1 | //(GMG/BM, last modified 22.06.96) |
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2 | /////////////////////////////////////////////////////////////////////////////// |
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3 | version="$Id: ringgb.lib,v 1.16 2001/01/16 13:48:40 Singular Exp $"; |
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4 | category="Beta Testing"; |
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5 | info=" |
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6 | LIBRARY: ringgb.lib Examples and tests for ringgb development |
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7 | |
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8 | PROCEDURES: |
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9 | findZeroPoly(f); finds a zero polynomial for reducing f |
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10 | zeroReduce(f); normal form of f concerning the ideal of zero polynomials |
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11 | zeroReduceProt(f); normal form of f concerning the ideal of zero polynomials |
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12 | "; |
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13 | |
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14 | LIB "general.lib"; |
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15 | /////////////////////////////////////////////////////////////////////////////// |
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16 | |
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17 | proc findZeroPoly (poly f) |
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18 | "USAGE: findZeroPolyWrap(f); f - a polynomial |
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19 | RETURN: zero polynomial with the same leading term as f if exists, otherwise 0 |
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20 | NOTE: just a wrapper, work only in Z/2^n with n < int_machine_size - 1 |
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21 | EXAMPLE: example findZeroPoly; shows an example |
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22 | " |
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23 | { |
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24 | return(system("findZeroPoly", f)); |
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25 | } |
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26 | example |
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27 | { "EXAMPLE:"; echo = 2; |
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28 | option(teach); |
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29 | ring r = 65536, (y,x), dp; |
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30 | poly f = 1024*x^8*y^2+11264*x^8*y+28672*x^8+45056*x^7*y^2+36864*x^7*y+16384*x^7+40960*x^6*y^2+57344*x^6*y+32768*x^6+30720*x^5*y^2+10240*x^5*y+8192*x^5+35840*x^4*y^2+1024*x^4*y+20480*x^4+30720*x^3*y^2+10240*x^3*y+8192*x^3+4096*x^2*y^2+45056*x^2*y+49152*x^2+40960*x*y^2+57344*x*y+32768*x; |
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31 | findZeroPoly(f); |
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32 | } |
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33 | |
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34 | proc findZeroPoly (poly f) |
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35 | "USAGE: findZeroPoly(f); f - a polynomial |
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36 | RETURN: zero polynomial with the same leading term as f if exists, otherwise 0 |
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37 | EXAMPLE: example findZeroPoly; shows an example |
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38 | " |
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39 | { |
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40 | list data = getZeroCoef(f); |
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41 | if (data[1] == 0) |
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42 | { |
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43 | return(0); |
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44 | } |
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45 | number q = leadcoef(f) / data[1]; |
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46 | if (q == 0) |
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47 | { |
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48 | return(0); |
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49 | } |
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50 | poly g = getZeroPolyRaw(data[2]); |
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51 | g = leadmonom(f) / leadmonom(g) * g; |
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52 | return(q * data[1] * g); |
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53 | //return(system("findZeroPoly", f)); |
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54 | } |
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55 | example |
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56 | { "EXAMPLE:"; echo = 2; |
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57 | option(teach); |
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58 | ring r = 65536, (y,x), dp; |
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59 | poly f = 1024*x^8*y^2+11264*x^8*y+28672*x^8+45056*x^7*y^2+36864*x^7*y+16384*x^7+40960*x^6*y^2+57344*x^6*y+32768*x^6+30720*x^5*y^2+10240*x^5*y+8192*x^5+35840*x^4*y^2+1024*x^4*y+20480*x^4+30720*x^3*y^2+10240*x^3*y+8192*x^3+4096*x^2*y^2+45056*x^2*y+49152*x^2+40960*x*y^2+57344*x*y+32768*x; |
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60 | findZeroPoly(f); |
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61 | } |
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62 | |
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63 | proc zeroReduceExt (poly f , int i) |
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64 | "USAGE: zeroReduceExt(f, i); f - a polynomial, i - noisy level |
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65 | RETURN: reduced normal form of f modulo zero polynomials |
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66 | EXAMPLE: example zeroReduceExt; shows an example |
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67 | " |
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68 | { |
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69 | poly h = f; |
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70 | poly n = 0; |
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71 | poly g = findZeroPoly(h); |
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72 | while ( h <> 0 ) { |
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73 | while ( g <> 0 ) { |
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74 | h = h - g; |
