1 | ////////////////////////////////////////////////////////////////////// |
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2 | version="version JMBTest.lib 4.0.0.0 Jun_2013 "; // $Id$ |
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3 | category="Algebraic Geometry"; |
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4 | // summary description of the library |
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5 | info=" |
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6 | LIBRARY: JMBTest.lib A library for Singular which performs JM basis test. |
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7 | AUTHOR: Michela Ceria, email: michela.ceria@unito.it |
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8 | |
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9 | SEE ALSO: JMSConst_lib |
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10 | KEYWORDS: J-marked schemes |
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11 | |
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12 | OVERVIEW: |
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13 | The library performs the J-marked basis test, as described in [CR], [BCLR]. |
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14 | Such a test is performed via the criterion explained in [BCLR], |
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15 | concerning Eliahou-Kervaire polynomials (EK from now on). |
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16 | We point out that all the polynomials are homogeneous |
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17 | and they must be arranged by degree. |
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18 | The fundamental steps are the following:@* |
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19 | -construct the Vm polynomials, via the algorithm VConstructor |
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20 | explained in [CR];@* |
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21 | -construct the Eliahou-Kervaire polynomials defined in [BCLR];@* |
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22 | -reduce the Eliahou-Kervaire polynomials using the Vm's;@* |
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23 | -if it exist an Eliahou-Kervaire polynomial such that its reduction |
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24 | mod Vm is different from zero, the given one is not a J-Marked basis. |
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25 | |
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26 | The algorithm terminates only if the ordering is rp. |
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27 | Anyway, the number of reduction steps is bounded. |
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28 | |
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29 | REFERENCES: |
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30 | [CR] Francesca Cioffi, Margherita Roggero,Flat Families by Strongly |
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31 | Stable Ideals and a Generalization of Groebner Bases, |
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32 | J. Symbolic Comput. 46, 1070-1084, (2011).@* |
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33 | [BCLR] Cristina Bertone, Francesca Cioffi, Paolo Lella, |
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34 | Margherita Roggero, Upgraded methods for the effective |
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35 | computation of marked schemes on a strongly stable ideal, |
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36 | Journal of Symbolic Computation |
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37 | (2012), http://dx.doi.org/10.1016/j.jsc.2012.07.006 @* |
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38 | |
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39 | PROCEDURES: |
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40 | Minimus(ideal) minimal variable in an ideal |
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41 | Maximus(ideal) maximal variable in an ideal |
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42 | StartOrderingV(list,list) ordering of polynomials as in [BCLR] |
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43 | TestJMark(list) tests whether we have a J-marked basis |
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44 | "; |
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45 | LIB "qhmoduli.lib"; |
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46 | LIB "monomialideal.lib"; |
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47 | LIB "ring.lib"; |
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48 | //////////////////////////////////////////////////////////////////// |
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49 | proc mod_init() |
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50 | "USAGE: mod_init(); |
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51 | RETURN: struct: jmp |
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52 | EXAMPLE: example mod_init; shows an example" |
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53 | { |
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54 | newstruct("jmp", "poly h, poly t"); |
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55 | } |
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56 | example |
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57 | { "EXAMPLE:"; echo = 2; |
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58 | mod_init(); |
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59 | } |
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60 | //////////////////////////////////////////////////////////////////// |
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61 | proc Terns(list G, int c) |
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62 | "USAGE: Terns(G,c); G list, c int |
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63 | RETURN: list: T |
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64 | NOTE: Input is a list of J-marked polynomials |
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65 | (arranged by degree) and an integer. |
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66 | EXAMPLE: example Terns; shows an example" |
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67 | { |
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68 | list T=list(); |
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69 | int z; |
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70 | for(int k=1; k<=size(G[c]);k=k+1) |
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71 | { |
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72 | //Loop on G[c] making positions of polynomials in G[c] |
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73 | z=size(T); |
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74 | T=insert(T,list(1,c,k) ,size(T)); |
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75 | } |
