1 | // Singular-library |
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2 | /////////////////////////////////////////////////////////////////////////////// |
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3 | // version string automatically expanded by CVS |
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4 | version="$Id: ringgb.lib,v 1.16 2001/01/16 13:48:40 Singular Exp $"; |
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5 | category="Miscellaneous"; |
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6 | info=" |
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7 | LIBRARY: ringgb.lib Functions for coefficient rings |
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8 | AUTHOR: Oliver Wienand, email: wienand@mathematik.uni-kl.de |
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9 | |
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10 | KEYWORDS: vanishing polynomial; zeroreduce; polynomial functions; library, ringgb.lib; ringgb.lib, functions for coefficient rings |
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11 | |
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12 | PROCEDURES: |
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13 | findZeroPoly(f); finds a vanishing polynomial for reducing f |
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14 | zeroReduce(f); normal form of f concerning the ideal of vanishing polynomials |
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15 | testZero(poly f); tests f defines the constant zero function |
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16 | noElements(def r); the number of elements of the coefficient ring, if of type (integer, ...) |
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17 | "; |
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18 | |
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19 | LIB "general.lib"; |
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20 | /////////////////////////////////////////////////////////////////////////////// |
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21 | |
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22 | proc findZeroPoly (poly f) |
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23 | "USAGE: findZeroPoly(f); f - a polynomial |
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24 | RETURN: zero polynomial with the same leading term as f if exists, otherwise 0 |
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25 | EXAMPLE: example findZeroPoly; shows an example |
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26 | " |
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27 | { |
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28 | list data = getZeroCoef(f); |
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29 | if (data[1] == 0) |
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30 | { |
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31 | return(0); |
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32 | } |
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33 | number q = leadcoef(f) / data[1]; |
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34 | if (q == 0) |
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35 | { |
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36 | return(0); |
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37 | } |
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38 | poly g = getZeroPolyRaw(data[2]); |
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39 | g = leadmonom(f) / leadmonom(g) * g; |
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40 | return(q * data[1] * g); |
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41 | //return(system("findZeroPoly", f)); |
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42 | } |
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43 | example |
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44 | { "EXAMPLE:"; echo = 2; |
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45 | ring r = (integer, 65536), (y,x), dp; |
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46 | 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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47 | findZeroPoly(f); |
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48 | } |
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49 | |
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50 | proc zeroReduce(poly f, list #) |
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51 | "USAGE: zeroReduce(f, [i = 0]); f - a polynomial, i - noise level (if != 0 prints all steps) |
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52 | RETURN: reduced normal form of f modulo zero polynomials |
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53 | EXAMPLE: example zeroReduce; shows an example |
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54 | " |
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55 | { |
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56 | int i = 0; |
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57 | if (size(#) > 0) |
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58 | { |
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59 | i = #[1]; |
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60 | } |
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61 | poly h = f; |
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62 | poly n = 0; |
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63 | poly g = findZeroPoly(h); |
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64 | if (i <> 0) { |
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65 | printf("reducing polyfct : %s", h); |
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66 | } |
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67 | while ( h <> 0 ) { |
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68 | while ( g <> 0 ) { |
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69 | h = h - g; |
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70 | if (i <> 0) { |
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71 | printf(" reduce with: %s", g); |
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72 | printf(" to: %s", h); |
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73 | } |
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74 | g = findZeroPoly(h); |
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75 | } |
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76 | n = lead(h) + n; |
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77 | if (i <> 0) { |
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78 | printf("head irreducible : %s", lead(h)); |
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79 | printf("irreducible start : %s", n); |
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80 | printf("remains to check : %s", h - lead(h)); |
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81 | } |
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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 | ring r = (integer, 65536), (y,x), dp; |
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90 | 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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91 | zeroReduce(f); |
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92 | kill r; |
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93 | ring r = (integer, 2, 32), (x,y,z), dp; |
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94 | // Polynomial 1: |
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95 | poly p1 = 3795162112*x^3+587202566*x^2*y+2936012853*x*y*z+2281701376*x+548767119*y^3+16777216*y^2+268435456*y*z+1107296256*y+4244635648*z^3+4244635648*z^2+16777216*z; |
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96 | // Polynomial 2: |
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97 | poly p2 = 1647678464*x^3+587202566*x^2*y+2936012853*x*y*z+134217728*x+548767119*y^3+16777216*y^2+268435456*y*z+1107296256*y+2097152000*z^3+2097152000*z^2+16777216*z; |
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98 | zeroReduce(p1-p2); |
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99 | } |
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100 | |
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101 | proc testZero(poly f) |
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102 | "USAGE: testZero(f); f - a polynomial |
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103 | RETURN: returns 1 if f is zero as a function and otherwise a counterexample as a list [f(x_1, ..., x_n), x_1, ..., x_n] |
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104 | EXAMPLE: example testZero; shows an example |
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105 | " |
