
2612 | } |
---|
2613 | } |
---|
2614 | for (j=NR+1;j<=RR;j=j+1) |
---|
2615 | { |
---|
2616 | for (k=1;k<=N;k=k+1) |
---|
2617 | { |
---|
2618 | partM[j,k]=number(0); |
---|
2619 | } |
---|
2620 | } |
---|
2621 | } |
---|
2622 | matrix localM=partM; |
---|
2623 | matrix conjM=partM; |
---|
2624 | for (j=2;j<=id;j=j+1) |
---|
2625 | { |
---|
2626 | conjM=Frobenius(conjM,1); |
---|
2627 | localM=transpose(concat(transpose(localM),transpose(conjM))); |
---|
2628 | } |
---|
2629 | localM=gauss_row(localM); |
---|
2630 | setring BR; |
---|
2631 | totalM=transpose(concat(transpose(totalM),transpose(imap(SS,localM)))); |
---|
2632 | totalM=transpose(compress(transpose(totalM))); |
---|
2633 | setring SS; |
---|
2634 | kill(xt,yt,Mt,nFORMS(n),partM,auxMC,conjM,localM); |
---|
2635 | setring BR; |
---|
2636 | kill(SS); |
---|
2637 | } |
---|
2638 | } |
---|
2639 | k=s-NPls; |
---|
2640 | int l; |
---|
2641 | for (i=1;i<=k;i=i+1) |
---|
2642 | { |
---|
2643 | // in this case D[NPls+i]>0 is assumed to be true during the |
---|
2644 | // Brill-Noether algorithm |
---|
2645 | RR=D[NPls+i]; |
---|
2646 | setring aff_r; |
---|
2647 | def SS=@EXTRA@[2][i]; |
---|
2648 | def extra_dgs=@EXTRA@[3]; |
---|
2649 | setring SS; |
---|
2650 | poly xt=PARAMETRIZATION[1][1]; |
---|
2651 | poly yt=PARAMETRIZATION[1][2]; |
---|
2652 | ordx=PARAMETRIZATION[2][1]; |
---|
2653 | ordy=PARAMETRIZATION[2][2]; |
---|
2654 | nr=ncols(BRANCH[1]); |
---|
2655 | if ( ordx>-1 || ordy>-1 ) |
---|
2656 | { |
---|
2657 | while ( ( RR>ordx && ordx>-1 ) || ( RR>ordy && ordy>-1 ) ) |
---|
2658 | { |
---|
2659 | BRANCH[1]=extdevelop(BRANCH,2*nr); |
---|
2660 | nr=ncols(BRANCH[1]); |
---|
2661 | PARAMETRIZATION=param(BRANCH,0); |
---|
2662 | xt=PARAMETRIZATION[1][1]; |
---|
2663 | yt=PARAMETRIZATION[1][2]; |
---|
2664 | ordx=PARAMETRIZATION[2][1]; |
---|
2665 | ordy=PARAMETRIZATION[2][2]; |
---|
2666 | } |
---|
2667 | } |
---|
2668 | number A=POINT[1]; |
---|
2669 | number B=POINT[2]; |
---|
2670 | map Mt=BR,A+xt,B+yt,1; |
---|
2671 | kill(A,B); |
---|
2672 | ideal nFORMS(n)=Mt(nFORMS(n)); |
---|
2673 | // rewrite properly the above conditions to obtain the local equations |
---|
2674 | matrix partM[RR][N]; |
---|
2675 | matrix auxMC=coeffs(nFORMS(n),t); |
---|
2676 | NR=nrows(auxMC); |
---|
2677 | if (RR<=NR) |
---|
2678 | { |
---|
2679 | for (j=1;j<=RR;j=j+1) |
---|
2680 | { |
---|
2681 | for (kk=1;kk<=N;kk=kk+1) |
---|
2682 | { |
---|
2683 | partM[j,kk]=number(auxMC[j,kk]); |
---|
2684 | } |
---|
2685 | } |
---|
2686 | } |
---|
2687 | else |
---|
2688 | { |
---|
2689 | for (j=1;j<=NR;j=j+1) |
---|
2690 | { |
---|
2691 | for (kk=1;kk<=N;kk=kk+1) |
---|
2692 | { |
---|
2693 | partM[j,kk]=number(auxMC[j,kk]); |
---|
2694 | } |
---|
2695 | } |
---|
2696 | for (j=NR+1;j<=RR;j=j+1) |
---|
2697 | { |
---|
2698 | for (kk=1;kk<=N;kk=kk+1) |
---|
2699 | { |
---|
2700 | partM[j,kk]=number(0); |
---|
2701 | } |
---|
2702 | } |
---|
2703 | } |
---|
2704 | matrix localM=partM; |
---|
2705 | matrix conjM=partM; |
---|
2706 | l=extra_dgs[i]; |
---|
2707 | for (j=2;j<=l;j=j+1) |
---|
2708 | { |
---|
2709 | conjM=Frobenius(conjM,1); |
---|
2710 | localM=transpose(concat(transpose(localM),transpose(conjM))); |
---|
2711 | } |
---|
2712 | localM=gauss_row(localM); |
---|
2713 | setring BR; |
---|
2714 | totalM=transpose(concat(transpose(totalM),transpose(imap(SS,localM)))); |
---|
2715 | totalM=transpose(compress(transpose(totalM))); |
---|
2716 | setring SS; |
---|
2717 | kill(xt,yt,Mt,nFORMS(n),partM,auxMC,conjM,localM); |
---|
2718 | setring BR; |
---|
2719 | kill(SS); |
---|
2720 | } |
---|
2721 | } |
---|
2722 | return(Ker(totalM)); |
---|
2723 | } |
---|
2724 | /////////////////////////////////////////////////////////////////////////////// |
---|
2725 | static proc local_IN (poly h,int m) |
---|
2726 | { |
---|
2727 | // computes the intersection number of h and the curve CHI at a certain place |
---|
2728 | // returns a list with the intersection number and the "leading coefficient" |
---|
2729 | // the procedure must be called inside a local ring, h must be a local |
---|
2730 | // equation with respect to the desired place, and m indicates the |
---|
2731 | // number of place inside that local ring, containing lists |
---|
2732 | // POINT(S)/BRANCH(ES)/PARAMETRIZATION(S) when m=0 an "extra place" is |
---|
2733 | // considered |
---|
2734 | def BR=basering; |
---|
2735 | if (m>0) |
---|
2736 | { |
---|
2737 | int nr=ncols(BRANCHES[m][1]); |
---|
2738 | poly xt=PARAMETRIZATIONS[m][1][1]; |
---|
2739 | poly yt=PARAMETRIZATIONS[m][1][2]; |
---|
2740 | int ordx=PARAMETRIZATIONS[m][2][1]; |
---|
2741 | int ordy=PARAMETRIZATIONS[m][2][2]; |
---|
2742 | map phi=BR,xt,yt,1; |
---|
2743 | poly ht=phi(h); |
---|
2744 | int inum=mindeg(ht); |
---|
2745 | if ( ordx>-1 || ordy>-1 ) |
---|
2746 | { |
---|
2747 | while ( ( inum>ordx && ordx>-1 ) || ( inum>ordy && ordy>-1 ) ) |
---|
2748 | { |
---|
2749 | BRANCHES[m]=extdevelop(BRANCHES[m],2*nr); |
---|
2750 | nr=ncols(BRANCHES[m][1]); |
---|
2751 | PARAMETRIZATIONS[m]=param(BRANCHES[m],0); |
---|
2752 | xt=PARAMETRIZATIONS[m][1][1]; |
---|
2753 | yt=PARAMETRIZATIONS[m][1][2]; |
---|
2754 | ordx=PARAMETRIZATIONS[m][2][1]; |
---|
2755 | ordy=PARAMETRIZATIONS[m][2][2]; |
---|
2756 | ht=phi(h); |
---|
2757 | inum=mindeg(ht); |
---|
2758 | } |
---|
2759 | } |
---|
2760 | } |
---|
2761 | else |
---|
2762 | { |
---|
2763 | int nr=ncols(BRANCH[1]); |
---|
2764 | poly xt=PARAMETRIZATION[1][1]; |
---|
2765 | poly yt=PARAMETRIZATION[1][2]; |
---|
2766 | int ordx=PARAMETRIZATION[2][1]; |
---|
2767 | int ordy=PARAMETRIZATION[2][2]; |
---|
2768 | map phi=basering,xt,yt,1; |
---|
2769 | poly ht=phi(h); |
---|
2770 | int inum=mindeg(ht); |
---|
2771 | if ( ordx>-1 || ordy>-1 ) |
---|
2772 | { |
---|
2773 | while ( ( inum>ordx && ordx>-1 ) || ( inum>ordy && ordy>-1 ) ) |
---|
2774 | { |
---|
2775 | BRANCH=extdevelop(BRANCH,2*nr); |
---|
2776 | nr=ncols(BRANCH[1]); |
---|
2777 | PARAMETRIZATION=param(BRANCH,0); |
---|
2778 | xt=PARAMETRIZATION[1][1]; |
---|
2779 | yt=PARAMETRIZATION[1][2]; |
---|
2780 | ordx=PARAMETRIZATION[2][1]; |
---|
2781 | ordy=PARAMETRIZATION[2][2]; |
---|
2782 | ht=phi(h); |
---|
2783 | inum=mindeg(ht); |
---|
2784 | } |
---|
2785 | } |
---|
2786 | } |
---|
2787 | list answer=list(); |
---|
2788 | answer[1]=inum; |
---|
2789 | number AA=number(coeffs(ht,t)[inum+1,1]); |
---|
2790 | answer[2]=AA; |
---|
2791 | return(answer); |
---|
2792 | } |
---|
2793 | /////////////////////////////////////////////////////////////////////////////// |
---|
2794 | static proc extra_place (ideal P) |
---|
2795 | { |
---|
2796 | // computes the "rational" place which is defined over a (closed) "extra" |
---|
2797 | // point |
---|
2798 | // an "extra" point will be necessarily affine, non-singular and non-rational |
---|
2799 | // creates : a specific local ring to deal with the (unique) place above it |
---|
2800 | // returns : list with the above local ring and the degree of the point/place |
---|
2801 | // warning : the procedure must be called inside the affine ring aff_r |
---|
2802 | def base_r=basering; |
---|
2803 | int ext=deg(P[1]); |
---|
2804 | poly aux=subst(P[2],y,1); |
---|
2805 | ext=ext*deg(aux); |
---|
2806 | // P is assumed to be a std. resp. "(x,y),lp" and thus P[1] depends only |
---|
2807 | // on "y" |
---|
2808 | if (deg(P[1])==1) |
---|
2809 | { |
---|
2810 | // the point is non-rational but the second component needs no field |
---|
2811 | // extension |
---|
2812 | number B=-number(subst(P[1],y,0)); |
---|
2813 | poly aux2=subst(P[2],y,B); |
---|
2814 | // careful : the parameter will be called "a" anyway |
---|
2815 | ring ES=(char(basering),a),(x,y,t),ls; |
---|
2816 | map psi=base_r,a,0; |
---|
2817 | minpoly=number(psi(aux2)); |
---|
2818 | kill(psi); |
---|
2819 | number A=a; |
---|
2820 | number B=imap(base_r,B); |
---|
2821 | } |
---|
2822 | else |
---|
2823 | { |
---|
2824 | if (deg(subst(P[2],y,1))==1) |
---|
2825 | { |
---|
2826 | // the point is non-rational but the needed minpoly is just P[1] |
---|
2827 | // careful : the parameter will be called "a" anyway |
---|
2828 | poly P1=P[1]; |
---|
2829 | poly P2=P[2]; |
---|
2830 | ring ES=(char(basering),a),(x,y,t),ls; |
---|
2831 | map psi=base_r,0,a; |
---|
2832 | minpoly=number(psi(P1)); |
---|
2833 | kill(psi); |
---|
2834 | poly aux1=imap(base_r,P2); |
---|
2835 | poly aux2=subst(aux1,y,a); |
---|
2836 | number A=-number(subst(aux2,x,0)); |
---|
2837 | number B=a; |
---|
2838 | } |
---|
2839 | else |
---|
2840 | { |
---|
2841 | // this is the most complicated case of non-rational point |
---|
2842 | poly P1=P[1]; |
---|
2843 | poly P2=P[2]; |
---|
2844 | int p=char(basering); |
---|
2845 | int Q=p^ext; |
---|
2846 | ring aux_r=(Q,a),(x,y,t),ls; |
---|
2847 | string minpoly_string=string(minpoly); |
---|
2848 | ring ES=(char(basering),a),(x,y,t),ls; |
---|
2849 | execute("minpoly="+minpoly_string+";"); |
---|
2850 | poly P_1=imap(base_r,P1); |
---|
2851 | poly P_2=imap(base_r,P2); |
---|
2852 | ideal factors1=factorize(P_1,1); |
---|
2853 | number B=-number(subst(factors1[1],y,0)); |
---|
2854 | poly P_0=subst(P_2,y,B); |
---|
2855 | ideal factors2=factorize(P_0,1); |
---|
2856 | number A=-number(subst(factors2[1],x,0)); |
---|
2857 | kill(aux_r); |
---|
2858 | } |
---|
2859 | } |
---|
2860 | list POINT=list(); |
---|
2861 | POINT[1]=A; |
---|
2862 | POINT[2]=B; |
---|
2863 | export(POINT); |
---|
2864 | map phi=base_r,x+A,y+B; |
---|
2865 | poly LOC_EQ=phi(CHI); |
---|
2866 | kill(A,B,phi); |
---|
2867 | list L@HNE=essdevelop(LOC_EQ)[1]; |
---|
2868 | export(L@HNE); |
---|
2869 | int m=nrows(L@HNE[1]); |
---|
2870 | int n=ncols(L@HNE[1]); |
---|
2871 | intvec Li2=L@HNE[2]; |
---|
2872 | int Li3=L@HNE[3]; |
---|
2873 | setring ES; |
---|
2874 | string newa=subfield(HNEring); |
---|
2875 | poly paux=importdatum(HNEring,"L@HNE[4]",newa); |
---|
2876 | matrix Maux[m][n]; |
---|
2877 | int i,j; |
---|
2878 | for (i=1;i<=m;i=i+1) |
---|
2879 | { |
---|
2880 | for (j=1;j<=n;j=j+1) |
---|
2881 | { |
---|
2882 | Maux[i,j]=importdatum(HNEring,"L@HNE[1]["+string(i)+","+ |
---|
2883 | string(j)+"]",newa); |
---|
2884 | } |
---|
2885 | } |
---|
2886 | list BRANCH=list(); |
---|
2887 | BRANCH[1]=Maux; |
---|
2888 | BRANCH[2]=Li2; |
---|
2889 | BRANCH[3]=Li3; |
---|
2890 | BRANCH[4]=paux; |
---|
2891 | export(BRANCH); |
---|
2892 | list PARAMETRIZATION=param(BRANCH,0); |
---|
2893 | export(PARAMETRIZATION); |
---|
2894 | kill(HNEring); |
---|
2895 | setring base_r; |
---|
2896 | list answer=list(); |
---|
2897 | answer[1]=ES; |
---|
2898 | answer[2]=ext; |
---|
2899 | kill(ES); |
---|
2900 | return(answer); |
---|
2901 | } |
---|
2902 | /////////////////////////////////////////////////////////////////////////////// |
---|
2903 | static proc intersection_div (poly H,list CURVE) |
---|
2904 | "USAGE: intersection_div(H,CURVE), where H is a homogeneous polynomial |
---|
2905 | in ring Proj_R=p,(x,y,z),lp and CURVE is the list of data for |
---|
2906 | the given curve |
---|
2907 | CREATE: new places which had not been computed in advance if necessary |
---|
2908 | those places are stored each one in a local ring where you find |
---|
2909 | lists POINT,BRANCH,PARAMETRIZATION for the place living in that |
---|
2910 | ring; the degree of the point/place in such a ring is stored in |
---|
2911 | an intvec, and the base points in the remaining list |
---|
2912 | Everything is exported in a list @EXTRA@ inside the ring |
---|
2913 | aff_r=CURVE[1][1] and returned with the updated CURVE |
---|
2914 | RETURN: list with the intersection divisor (intvec) between the underlying |
---|
2915 | curve and H=0, and the list CURVE updated |
---|
2916 | SEE ALSO: Adj_div, NSplaces, closed_points |
---|