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75 | if (i == 1) { |
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76 | printf("reduce with: %s", g); |
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77 | printf("to: %s", h); |
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78 | } |
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79 | g = findZeroPoly(h); |
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80 | } |
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81 | n = lead(h) + n; |
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82 | h = h - lead(h); |
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83 | g = findZeroPoly(h); |
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84 | } |
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85 | return(n); |
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86 | } |
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87 | example |
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88 | { "EXAMPLE:"; echo = 2; |
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89 | option(teach); |
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90 | ring r = 65536, (y,x), dp; |
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91 | poly f = 1024*x^8*y^2+11264*x^8*y+28672*x^8+45056*x^7*y^2+36864*x^7*y+16384*x^7+40960*x^6*y^2+57344*x^6*y+32768*x^6+30720*x^5*y^2+10240*x^5*y+8192*x^5+35840*x^4*y^2+1024*x^4*y+20480*x^4+30720*x^3*y^2+10240*x^3*y+8192*x^3+4096*x^2*y^2+45056*x^2*y+49152*x^2+40960*x*y^2+57344*x*y+32768*x; |
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92 | zeroReduceExt(f,0); |
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93 | zeroReduceExt(f,1); |
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94 | } |
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95 | |
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96 | proc zeroReduce (poly f) |
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97 | "USAGE: zeroReduce(f); f - a polynomial |
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98 | RETURN: reduced normal form of f modulo zero polynomials |
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99 | EXAMPLE: example zeroReduce; shows an example |
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100 | " |
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101 | { |
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102 | return(zeroReduceExt(f, 0)); |
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103 | } |
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104 | example |
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105 | { "EXAMPLE:"; echo = 2; |
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106 | option(teach); |
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107 | ring r = 65536, (y,x), dp; |
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108 | poly f = 1024*x^8*y^2+11264*x^8*y+28672*x^8+45056*x^7*y^2+36864*x^7*y+16384*x^7+40960*x^6*y^2+57344*x^6*y+32768*x^6+30720*x^5*y^2+10240*x^5*y+8192*x^5+35840*x^4*y^2+1024*x^4*y+20480*x^4+30720*x^3*y^2+10240*x^3*y+8192*x^3+4096*x^2*y^2+45056*x^2*y+49152*x^2+40960*x*y^2+57344*x*y+32768*x; |
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109 | zeroReduce(f); |
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110 | } |
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111 | |
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112 | proc zeroReduceProt (poly f) |
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113 | "USAGE: zeroReduceProt(f); f - a polynomial |
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114 | RETURN: reduced normal form of f modulo zero polynomials and describes the way *g* |
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115 | EXAMPLE: example zeroReduceProt; shows an example |
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116 | " |
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117 | { |
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118 | return(zeroReduceExt(f, 1)); |
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119 | } |
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120 | example |
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121 | { "EXAMPLE:"; echo = 2; |
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122 | option(teach); |
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123 | ring r = 65536, (y,x), dp; |
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124 | poly f = 1024*x^8*y^2+11264*x^8*y+28672*x^8+45056*x^7*y^2+36864*x^7*y+16384*x^7+40960*x^6*y^2+57344*x^6*y+32768*x^6+30720*x^5*y^2+10240*x^5*y+8192*x^5+35840*x^4*y^2+1024*x^4*y+20480*x^4+30720*x^3*y^2+10240*x^3*y+8192*x^3+4096*x^2*y^2+45056*x^2*y+49152*x^2+40960*x*y^2+57344*x*y+32768*x; |
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125 | zeroReduceProt(f); |
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126 | } |
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127 | |
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128 | proc getZeroCoef(poly f) |
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129 | { |
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130 | if (f == 0) |
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131 | { |
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132 | return(0, leadexp(f)) |
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133 | } |
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134 | list data = sort(leadexp(f)); |
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135 | intvec exp = data[1]; |
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136 | intvec index = data[2]; |
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137 | intvec nec = 0:size(exp); |
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138 | int i = 1; |
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139 | int j = 2; |
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140 | bigint g; |
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141 | bigint G = 1; |
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142 | bigint B = noElements(basering); |
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143 | |
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144 | for (; exp[i] < 2; i++) {if (i == size(exp)) break;} |
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145 | for (; i <= size(exp); i++) |
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146 | { |
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147 | g = gcd(B, G); |
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148 | G = G * g; |
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149 | B = B / g; |
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150 | if (g != 1) |
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151 | { |