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106 | { |
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107 | poly g; |
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108 | int j; |
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109 | bigint i = 0; |
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110 | bigint modul = noElements(basering); |
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111 | printf("Teste %s Belegungen ...", modul^nvars(basering)); |
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112 | for (; i < modul^nvars(basering); i = i + 1) |
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113 | { |
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114 | if ((i + 1) % modul^(nvars(basering)/2) == 0) |
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115 | { |
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116 | printf("bisher: %s", i); |
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117 | } |
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118 | g = f; |
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119 | for (j = 1; j <= nvars(basering); j++) |
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120 | { |
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121 | g = subst(g, var(j), number((i / modul^(j-1)) % modul)); |
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122 | } |
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123 | if (g != 0) |
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124 | { |
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125 | list counter = g; |
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126 | for (j = 1; j <= nvars(basering); j++) |
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127 | { |
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128 | counter = insert(counter, (i / modul^(j-1)) % modul); |
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129 | } |
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130 | return(counter); |
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131 | } |
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132 | } |
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133 | return(1); |
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134 | } |
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135 | example |
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136 | { "EXAMPLE:"; echo = 2; |
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137 | ring r = (integer, 12), (y,x), dp; |
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138 | 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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139 | zeroReduce(f); |
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140 | testZero(f); |
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141 | poly g = findZeroPoly(x2y3); |
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142 | g; |
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143 | testZero(g); |
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144 | } |
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145 | |
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146 | proc noElements(def r) |
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147 | "USAGE: noElements(r); r - a ring with a finite coefficient ring of type integer |
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148 | RETURN: returns the number of elements of the coefficient ring of r |
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149 | EXAMPLE: example noElements; shows an example |
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150 | " |
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151 | |
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152 | { |
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153 | list l = ringlist(basering); |
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154 | return(l[1][2][1]^l[1][2][2]); |
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155 | } |
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156 | example |
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157 | { "EXAMPLE:"; echo = 2; |
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158 | ring r = (integer, 233,6), (y,x), dp; |
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159 | noElements(r); |
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160 | } |
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161 | |
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162 | static proc getZeroCoef(poly f) |
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163 | { |
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164 | if (f == 0) |
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165 | { |
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166 | return(0, leadexp(f)) |
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167 | } |
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168 | list data = sort(leadexp(f)); |
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169 | intvec exp = data[1]; |
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170 | intvec index = data[2]; |
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171 | intvec nec = 0:size(exp); |
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172 | int i = 1; |
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173 | int j = 2; |
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174 | bigint g; |
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175 | bigint G = 1; |
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176 | bigint modul = noElements(basering); |
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177 | bigint B = modul; |
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178 | |
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179 | for (; exp[i] < 2; i++) {if (i == size(exp)) break;} |
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180 | for (; i <= size(exp); i++) |
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181 | { |
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182 | g = gcd(B, G); |
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183 | G = G * g; |
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184 | B = B / g; |
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185 | if (g != 1) |
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186 | { |
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187 | nec[index[i]] = j - 1; |
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188 | } |
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189 | if (B == 1) |
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190 | { |
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191 | return(B, nec); |
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192 | } |
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193 | for (; j <= exp[i]; j++) |
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194 | { |
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195 | g = gcd(B, bigint(j)); |
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196 | G = G * g; |
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197 | B = B / g; |
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198 | if (g != 1) |
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199 | { |
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200 | nec[index[i]] = j; |
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201 | } |
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202 | if (B == 1) |
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203 | { |
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204 | return(B, nec); |
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205 | } |
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206 | } |
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207 | } |
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208 | if (B == modul) |
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209 | { |
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210 | nec = 0; |
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211 | return(0, nec); |
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212 | } |
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213 | return(B, nec); |
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214 | } |
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215 | |
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216 | static proc getZeroPolyRaw(intvec fexp) |
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217 | { |
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218 | list data = sort(fexp); |
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219 | intvec exp = data[1]; |
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220 | intvec index = data[2]; |