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76 | return(T); |
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77 | } |
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78 | example |
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79 | { "EXAMPLE:"; echo = 2; |
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80 | ring r=0, (x,y,z), rp; |
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81 | jmp r1; |
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82 | r1.h=z^3; |
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83 | r1.t=poly(0); |
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84 | jmp r2; |
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85 | r2.h=z^2*y; |
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86 | r2.t=poly(0); |
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87 | jmp r3; |
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88 | r3.h=z*y^2 ; |
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89 | r3.t=-x^2*y; |
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90 | jmp r4; |
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91 | r4.h=y^5; |
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92 | r4.t=poly(0); |
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93 | list G2F=list(list(r1,r2,r3),list(r4)); |
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94 | Terns(G2F, 1); |
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95 | Terns(G2F, 2); |
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96 | } |
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97 | //////////////////////////////////////////////////////////////////// |
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98 | proc VConst(list G, int c) |
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99 | "USAGE: VConst(G, c); G list, c int |
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100 | RETURN: list: V |
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101 | NOTES: this procedure computes the Vm polynomials following the |
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102 | algorithm in [CR],but it only keeps in memory the monomials by |
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103 | which the G's must be multplied and their positions. |
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104 | EXAMPLE: example VConst; shows an example" |
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105 | { |
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106 | jmp f=G[1][1]; |
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107 | int aJ=deg(f.h); |
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108 | // minimal degree of polynomials in G |
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109 | //print(aJ); |
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110 | list V=list(); |
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111 | V[1]=Terns(G,1); |
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112 | // V[1]=G[1] (keeping in memory only [head, position]) |
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113 | //print(c-aJ+1); |
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114 | int i; |
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115 | int j; |
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116 | int m; |
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117 | list OO; |
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118 | jmp p; |
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119 | for(m=2; m<=c-aJ+1; m=m+1) |
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120 | { |
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121 | //print("entro nel form"); |
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122 | if(m>size(G)) |
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123 | {V[m]=list(); |
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124 | //If we have not G[m] we insert a list() |
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125 | //print("vuota prima"); |
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126 | } |
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127 | else |
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128 | {V[m]=Terns(G,m); |
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129 | //print("piena prima"); |
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130 | } |
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131 | for(i=1; i<nvars(basering)+1; i=i+1) |
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132 | { |
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133 | //print("entrata fori"); |
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134 | //print(i); |
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135 | for(j=1; j<=size(V[m-1]); j=j+1) |
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136 | { |
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137 | p=G[V[m-1][j][2]][V[m-1][j][3]]; |
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138 | //print(p.h); |
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139 | //print(p.t); |
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140 | //print(var(i)); |
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141 | //print(Minimus(V[m-1][j][1]*p.h)); |
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142 | if(var(i)<=Minimus(variables(V[m-1][j][1]*p.h))) |
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143 | { |
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144 | //Can I multiply by the current variable? |
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145 | //print("minoremin"); |
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146 | //print("fin qui ci sono"); |
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147 | //print(V[m-1][j][1]); |
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148 | OO=list(var(i)*V[m-1][j][1],V[m-1][j][2],V[m-1][j][3]); |
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149 | V[m]=insert(V[m], OO ,size(V[m])); |
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150 | } |
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151 | } |
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152 | } |
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153 | } |
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154 | return (V);} |
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155 | example |
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156 | { "EXAMPLE:"; echo = 2; |
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157 | ring r=0, (x,y,z), rp; |
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158 | jmp r1; |
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159 | r1.h=z^3; |
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160 | r1.t=poly(0); |
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161 | jmp r2; |
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162 | r2.h=z^2*y; |
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163 | r2.t=poly(0); |