2917 | NOTE: The procedure must be called from the ring Proj_R=CURVE[1][2] |
---|
2918 | (projective). |
---|
2919 | If @EXTRA@ already exists, the new places are added to the |
---|
2920 | previous data. |
---|
2921 | " |
---|
2922 | { |
---|
2923 | // computes the intersection divisor of H and the curve CHI |
---|
2924 | // returns a list with (possibly) "extra places" and it must be called |
---|
2925 | // inside Proj_R |
---|
2926 | // in case of extra places, some local rings ES(1) ... ES(m) are created |
---|
2927 | // together with an integer list "extra_dgs" containing the degrees of |
---|
2928 | // those places |
---|
2929 | intvec opgt=option(get); |
---|
2930 | option(redSB); |
---|
2931 | intvec interdiv; |
---|
2932 | def BRing=basering; |
---|
2933 | int Tz1=deg(H); |
---|
2934 | list Places=CURVE[3]; |
---|
2935 | int N=size(Places); |
---|
2936 | def aff_r=CURVE[1][1]; |
---|
2937 | setring aff_r; |
---|
2938 | if (defined(@EXTRA@)==0) |
---|
2939 | { |
---|
2940 | list @EXTRA@=list(); |
---|
2941 | list EP=list(); |
---|
2942 | list ES=list(); |
---|
2943 | list extra_dgs=list(); |
---|
2944 | } |
---|
2945 | else |
---|
2946 | { |
---|
2947 | list EP=@EXTRA@[1]; |
---|
2948 | list ES=@EXTRA@[2]; |
---|
2949 | list extra_dgs=@EXTRA@[3]; |
---|
2950 | } |
---|
2951 | int NN=size(extra_dgs); |
---|
2952 | int counterEPl=0; |
---|
2953 | setring BRing; |
---|
2954 | poly h=subst(H,z,1); |
---|
2955 | int Tz2=deg(h); |
---|
2956 | int Tz3=Tz1-Tz2; |
---|
2957 | int i,j,k,l,m,n,s,np,NP,I_N; |
---|
2958 | if (Tz3==0) |
---|
2959 | { |
---|
2960 | // if this still does not work -> try always with ALL points in |
---|
2961 | // Inf_Points !!!! |
---|
2962 | poly Hinf=subst(H,z,0); |
---|
2963 | setring aff_r; |
---|
2964 | // compute the points at infinity of H and see which of them are in |
---|
2965 | // Inf_Points |
---|
2966 | poly h=imap(BRing,h); |
---|
2967 | poly hinf=imap(BRing,Hinf); |
---|
2968 | ideal pinf=factorize(hinf,1); |
---|
2969 | list TIP=Inf_Points[1]+Inf_Points[2]; |
---|
2970 | s=size(TIP); |
---|
2971 | NP=size(pinf); |
---|
2972 | for (i=1;i<=NP;i=i+1) |
---|
2973 | { |
---|
2974 | for (j=1;j<=s;j=j+1) |
---|
2975 | { |
---|
2976 | if (pinf[i]==TIP[j][1]) |
---|
2977 | { |
---|
2978 | np=size(TIP[j][2]); |
---|
2979 | for (k=1;k<=np;k=k+1) |
---|
2980 | { |
---|
2981 | n=TIP[j][2][k]; |
---|
2982 | l=Places[n][1]; |
---|
2983 | m=Places[n][2]; |
---|
2984 | def SS=CURVE[5][l][1]; |
---|
2985 | setring SS; |
---|
2986 | // local equation h of H |
---|
2987 | if (POINTS[m][2]==number(1)) |
---|
2988 | { |
---|
2989 | number A=POINTS[m][1]; |
---|
2990 | map psi=BRing,x+A,1,y; |
---|
2991 | kill(A); |
---|
2992 | } |
---|
2993 | else |
---|
2994 | { |
---|
2995 | map psi=BRing,1,x,y; |
---|
2996 | } |
---|
2997 | poly h=psi(H); |
---|
2998 | I_N=local_IN(h,m)[1]; |
---|
2999 | interdiv[n]=I_N; |
---|
3000 | kill(h,psi); |
---|
3001 | setring aff_r; |
---|
3002 | kill(SS); |
---|
3003 | } |
---|
3004 | break; |
---|
3005 | } |
---|
3006 | } |
---|
3007 | } |
---|
3008 | kill(hinf,pinf); |
---|
3009 | } |
---|
3010 | else |
---|
3011 | { |
---|
3012 | // H is a multiple of z and hence all the points in Inf_Points intersect |
---|
3013 | // with H |
---|
3014 | setring aff_r; |
---|
3015 | poly h=imap(BRing,h); |
---|
3016 | list TIP=Inf_Points[1]+Inf_Points[2]; |
---|
3017 | s=size(TIP); |
---|
3018 | for (j=1;j<=s;j=j+1) |
---|
3019 | { |
---|
3020 | np=size(TIP[j][2]); |
---|
3021 | for (k=1;k<=np;k=k+1) |
---|
3022 | { |
---|
3023 | n=TIP[j][2][k]; |
---|
3024 | l=Places[n][1]; |
---|
3025 | m=Places[n][2]; |
---|
3026 | def SS=CURVE[5][l][1]; |
---|
3027 | setring SS; |
---|
3028 | // local equation h of H |
---|
3029 | if (POINTS[m][2]==number(1)) |
---|
3030 | { |
---|
3031 | number A=POINTS[m][1]; |
---|
3032 | map psi=BRing,x+A,1,y; |
---|
3033 | kill(A); |
---|
3034 | } |
---|
3035 | else |
---|
3036 | { |
---|
3037 | map psi=BRing,1,x,y; |
---|
3038 | } |
---|
3039 | poly h=psi(H); |
---|
3040 | I_N=local_IN(h,m)[1]; |
---|
3041 | interdiv[n]=I_N; |
---|
3042 | kill(h,psi); |
---|
3043 | setring aff_r; |
---|
3044 | kill(SS); |
---|
3045 | } |
---|
3046 | } |
---|
3047 | } |
---|
3048 | // compute common affine points of H and CHI |
---|
3049 | ideal CAL=h,CHI; |
---|
3050 | CAL=std(CAL); |
---|
3051 | if (CAL<>1) |
---|
3052 | { |
---|
3053 | list TAP=list(); |
---|
3054 | TAP=closed_points(CAL); |
---|
3055 | NP=size(TAP); |
---|
3056 | list auxP=list(); |
---|
3057 | int dP; |
---|
3058 | for (i=1;i<=NP;i=i+1) |
---|
3059 | { |
---|
3060 | if (belongs(TAP[i],Aff_SLocus)==1) |
---|
3061 | { |
---|
3062 | // search the point in the list Aff_SPoints |
---|
3063 | j=isInLP(TAP[i],Aff_SPoints); |
---|
3064 | np=size(Aff_SPoints[j][2]); |
---|
3065 | for (k=1;k<=np;k=k+1) |
---|
3066 | { |
---|
3067 | n=Aff_SPoints[j][2][k]; |
---|
3068 | l=Places[n][1]; |
---|
3069 | m=Places[n][2]; |
---|
3070 | def SS=CURVE[5][l][1]; |
---|
3071 | setring SS; |
---|
3072 | // local equation h of H |
---|
3073 | number A=POINTS[m][1]; |
---|
3074 | number B=POINTS[m][2]; |
---|
3075 | map psi=BRing,x+A,y+B,1; |
---|
3076 | poly h=psi(H); |
---|
3077 | I_N=local_IN(h,m)[1]; |
---|
3078 | interdiv[n]=I_N; |
---|
3079 | kill(A,B,h,psi); |
---|
3080 | setring aff_r; |
---|
3081 | kill(SS); |
---|
3082 | } |
---|
3083 | } |
---|
3084 | else |
---|
3085 | { |
---|
3086 | auxP=list(); |
---|
3087 | auxP[1]=TAP[i]; |
---|
3088 | dP=degree_P(auxP); |
---|
3089 | if (defined(Aff_Points(dP))<>0) |
---|
3090 | { |
---|
3091 | // search the point in the list Aff_Points(dP) |
---|
3092 | j=isInLP(TAP[i],Aff_Points(dP)); |
---|
3093 | n=Aff_Points(dP)[j][2][1]; |
---|
3094 | l=Places[n][1]; |
---|
3095 | m=Places[n][2]; |
---|
3096 | def SS=CURVE[5][l][1]; |
---|
3097 | setring SS; |
---|
3098 | // local equation h of H |
---|
3099 | number A=POINTS[m][1]; |
---|
3100 | number B=POINTS[m][2]; |
---|
3101 | map psi=BRing,x+A,y+B,1; |
---|
3102 | poly h=psi(H); |
---|
3103 | I_N=local_IN(h,m)[1]; |
---|
3104 | interdiv[n]=I_N; |
---|
3105 | kill(A,B,h,psi); |
---|
3106 | setring aff_r; |
---|
3107 | kill(SS); |
---|
3108 | } |
---|
3109 | else |
---|
3110 | { |
---|
3111 | // previously check if it is an existing "extra place" |
---|
3112 | j=isInLP(TAP[i],EP); |
---|
3113 | if (j>0) |
---|
3114 | { |
---|
3115 | def SS=ES[j]; |
---|
3116 | setring SS; |
---|
3117 | // local equation h of H |
---|
3118 | number A=POINT[1]; |
---|
3119 | number B=POINT[2]; |
---|
3120 | map psi=BRing,x+A,y+B,1; |
---|
3121 | poly h=psi(H); |
---|
3122 | I_N=local_IN(h,0)[1]; |
---|
3123 | interdiv[N+j]=I_N; |
---|
3124 | setring aff_r; |
---|
3125 | kill(SS); |
---|
3126 | } |
---|
3127 | else |
---|
3128 | { |
---|
3129 | // then we must create a new "extra place" |
---|
3130 | counterEPl=counterEPl+1; |
---|
3131 | list EXTRA_PLACE=extra_place(TAP[i]); |
---|
3132 | def SS=EXTRA_PLACE[1]; |
---|
3133 | ES[NN+counterEPl]=SS; |
---|
3134 | extra_dgs[NN+counterEPl]=EXTRA_PLACE[2]; |
---|
3135 | EP[NN+counterEPl]=list(); |
---|
3136 | EP[NN+counterEPl][1]=TAP[i]; |
---|
3137 | EP[NN+counterEPl][2]=0; |
---|
3138 | setring SS; |
---|
3139 | // local equation h of H |
---|
3140 | number A=POINT[1]; |
---|
3141 | number B=POINT[2]; |
---|
3142 | map psi=BRing,x+A,y+B,1; |
---|
3143 | poly h=psi(H); |
---|
3144 | I_N=local_IN(h,0)[1]; |
---|
3145 | kill(A,B,h,psi); |
---|
3146 | interdiv[N+NN+counterEPl]=I_N; |
---|
3147 | setring aff_r; |
---|
3148 | kill(SS); |
---|
3149 | } |
---|
3150 | } |
---|
3151 | } |
---|
3152 | } |
---|
3153 | kill(TAP,auxP); |
---|
3154 | } |
---|
3155 | kill(h,CAL,TIP); |
---|
3156 | @EXTRA@[1]=EP; |
---|
3157 | @EXTRA@[2]=ES; |
---|
3158 | @EXTRA@[3]=extra_dgs; |
---|
3159 | kill(EP); |
---|
3160 | list update_CURVE=CURVE; |
---|
3161 | if (size(extra_dgs)>0) |
---|
3162 | { |
---|
3163 | export(@EXTRA@); |
---|
3164 | update_CURVE[1][1]=basering; |
---|
3165 | } |
---|
3166 | else |
---|
3167 | { |
---|
3168 | kill(@EXTRA@); |
---|
3169 | } |
---|
3170 | setring BRing; |
---|
3171 | kill(h); |
---|
3172 | kill(aff_r); |
---|
3173 | list answer=list(); |
---|
3174 | answer[1]=interdiv; |
---|
3175 | answer[2]=update_CURVE; |
---|
3176 | option(set,opgt); |
---|
3177 | return(answer); |
---|
3178 | } |
---|
3179 | /////////////////////////////////////////////////////////////////////////////// |
---|
3180 | static proc local_eq (poly H,SS,int m) |
---|
3181 | { |
---|
3182 | // computes a local equation of poly H in the ring SS related to the place |
---|
3183 | // "m" |
---|
3184 | // list POINT/POINTS is searched depending on wether m=0 or m>0 respectively |
---|
3185 | // warning : the procedure must be called from ring "Proj_R" and returns a |
---|
3186 | // string |
---|
3187 | def BRing=basering; |
---|
3188 | setring SS; |
---|
3189 | if (m>0) |
---|
3190 | { |
---|
3191 | if (POINTS[m][3]==number(1)) |
---|
3192 | { |
---|
3193 | number A=POINTS[m][1]; |
---|
3194 | number B=POINTS[m][2]; |
---|
3195 | map psi=BRing,x+A,y+B,1; |
---|
3196 | } |
---|
3197 | else |
---|
3198 | { |
---|
3199 | if (POINTS[m][2]==number(1)) |
---|
3200 | { |
---|
3201 | number A=POINTS[m][1]; |
---|
3202 | map psi=BRing,x+A,1,y; |
---|
3203 | } |
---|
3204 | else |
---|
3205 | { |
---|
3206 | map psi=BRing,1,x,y; |
---|
3207 | } |
---|
3208 | } |
---|
3209 | } |
---|
3210 | else |
---|
3211 | { |
---|
3212 | number A=POINT[1]; |
---|
3213 | number B=POINT[2]; |
---|
3214 | map psi=BRing,x+A,y+B,1; |
---|
3215 | } |
---|
3216 | poly h=psi(H); |
---|
3217 | string str_h=string(h); |
---|
3218 | setring BRing; |
---|
3219 | return(str_h); |
---|
3220 | } |
---|
3221 | /////////////////////////////////////////////////////////////////////////////// |
---|
3222 | static proc min_wt_rmat (matrix M) |
---|
3223 | { |
---|
3224 | // finds the row of M with minimum non-zero entries, i.e. minimum |
---|
3225 | // "Hamming-weight" |
---|
3226 | int m=nrows(M); |
---|
3227 | int n=ncols(M); |
---|
3228 | int i,j; |
---|
3229 | int Hwt=0; |
---|
3230 | for (j=1;j<=n;j=j+1) |
---|
3231 | { |
---|
3232 | if (M[1,j]<>0) |
---|
3233 | { |
---|
3234 | Hwt=Hwt+1; |
---|
3235 | } |
---|
3236 | } |
---|
3237 | int minHwt=Hwt; |
---|
3238 | int k=1; |
---|
3239 | for (i=2;i<=m;i=i+1) |
---|
3240 | { |
---|
3241 | Hwt=0; |
---|
3242 | for (j=1;j<=n;j=j+1) |
---|
3243 | { |
---|
3244 | if (M[i,j]<>0) |
---|
3245 | { |
---|
3246 | Hwt=Hwt+1; |
---|
3247 | } |
---|
3248 | } |
---|
3249 | if (Hwt<minHwt) |
---|
3250 | { |
---|
3251 | minHwt=Hwt; |
---|
3252 | k=i; |
---|
3253 | } |
---|
3254 | } |
---|
3255 | return(k); |
---|
3256 | } |
---|
3257 | /////////////////////////////////////////////////////////////////////////////// |
---|
3258 | |
---|
3259 | // ============================================================================ |
---|
3260 | // ******* MAIN PROCEDURE : the Brill-Noether algorithm ******** |
---|
3261 | // ============================================================================ |
---|
3262 | |
---|
3263 | proc BrillNoether (intvec G,list CURVE) |
---|
3264 | "USAGE: BrillNoether(G,CURVE); G an intvec, CURVE a list |
---|
3265 | RETURN: list of ideals (each of them with two homogeneous generators, |
---|
3266 | which represent the numerator, resp. denominator, of a rational |
---|
3267 | function).@* |
---|
3268 | The corresponding rational functions form a vector basis of the |
---|
3269 | linear system L(G), G a rational divisor over a non-singular curve. |
---|
3270 | NOTE: The procedure must be called from the ring CURVE[1][2], where |
---|
3271 | CURVE is the output of the procedure @code{NSplaces}. @* |
---|
3272 | The intvec G represents a rational divisor supported on the closed |
---|
3273 | places of CURVE[3] (e.g. @code{G=2,0,-1;} means 2 times the closed |