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152 | nec[index[i]] = j - 1; |
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153 | } |
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154 | if (B == 1) |
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155 | { |
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156 | return(B, nec); |
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157 | } |
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158 | for (; j <= exp[i]; j++) |
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159 | { |
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160 | g = gcd(B, bigint(j)); |
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161 | G = G * g; |
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162 | B = B / g; |
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163 | if (g != 1) |
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164 | { |
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165 | nec[index[i]] = j; |
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166 | } |
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167 | if (B == 1) |
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168 | { |
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169 | return(B, nec); |
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170 | } |
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171 | } |
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172 | } |
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173 | return(B, nec); |
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174 | } |
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175 | |
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176 | proc getZeroPolyRaw(intvec fexp) |
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177 | { |
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178 | list data = sort(fexp); |
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179 | intvec exp = data[1]; |
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180 | intvec index = data[2]; |
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181 | int j = 0; |
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182 | poly res = 1; |
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183 | poly tillnow = 1; |
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184 | int i = 1; |
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185 | for (; exp[i] < 2; i++) {if (i == size(exp)) break;} |
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186 | for (; i <= size(exp); i++) |
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187 | { |
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188 | for (; j < exp[i]; j++) |
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189 | { |
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190 | tillnow = tillnow * (var(1) - j); |
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191 | } |
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192 | res = res * subst(tillnow, var(1), var(index[i])); |
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193 | } |
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194 | return(res); |
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195 | } |
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196 | |
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197 | proc getZeroPoly(poly f) |
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198 | { |
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199 | list data = getZeroCoef(f); |
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200 | poly g = getZeroPolyRaw(data[2]); |
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201 | g = leadmonom(f) / leadmonom(g) * g; |
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202 | return(data[1] * g); |
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203 | } |
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204 | |
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205 | proc testZero(poly f) |
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206 | { |
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207 | poly g; |
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208 | int j; |
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209 | bigint i = 0; |
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210 | bigint modul = noElements(basering); |
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211 | printf("Teste %s Belegungen ...", modul^nvars(basering)); |
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212 | for (; i < modul^nvars(basering); i = i + 1) |
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213 | { |
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214 | if (i % modul == 0) |
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215 | { |
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216 | printf("bisher: %s", i); |
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217 | } |
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218 | g = f; |
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219 | for (j = 1; j <= nvars(basering); j++) |
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220 | { |
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221 | g = subst(g, var(j), number((i / modul^(j-1)) % modul)); |
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222 | } |
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223 | if (g != 0) |
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224 | { |
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225 | list counter = g; |
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226 | for (j = 1; j <= nvars(basering); j++) |
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227 | { |
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228 | counter = insert(counter, (i / modul^(j-1)) % modul); |
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229 | } |
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230 | return(counter); |
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231 | } |
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232 | } |
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233 | return(1); |
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234 | } |
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235 | |
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236 | proc noElements(def r) |
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237 | { |
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238 | list l = ringlist(basering); |
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239 | return(l[1][2][1]^l[1][2][2]); |
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240 | } |
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