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164 | jmp r3; |
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165 | r3.h=z*y^2 ; |
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166 | r3.t=-x^2*y; |
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167 | jmp r4; |
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168 | r4.h=y^5; |
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169 | r4.t=poly(0); |
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170 | list G2F=list(list(r1,r2,r3),list(r4)); |
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171 | VConst(G2F,4,basering);} |
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172 | //////////////////////////////////////////////////////////////////// |
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173 | proc Minimus(ideal L) |
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174 | "USAGE: Minimus(L); G list, c int |
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175 | RETURN: list: V |
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176 | NOTES: it returns the minimal variable generating the ideal L.@* |
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177 | The input must be an ideal generated by variables. |
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178 | EXAMPLE: example Minimus; shows an example" |
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179 | { |
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180 | poly min=L[1]; |
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181 | int i; |
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182 | for(i=2;i<=size(L); i++) |
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183 | { |
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184 | if(L[i]<min){min=L[i];} |
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185 | } |
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186 | return(min); |
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187 | } |
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188 | example |
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189 | { "EXAMPLE:"; echo = 2; |
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190 | ring r=0, (x,y,z), rp; |
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191 | ideal I=y,x,z; |
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192 | Minimus(I); |
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193 | } |
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194 | //////////////////////////////////////////////////////////////////// |
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195 | proc Maximus(ideal L) |
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196 | "USAGE: Maximus(L); G list, c int |
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197 | RETURN: list: V |
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198 | NOTES: it returns the maximal variable generating the ideal L.@* |
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199 | The input must be an ideal generated by variables. |
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200 | EXAMPLE: example Maximus; shows an example" |
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201 | { |
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202 | poly max=L[1]; |
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203 | int i; |
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204 | for(i=2;i<=size(L); i++) |
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205 | { |
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206 | if(L[i]>max){max=L[i];} |
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207 | } |
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208 | return(max); |
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209 | } |
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210 | example |
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211 | { "EXAMPLE:"; echo = 2; |
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212 | ring r=0, (x,y,z), rp; |
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213 | ideal I=y,x,z; |
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214 | Maximus(I); |
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215 | } |
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216 | //////////////////////////////////////////////////////////////////// |
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217 | proc GJmpMins(jmp P, jmp Q) |
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218 | "USAGE: GJmpMins(P,Q); P jmp, Q jmp |
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219 | RETURN: int: d |
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220 | EXAMPLE: example GJmpMins; shows an example" |
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221 | { |
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222 | int d=1; |
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223 | //-1=lower, 0=equal, 1=higher |
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224 | //At the beginning suppose Q is higher |
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225 | if(deg(P.h)<deg(Q.h)) |
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226 | { |
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227 | //Compare degrees; |
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228 | d=-1; |
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229 | //print("Per Grado"); |
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230 | } |
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231 | if(deg(P.h)==deg(Q.h)) |
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232 | { |
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233 | if(P.h==Q.h) |
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234 | { |
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235 | if(P.t==Q.t) |
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236 | { |
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237 | //head=tail |
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238 | d=0; |
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239 | //print("Uguali"); |
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240 | } |
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241 | } |
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242 | else |
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243 | { |
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244 | //print(Minimus(variables(P.h/gcdMon(P.h,Q.h)))); |
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245 | //print(Minimus(variables(Q.h/gcdMon(P.h,Q.h)))); |
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246 | if(Minimus(variables(P.h/gcdMon(P.h,Q.h)))<Minimus(variables(Q.h/gcdMon(P.h,Q.h)))) |
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247 | { |
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248 | d=-1; |
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249 | //print("Per Indice"); |