---|
3274 | place 1 minus 1 times the closed place 3). |
---|
3275 | SEE ALSO: Adj_div, NSplaces, Weierstrass |
---|
3276 | EXAMPLE: example BrillNoether; shows an example |
---|
3277 | " |
---|
3278 | { |
---|
3279 | // computes a vector basis for the space L(G), |
---|
3280 | // where G is a given rational divisor over the non-singular curve |
---|
3281 | // returns : list of ideals in R each with 2 elements H,Ho such that |
---|
3282 | // the set of functions {H/Ho} is the searched basis |
---|
3283 | // warning : the conductor and sufficiently many points of the plane |
---|
3284 | // curve should be computed in advance, in list CURVE |
---|
3285 | // the algorithm of Brill-Noether is carried out in the procedure |
---|
3286 | def BRing=basering; |
---|
3287 | int degX=CURVE[2][1]; |
---|
3288 | list Places=CURVE[3]; |
---|
3289 | intvec Conductor=CURVE[4]; |
---|
3290 | if (deg_D(G,Places)<0) |
---|
3291 | { |
---|
3292 | return(list()); |
---|
3293 | } |
---|
3294 | intvec nuldiv; |
---|
3295 | if (G==nuldiv) |
---|
3296 | { |
---|
3297 | list quickL=list(); |
---|
3298 | ideal quickId; |
---|
3299 | quickId[1]=1; |
---|
3300 | quickId[2]=1; |
---|
3301 | quickL[1]=quickId; |
---|
3302 | return(quickL); |
---|
3303 | } |
---|
3304 | intvec J=max_D(G,nuldiv)+Conductor; |
---|
3305 | int n=estim_n(J,degX,Places); |
---|
3306 | dbprint(printlevel+1,"Forms of degree "+string(n)+" : "); |
---|
3307 | matrix W=nmultiples(n,degX,CHI); |
---|
3308 | kill(nFORMS(n-degX)); |
---|
3309 | list update_CURVE=CURVE; |
---|
3310 | matrix V=interpolating_forms(J,n,update_CURVE); |
---|
3311 | matrix VmW=supplement(W,V); |
---|
3312 | int k=min_wt_rmat(VmW); |
---|
3313 | int N=size(nFORMS(n)); |
---|
3314 | matrix H0[1][N]; |
---|
3315 | int i,j; |
---|
3316 | for (i=1;i<=N;i=i+1) |
---|
3317 | { |
---|
3318 | H0[1,i]=VmW[k,i]; |
---|
3319 | } |
---|
3320 | poly Ho; |
---|
3321 | for (i=1;i<=N;i=i+1) |
---|
3322 | { |
---|
3323 | Ho=Ho+(H0[1,i]*nFORMS(n)[i]); |
---|
3324 | } |
---|
3325 | list INTERD=intersection_div(Ho,update_CURVE); |
---|
3326 | intvec NHo=INTERD[1]; |
---|
3327 | update_CURVE=INTERD[2]; |
---|
3328 | intvec AR=NHo-G; |
---|
3329 | matrix V2=interpolating_forms(AR,n,update_CURVE); |
---|
3330 | def aux_RING=update_CURVE[1][1]; |
---|
3331 | setring aux_RING; |
---|
3332 | if (defined(@EXTRA@)<>0) |
---|
3333 | { |
---|
3334 | kill(@EXTRA@); |
---|
3335 | } |
---|
3336 | setring BRing; |
---|
3337 | update_CURVE[1][1]=aux_RING; |
---|
3338 | kill(aux_RING); |
---|
3339 | matrix B0=supplement(W,V2); |
---|
3340 | if (Hamming_wt(B0)==0) |
---|
3341 | { |
---|
3342 | return(list()); |
---|
3343 | } |
---|
3344 | int ld=nrows(B0); |
---|
3345 | list Bres=list(); |
---|
3346 | ideal HH; |
---|
3347 | poly H; |
---|
3348 | for (j=1;j<=ld;j=j+1) |
---|
3349 | { |
---|
3350 | H=0; |
---|
3351 | for (i=1;i<=N;i=i+1) |
---|
3352 | { |
---|
3353 | H=H+(B0[j,i]*nFORMS(n)[i]); |
---|
3354 | } |
---|
3355 | HH=H,Ho; |
---|
3356 | Bres[j]=simplifyRF(HH); |
---|
3357 | } |
---|
3358 | kill(nFORMS(n)); |
---|
3359 | dbprint(printlevel+1," "); |
---|
3360 | dbprint(printlevel+2,"Vector basis successfully computed "); |
---|
3361 | dbprint(printlevel+1," "); |
---|
3362 | return(Bres); |
---|
3363 | } |
---|
3364 | example |
---|
3365 | { |
---|
3366 | "EXAMPLE:"; echo = 2; |
---|
3367 | int plevel=printlevel; |
---|
3368 | printlevel=-1; |
---|
3369 | ring s=2,(x,y),lp; |
---|
3370 | list C=Adj_div(x3y+y3+x); |
---|
3371 | C=NSplaces(3,C); |
---|
3372 | // the first 3 Places in C[3] are of degree 1. |
---|
3373 | // we define the rational divisor G = 4*C[3][1]+4*C[3][3] (of degree 8): |
---|
3374 | intvec G=4,0,4; |
---|
3375 | def R=C[1][2]; |
---|
3376 | setring R; |
---|
3377 | list LG=BrillNoether(G,C); |
---|
3378 | // here is the vector basis of L(G): |
---|
3379 | LG; |
---|
3380 | printlevel=plevel; |
---|
3381 | } |
---|
3382 | /////////////////////////////////////////////////////////////////////////////// |
---|
3383 | |
---|
3384 | // *** procedures for dealing with "RATIONAL FUNCTIONS" over a plane curve *** |
---|
3385 | // a rational function F may be given by (homogeneous) ideal or (affine) poly |
---|
3386 | // (or number) |
---|
3387 | |
---|
3388 | static proc polytoRF (F) |
---|
3389 | { |
---|
3390 | // converts a poly (or number) into a "rational function" of type "ideal" |
---|
3391 | // warning : it must be called inside "R" and poly should be affine |
---|
3392 | ideal RF; |
---|
3393 | RF[1]=homog(F,z); |
---|
3394 | RF[2]=z^(deg(F)); |
---|
3395 | return(RF); |
---|
3396 | } |
---|
3397 | /////////////////////////////////////////////////////////////////////////////// |
---|
3398 | static proc simplifyRF (ideal F) |
---|
3399 | { |
---|
3400 | // simplifies a rational function f/g extracting the gcd(f,g) |
---|
3401 | // maybe add a "restriction" to the curve "CHI" but it is not easy to |
---|
3402 | // programm |
---|
3403 | poly auxp=gcd(F[1],F[2]); |
---|
3404 | return(ideal(division(F,auxp)[1])); |
---|
3405 | } |
---|
3406 | /////////////////////////////////////////////////////////////////////////////// |
---|
3407 | static proc sumRF (F,G) |
---|
3408 | { |
---|
3409 | // sum of two "rational functions" F,G given by either a pair |
---|
3410 | // numerator/denominator or a poly |
---|
3411 | if ( typeof(F)=="ideal" && typeof(G)=="ideal" ) |
---|
3412 | { |
---|
3413 | ideal FG; |
---|
3414 | FG[1]=F[1]*G[2]+F[2]*G[1]; |
---|
3415 | FG[2]=F[2]*G[2]; |
---|
3416 | return(simplifyRF(FG)); |
---|
3417 | } |
---|
3418 | else |
---|
3419 | { |
---|
3420 | if (typeof(F)=="ideal") |
---|
3421 | { |
---|
3422 | ideal GG=polytoRF(G); |
---|
3423 | ideal FG; |
---|
3424 | FG[1]=F[1]*GG[2]+F[2]*GG[1]; |
---|
3425 | FG[2]=F[2]*GG[2]; |
---|
3426 | return(simplifyRF(FG)); |
---|
3427 | } |
---|
3428 | else |
---|
3429 | { |
---|
3430 | if (typeof(G)=="ideal") |
---|
3431 | { |
---|
3432 | ideal FF=polytoRF(F); |
---|
3433 | ideal FG; |
---|
3434 | FG[1]=FF[1]*G[2]+FF[2]*G[1]; |
---|
3435 | FG[2]=FF[2]*G[2]; |
---|
3436 | return(simplifyRF(FG)); |
---|
3437 | } |
---|
3438 | else |
---|
3439 | { |
---|
3440 | return(F+G); |
---|
3441 | } |
---|
3442 | } |
---|
3443 | } |
---|
3444 | } |
---|
3445 | /////////////////////////////////////////////////////////////////////////////// |
---|
3446 | static proc negRF (F) |
---|
3447 | { |
---|
3448 | // returns -F as rational function |
---|
3449 | if (typeof(F)=="ideal") |
---|
3450 | { |
---|
3451 | ideal FF=F; |
---|
3452 | FF[1]=-F[1]; |
---|
3453 | return(FF); |
---|
3454 | } |
---|
3455 | else |
---|
3456 | { |
---|
3457 | return(-F); |
---|
3458 | } |
---|
3459 | } |
---|
3460 | /////////////////////////////////////////////////////////////////////////////// |
---|
3461 | static proc escprodRF (l,F) |
---|
3462 | { |
---|
3463 | // computes l*F as rational function |
---|
3464 | // l should be either a number or a poly of degree zero |
---|
3465 | if (typeof(F)=="ideal") |
---|
3466 | { |
---|
3467 | ideal lF=F; |
---|
3468 | lF[1]=l*F[1]; |
---|
3469 | return(lF); |
---|
3470 | } |
---|
3471 | else |
---|
3472 | { |
---|
3473 | return(l*F); |
---|
3474 | } |
---|
3475 | } |
---|
3476 | /////////////////////////////////////////////////////////////////////////////// |
---|
3477 | |
---|
3478 | // ******** procedures to compute Weierstrass semigroups ******** |
---|
3479 | |
---|
3480 | static proc orderRF (ideal F,SS,int m) |
---|
3481 | "USAGE: orderRF(F,SS,m); F an ideal, SS a ring and m an integer |
---|
3482 | RETURN: list with the order (int) and the leading coefficient (number) |
---|
3483 | NOTE: F represents a rational function, thus the procedure must be |
---|
3484 | called from global ring R or R(d). |
---|
3485 | SS contains the name of a local ring where rational places are |
---|
3486 | stored, and then we take that which is in position m in the |
---|
3487 | corresponding lists of data. |
---|
3488 | The order of F at the place given by SS,m is returned together |
---|
3489 | with the coefficient of minimum degree in the corresponding power |
---|
3490 | series. |
---|
3491 | " |
---|
3492 | { |
---|
3493 | // computes the order of a rational function F at a RATIONAL place given by |
---|
3494 | // a local ring SS and a position "m" inside SS |
---|
3495 | // warning : the procedure must be called from global projective ring "R" or |
---|
3496 | // "R(i)" |
---|
3497 | // returns a list with the order (int) and the "leading coefficient" (number) |
---|
3498 | def BR=basering; |
---|
3499 | poly f=F[1]; |
---|
3500 | string sf=local_eq(f,SS,m); |
---|
3501 | poly g=F[2]; |
---|
3502 | string sg=local_eq(g,SS,m); |
---|
3503 | setring SS; |
---|
3504 | execute("poly ff="+sf+";"); |
---|
3505 | execute("poly gg="+sg+";"); |
---|
3506 | list o1=local_IN(ff,m); |
---|
3507 | list o2=local_IN(gg,m); |
---|
3508 | int oo=o1[1]-o2[1]; |
---|
3509 | number lc=o1[2]/o2[2]; |
---|
3510 | setring BR; |
---|
3511 | number LC=number(imap(SS,lc)); |
---|
3512 | return(list(oo,LC)); |
---|
3513 | } |
---|
3514 | /////////////////////////////////////////////////////////////////////////////// |
---|
3515 | |
---|
3516 | proc Weierstrass (int P,int m,list CURVE) |
---|
3517 | "USAGE: Weierstrass( i, m, CURVE ); i,m integers and CURVE a list |
---|
3518 | RETURN: list WS of two lists: |
---|
3519 | @format |
---|
3520 | WS[1] list of integers (Weierstr. semigroup of the curve at place i up to m) |
---|
3521 | WS[2] list of ideals (the associated rational functions) |
---|
3522 | @end format |
---|
3523 | NOTE: The procedure must be called from the ring CURVE[1][2], |
---|
3524 | where CURVE is the output of the procedure @code{NSplaces}. |
---|
3525 | @* i represents the place CURVE[3][i]. |
---|
3526 | @* Rational functions are represented by numerator/denominator |
---|
3527 | in form of ideals with two homogeneous generators. |
---|
3528 | WARNING: The place must be rational, i.e., necessarily CURVE[3][i][1]=1. @* |
---|
3529 | SEE ALSO: Adj_div, NSplaces, BrillNoether |
---|
3530 | EXAMPLE: example Weierstrass; shows an example |
---|
3531 | " |
---|
3532 | { |
---|
3533 | // computes the Weierstrass semigroup at a RATIONAL place P up to a bound "m" |
---|
3534 | // together with the functions achieving each value up to m, via |
---|
3535 | // Brill-Noether |
---|
3536 | // returns 2 lists : the first consists of all the poles up to m in |
---|
3537 | // increasing order and the second consists of the corresponging rational |
---|
3538 | // functions |
---|
3539 | list Places=CURVE[3]; |
---|
3540 | intvec pl=Places[P]; |
---|
3541 | int dP=pl[1]; |
---|
3542 | int nP=pl[2]; |
---|
3543 | if (dP<>1) |
---|
3544 | { |
---|
3545 | ERROR("the given place is not defined over the prime field"); |
---|
3546 | } |
---|
3547 | if (m<=0) |
---|
3548 | { |
---|
3549 | if (m==0) |
---|
3550 | { |
---|
3551 | list semig=list(); |
---|
3552 | int auxint=0; |
---|
3553 | semig[1]=auxint; |
---|
3554 | list funcs=list(); |
---|
3555 | ideal auxF; |
---|
3556 | auxF[1]=1; |
---|
3557 | auxF[2]=1; |
---|
3558 | funcs[1]=auxF; |
---|
3559 | return(list(semig,funcs)); |
---|
3560 | } |
---|
3561 | else |
---|
3562 | { |
---|
3563 | ERROR("second argument must be non-negative"); |
---|
3564 | } |
---|
3565 | } |
---|
3566 | int auxint=0; |
---|
3567 | ideal auxF; |
---|
3568 | auxF[1]=1; |
---|
3569 | auxF[2]=1; |
---|
3570 | // Brill-Noether algorithm |
---|
3571 | intvec mP; |
---|
3572 | mP[P]=m; |
---|
3573 | list LmP=BrillNoether(mP,CURVE); |
---|
3574 | int lmP=size(LmP); |
---|
3575 | if (lmP==1) |
---|
3576 | { |
---|
3577 | list semig=list(); |
---|
3578 | semig[1]=auxint; |
---|
3579 | list funcs=list(); |
---|
3580 | funcs[1]=auxF; |
---|
3581 | return(list(semig,funcs)); |
---|