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250 | } |
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251 | } |
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252 | } |
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253 | return(d); |
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254 | } |
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255 | example |
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256 | { "EXAMPLE:"; echo = 2; |
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257 | ring r=0, (x,y,z), rp; |
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258 | jmp p1; |
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259 | p1.h=poly(1); |
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260 | p1.t=poly(1); |
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261 | jmp p2; |
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262 | p2.h=x^2; |
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263 | p2.t=poly(0); |
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264 | jmp p3; |
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265 | p3.h=x; |
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266 | p3.t=poly(0); |
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267 | GJmpMins(p1, p2); |
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268 | GJmpMins(p2, p3); |
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269 | GJmpMins(p1,p1); |
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270 | } |
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271 | //////////////////////////////////////////////////////////////////// |
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272 | proc TernCompare(list A, list B, list G) |
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273 | "USAGE: TernCompare(A,B,C); A list, B list, G list |
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274 | RETURN: int: d |
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275 | NOTE: A and B are terns, while G is the given list of |
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276 | J-marked polynomials. |
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277 | EXAMPLE: example TernCompare; shows an example" |
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278 | { |
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279 | int d=-1; |
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280 | //Start: A<B |
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281 | if(A[1]==B[1]) |
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282 | { |
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283 | if(A[2]==B[2]&& A[3]==B[3]) |
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284 | { |
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285 | //print("Uguali"); |
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286 | d=0; |
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287 | } |
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288 | else |
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289 | { |
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290 | jmp g1=G[A[2]][A[3]]; |
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291 | jmp g2=G[B[2]][B[3]]; |
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292 | if(GJmpMins(g1, g2)==1) |
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293 | { |
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294 | //print("Maggiore per il G"); |
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295 | d=1; |
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296 | } |
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297 | } |
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298 | } |
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299 | else |
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300 | { |
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301 | if(A[1]>B[1]) |
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302 | { |
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303 | //the ordering MUST be rp |
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304 | //print("Maggiore per Lex"); |
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305 | d=1; |
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306 | } |
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307 | } |
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308 | return(d); |
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309 | } |
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310 | example |
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311 | { "EXAMPLE:"; echo = 2; |
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312 | ring r=0, (x,y,z), rp; |
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313 | jmp r1; |
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314 | r1.h=z^3; |
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315 | r1.t=poly(0); |
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316 | jmp r2; |
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317 | r2.h=z^2*y; |
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318 | r2.t=poly(0); |
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319 | jmp r3; |
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320 | r3.h=z*y^2 ; |
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321 | r3.t=-x^2*y; |
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322 | jmp r4; |
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323 | r4.h=y^5; |
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324 | r4.t=poly(0); |
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325 | list G2F=list(list(r1,r2,r3),list(r4)); |
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326 | TernCompare([1,1,1],[x,1,1],G2F); |
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327 | } |
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328 | //////////////////////////////////////////////////////////////////// |
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329 | proc MinOfV(list V, list G) |
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330 | "USAGE: Minimal(V,G); V list, G list |
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331 | RETURN: int: R |
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332 | NOTE: Input=lista(terne), G. |
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333 | EXAMPLE: example Minimal; shows an example" |
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334 | { |
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335 | //Minimal element for a given degree |
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336 | list R=list(); |
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337 | list MIN=V[1]; |
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338 | int h=1; |
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339 | int i; |
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340 | for(i=2; i<=size(V); i++) |