3582 | } |
---|
3583 | def SS=CURVE[5][dP][1]; |
---|
3584 | list ordLmP=list(); |
---|
3585 | list polLmP=list(); |
---|
3586 | list sortpol=list(); |
---|
3587 | int maxpol; |
---|
3588 | int i,j,k; |
---|
3589 | for (i=1;i<=lmP-1;i=i+1) |
---|
3590 | { |
---|
3591 | for (j=1;j<=lmP-i+1;j=j+1) |
---|
3592 | { |
---|
3593 | ordLmP[j]=orderRF(LmP[j],SS,nP); |
---|
3594 | polLmP[j]=-ordLmP[j][1]; |
---|
3595 | } |
---|
3596 | sortpol=sort(polLmP); |
---|
3597 | polLmP=sortpol[1]; |
---|
3598 | maxpol=polLmP[lmP-i+1]; |
---|
3599 | LmP=permute_L(LmP,sortpol[2]); |
---|
3600 | ordLmP=permute_L(ordLmP,sortpol[2]); |
---|
3601 | // triangulate LmP |
---|
3602 | for (k=1;k<=lmP-i;k=k+1) |
---|
3603 | { |
---|
3604 | if (polLmP[lmP-i+1-k]==maxpol) |
---|
3605 | { |
---|
3606 | LmP[lmP-i+1-k]=sumRF(LmP[lmP-i+1-k],negRF(escprodRF( |
---|
3607 | ordLmP[lmP-i+1-k][2]/ordLmP[lmP-i+1][2],LmP[lmP-i+1]))); |
---|
3608 | } |
---|
3609 | else |
---|
3610 | { |
---|
3611 | break; |
---|
3612 | } |
---|
3613 | } |
---|
3614 | } |
---|
3615 | polLmP[1]=auxint; |
---|
3616 | LmP[1]=auxF; |
---|
3617 | return(list(polLmP,LmP)); |
---|
3618 | } |
---|
3619 | example |
---|
3620 | { |
---|
3621 | "EXAMPLE:"; echo = 2; |
---|
3622 | int plevel=printlevel; |
---|
3623 | printlevel=-1; |
---|
3624 | ring s=2,(x,y),lp; |
---|
3625 | list C=Adj_div(x3y+y3+x); |
---|
3626 | C=NSplaces(3,C); |
---|
3627 | def R=C[1][2]; |
---|
3628 | setring R; |
---|
3629 | // Place C[3][1] has degree 1 (i.e it is rational); |
---|
3630 | list WS=Weierstrass(1,7,C); |
---|
3631 | // the first part of the list is the Weierstrass semigroup up to 7 : |
---|
3632 | WS[1]; |
---|
3633 | // and the second part are the corresponding functions : |
---|
3634 | WS[2]; |
---|
3635 | printlevel=plevel; |
---|
3636 | } |
---|
3637 | /////////////////////////////////////////////////////////////////////////////// |
---|
3638 | |
---|
3639 | // axiliary procedure for permuting a list or intvec |
---|
3640 | |
---|
3641 | proc permute_L (L,P) |
---|
3642 | "USAGE: permute_L( L, P ); L,P either intvecs or lists |
---|
3643 | RETURN: list obtained from L by applying the permutation given by P. |
---|
3644 | NOTE: If P is a list, all entries must be integers. |
---|
3645 | SEE ALSO: sys_code, AGcode_Omega, prepSV |
---|
3646 | EXAMPLE: example permute_L; shows an example |
---|
3647 | " |
---|
3648 | { |
---|
3649 | // permutes the list L according to the permutation P (both intvecs or |
---|
3650 | // lists of integers) |
---|
3651 | int s=size(L); |
---|
3652 | int n=size(P); |
---|
3653 | int i; |
---|
3654 | if (s<n) |
---|
3655 | { |
---|
3656 | for (i=s+1;i<=n;i=i+1) |
---|
3657 | { |
---|
3658 | L[i]=0; |
---|
3659 | } |
---|
3660 | s=size(L); |
---|
3661 | } |
---|
3662 | list auxL=L; |
---|
3663 | for (i=1;i<=n;i=i+1) |
---|
3664 | { |
---|
3665 | auxL[i]=L[P[i]]; |
---|
3666 | } |
---|
3667 | return(auxL); |
---|
3668 | } |
---|
3669 | example |
---|
3670 | { |
---|
3671 | "EXAMPLE:"; echo = 2; |
---|
3672 | list L=list(); |
---|
3673 | L[1]="a"; |
---|
3674 | L[2]="b"; |
---|
3675 | L[3]="c"; |
---|
3676 | L[4]="d"; |
---|
3677 | intvec P=1,3,4,2; |
---|
3678 | // the list L is permuted according to P : |
---|
3679 | permute_L(L,P); |
---|
3680 | } |
---|
3681 | /////////////////////////////////////////////////////////////////////////////// |
---|
3682 | static proc evalRF (ideal F,SS,int m) |
---|
3683 | "USAGE: evalRF(F,SS,m), where F is an ideal, SS is a ring and m is an |
---|
3684 | integer |
---|
3685 | RETURN: the evaluation (number) of F at the place given by SS,m if it is |
---|
3686 | well-defined |
---|
3687 | NOTE: F represents a rational function, thus the procedure must be |
---|
3688 | called from R or R(d). |
---|
3689 | SS contains the name of a local ring where rational places are |
---|
3690 | stored, and then we take that which is in position m in the |
---|
3691 | corresponding lists of data. |
---|
3692 | " |
---|
3693 | { |
---|
3694 | // evaluates a rational function F at a RATIONAL place given by |
---|
3695 | // a local ring SS and a position "m" inside SS |
---|
3696 | list olc=orderRF(F,SS,m); |
---|
3697 | int oo=olc[1]; |
---|
3698 | if (oo==0) |
---|
3699 | { |
---|
3700 | return(olc[2]); |
---|
3701 | } |
---|
3702 | else |
---|
3703 | { |
---|
3704 | if (oo>0) |
---|
3705 | { |
---|
3706 | return(number(0)); |
---|
3707 | } |
---|
3708 | else |
---|
3709 | { |
---|
3710 | ERROR("the function is not well-defined at the given place"); |
---|
3711 | } |
---|
3712 | } |
---|
3713 | } |
---|
3714 | /////////////////////////////////////////////////////////////////////////////// |
---|
3715 | // |
---|
3716 | // ******** procedures for constructing AG codes ******** |
---|
3717 | // |
---|
3718 | /////////////////////////////////////////////////////////////////////////////// |
---|
3719 | |
---|
3720 | static proc gen_mat (list LF,intvec LP,RP) |
---|
3721 | "USAGE: gen_mat(LF,LP,RP); LF list of rational functions, |
---|
3722 | LP intvec of rational places and RP a local ring |
---|
3723 | RETURN: a generator matrix of the evaluation code given by LF and LP |
---|
3724 | SEE ALSO: extcurve |
---|
3725 | KEYWORDS: evaluation codes |
---|
3726 | NOTE: Rational places are searched in local ring RP |
---|
3727 | The procedure must be called from R or R(d) fromlist CURVE |
---|
3728 | after having executed extcurve(d,CURVE) |
---|
3729 | " |
---|
3730 | { |
---|
3731 | // computes a generator matrix (with numbers) of the evaluation code given |
---|
3732 | // by a list of rational functions LF and a list of RATIONAL places LP |
---|
3733 | int m=size(LF); |
---|
3734 | int n=size(LP); |
---|
3735 | matrix GM[m][n]; |
---|
3736 | int i,j; |
---|
3737 | for (i=1;i<=m;i=i+1) |
---|
3738 | { |
---|
3739 | for (j=1;j<=n;j=j+1) |
---|
3740 | { |
---|
3741 | GM[i,j]=evalRF(LF[i],RP,LP[j]); |
---|
3742 | } |
---|
3743 | } |
---|
3744 | return(GM); |
---|
3745 | } |
---|
3746 | /////////////////////////////////////////////////////////////////////////////// |
---|
3747 | |
---|
3748 | proc dual_code (matrix G) |
---|
3749 | "USAGE: dual_code(G); G a matrix of numbers |
---|
3750 | RETURN: a generator matrix of the dual code generated by G |
---|
3751 | NOTE: The input should be a matrix G of numbers. @* |
---|
3752 | The output is also a parity check matrix for the code defined by G |
---|
3753 | KEYWORDS: linear code, dual |
---|
3754 | EXAMPLE: example dual_code; shows an example |
---|
3755 | " |
---|
3756 | { |
---|
3757 | // computes the dual code of C given by a generator matrix G |
---|
3758 | // i.e. computes a parity-check matrix H of C |
---|
3759 | // conversely : computes also G if H is given |
---|
3760 | return(Ker(G)); |
---|
3761 | } |
---|
3762 | example |
---|
3763 | { |
---|
3764 | "EXAMPLE:"; echo = 2; |
---|
3765 | ring s=2,T,lp; |
---|
3766 | // here is the Hamming code of length 7 and dimension 3 |
---|
3767 | matrix G[3][7]=1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0,0,1,1,1,1; |
---|
3768 | print(G); |
---|
3769 | matrix H=dual_code(G); |
---|
3770 | print(H); |
---|
3771 | } |
---|
3772 | /////////////////////////////////////////////////////////////////////////////// |
---|
3773 | |
---|
3774 | // ====================================================================== |
---|
3775 | // *********** initial test for disjointness *************** |
---|
3776 | // ====================================================================== |
---|
3777 | |
---|
3778 | static proc disj_divs (intvec H,intvec P,list EC) |
---|
3779 | { |
---|
3780 | int s1=size(H); |
---|
3781 | int s2=size(P); |
---|
3782 | list PLACES=EC[3]; |
---|
3783 | def BRing=basering; |
---|
3784 | def auxR=EC[1][5]; |
---|
3785 | setring auxR; |
---|
3786 | int s=res_deg(); |
---|
3787 | setring BRing; |
---|
3788 | kill(auxR); |
---|
3789 | int i,j,k,d,l,N,M; |
---|
3790 | intvec auxIV,iw; |
---|
3791 | for (i=1;i<=s;i=i+1) |
---|
3792 | { |
---|
3793 | if ( (s mod i) == 0 ) |
---|
3794 | { |
---|
3795 | def auxR=EC[5][i][1]; |
---|
3796 | setring auxR; |
---|
3797 | auxIV[i]=size(POINTS); |
---|
3798 | setring BRing; |
---|
3799 | kill(auxR); |
---|
3800 | } |
---|
3801 | else |
---|
3802 | { |
---|
3803 | auxIV[i]=0; |
---|
3804 | } |
---|
3805 | } |
---|
3806 | for (i=1;i<=s1;i=i+1) |
---|
3807 | { |
---|
3808 | if (H[i]<>0) |
---|
3809 | { |
---|
3810 | iw=PLACES[i]; |
---|
3811 | d=iw[1]; |
---|
3812 | if ( (s mod d) == 0 ) |
---|
3813 | { |
---|
3814 | l=iw[2]; |
---|
3815 | // check that this place(s) are not in sup(D) |
---|
3816 | if (d==1) |
---|
3817 | { |
---|
3818 | for (j=1;j<=s2;j=j+1) |
---|
3819 | { |
---|
3820 | if (l==P[j]) |
---|
3821 | { |
---|
3822 | return(0); |
---|
3823 | } |
---|
3824 | } |
---|
3825 | } |
---|
3826 | else |
---|
3827 | { |
---|
3828 | N=0; |
---|
3829 | for (j=1;j<d;j=j+1) |
---|
3830 | { |
---|
3831 | N=N+j*auxIV[j]; |
---|
3832 | } |
---|
3833 | N=N+d*(l-1); |
---|
3834 | M=N+d; |
---|
3835 | for (k=N+1;k<=M;k=k+1) |
---|
3836 | { |
---|
3837 | for (j=1;j<=s2;j=j+1) |
---|
3838 | { |
---|
3839 | if (k==P[j]) |
---|
3840 | { |
---|
3841 | return(0); |
---|
3842 | } |
---|
3843 | } |
---|
3844 | } |
---|
3845 | } |
---|
3846 | } |
---|
3847 | } |
---|
3848 | } |
---|
3849 | kill(auxIV,iw); |
---|
3850 | return(1); |
---|
3851 | } |
---|
3852 | /////////////////////////////////////////////////////////////////////////////// |
---|
3853 | |
---|
3854 | proc AGcode_L (intvec G,intvec D,list EC) |
---|
3855 | "USAGE: AGcode_L( G, D, EC ); G,D intvec, EC a list |
---|
3856 | RETURN: a generator matrix for the evaluation AG code defined by the |
---|
3857 | divisors G and D. |
---|
3858 | NOTE: The procedure must be called within the ring EC[1][4], |
---|
3859 | where EC is the output of @code{extcurve(d)} (or within |
---|
3860 | the ring EC[1][2] if d=1). @* |
---|
3861 | The entry i in the intvec D refers to the i-th rational |
---|
3862 | place in EC[1][5] (i.e., to POINTS[i], etc., see @ref{extcurve}).@* |
---|
3863 | The intvec G represents a rational divisor (see @ref{BrillNoether} |
---|
3864 | for more details).@* |
---|
3865 | The code evaluates the vector basis of L(G) at the rational |
---|
3866 | places given by D. |
---|
3867 | WARNINGS: G should satisfy @math{ 2*genus-2 < deg(G) < size(D) }, which is |
---|
3868 | not checked by the algorithm. |
---|
3869 | G and D should have disjoint supports (checked by the algorithm). |
---|
3870 | SEE ALSO: Adj_div, BrillNoether, extcurve, AGcode_Omega |
---|
3871 | EXAMPLE: example AGcode_L; shows an example |
---|
3872 | " |
---|
3873 | { |
---|
3874 | // returns a generator matrix for the evaluation AG code given by G and D |
---|
3875 | // G must be a divisor defined over the prime field and D an intvec of |
---|
3876 | // "rational places" |
---|
3877 | // it must be called inside R or R(d) and requires previously "extcurve(d)" |
---|
3878 | def BR=basering; |
---|
3879 | if (disj_divs(G,D,EC)==0) |
---|
3880 | { |
---|
3881 | dbprint(printlevel+3,"? the divisors do not have disjoint supports, |
---|
3882 | 0-matrix returned ?"); |
---|
3883 | matrix answer; |
---|
3884 | return(answer); |
---|
3885 | } |
---|
3886 | if (res_deg()>1) |
---|
3887 | { |
---|
3888 | def R=EC[1][2]; |
---|
3889 | setring R; |
---|
3890 | list LG=BrillNoether(G,EC); |
---|
3891 | setring BR; |
---|
3892 | list LG=imap(R,LG); |
---|
3893 | setring R; |
---|
3894 | kill(LG); |
---|
3895 | setring BR; |
---|
3896 | kill(R); |
---|
3897 | } |
---|
3898 | else |
---|
3899 | { |
---|
3900 | list LG=BrillNoether(G,EC); |
---|
3901 | } |
---|
3902 | def RP=EC[1][5]; |
---|
3903 | matrix M=gen_mat(LG,D,RP); |
---|
3904 | kill(LG); |
---|
3905 | return(M); |
---|
3906 | } |
---|
3907 | example |
---|
3908 | { |
---|
3909 | "EXAMPLE:"; echo = 2; |
---|
3910 | int plevel=printlevel; |
---|
3911 | printlevel=-1; |
---|
3912 | ring s=2,(x,y),lp; |
---|
3913 | list HC=Adj_div(x3+y2+y); |
---|
3914 | HC=NSplaces(1,HC); |
---|
3915 | HC=extcurve(2,HC); |
---|
3916 | def ER=HC[1][4]; |
---|
3917 | setring ER; |
---|
3918 | intvec G=5; // the rational divisor G = 5*HC[3][1] |
---|
3919 | intvec D=2..9; // D = sum of the rational places no. 2..9 over F_4 |
---|
3920 | // let us construct the corresponding evaluation AG code : |
---|
3921 | matrix C=AGcode_L(G,D,HC); |
---|
3922 | // here is a linear code of type [8,5,>=3] over F_4 |
---|
3923 | print(C); |
---|
3924 | printlevel=plevel; |
---|
3925 | } |
---|
3926 | /////////////////////////////////////////////////////////////////////////////// |
---|
3927 | |
---|
3928 | proc AGcode_Omega (intvec G,intvec D,list EC) |
---|
3929 | "USAGE: AGcode_Omega( G, D, EC ); G,D intvec, EC a list |
---|
3930 | RETURN: a generator matrix for the residual AG code defined by the |
---|
3931 | divisors G and D. |
---|
3932 | NOTE: The procedure must be called within the ring EC[1][4], |
---|
3933 | where EC is the output of @code{extcurve(d)} (or within |
---|
3934 | the ring EC[1][2] if d=1). @* |
---|
3935 | The entry i in the intvec D refers to the i-th rational |
---|
3936 | place in EC[1][5] (i.e., to POINTS[i], etc., see @ref{extcurve}).@* |
---|
3937 | The intvec G represents a rational divisor (see @ref{BrillNoether} |
---|
3938 | for more details).@* |
---|
3939 | The code computes the residues of a vector space basis of |
---|
3940 | @math{\Omega(G-D)} at the rational places given by D. |
---|
3941 | WARNINGS: G should satisfy @math{ 2*genus-2 < deg(G) < size(D) }, which is |
---|
3942 | not checked by the algorithm. |
---|
3943 | G and D should have disjoint supports (checked by the algorithm). |
---|
3944 | SEE ALSO: Adj_div, BrillNoether, extcurve, AGcode_L |
---|
3945 | EXAMPLE: example AGcode_Omega; shows an example |
---|
3946 | " |
---|
3947 | { |
---|
3948 | // returns a generator matrix for the residual AG code given by G and D |
---|
3949 | // G must be a divisor defined over the prime field and D an intvec or |
---|
3950 | // "rational places" |
---|
3951 | // it must be called inside R or R(d) and requires previously "extcurve(d)" |
---|
3952 | return(dual_code(AGcode_L(G,D,EC))); |
---|
3953 | } |
---|
3954 | example |
---|
3955 | { |
---|
3956 | "EXAMPLE:"; echo = 2; |
---|
3957 | int plevel=printlevel; |
---|
3958 | printlevel=-1; |
---|
3959 | ring s=2,(x,y),lp; |
---|
3960 | list HC=Adj_div(x3+y2+y); |
---|
3961 | HC=NSplaces(1,HC); |
---|
3962 | HC=extcurve(2,HC); |
---|
3963 | def ER=HC[1][4]; |
---|
3964 | setring ER; |
---|
3965 | intvec G=5; // the rational divisor G = 5*HC[3][1] |
---|
3966 | intvec D=2..9; // D = sum of the rational places no. 2..9 over F_4 |
---|
3967 | // let us construct the corresponding residual AG code : |
---|
3968 | matrix C=AGcode_Omega(G,D,HC); |
---|
3969 | // here is a linear code of type [8,3,>=5] over F_4 |
---|
3970 | print(C); |
---|
3971 | printlevel=plevel; |
---|
3972 | } |
---|
3973 | /////////////////////////////////////////////////////////////////////////////// |
---|
3974 | |
---|
3975 | // ============================================================================ |
---|
3976 | // ******* auxiliary procedure to define AG codes over extensions ******** |
---|
3977 | // ============================================================================ |
---|
3978 | |
---|
3979 | proc extcurve (int d,list CURVE) |
---|
3980 | "USAGE: extcurve( d, CURVE ); d an integer, CURVE a list |
---|
3981 | RETURN: list L which is the update of the list CURVE with additional entries |
---|
3982 | @format |
---|
3983 | L[1][3]: ring (p,a),(x,y),lp (affine), |
---|
3984 | L[1][4]: ring (p,a),(x,y,z),lp (projective), |
---|
3985 | L[1][5]: ring (p,a),(x,y,t),ls (local), |
---|
3986 | L[2][3]: int (the number of rational places), |
---|
3987 | @end format |
---|
3988 | the rings being defined over a field extension of degree d. @* |
---|
3989 | If d<2 then @code{extcurve(d,CURVE);} creates a list L which |
---|
3990 | is the update of the list CURVE with additional entries |
---|
3991 | @format |
---|
3992 | L[1][5]: ring p,(x,y,t),ls, |
---|
3993 | L[2][3]: int (the number of places over the base field). |
---|
3994 | @end format |
---|
3995 | In both cases, in the ring L[1][5] lists with the data for all the |
---|
3996 | rational places (after a field extension of degree d) are |
---|
3997 | created (see @ref{Adj_div}): |
---|
3998 | @format |
---|
3999 | lists POINTS, LOC_EQS, BRANCHES, PARAMETRIZATIONS. |
---|
4000 | @end format |
---|
4001 | NOTE: The list CURVE should be the output of @code{NSplaces} and has |
---|
4002 | to contain (at least) all places up to degree d. @* |
---|
4003 | This procedure must be executed before constructing AG codes, |
---|
4004 | even if no extension is needed. The ring L[1][4] must be active |
---|
4005 | when constructing codes over the field extension.@* |
---|
4006 | SEE ALSO: closed_points, Adj_div, NSplaces, AGcode_L, AGcode_Omega |
---|
4007 | EXAMPLE: example extcurve; shows an example |
---|
4008 | " |
---|
4009 | { |
---|
4010 | // extends the underlying curve and all its associated objects to a larger |
---|
4011 | // base field in order to evaluate points over such a extension |
---|
4012 | // if d<2 then the only change is that a local ring "RatPl" (which is a |
---|
4013 | // copy of "S(1)") is created in order to store the rational places |
---|
4014 | // where we can do evaluations |
---|
4015 | // otherwise, such a ring contains all places which are rational over the |
---|
4016 | // extension |
---|
4017 | // warning : list Places does not change so that only divisors which are |
---|
4018 | // "rational over the prime field" are allowed; this probably will |
---|
4019 | // change in the future |
---|
4020 | // warning : the places in RatPl are ranged in increasing degree, respecting |
---|
4021 | // the order from list Places and placing the conjugate branches all |
---|
4022 | // together |
---|
4023 | def BR=basering; |
---|
4024 | list ext_CURVE=CURVE; |
---|
4025 | if (d<2) |
---|
4026 | { |
---|
4027 | def SS=CURVE[5][1][1]; |
---|
4028 | ring RatPl=char(basering),(x,y,t),ls; |
---|
4029 | list POINTS=imap(SS,POINTS); |
---|
4030 | list LOC_EQS=imap(SS,LOC_EQS); |
---|
4031 | list BRANCHES=imap(SS,BRANCHES); |
---|
4032 | list PARAMETRIZATIONS=imap(SS,PARAMETRIZATIONS); |
---|
4033 | export(POINTS); |
---|
4034 | export(LOC_EQS); |
---|
4035 | export(BRANCHES); |
---|
4036 | export(PARAMETRIZATIONS); |
---|
4037 | int NrRatPl=size(POINTS); |
---|
4038 | ext_CURVE[2][3]=NrRatPl; |
---|
4039 | setring BR; |
---|
4040 | ext_CURVE[1][5]=RatPl; |
---|
4041 | dbprint(printlevel+1,""); |
---|
4042 | dbprint(printlevel+2,"Total number of rational places : "+string(NrRatPl)); |
---|
4043 | dbprint(printlevel+1,""); |
---|
4044 | kill(RatPl); |
---|
4045 | return(ext_CURVE); |
---|
4046 | } |
---|
4047 | else |
---|
4048 | { |
---|
4049 | if (size(CURVE[5])>=d) |
---|
4050 | { |
---|
4051 | int i,j,k; |
---|
4052 | if (typeof(CURVE[5][d])=="list") |
---|
4053 | { |
---|
4054 | def S(d)=CURVE[5][d][1]; |
---|
4055 | setring S(d); |
---|
4056 | } |
---|
4057 | else |
---|
4058 | { |
---|
4059 | ring S(d)=char(basering),t,ls; |
---|
4060 | list POINTS=list(); |
---|
4061 | } |
---|
4062 | string smin=string(minpoly); |
---|
4063 | setring BR; |
---|
4064 | ring RatPl=(char(basering),a),(x,y,t),ls; |
---|
4065 | execute("minpoly="+smin+";"); |
---|
4066 | list POINTS=imap(S(d),POINTS); |
---|
4067 | list LOC_EQS=imap(S(d),LOC_EQS); |
---|
4068 | list BRANCHES=imap(S(d),BRANCHES); |
---|
4069 | list PARAMETRIZATIONS=imap(S(d),PARAMETRIZATIONS); |
---|
4070 | kill(S(d)); |
---|
4071 | int s=size(POINTS); |
---|
4072 | int counter=0; |
---|
4073 | int piv=0; |
---|
4074 | for (j=1;j<=s;j=j+1) |
---|
4075 | { |
---|
4076 | counter=counter+1; |
---|
4077 | piv=counter; |
---|
4078 | for (k=1;k<d;k=k+1) |
---|
4079 | { |
---|
4080 | POINTS=insert(POINTS,Frobenius(POINTS[piv],1),counter); |
---|
4081 | LOC_EQS=insert(LOC_EQS,Frobenius(LOC_EQS[piv],1),counter); |
---|
4082 | BRANCHES=insert(BRANCHES,conj_b(BRANCHES[piv],1),counter); |
---|
4083 | PARAMETRIZATIONS=insert(PARAMETRIZATIONS,Frobenius( |
---|
4084 | PARAMETRIZATIONS[piv],1),counter); |
---|
4085 | counter=counter+1; |
---|
4086 | piv=counter; |
---|
4087 | } |
---|
4088 | } |
---|
4089 | string olda; |
---|
4090 | poly paux; |
---|
4091 | intvec iv,iw; |
---|
4092 | int ii,jj,kk,m,n; |
---|
4093 | for (i=d-1;i>=2;i=i-1) |
---|
4094 | { |
---|
4095 | if ( (d mod i)==0 ) |
---|
4096 | { |
---|
4097 | if (typeof(CURVE[5][i])=="list") |
---|
4098 | { |
---|
4099 | def S(i)=CURVE[5][i][1]; |
---|
4100 | } |
---|
4101 | else |
---|
4102 | { |
---|
4103 | ring S(i)=char(basering),t,ls; |
---|
4104 | list POINTS=list(); |
---|
4105 | setring BR; |
---|
4106 | } |
---|
4107 | olda=subfield(S(i)); |
---|
4108 | setring S(i); |
---|
4109 | s=size(POINTS); |
---|
4110 | setring RatPl; |
---|
4111 | for (j=s;j>=1;j=j-1) |
---|
4112 | { |
---|
4113 | counter=0; |
---|
4114 | POINTS=insert(POINTS,list(),0); |
---|
4115 | POINTS[1][1]=number(importdatum(S(i),"POINTS["+string(j) |
---|
4116 | +"][1]",olda)); |
---|
4117 | POINTS[1][2]=number(importdatum(S(i),"POINTS["+string(j) |
---|
4118 | +"][2]",olda)); |
---|
4119 | POINTS[1][3]=number(importdatum(S(i),"POINTS["+string(j) |
---|
4120 | +"][3]",olda)); |
---|
4121 | LOC_EQS=insert(LOC_EQS,importdatum(S(i),"LOC_EQS["+string(j) |
---|
4122 | +"]",olda),0); |
---|
4123 | BRANCHES=insert(BRANCHES,list(),0); |
---|
4124 | setring S(i); |
---|
4125 | m=nrows(BRANCHES[j][1]); |
---|
4126 | n=ncols(BRANCHES[j][1]); |
---|
4127 | iv=BRANCHES[j][2]; |
---|
4128 | kk=BRANCHES[j][3]; |
---|
4129 | poly par@1=subst(PARAMETRIZATIONS[j][1][1],t,x); |
---|
4130 | poly par@2=subst(PARAMETRIZATIONS[j][1][2],t,x); |
---|
4131 | export(par@1); |
---|
4132 | export(par@2); |
---|
4133 | iw=PARAMETRIZATIONS[j][2]; |
---|
4134 | setring RatPl; |
---|
4135 | paux=importdatum(S(i),"BRANCHES["+string(j)+"][4]",olda); |
---|
4136 | matrix Maux[m][n]; |
---|
4137 | for (ii=1;ii<=m;ii=ii+1) |
---|
4138 | { |
---|
4139 | for (jj=1;jj<=n;jj=jj+1) |
---|
4140 | { |
---|
4141 | Maux[ii,jj]=importdatum(S(i),"BRANCHES["+string(j) |
---|
4142 | +"][1]["+string(ii)+","+string(jj)+"]",olda); |
---|
4143 | } |
---|
4144 | } |
---|
4145 | BRANCHES[1][1]=Maux; |
---|
4146 | BRANCHES[1][2]=iv; |
---|
4147 | BRANCHES[1][3]=kk; |
---|
4148 | BRANCHES[1][4]=paux; |
---|
4149 | kill(Maux); |
---|
4150 | PARAMETRIZATIONS=insert(PARAMETRIZATIONS,list(),0); |
---|
4151 | PARAMETRIZATIONS[1][1]=ideal(0); |
---|
4152 | PARAMETRIZATIONS[1][1][1]=importdatum(S(i),"par@1",olda); |
---|
4153 | PARAMETRIZATIONS[1][1][2]=importdatum(S(i),"par@2",olda); |
---|
4154 | PARAMETRIZATIONS[1][1][1]=subst(PARAMETRIZATIONS[1][1][1],x,t); |
---|
4155 | PARAMETRIZATIONS[1][1][2]=subst(PARAMETRIZATIONS[1][1][2],x,t); |
---|
4156 | PARAMETRIZATIONS[1][2]=iw; |
---|
4157 | for (k=1;k<i;k=k+1) |
---|
4158 | { |
---|
4159 | counter=counter+1; |
---|
4160 | piv=counter; |
---|
4161 | POINTS=insert(POINTS,Frobenius(POINTS[piv],1),counter); |
---|
4162 | LOC_EQS=insert(LOC_EQS,Frobenius(LOC_EQS[piv],1),counter); |
---|
4163 | BRANCHES=insert(BRANCHES,conj_b(BRANCHES[piv],1),counter); |
---|
4164 | PARAMETRIZATIONS=insert(PARAMETRIZATIONS,Frobenius( |
---|
4165 | PARAMETRIZATIONS[piv],1),counter); |
---|
4166 | } |
---|
4167 | setring S(i); |
---|
4168 | kill(par@1,par@2); |
---|
4169 | setring RatPl; |
---|
4170 | } |
---|
4171 | kill(S(i)); |
---|
4172 | } |
---|
4173 | } |
---|
4174 | kill(iw); |
---|