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341 | { |
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342 | //I consider the first as minimum |
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343 | //If I find something smaller I change minimum |
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344 | if(TernCompare(V[i],MIN,G)<=0) |
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345 | { |
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346 | MIN=V[i]; |
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347 | h=i; |
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348 | } |
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349 | } |
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350 | //Return: [minimum,position of the minimum] |
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351 | R=MIN,h; |
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352 | return(R); |
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353 | } |
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354 | example |
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355 | { "EXAMPLE:"; echo = 2; |
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356 | ring r=0, (x,y,z), rp; |
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357 | jmp r1; |
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358 | r1.h=z^3; |
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359 | r1.t=poly(0); |
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360 | jmp r2; |
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361 | r2.h=z^2*y; |
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362 | r2.t=poly(0); |
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363 | jmp r3; |
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364 | r3.h=z*y^2 ; |
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365 | r3.t=-x^2*y; |
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366 | jmp r4; |
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367 | r4.h=y^5; |
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368 | r4.t=poly(0); |
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369 | list G2F=list(list(r1,r2,r3),list(r4)); |
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370 | MinOfV(VConst(G2F,4,basering)[1],G2F); |
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371 | } |
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372 | //////////////////////////////////////////////////////////////////// |
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373 | proc OrderingV(list V,list G,list R) |
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374 | "USAGE: OrderingV(V,G,R); V list, G list, R list |
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375 | RETURN: list: R |
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376 | NOTE: Input: Vm,G,emptylist |
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377 | EXAMPLE: example OrderingV; shows an example" |
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378 | { |
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379 | //Order V[m] |
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380 | //R will contain results but at the beginning it is empty |
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381 | list M=list(); |
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382 | if(size(V)==1) |
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383 | { |
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384 | R=insert(R,V[1],size(R)); |
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385 | } |
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386 | else |
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387 | { |
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388 | M=MinOfV(V,G); |
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389 | R=insert(R,M[1],size(R)); |
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390 | V=delete(V,M[2]); |
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391 | //recursive call |
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392 | R=OrderingV(V,G,R); |
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393 | } |
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394 | return(R); |
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395 | } |
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396 | example |
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397 | { "EXAMPLE:"; echo = 2; |
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398 | ring r=0, (x,y,z), rp; |
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399 | jmp r1; |
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400 | r1.h=z^3; |
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401 | r1.t=poly(0); |
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402 | jmp r2; |
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403 | r2.h=z^2*y; |
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404 | r2.t=poly(0); |
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405 | jmp r3; |
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406 | r3.h=z*y^2; |
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407 | r3.t=-x^2*y; |
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408 | jmp r4; |
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409 | r4.h=y^5; |
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410 | r4.t=poly(0); |
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411 | list G2F=list(list(r1,r2,r3),list(r4)); |
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412 | OrderingV(VConst(G2F,4,basering)[1],G2F,list()); |
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413 | } |
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414 | //////////////////////////////////////////////////////////////////// |
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415 | proc StartOrderingV(list V,list G) |
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416 | "USAGE: StartOrdina(V,G); V list, G list |
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417 | RETURN: list: R |
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418 | NOTE: Input Vm,G. This procedure uses OrderingV to get |
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419 | the ordered polynomials as in [BCLR]. |
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420 | EXAMPLE: example StartOrderingV; shows an example" |
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421 | { |
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422 | return(OrderingV(V,G, list())); |
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423 | } |
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424 | example |
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425 | { "EXAMPLE:"; echo = 2; |
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426 | ring r=0, (x,y,z), rp; |
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427 | jmp r1; |
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428 | r1.h=z^3; |