4175 | kill(paux); |
---|
4176 | if (typeof(CURVE[5][1])=="list") |
---|
4177 | { |
---|
4178 | def S(1)=CURVE[5][1][1]; |
---|
4179 | } |
---|
4180 | else |
---|
4181 | { |
---|
4182 | ring S(1)=char(basering),t,ls; |
---|
4183 | list POINTS=list(); |
---|
4184 | setring RatPl; |
---|
4185 | } |
---|
4186 | POINTS=imap(S(1),POINTS)+POINTS; |
---|
4187 | LOC_EQS=imap(S(1),LOC_EQS)+LOC_EQS; |
---|
4188 | BRANCHES=imap(S(1),BRANCHES)+BRANCHES; |
---|
4189 | PARAMETRIZATIONS=imap(S(1),PARAMETRIZATIONS)+PARAMETRIZATIONS; |
---|
4190 | export(POINTS); |
---|
4191 | export(LOC_EQS); |
---|
4192 | export(BRANCHES); |
---|
4193 | export(PARAMETRIZATIONS); |
---|
4194 | int NrRatPl=size(POINTS); |
---|
4195 | ext_CURVE[2][3]=NrRatPl; |
---|
4196 | setring BR; |
---|
4197 | ext_CURVE[1][5]=RatPl; |
---|
4198 | ring r(d)=(char(basering),a),(x,y),lp; |
---|
4199 | execute("minpoly="+smin+";"); |
---|
4200 | setring BR; |
---|
4201 | ext_CURVE[1][3]=r(d); |
---|
4202 | ring R(d)=(char(basering),a),(x,y,z),lp; |
---|
4203 | execute("minpoly="+smin+";"); |
---|
4204 | setring BR; |
---|
4205 | ext_CURVE[1][4]=R(d); |
---|
4206 | dbprint(printlevel+1,""); |
---|
4207 | dbprint(printlevel+2,"Total number of rational places : NrRatPl = " |
---|
4208 | +string(NrRatPl)); |
---|
4209 | dbprint(printlevel+1,""); |
---|
4210 | kill(S(1)); |
---|
4211 | kill(R(d)); |
---|
4212 | kill(RatPl); |
---|
4213 | return(ext_CURVE); |
---|
4214 | } |
---|
4215 | else |
---|
4216 | { |
---|
4217 | ERROR("you must compute first all the places up to degree "+string(d)); |
---|
4218 | return(); |
---|
4219 | } |
---|
4220 | } |
---|
4221 | } |
---|
4222 | example |
---|
4223 | { |
---|
4224 | "EXAMPLE:"; echo = 2; |
---|
4225 | int plevel=printlevel; |
---|
4226 | printlevel=-1; |
---|
4227 | ring s=2,(x,y),lp; |
---|
4228 | list C=Adj_div(x5+y2+y); |
---|
4229 | C=NSplaces(3,C); |
---|
4230 | // since we have all points up to degree 4, we can extend the curve |
---|
4231 | // to that extension, in order to get rational points over F_16; |
---|
4232 | C=extcurve(4,C); |
---|
4233 | // e.g., display the basepoint of place no. 32: |
---|
4234 | def R=C[1][5]; |
---|
4235 | setring R; |
---|
4236 | POINTS[32]; |
---|
4237 | printlevel=plevel; |
---|
4238 | } |
---|
4239 | /////////////////////////////////////////////////////////////////////////////// |
---|
4240 | |
---|
4241 | // specific procedures for linear/AG codes |
---|
4242 | |
---|
4243 | static proc Hamming_wt (matrix A) |
---|
4244 | "USAGE: Hamming_wt(A), where A is any matrix |
---|
4245 | RETURN: the Hamming weight (number of non-zero entries) of the matrix A |
---|
4246 | " |
---|
4247 | { |
---|
4248 | // computes the Hamming weight (number of non-zero entries) of any matrix |
---|
4249 | // notice that "words" are represented by matrices of size 1xn |
---|
4250 | // computing the Hamming distance between two matrices can be done by |
---|
4251 | // Hamming_wt(A-B) |
---|
4252 | int m=nrows(A); |
---|
4253 | int n=ncols(A); |
---|
4254 | int i,j; |
---|
4255 | int w=0; |
---|
4256 | for (i=1;i<=m;i=i+1) |
---|
4257 | { |
---|
4258 | for (j=1;j<=n;j=j+1) |
---|
4259 | { |
---|
4260 | if (A[i,j]<>0) |
---|
4261 | { |
---|
4262 | w=w+1; |
---|
4263 | } |
---|
4264 | } |
---|
4265 | } |
---|
4266 | return(w); |
---|
4267 | } |
---|
4268 | /////////////////////////////////////////////////////////////////////////////// |
---|
4269 | |
---|
4270 | // Basic Algorithm of Skorobogatov and Vladut for decoding AG codes |
---|
4271 | // warning : the user must choose carefully the parameters for the code and |
---|
4272 | // the decoding since they will never be checked by the procedures |
---|
4273 | |
---|
4274 | proc prepSV (intvec G,intvec D,intvec F,list EC) |
---|
4275 | "USAGE: prepSV( G, D, F, EC ); G,D,F intvecs and EC a list |
---|
4276 | RETURN: list E of size n+3, where n=size(D). All its entries but E[n+3] |
---|
4277 | are matrices: |
---|
4278 | @format |
---|
4279 | E[1]: parity check matrix for the current AG code |
---|
4280 | E[2] ... E[n+2]: matrices used in the procedure decodeSV |
---|
4281 | E[n+3]: intvec with |
---|
4282 | E[n+3][1]: correction capacity @math{epsilon} of the algorithm |
---|
4283 | E[n+3][2]: designed Goppa distance @math{delta} of the current AG code |
---|
4284 | @end format |
---|
4285 | NOTE: Computes the preprocessing for the basic (Skorobogatov-Vladut) |
---|
4286 | decoding algorithm.@* |
---|
4287 | The procedure must be called within the ring EC[1][4], where EC is |
---|
4288 | the output of @code{extcurve(d)} (or in the ring EC[1][2] if d=1) @* |
---|
4289 | The intvec G and F represent rational divisors (see |
---|
4290 | @ref{BrillNoether} for more details).@* |
---|
4291 | The intvec D refers to rational places (see @ref{AGcode_Omega} |
---|
4292 | for more details.). |
---|
4293 | The current AG code is @code{AGcode_Omega(G,D,EC)}.@* |
---|
4294 | If you know the exact minimum distance d and you want to use it in |
---|
4295 | @code{decodeSV} instead of @math{delta}, you can change the value |
---|
4296 | of E[n+3][2] to d before applying decodeSV. |
---|
4297 | If you have a systematic encoding for the current code and want to |
---|
4298 | keep it during the decoding, you must previously permute D (using |
---|
4299 | @code{permute_L(D,P);}), e.g., according to the permutation |
---|
4300 | P=L[3], L being the output of @code{sys_code}. |
---|
4301 | WARNINGS: F must be a divisor with support disjoint from the support of D and |
---|
4302 | with degree @math{epsilon + genus}, where |
---|
4303 | @math{epsilon:=[(deg(G)-3*genus+1)/2]}.@* |
---|
4304 | G should satisfy @math{ 2*genus-2 < deg(G) < size(D) }, which is |
---|
4305 | not checked by the algorithm. |
---|
4306 | G and D should also have disjoint supports (checked by the |
---|
4307 | algorithm). |
---|
4308 | KEYWORDS: SV-decoding algorithm, preprocessing |
---|
4309 | SEE ALSO: extcurve, AGcode_Omega, decodeSV, sys_code, permute_L |
---|
4310 | EXAMPLE: example prepSV; shows an example |
---|
4311 | " |
---|
4312 | { |
---|
4313 | if (disj_divs(F,D,EC)==0) |
---|
4314 | { |
---|
4315 | dbprint(printlevel+3,"? the divisors do not have disjoint supports, |
---|
4316 | empty list returned ?"); |
---|
4317 | return(list()); |
---|
4318 | } |
---|
4319 | list E=list(); |
---|
4320 | list Places=EC[3]; |
---|
4321 | int m=deg_D(G,Places); |
---|
4322 | int genusX=EC[2][2]; |
---|
4323 | int e=(m+1-3*genusX)/2; |
---|
4324 | if (e<1) |
---|
4325 | { |
---|
4326 | dbprint(printlevel+3,"? the correction capacity of the basic algorithm |
---|
4327 | is zero, empty list returned ?"); |
---|
4328 | return(list()); |
---|
4329 | } |
---|
4330 | // deg(F)==e+genusX should be satisfied, and sup(D),sup(F) should be |
---|
4331 | // disjoint !!!! |
---|
4332 | int n=size(D); |
---|
4333 | // 2*genusX-2<m<n should also be satisfied !!!! |
---|
4334 | matrix EE=AGcode_L(G,D,EC); |
---|
4335 | int l=nrows(EE); |
---|
4336 | E[1]=EE; |
---|
4337 | matrix GP=AGcode_L(F,D,EC); |
---|
4338 | int r=nrows(GP); |
---|
4339 | intvec H=G-F; |
---|
4340 | matrix HP=AGcode_L(H,D,EC); |
---|
4341 | int s=nrows(HP); |
---|
4342 | int i,j,k; |
---|
4343 | kill(EE); |
---|
4344 | for (k=1;k<=n;k=k+1) |
---|
4345 | { |
---|
4346 | E[k+1]=GP[1,k]*submat(HP,1..s,k..k); |
---|
4347 | for (i=2;i<=r;i=i+1) |
---|
4348 | { |
---|
4349 | E[k+1]=concat(E[k+1],GP[i,k]*submat(HP,1..s,k..k)); |
---|
4350 | } |
---|
4351 | } |
---|
4352 | E[n+2]=GP; |
---|
4353 | intvec IW=e,m+2-2*genusX; |
---|
4354 | E[n+3]=IW; |
---|
4355 | kill(IW); |
---|
4356 | return(E); |
---|
4357 | } |
---|
4358 | example |
---|
4359 | { |
---|
4360 | "EXAMPLE:"; echo = 2; |
---|
4361 | int plevel=printlevel; |
---|
4362 | printlevel=-1; |
---|
4363 | ring s=2,(x,y),lp; |
---|
4364 | list HC=Adj_div(x3+y2+y); |
---|
4365 | HC=NSplaces(1,HC); |
---|
4366 | HC=extcurve(2,HC); |
---|
4367 | def ER=HC[1][4]; |
---|
4368 | setring ER; |
---|
4369 | intvec G=5; // the rational divisor G = 5*HC[3][1] |
---|
4370 | intvec D=2..9; // D = sum of the rational places no. 2..9 over F_4 |
---|
4371 | // construct the corresp. residual AG code of type [8,3,>=5] over F_4: |
---|
4372 | matrix C=AGcode_Omega(G,D,HC); |
---|
4373 | // we can correct 1 error and the genus is 1, thus F must have degree 2 |
---|
4374 | // and support disjoint from that of D; |
---|
4375 | intvec F=2; |
---|
4376 | list SV=prepSV(G,D,F,HC); |
---|
4377 | // now everything is prepared to decode with the basic algorithm; |
---|
4378 | // for example, here is a parity check matrix to compute the syndrome : |
---|
4379 | print(SV[1]); |
---|
4380 | // and here you have the correction capacity of the algorithm : |
---|
4381 | int epsilon=SV[size(D)+3][1]; |
---|
4382 | epsilon; |
---|
4383 | printlevel=plevel; |
---|
4384 | } |
---|
4385 | /////////////////////////////////////////////////////////////////////////////// |
---|
4386 | |
---|
4387 | proc decodeSV (matrix y,list K) |
---|
4388 | "USAGE: decodeSV( y, K ); y a row-matrix and K a list |
---|
4389 | RETURN: a codeword (row-matrix) if possible, resp. the 0-matrix (of size |
---|
4390 | 1) if decoding is impossible. |
---|
4391 | For decoding the basic (Skorobogatov-Vladut) decoding algorithm |
---|
4392 | is applied. |
---|
4393 | NOTE: The list_expression should be the output K of the procedure |
---|
4394 | @code{prepSV}.@* |
---|
4395 | The matrix_expression should be a (1 x n)-matrix, where |
---|
4396 | n = ncols(K[1]).@* |
---|
4397 | The decoding may fail if the number of errors is greater than |
---|
4398 | the correction capacity of the algorithm. |
---|
4399 | KEYWORDS: SV-decoding algorithm |
---|
4400 | SEE ALSO: extcurve, AGcode_Omega, prepSV |
---|
4401 | EXAMPLE: example decodeSV; shows an example |
---|
4402 | " |
---|
4403 | { |
---|
4404 | // decodes y with the "basic decoding algorithm", if possible |
---|
4405 | // requires the preprocessing done by the procedure "prepSV" |
---|
4406 | // the procedure must be called from ring R or R(d) |
---|
4407 | // returns either a codeword (matrix) of none (in case of too many errors) |
---|
4408 | matrix syndr=K[1]*transpose(y); |
---|
4409 | if (Hamming_wt(syndr)==0) |
---|
4410 | { |
---|
4411 | return(y); |
---|
4412 | } |
---|
4413 | matrix Ey=y[1,1]*K[2]; |
---|
4414 | int n=ncols(y); |
---|
4415 | int i; |
---|
4416 | for (i=2;i<=n;i=i+1) |
---|
4417 | { |
---|
4418 | Ey=Ey+y[1,i]*K[i+1]; |
---|
4419 | } |
---|
4420 | matrix Ky=get_NZsol(Ey); |
---|
4421 | if (Hamming_wt(Ky)==0) |
---|
4422 | { |
---|
4423 | dbprint(printlevel+3,"? no error-locator found ?"); |
---|
4424 | dbprint(printlevel+3,"? too many errors occur, 0-matrix returned ?"); |
---|
4425 | matrix answer; |
---|
4426 | return(answer); |
---|
4427 | } |
---|
4428 | int r=nrows(K[n+2]); |
---|
4429 | matrix ErrLoc[1][n]; |
---|
4430 | list Z=list(); |
---|
4431 | list NZ=list(); |
---|
4432 | int j; |
---|
4433 | for (j=1;j<=n;j=j+1) |
---|
4434 | { |
---|
4435 | for (i=1;i<=r;i=i+1) |
---|
4436 | { |
---|
4437 | ErrLoc[1,j]=ErrLoc[1,j]+K[n+2][i,j]*Ky[1,i]; |
---|
4438 | } |
---|
4439 | if (ErrLoc[1,j]==0) |
---|
4440 | { |
---|
4441 | Z=insert(Z,j,size(Z)); |
---|
4442 | } |
---|
4443 | else |
---|
4444 | { |
---|
4445 | NZ=insert(NZ,j,size(NZ)); |
---|
4446 | } |
---|
4447 | } |
---|
4448 | int k=size(NZ); |
---|
4449 | int l=nrows(K[1]); |
---|
4450 | int s=l+k; |
---|
4451 | matrix A[s][n]; |
---|
4452 | matrix b[s][1]; |
---|
4453 | for (i=1;i<=l;i=i+1) |
---|
4454 | { |
---|
4455 | for (j=1;j<=n;j=j+1) |
---|
4456 | { |
---|
4457 | A[i,j]=K[1][i,j]; |
---|
4458 | } |
---|
4459 | b[i,1]=syndr[i,1]; |
---|
4460 | } |
---|
4461 | for (i=1;i<=k;i=i+1) |
---|
4462 | { |
---|
4463 | A[l+i,NZ[i]]=number(1); |
---|
4464 | } |
---|