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429 | r1.t=poly(0); |
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430 | jmp r2; |
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431 | r2.h=z^2*y; |
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432 | r2.t=poly(0); |
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433 | jmp r3; |
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434 | r3.h=z*y^2; |
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435 | r3.t=-x^2*y; |
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436 | jmp r4; |
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437 | r4.h=y^5; |
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438 | r4.t=poly(0); |
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439 | list G2F=list(list(r1,r2,r3),list(r4)); |
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440 | StartOrderingV(VConst(G2F,4,basering)[1],G2F); |
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441 | } |
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442 | //////////////////////////////////////////////////////////////////// |
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443 | proc Multiply(list L, list G) |
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444 | "USAGE: moltiplica(L,G); L list, G list |
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445 | RETURN: jmp: K |
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446 | NOTE: Input: a 3-ple,G. It performs the product associated |
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447 | to the 3-uple. |
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448 | EXAMPLE: example Multiply; shows an example" |
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449 | { |
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450 | jmp g=G[L[2]][L[3]]; |
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451 | jmp K; |
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452 | K.h=L[1]*g.h; |
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453 | K.t=L[1]*g.t; |
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454 | return(K); |
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455 | } |
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456 | example |
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457 | { "EXAMPLE:"; echo = 2; |
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458 | ring r=0, (x,y,z), rp; |
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459 | list P=x^2,1,1; |
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460 | jmp r1; |
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461 | r1.h=z^3; |
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462 | r1.t=poly(0); |
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463 | jmp r2; |
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464 | r2.h=z^2*y; |
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465 | r2.t=poly(0); |
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466 | jmp r3; |
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467 | r3.h=z*y^2 ; |
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468 | r3.t=-x^2*y; |
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469 | jmp r4; |
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470 | r4.h=y^5; |
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471 | r4.t=poly(0); |
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472 | list G2F=list(list(r1,r2,r3),list(r4)); |
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473 | Multiply(P,G2F); |
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474 | } |
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475 | //////////////////////////////////////////////////////////////////// |
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476 | proc IdealOfV(list V) |
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477 | "USAGE: IdealOfV(V); V list |
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478 | RETURN: ideal: I |
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479 | NOTES: this procedure takes a list of Vm's of a certain degree |
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480 | and construct their ideal, multiplying the head by the weighted |
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481 | variable t. |
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482 | EXAMPLE: example IdealOfV; shows an example" |
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483 | { |
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484 | ideal I=0; |
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485 | int i; |
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486 | if (size(V)!=0) |
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487 | { |
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488 | list M=list(); |
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489 | jmp g; |
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490 | for(i=1; i<= size(V); i++) |
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491 | { |
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492 | g=V[i]; |
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493 | g.h=t*g.h; |
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494 | M[i]=g.h+g.t; |
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495 | } |
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496 | I=M[1..size(M)]; |
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497 | //print("IdealOfV"); |
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498 | //I=std(I); |
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499 | } |
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500 | return(I); |
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501 | } |
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502 | example |
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503 | { "EXAMPLE:"; echo = 2; |
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504 | ring r=0, (x,y,z,t), rp; |
---|
505 | jmp r1; |
---|
506 | r1.h=z^3; |
---|
507 | r1.t=poly(0); |
---|
508 | jmp r2; |
---|
509 | r2.h=z^2*y; |
---|
510 | r2.t=poly(0); |
---|
511 | jmp r3; |
---|
512 | r3.h=z*y^2 ; |
---|
513 | r3.t=-x^2*y; |
---|
514 | jmp r4; |
---|
515 | r4.h=y^5; |
---|
516 | r4.t=poly(0); |
---|
517 | list G2F=list(list(r1,r2,r3),list(r4)); |
---|
518 | IdealOfV(G2F[1]); |
---|
519 | } |
---|
520 | //////////////////////////////////////////////////////////////////// |
---|
521 | proc NewWeight(int n) |
---|
522 | "USAGE: NewWeight(n); n int |
---|
523 | RETURN: intvec: u |
---|
524 | EXAMPLE: example NewWeight; shows an example" |
---|
525 | { |
---|
526 | intvec u=0; |
---|
527 | u[n]=1; |
---|
528 | return(u); |
---|
529 | } |
---|
530 | example |
---|
531 | { "EXAMPLE:"; echo = 2; |
---|
532 | NewWeight(3); |
---|
533 | } |
---|
534 | //////////////////////////////////////////////////////////////////// |
---|
535 | proc FinalVm(list V1 , list G1 , r) |
---|