4465 | intvec opgt=option(get); |
---|
4466 | option(redSB); |
---|
4467 | matrix L=transpose(syz(concat(A,-b))); |
---|
4468 | if (nrows(L)==1) |
---|
4469 | { |
---|
4470 | if (L[1,n+1]<>0) |
---|
4471 | { |
---|
4472 | poly pivote=L[1,n+1]; |
---|
4473 | matrix sol=submat(L,1..1,1..n); |
---|
4474 | if (pivote<>1) |
---|
4475 | { |
---|
4476 | sol=(number(1)/number(pivote))*sol; |
---|
4477 | } |
---|
4478 | // check at least that the number of comitted errors is less than half |
---|
4479 | // the Goppa distance |
---|
4480 | // imposing Hamming_wt(sol)<=K[n+3][1] would be more correct, but maybe |
---|
4481 | // is too strong |
---|
4482 | // on the other hand, if Hamming_wt(sol) is too large the decoding may |
---|
4483 | // not be acceptable |
---|
4484 | if ( Hamming_wt(sol) <= (K[n+3][2]-1)/2 ) |
---|
4485 | { |
---|
4486 | option(set,opgt); |
---|
4487 | return(y-sol); |
---|
4488 | } |
---|
4489 | else |
---|
4490 | { |
---|
4491 | dbprint(printlevel+3,"? non-acceptable decoding ?"); |
---|
4492 | } |
---|
4493 | } |
---|
4494 | else |
---|
4495 | { |
---|
4496 | dbprint(printlevel+3,"? no solution found ?"); |
---|
4497 | } |
---|
4498 | } |
---|
4499 | else |
---|
4500 | { |
---|
4501 | dbprint(printlevel+3,"? non-unique solution ?"); |
---|
4502 | } |
---|
4503 | option(set,opgt); |
---|
4504 | dbprint(printlevel+3,"? too many errors occur, 0-matrix returned ?"); |
---|
4505 | matrix answer; |
---|
4506 | return(answer); |
---|
4507 | } |
---|
4508 | example |
---|
4509 | { |
---|
4510 | "EXAMPLE:"; echo = 2; |
---|
4511 | int plevel=printlevel; |
---|
4512 | printlevel=-1; |
---|
4513 | ring s=2,(x,y),lp; |
---|
4514 | list HC=Adj_div(x3+y2+y); |
---|
4515 | HC=NSplaces(1,HC); |
---|
4516 | HC=extcurve(2,HC); |
---|
4517 | def ER=HC[1][4]; |
---|
4518 | setring ER; |
---|
4519 | intvec G=5; // the rational divisor G = 5*HC[3][1] |
---|
4520 | intvec D=2..9; // D = sum of the rational places no. 2..9 over F_4 |
---|
4521 | // construct the corresp. residual AG code of type [8,3,>=5] over F_4: |
---|
4522 | matrix C=AGcode_Omega(G,D,HC); |
---|
4523 | // we can correct 1 error and the genus is 1, thus F must have degree 2 |
---|
4524 | // and support disjoint from that of D |
---|
4525 | intvec F=2; |
---|
4526 | list SV=prepSV(G,D,F,HC); |
---|
4527 | // now we produce 1 error on the zero-codeword : |
---|
4528 | matrix y[1][8]; |
---|
4529 | y[1,3]=a; |
---|
4530 | // and then we decode : |
---|
4531 | print(decodeSV(y,SV)); |
---|
4532 | printlevel=plevel; |
---|
4533 | } |
---|
4534 | /////////////////////////////////////////////////////////////////////////////// |
---|
4535 | |
---|
4536 | proc sys_code (matrix C) |
---|
4537 | "USAGE: sys_code(C); C is a matrix of constants |
---|
4538 | RETURN: list L with: |
---|
4539 | @format |
---|
4540 | L[1] is the generator matrix in standard form of an equivalent code, |
---|
4541 | L[2] is the parity check matrix in standard form of such code, |
---|
4542 | L[3] is an intvec which represents the needed permutation. |
---|
4543 | @end format |
---|
4544 | NOTE: Computes a systematic code which is equivalent to the given one.@* |
---|
4545 | The input should be a matrix of numbers.@* |
---|
4546 | The output has to be interpreted as follows: if the input was |
---|
4547 | the generator matrix of an AG code then one should apply the |
---|
4548 | permutation L[3] to the divisor D of rational points by means |
---|
4549 | of @code{permute_L(D,L[3]);} before continuing to work with the |
---|
4550 | code (for instance, if you want to use the systematic encoding |
---|
4551 | together with a decoding algorithm). |
---|
4552 | KEYWORDS: linear code, systematic |
---|
4553 | SEE ALSO: permute_L, AGcode_Omega, prepSV |
---|
4554 | EXAMPLE: example sys_code; shows an example |
---|
4555 | " |
---|
4556 | { |
---|
4557 | // computes a systematic code which is equivalent to that given by C |
---|
4558 | int i,j,k,l,h,r; |
---|
4559 | int m=nrows(C); |
---|
4560 | int n=ncols(C); |
---|
4561 | int mr=m; |
---|
4562 | matrix A[m][n]=C; |
---|
4563 | poly c,p; |
---|
4564 | list corners=list(); |
---|
4565 | if(m>n) |
---|
4566 | { |
---|
4567 | mr=n; |
---|
4568 | } |
---|
4569 | // first the matrix A will be reduced with elementary operations by rows |
---|
4570 | for(i=1;i<=mr;i=i+1) |
---|
4571 | { |
---|
4572 | if((i+l)>n) |
---|
4573 | { |
---|
4574 | // the process is finished |
---|
4575 | break; |
---|
4576 | } |
---|
4577 | // look for a non-zero element |
---|
4578 | if(A[i,i+l]==0) |
---|
4579 | { |
---|
4580 | h=i; |
---|
4581 | p=0; |
---|
4582 | for (j=i+1;j<=m;j=j+1) |
---|
4583 | { |
---|
4584 | c=A[j,i+l]; |
---|
4585 | if (c!=0) |
---|
4586 | { |
---|
4587 | p=c; |
---|
4588 | h=j; |
---|
4589 | break; |
---|
4590 | } |
---|
4591 | } |
---|
4592 | if (h!=i) |
---|
4593 | { |
---|
4594 | // permutation of rows i and h |
---|
4595 | for (j=1;j<=n;j=j+1) |
---|
4596 | { |
---|
4597 | c=A[i,j]; |
---|
4598 | A[i,j]=A[h,j]; |
---|
4599 | A[h,j]=c; |
---|
4600 | } |
---|
4601 | } |
---|
4602 | if(p==0) |
---|
4603 | { |
---|
4604 | // non-zero element not found in the current column |
---|
4605 | l=l+1; |
---|
4606 | continue; |
---|
4607 | } |
---|
4608 | } |
---|
4609 | // non-zero element was found in "strategic" position |
---|
4610 | corners[i]=i+l; |
---|
4611 | // make zeros below that position |
---|
4612 | for(j=i+1;j<=m;j=j+1) |
---|
4613 | { |
---|
4614 | c=A[j,i+l]/A[i,i+l]; |
---|
4615 | for(k=i+l+1;k<=n;k=k+1) |
---|
4616 | { |
---|
4617 | A[j,k]=A[j,k]-A[i,k]*c; |
---|
4618 | } |
---|
4619 | A[j,i+l]=0; |
---|
4620 | A[j,i]=0; |
---|
4621 | } |
---|
4622 | // the rank is at least r |
---|
4623 | // when the process stops the last r is actually the true rank of A=a |
---|
4624 | r=i; |
---|
4625 | } |
---|
4626 | if (r<m) |
---|
4627 | { |
---|
4628 | ERROR("the given matrix does not have maximum rank"); |
---|
4629 | } |
---|
4630 | // set the corners to the beginning and construct the permutation |
---|
4631 | intvec PCols=1..n; |
---|
4632 | for (j=1;j<=m;j=j+1) |
---|
4633 | { |
---|
4634 | if (corners[j]>j) |
---|
4635 | { |
---|
4636 | // interchange columns j and corners[j] |
---|
4637 | for (i=1;i<=m;i=i+1) |
---|
4638 | { |
---|
4639 | c=A[i,j]; |
---|
4640 | A[i,j]=A[i,corners[j]]; |
---|
4641 | A[i,corners[j]]=c; |
---|
4642 | } |
---|
4643 | k=PCols[j]; |
---|
4644 | PCols[j]=PCols[corners[j]]; |
---|
4645 | PCols[corners[j]]=k; |
---|
4646 | } |
---|
4647 | } |
---|
4648 | // convert the diagonal into ones |
---|
4649 | for (i=1;i<=m;i=i+1) |
---|
4650 | { |
---|
4651 | for (j=i;j<=n;j=j+1) |
---|
4652 | { |
---|
4653 | A[i,j]=A[i,j]/A[i,i]; |
---|
4654 | } |
---|
4655 | } |
---|
4656 | // complete a block with the identity matrix |
---|
4657 | for (k=1;k<m;k=k+1) |
---|
4658 | { |
---|
4659 | for (i=k+1;i<=m;i=i+1) |
---|
4660 | { |
---|
4661 | for (j=i;j<=n;j=j+1) |
---|
4662 | { |
---|
4663 | A[k,j]=A[k,j]-A[i,j]*A[k,i]; |
---|
4664 | } |
---|
4665 | } |
---|
4666 | } |
---|
4667 | // compute a parity-check matrix in standard form |
---|
4668 | matrix B=concat(-transpose(submat(A,1..m,m+1..n)),diag(1,n-m)); |
---|
4669 | list L=list(); |
---|
4670 | L[1]=A; |
---|
4671 | L[2]=B; |
---|
4672 | L[3]=PCols; |
---|
4673 | return(L); |
---|
4674 | } |
---|
4675 | example |
---|
4676 | { |
---|
4677 | "EXAMPLE:"; echo = 2; |
---|
4678 | ring s=3,T,lp; |
---|
4679 | matrix C[2][5]=0,1,0,1,1,0,1,0,0,1; |
---|
4680 | print(C); |
---|
4681 | list L=sys_code(C); |
---|
4682 | L[3]; |
---|
4683 | // here is the generator matrix in standard form |
---|
4684 | print(L[1]); |
---|
4685 | // here is the control matrix in standard form |
---|
4686 | print(L[2]); |
---|
4687 | // we can check that both codes are dual to each other |
---|
4688 | print(L[1]*transpose(L[2])); |
---|
4689 | } |
---|
4690 | /////////////////////////////////////////////////////////////////////////////// |
---|
4691 | |
---|
4692 | /* |
---|
4693 | // ============================================================================ |
---|
4694 | // ******* ADDITIONAL INFORMATION ABOUT THE LIBRARY ******** |
---|
4695 | // ============================================================================ |
---|
4696 | |
---|
4697 | A SINGULAR library for plane curves, Weierstrass semigroups and AG codes |
---|
4698 | Also available via http://wmatem.eis.uva.es/~ignfar/singular/ |
---|
4699 | |
---|
4700 | PREVIOUS WARNINGS : |
---|
4701 | |
---|
4702 | (1) The procedures will work only in positive characteristic |
---|
4703 | (2) The base field must be prime (this may change in the future) |
---|
4704 | This limitation is not too serious, since in coding theory |
---|
4705 | the used curves are usually (if not always) defined over a |
---|
4706 | prime field, and extensions are only considered for |
---|
4707 | evaluating functions in a field with many points; |
---|
4708 | by the same reason, auxiliary divisors are usually defined |
---|
4709 | over the prime field, |
---|
4710 | with the exception of that consisting of "rational points" |
---|
4711 | (3) The curve must be absolutely irreducible (but it is not checked) |
---|
4712 | (4) Only (algebraic projective) plane curves are considered |
---|
4713 | |
---|
4714 | GENERAL CONCEPTS : |
---|
4715 | |
---|
4716 | (1) An affine point P is represented by a std of a prime ideal, |
---|
4717 | and an intvec containing the position of the places above P |
---|
4718 | in the list of Places; if the point is at infinity, the ideal is |
---|
4719 | changed by a homogeneous irreducible polynomial in two variables |
---|
4720 | (2) A place is represented by : |
---|
4721 | a base point (list of homogeneous coordinates), |
---|
4722 | a local equation for the curve at the base point, |
---|
4723 | a Hamburger-Noether expansion of the corresponding branch, |
---|
4724 | and a local parametrization (in "t") of such branch; everything is |
---|
4725 | stored in a local ring "_[5][d][1]", d being the degree of the place, |
---|
4726 | by means of lists "POINTS,LOC_EQS,BRANCHES,PARAMETRIZATIONS", and |
---|
4727 | the degrees of the base points corresponding to the places in the |
---|
4728 | ring "_[5][d][1]" are stored in an intvec "_[5][d][2]" |
---|
4729 | (3) A divisor is represented by an intvec, where the integer at the |
---|
4730 | position i means the coefficient of the i-th place in the divisor |
---|
4731 | (4) Rational functions are represented by numerator/denominator |
---|
4732 | in form of ideals with two homogeneous generators |
---|
4733 | |
---|
4734 | OUTLINE/EXAMPLE OF THE USE OF THE LIBRARY : |
---|
4735 | |
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4736 | Plane curves : |
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4737 | |
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4738 | (1.0) ring s=p,(x,y[,z]),lp; |
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4739 | |
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4740 | Notice that if you use 3 variables, then the equation |
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4741 | of the curve is assumed to be a homogeneous polynomial. |
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4742 | |
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4743 | (1.1) list CURVE=Adj_div(equation); |
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4744 | |
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4745 | In CURVE[3] are listed all the (singular closed) places |
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4746 | with their corresponding degrees; thus, you can now decide |
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4747 | how many other points you want to compute with NSplaces. |
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4748 | |