536 | "USAGE: FinalVm(V1, G1, r); V1 list, G1 list , r |
---|
537 | RETURN: intvec: u |
---|
538 | EXAMPLE: example NewWeight; shows an example" |
---|
539 | { |
---|
540 | //multiply and reduce, degree by degree |
---|
541 | intvec u=NewWeight(nvars(r)+1); |
---|
542 | list L=ringlist(r); |
---|
543 | L[2]=insert(L[2],"t",size(L[2])); |
---|
544 | //print(L[2]); |
---|
545 | list ordlist="a",u; |
---|
546 | L[3]=insert(L[3],ordlist,0); |
---|
547 | def H=ring(L); |
---|
548 | //print(V1); |
---|
549 | //print(G1); |
---|
550 | list M=list(); |
---|
551 | jmp p; |
---|
552 | list N; |
---|
553 | poly q; |
---|
554 | poly s; |
---|
555 | int i; |
---|
556 | int j; |
---|
557 | for(i=1; i<=size(G1); i++) |
---|
558 | { |
---|
559 | N=list(); |
---|
560 | for(j=1; j<=size(G1[i]); j++) |
---|
561 | { |
---|
562 | p=G1[i][j]; |
---|
563 | q=p.h; |
---|
564 | s=p.t; |
---|
565 | N[j]=list(q,s); |
---|
566 | } |
---|
567 | M[i]=N; |
---|
568 | } |
---|
569 | p.h=poly(0); |
---|
570 | p.t=poly(0); |
---|
571 | setring H; |
---|
572 | list R=list(); |
---|
573 | list S=list(); |
---|
574 | //print("anello definito"); |
---|
575 | def V=imap(r,V1); |
---|
576 | //def G=imap(r,G1); |
---|
577 | //print(V); |
---|
578 | def MM=imap(r,M); |
---|
579 | list G=list(); |
---|
580 | list N=list(); |
---|
581 | for(i=1; i<=size(MM); i++) |
---|
582 | { |
---|
583 | for(j=1; j<=size(MM[i]); j++) |
---|
584 | { |
---|
585 | p.h=MM[i][j][1]; |
---|
586 | p.t=MM[i][j][2]; |
---|
587 | N[j]=p; |
---|
588 | } |
---|
589 | G[i]=N; |
---|
590 | } |
---|
591 | ideal I=0; |
---|
592 | jmp LL; |
---|
593 | jmp UU; |
---|
594 | for(i=1; i<=size(V);i++) |
---|
595 | { |
---|
596 | R[i]=list(); |
---|
597 | S[i]=list(); |
---|
598 | I=0; |
---|
599 | for(j=1;j<=size(V[i]); j++) |
---|
600 | { |
---|
601 | LL=Multiply(V[i][j],G); |
---|
602 | LL.t=reduce(t*LL.t,I); |
---|
603 | //I only reduce the tail |
---|
604 | LL.t=subst(LL.t,t,1); |
---|
605 | S[i]=insert(S[i],LL,size(S[i])); |
---|
606 | LL.h=t*LL.h; |
---|
607 | R[i]=insert(R[i],LL,size(R[i])); |
---|
608 | UU=R[i][j]; |
---|
609 | I=I+ideal(UU.h+UU.t); |
---|
610 | attrib(I,"isSB",1); |
---|
611 | } |
---|
612 | } |
---|
613 | list M=list(); |
---|
614 | poly q; |
---|
615 | poly s; |
---|
616 | for(i=1; i<=size(S); i++) |
---|
617 | { |
---|
618 | N=list(); |
---|
619 | for(j=1; j<=size(S[i]); j++) |
---|
620 | { |
---|
621 | p=S[i][j]; |
---|
622 | q=p.h; |
---|
623 | s=p.t; |
---|
624 | N[j]=list(q,s); |
---|
625 | } |
---|
626 | M[i]=N; |
---|
627 | } |
---|
628 | p.h=poly(0); |
---|
629 | p.t=poly(0); |
---|
630 | setring r; |
---|
631 | def MM=imap(H,M); |
---|
632 | list MMM=list(); |
---|
633 | for(i=1; i<=size(MM); i++) |
---|
634 | { |
---|
635 | N=list(); |
---|
636 | for(j=1; j<=size(MM[i]); j++) |
---|
637 | { |
---|
638 | p.h=MM[i][j][1]; |
---|
639 | p.t=MM[i][j][2]; |
---|
640 | N[j]=p; |
---|
641 | } |
---|
642 | MMM[i]=N; |
---|
643 | } |
---|
644 | return(MMM); |
---|
645 | } |
---|
646 | example |
---|
647 | { "EXAMPLE:"; echo = 2; |
---|
648 | ring r=0, (x,y,z), rp; |
---|
649 | jmp r1; |
---|
650 | r1.h=z^3; |
---|
651 | r1.t=poly(0); |
---|
652 | jmp r2; |
---|
653 | r2.h=z^2*y; |
---|
654 | r2.t=poly(0); |
---|
655 | jmp r3; |
---|
656 | r3.h=z*y^2 ; |
---|
657 | r3.t=-x^2*y; |
---|
658 | jmp r4; |
---|
659 | r4.h=y^5; |
---|
660 | r4.t=poly(0); |
---|
661 | list G2F=list(list(r1,r2,r3),list(r4)); |
---|
662 | FinalVm(VConst(G2F,6,r) , G2F, r); |
---|
663 | } |
---|
664 | //////////////////////////////////////////////////////////////////// |
---|
665 | proc ConstructorMain(list G, int c,r) |
---|
666 | "USAGE: Costruttore(G,c); G list, c int |
---|
667 | RETURN: list: R |
---|
668 | NOTE: At the end separated by degree. |
---|
669 | EXAMPLE: example Costruttore; shows an example" |
---|
670 | { |
---|
671 | list V=list(); |
---|
672 | V= VConst(G,c); |
---|
673 | //print("VConst"); |
---|
674 | //V non ordered |
---|
675 | list L=list(); |
---|
676 | list R=list(); |
---|
677 | int i; |
---|
678 | // head, position |
---|
679 | //order the different degrees |
---|
680 | for(i=1; i<=size(V); i++) |
---|
681 | { |
---|
682 | L[i]=StartOrderingV(V[i], G); |
---|
683 | } |
---|
684 | //multiply and reduce |
---|
685 | //print("Ordinare"); |
---|
686 | R=FinalVm(L, G, r); |
---|
687 | //print("FinalVm"); |
---|
688 | return(R); |
---|
689 | } |
---|
690 | example |
---|
691 | { "EXAMPLE:"; echo = 2; |
---|
692 | ring r=0, (x,y,z), rp; |
---|
693 | jmp r1; |
---|
694 | r1.h=z^3; |
---|
695 | r1.t=poly(0); |
---|
696 | jmp r2; |
---|
697 | r2.h=z^2*y; |
---|
698 | r2.t=poly(0); |
---|
699 | jmp r3; |
---|
700 | r3.h=z*y^2 ; |
---|
701 | r3.t=-x^2*y; |
---|
702 | jmp r4; |
---|
703 | r4.h=y^5; |
---|
704 | r4.t=poly(0); |
---|
705 | list G2F=list(list(r1,r2,r3),list(r4)); |
---|
706 | ConstructorMain(G2F,6,r); |
---|
707 | } |
---|
708 | //////////////////////////////////////////////////////////////////// |
---|
709 | proc EKCouples(jmp A, jmp B) |
---|
710 | "USAGE: CoppiaEK(A,B); A list, B list |
---|
711 | RETURN: list: L |
---|
712 | NOTE: At the end the monomials involved by EK. |
---|
713 | EXAMPLE: example EKCouples; shows an example" |
---|
714 | { |
---|
715 | poly E; |
---|
716 | list L=0,0; |
---|
717 | string s=varstr(basering); |
---|
718 | list VVV=varstr(basering); |
---|
719 | //L will contain results |
---|
720 | poly h=Minimus(variables(A.h)); |
---|
721 | //print(h); |
---|
722 | int l=findvars(h,1)[2][1]; |
---|
723 | if(l!=nvars(basering)) |
---|
724 | { |
---|
725 | //print("vero"); |
---|
726 | //print(l); |
---|
727 | for(int j=l+1;j<=nvars(basering); j++) |
---|
728 | { |
---|
729 | //print("entrata"); |
---|
730 | //print(var(j)); |
---|
731 | E=var(j)*A.h/B.h; |
---|
732 | //Candidate for * product |
---|
733 | //print(E); |
---|
734 | if(E!=0) |
---|
735 | { |
---|
736 | //print("primo if passato"); |
---|
737 | if(Minimus(variables(B.h))>=Maximus(variables(E))) |
---|
738 | { |
---|
739 | //Does it work with * ? |
---|
740 | //print("secondo if passato"); |
---|
741 | L[1]=j; |
---|
742 | L[2]=E; |
---|
743 | break; |
---|
744 | } |
---|
745 | } |
---|
746 | } |
---|
747 | } |
---|
748 | return (L); |
---|
749 | } |
---|
750 | example |
---|
751 | { "EXAMPLE:"; echo = 2; |
---|
752 | ring r=0, (x,y,z), rp; |
---|
753 | jmp A; |
---|
754 | A.h=y*z^2; |
---|
755 | A.t=poly(0); |
---|
756 | jmp B; |
---|
757 | B.h=y^2*z; |
---|
758 | B.t=poly(0); |
---|
759 | EKCouples(A,B); |
---|
760 | EKCouples(B,A); |
---|
761 | } |
---|
762 | //////////////////////////////////////////////////////////////////// |
---|
763 | proc EKPolys(list G) |
---|
764 | "USAGE: PolysEK(G); G list |
---|
765 | RETURN: list: EK, list: D |
---|
766 | NOTE: At the end EK polynomials and their degrees |
---|
767 | |
---|
768 | EXAMPLE: example PolysEK; shows an example" |
---|
769 | { |
---|
770 | list D=list(); |
---|
771 | list C=list(); |
---|
772 | list N=0,0; |
---|
773 | list EK=list(); |
---|
774 | int i; |
---|
775 | int j; |
---|