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4749 | (1.2) CURVE=NSplaces(range,CURVE); |
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4750 | |
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4751 | See help NSplaces tp know the meaning of "range"; |
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4752 | for instance, if you have that the singular places |
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4753 | have degree 2 at most and you want all the places |
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4754 | up to degree 5, you must write range=3. |
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4755 | |
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4756 | (1.3) CURVE=extcurve(extension,CURVE); |
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4757 | |
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4758 | The rational places over the extension are ranged in |
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4759 | the ring CURVE[1][5] with the following rules: |
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4760 | |
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4761 | (i) all the representatives of the same closed point |
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4762 | are listed in consecutive positions; |
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4763 | (ii) if deg(P)<deg(Q), then the representatives of P |
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4764 | are listed before those of Q; |
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4765 | (iii) if two closed points P,Q have the same degree, |
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4766 | then the representatives of P are listed before |
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4767 | if P appears before in the list CURVE[3]. |
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4768 | |
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4769 | Rational functions : |
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4770 | |
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4771 | (2.0) def R=CURVE[1][2]; |
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4772 | setring R; |
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4773 | (2.1) list LG=BrillNoether(intvec divisor,CURVE); |
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4774 | (2.2) list WS=Weierstrass(int place,int bound,CURVE); |
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4775 | |
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4776 | Algebraic Geometry codes : |
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4777 | |
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4778 | (3.0) def ER=CURVE[1][4]; // if extension>1; else use R instead |
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4779 | setring ER; |
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4780 | |
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4781 | Now you have to decide the divisor G and the sequence of |
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4782 | rational points D to use for constructing the codes; |
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4783 | first notice that the syntax for G and D is different: |
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4784 | |
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4785 | (a) G is a divisor supported on the closed places of |
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4786 | CURVE[3], and you must say just the coefficient |
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4787 | of each such a point; for example, G=2,0,-1 means |
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4788 | 2 times the place 1 minus 1 times the place 3. |
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4789 | |
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4790 | (b) D is a sequence of rational points (all different |
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4791 | and counted 1 time each one), whose data are read |
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4792 | from the lists inside CURVE[1][5] and now you must |
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4793 | say just the order how you range the chosen point; |
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4794 | for example, D=2,4,1 means that you choose the |
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4795 | rational places 1,2,4 and you range them as 2,4,1. |
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4796 | |
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4797 | (3.1) matrix C=AGcode_L(divisor,places,CURVE); |
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4798 | |
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4799 | (3.2) AGcode_Omega(divisor,places,CURVE); |
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4800 | |
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4801 | In the same way as for defining the codes, the auxiliary |
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4802 | divisor F must have disjoint support to that of D, and |
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4803 | its degree has to be given by a formula (see help prepSV). |
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4804 | |
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4805 | (3.3) list SV=prepSV(divisor,places,auxiliary_divisor,CURVE); |
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4806 | |
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4807 | (3.4) decodeSV(word,SV); |
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4808 | |
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4809 | Special Issues : |
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4810 | |
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4811 | (I) AG codes with systematic encoding : |
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4812 | |
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4813 | matrix C=AGcode_Omega(G,D,CURVE); |
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4814 | list CODE=sys_code(G); |
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4815 | C=CODE[1]; // generator matrix in standard form |
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4816 | D=permute_L(D,CODE[3]); // suitable permutation of coordinates |
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4817 | list SV=prepSV(G,D,F,CURVE); |
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4818 | SV[1]=CODE[2]; // parity-check matrix in standard form |
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4819 | |
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4820 | (II) Using the true minimum distance d for decoding : |
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4821 | |
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4822 | matrix C=AGcode_Omega(G,D,CURVE); |
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4823 | int n=size(D); |
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4824 | list SV=prepSV(G,D,F,CURVE); |
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4825 | SV[n+3][2]=d; // then use decodeSV(y,SV); |
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4826 | |
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4827 | |
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4828 | // ============================================================================ |
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4829 | // *** Some "macros" with typical examples of curves in Coding Theory **** |
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4830 | // ============================================================================ |
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4831 | |
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4832 | |
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4833 | proc Klein () |
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4834 | { |
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4835 | list KLEIN=Adj_div(x3y+y3+x); |
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4836 | KLEIN=NSplaces(2,KLEIN); |
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4837 | KLEIN=extcurve(3,KLEIN); |
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4838 | dbprint(printlevel+1,"Klein quartic over F_8 successfully constructed"); |
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4839 | return(KLEIN); |
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4840 | } |
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4841 | |
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4842 | proc Hermite (int m) |
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4843 | { |
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4844 | int p=char(basering); |
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4845 | int r=p^m; |
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4846 | list HERMITE=Adj_div(y^r+y-x^(r+1)); |
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4847 | HERMITE=NSplaces(2*m-1,HERMITE); |
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4848 | HERMITE=extcurve(2*m,HERMITE); |
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4849 | dbprint(printlevel+1,"Hermitian curve over F_("+string(r)+"^2) |
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4850 | successfully constructed"); |
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4851 | return(HERMITE); |
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4852 | } |
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4853 | |
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4854 | proc Suzuki () |
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4855 | { |
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4856 | list SUZUKI=Adj_div(x10+x3+y8+y); |
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4857 | SUZUKI=NSplaces(2,SUZUKI); |
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4858 | SUZUKI=extcurve(3,SUZUKI); |
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4859 | dbprint(printlevel+1,"Suzuki curve over F_8 successfully constructed"); |
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4860 | return(SUZUKI); |
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4861 | } |
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4862 | |
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4863 | // **** Other interesting examples : |
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4864 | |
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4865 | // A hyperelliptic curve with 33 rational points over F_16 |
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4866 | |
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4867 | list CURVE=Adj_div(x5+y2+y); |
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4868 | CURVE=NSplaces(3,CURVE); |
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4869 | CURVE=extcurve(4,CURVE); |
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4870 | |
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4871 | // A nice curve with 113 rational points over F_64 |
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4872 | |
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4873 | list CURVE=Adj_div(y9+y8+xy6+x2y3+y2+x3); |
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4874 | CURVE=NSplaces(4,CURVE); |
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4875 | CURVE=extcurve(6,CURVE); |
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4876 | |
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4877 | */ |
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4878 | ; |
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