776 | int k; |
---|
777 | int l; |
---|
778 | jmp p; |
---|
779 | for(i=1; i<=size(G); i++) |
---|
780 | { |
---|
781 | for(j=1; j<=size(G[i]); j++) |
---|
782 | { |
---|
783 | for(k=1; k<=size(G); k++) |
---|
784 | { |
---|
785 | for(l=1; l<=size(G[k]); l++) |
---|
786 | { |
---|
787 | if(i!=k||j!=l) |
---|
788 | { |
---|
789 | //Loop on polynomials |
---|
790 | C=EKCouples(G[i][j], G[k][l]); |
---|
791 | //print("coppia"); |
---|
792 | if(C[2]!=0) |
---|
793 | { |
---|
794 | C=insert(C,list(i,j,k,l),size(C)); |
---|
795 | EK=insert(EK,C,size(EK)); |
---|
796 | p=G[k][l]; |
---|
797 | D=insert(D,deg(C[2]*p.h),size(D)); |
---|
798 | } |
---|
799 | } |
---|
800 | } |
---|
801 | } |
---|
802 | } |
---|
803 | } |
---|
804 | //Double Return |
---|
805 | return(EK, D); |
---|
806 | } |
---|
807 | example |
---|
808 | { "EXAMPLE:"; echo = 2; |
---|
809 | ring r=0, (x,y,z), rp; |
---|
810 | jmp r1; |
---|
811 | r1.h=z^3; |
---|
812 | r1.t=poly(0); |
---|
813 | jmp r2; |
---|
814 | r2.h=z^2*y; |
---|
815 | r2.t=poly(0); |
---|
816 | jmp r3; |
---|
817 | r3.h=z*y^2; |
---|
818 | r3.t=-x^2*y; |
---|
819 | jmp r4; |
---|
820 | r4.h=y^5; |
---|
821 | r4.t=poly(0); |
---|
822 | list G2F=list(list(r1,r2,r3),list(r4)); |
---|
823 | EKPolys(G2F); |
---|
824 | } |
---|
825 | //////////////////////////////////////////////////////////////////// |
---|
826 | proc EKPolynomials(list EK, list G) |
---|
827 | "USAGE: EKPolynomials(EK,G); EK list, G list |
---|
828 | RETURN: list: p |
---|
829 | NOTE: At the end I obtain the EK polynomials and |
---|
830 | their degrees. |
---|
831 | EXAMPLE: example SpolyEK; shows an example" |
---|
832 | { |
---|
833 | jmp u=G[EK[3][1]][EK[3][2]]; |
---|
834 | jmp q=G[EK[3][3]][EK[3][4]]; |
---|
835 | return(var(EK[1])*(u.h+u.t)-EK[2]*(q.h+q.t)); |
---|
836 | } |
---|
837 | example |
---|
838 | { "EXAMPLE:"; echo = 2; |
---|
839 | ring r=0, (x,y,z), rp; |
---|
840 | jmp r1; |
---|
841 | r1.h=z^3; |
---|
842 | r1.t=poly(0); |
---|
843 | jmp r2; |
---|
844 | r2.h=z^2*y; |
---|
845 | r2.t=poly(0); |
---|
846 | jmp r3; |
---|
847 | r3.h=z*y^2; |
---|
848 | r3.t=-x^2*y; |
---|
849 | jmp r4; |
---|
850 | r4.h=y^5; |
---|
851 | r4.t=poly(0); |
---|
852 | list G2F=list(list(r1,r2,r3),list(r4)); |
---|
853 | list EK,D=EKPolys(G2F); |
---|
854 | EKPolynomials(EK[1],G2F); |
---|
855 | } |
---|
856 | //////////////////////////////////////////////////////////////////// |
---|
857 | proc TestJMark(list G1,r) |
---|
858 | "USAGE: TestJMark(G); G list |
---|
859 | RETURN: int: i |
---|
860 | NOTE: |
---|
861 | This procedure performs J-marked basis test.@* |
---|
862 | The input is a list of J-marked polynomials (jmp) arranged |
---|
863 | by degree, so G1 is a list of list.@* |
---|
864 | The output is a boolean evaluation: |
---|
865 | True=1/False=0 |
---|
866 | EXAMPLE: example TestJMark; shows an example" |
---|
867 | {int flag=1; |
---|
868 | if(size(G1)==1 && size(G1[1])==1) |
---|
869 | { |
---|
870 | //Hypersurface |
---|
871 | print("Only One Polynomial"); |
---|
872 | flag=1; |
---|
873 | } |
---|
874 | else |
---|
875 | { |
---|
876 | int d=0; |
---|
877 | list EK,D=EKPolys(G1); |
---|
878 | //print("PolysEK"); |
---|
879 | //I found EK couples |
---|
880 | int massimo=Max(D); |
---|
881 | list V1=ConstructorMain(G1,massimo,r); |
---|
882 | //print("Costruttore"); |
---|
883 | //print(V1); |
---|
884 | jmp mi=V1[1][1]; |
---|
885 | int minimo=Min(deg(mi.h)); |
---|
886 | intvec u=NewWeight(nvars(r)+1); |
---|
887 | list L=ringlist(r); |
---|
888 | L[2]=insert(L[2],"t",size(L[2])); |
---|
889 | //print(L[2]); |
---|
890 | list ordlist="a",u; |
---|
891 | L[3]=insert(L[3],ordlist,0); |
---|
892 | def H=ring(L); |
---|
893 | list JJ=list(); |
---|
894 | jmp pp; |
---|
895 | jmp qq; |
---|
896 | int i; |
---|
897 | int j; |
---|
898 | list NN; |
---|
899 | for(i=size(V1);i>0;i--) |
---|
900 | { |
---|
901 | NN=list(); |
---|
902 | for(j=size(V1[i]);j>0;j--) |
---|
903 | { |
---|
904 | //print(j); |
---|
905 | pp=V1[i][j]; |
---|
906 | NN[j]=list(pp.h,pp.t); |
---|
907 | } |
---|
908 | //print(NN); |
---|
909 | JJ[i]=NN; |
---|
910 | //print(JJ[i]); |
---|
911 | //print(i); |
---|
912 | } |
---|
913 | //print(JJ); |
---|
914 | list KK=list(); |
---|
915 | list UU=list(); |
---|
916 | //jmp qq; |
---|
917 | for(i=size(G1);i>0;i--) |
---|
918 | { |
---|
919 | for(j=size(G1[i]);j>0;j--) |
---|
920 | { |
---|
921 | //print(j); |
---|
922 | qq=G1[i][j]; |
---|
923 | UU[j]=list(qq.h,qq.t); |
---|
924 | } |
---|
925 | //print(UU); |
---|
926 | KK[i]=UU; |
---|
927 | } |
---|
928 | setring H; |
---|
929 | //I defined the new ring with the weighted |
---|
930 | //variable t |
---|
931 | poly p; |
---|
932 | //print("anello definito"); |
---|
933 | def JJJ=imap(r,JJ); |
---|
934 | def EK=imap(r,EK); |
---|
935 | //print(flag); |
---|
936 | //imap(r,D); |
---|
937 | list V=list(); |
---|
938 | jmp fp; |
---|
939 | //int i; |
---|
940 | //int j; |
---|
941 | list N; |
---|
942 | for(i=size(JJJ); i>0; i--) |
---|
943 | { |
---|
944 | N=list(); |
---|
945 | for(j=size(JJJ[i]); j>0; j--) |
---|
946 | { |
---|
947 | fp.h=JJJ[i][j][1]; |
---|
948 | fp.t=JJJ[i][j][2]; |
---|
949 | N[j]=fp; |
---|
950 | } |
---|
951 | V[i]=N; |
---|
952 | } |
---|
953 | //print(V); |
---|
954 | def KKJ=imap(r,KK); |
---|
955 | list G=list(); |
---|
956 | list U=list(); |
---|
957 | for(i=1; i<=size(KKJ); i++) |
---|
958 | { |
---|
959 | for(j=1; j<=size(KKJ[i]); j++) |
---|
960 | { |
---|
961 | fp.h=KKJ[i][j][1]; |
---|
962 | fp.t=KKJ[i][j][2]; |
---|
963 | U[j]=fp; |
---|
964 | } |
---|
965 | G[i]=U; |
---|
966 | } |
---|
967 | // print(V); |
---|
968 | //print(G); |
---|
969 | //I imported in H everithing I need |
---|
970 | poly q; |
---|
971 | ideal I; |
---|
972 | for(j=1; j<=size(EK);j++) |
---|
973 | { |
---|
974 | d=D[j]; |
---|
975 | p=EKPolynomials(EK[j],G); |
---|
976 | //print("arrivo"); |
---|
977 | I=IdealOfV(V[d-minimo+1]); |
---|
978 | attrib(I,"isSB",1); |
---|
979 | //print(I); |
---|
980 | q=reduce(t*p,I); |
---|
981 | //print(I[1]); |
---|
982 | //print(t*p); |
---|
983 | q=subst(q,t,1); |
---|
984 | //I reduce all the EK polynomials |
---|
985 | // q=RiduzPoly(V[d-minimo+1], p); |
---|
986 | if(q!=0) |
---|
987 | { |
---|
988 | //check whether reduction is 0 |
---|
989 | print("NOT A BASIS"); |
---|
990 | flag=0; |
---|
991 | break; |
---|
992 | } |
---|
993 | } |
---|
994 | } |
---|
995 | //print(flag); |
---|
996 | setring r; |
---|
997 | //typeof(flag); |
---|
998 | return(flag); |
---|
999 | } |
---|
1000 | example |
---|
1001 | { "EXAMPLE:"; echo = 2; |
---|
1002 | ring r=0, (x,y,z), rp; |
---|
1003 | jmp r1; |
---|
1004 | r1.h=z^3; |
---|
1005 | r1.t=poly(0); |
---|
1006 | jmp r2; |
---|
1007 | r2.h=z^2*y; |
---|
1008 | r2.t=poly(0); |
---|
1009 | jmp r3; |
---|
1010 | r3.h=z*y^2 ; |
---|
1011 | r3.t=-x^2*y; |
---|
1012 | jmp r4; |
---|
1013 | r4.h=y^5; |
---|
1014 | r4.t=poly(0); |
---|
1015 | list G2F=list(list(r1,r2,r3),list(r4)); |
---|
1016 | TestJMark(G2F,r); |
---|
1017 | } |
---|