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221 | int j = 0; |
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222 | poly res = 1; |
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223 | poly tillnow = 1; |
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224 | int i = 1; |
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225 | for (; exp[i] < 2; i++) {if (i == size(exp)) break;} |
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226 | for (; i <= size(exp); i++) |
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227 | { |
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228 | for (; j < exp[i]; j++) |
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229 | { |
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230 | tillnow = tillnow * (var(1) - j); |
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231 | } |
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232 | res = res * subst(tillnow, var(1), var(index[i])); |
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233 | } |
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234 | return(res); |
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235 | } |
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236 | |
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237 | static proc getZeroPoly(poly f) |
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238 | { |
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239 | list data = getZeroCoef(f); |
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240 | poly g = getZeroPolyRaw(data[2]); |
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241 | g = leadmonom(f) / leadmonom(g) * g; |
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242 | return(data[1] * g); |
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243 | } |
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244 | |
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245 | static proc findZeroPolyWrap (poly f) |
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246 | "USAGE: findZeroPolyWrap(f); f - a polynomial |
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247 | RETURN: zero polynomial with the same leading term as f if exists, otherwise 0 |
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248 | NOTE: just a wrapper, work only in Z/2^n with n < int_machine_size - 1 |
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249 | EXAMPLE: example findZeroPoly; shows an example |
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250 | " |
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251 | { |
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252 | return(system("findZeroPoly", f)); |
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253 | } |
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254 | example |
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255 | { "EXAMPLE:"; echo = 2; |
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256 | ring r = (integer, 2, 16), (y,x), dp; |
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257 | 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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258 | findZeroPoly(f); |
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259 | } |
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260 | |
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261 | /////////////////////////////////////////////////////////////////////////////// |
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262 | |
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263 | /* |
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264 | Examples: |
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265 | |
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266 | |
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267 | // POLYNOMIAL EXAMPLES (Singular ready) |
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268 | // =================== |
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269 | // |
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270 | // For each of the examples below, there are three equivalent polynomials. 'm' indicates the bit-widths of the |
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271 | // input/output variables. For some of the polynomials, I have attached the RTL as well. |
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272 | // |
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273 | // |
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274 | // 1) VOLTERRA MODELS: |
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275 | // |
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276 | // A) CUBIC FILTER: (m = 32, 3 Vars) |
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277 | |
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278 | LIB "ringgb.lib"; |
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279 | ring r = (integer, 2, 32), (x,y,z), dp; |
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280 | poly p1 = 3795162112*x^3+587202566*x^2*y+2936012853*x*y*z+2281701376*x+548767119*y^3+16777216*y^2+268435456*y*z \ |
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281 | +1107296256*y+4244635648*z^3+4244635648*z^2+16777216*z; |
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282 | poly p2 = 1647678464*x^3+587202566*x^2*y+2936012853*x*y*z+134217728*x+548767119*y^3+16777216*y^2+268435456*y*z \ |
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283 | +1107296256*y+2097152000*z^3+2097152000*z^2+16777216*z; |
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284 | poly p3 = 1647678464*x^3+587202566*x^2*y+2936012853*x*y*z+134217728*x+548767119*y^3+16777216*y^2+268435456*y*z \ |
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285 | +1107296256*y+2097152000*z^3+2097152000*z^2+16777216*z; |
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286 | zeroReduce(p1-p2); |
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287 | zeroReduce(p1-p3); |
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288 | zeroReduce(p2-p3); |
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289 | |
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290 | // B) DEGREE-4 FILTER: (m=16 , 3 Vars) |
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291 | |
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292 | LIB "ringgb.lib"; |
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293 | ring r = (integer, 2, 16), (x,y,z), dp; |
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294 | poly p1 = 16384*x^4+y^4+57344*z^4+64767*x*y^3+16127*y^2*z^2+8965*x^3*z+19275*x^2*y*z+51903*x*y*z+32768*x^2*y \ |
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295 | +40960*z^2+32768*x*y^2+49152*x^2+4869*y; |
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296 | poly p2 = 8965*x^3*z+19275*x^2*y*z+31999*x*y^3+51903*x*y*z+32768*x*y+y^4+32768*y^3+16127*y^2*z^2+32768*y^2 \ |
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297 | +4869*y+57344*z^4+40960*z^2; |
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298 | poly p3 = 8965*x^3*z+19275*x^2*y*z+31999*x*y^3+51903*x*y*z+32768*x*y+y^4+16127*y^2*z^2+4869*y+16384*z^3+16384*z; |
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299 | zeroReduce(p1-p2); |
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300 | zeroReduce(p1-p3); |
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301 | zeroReduce(p2-p3); |
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302 | |
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303 | |
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304 | // 2) Savitzsky Golay filter(m=16,5 Vars) |
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305 | |
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306 | LIB "ringgb.lib"; |
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307 | ring r = (integer, 2, 16), (v,w,x,y,z), dp; |
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308 | poly p1 = 25000*v^2*y+37322*v^2+22142*v*w*z+50356*w^3+58627*w^2+17797*w+17797*x^3+62500*x^2*z+41667*x \ |
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309 | +22142*y^3+23870*y^2+59464*y+41667*z+58627; |
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310 | poly p2 = 25000*v^2*y+4554*v^2+22142*v*w*z+32768*v+17588*w^3+25859*w^2+17797*w+17797*x^3+29732*x^2*z+32768*x^2 \ |
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311 | +32768*x*z+8899*x+22142*y^3+23870*y^2+59464*y+41667*z+58627; |
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312 | poly p3 = 25000*v^2*y+4554*v^2+22142*v*w*z+32768*v+17588*w^3+25859*w^2+17797*w+17797*x^3+29732*x^2*z+32768*x*z \ |
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313 | +41667*x+22142*y^3+23870*y^2+59464*y+41667*z+58627; |
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314 | zeroReduce(p1-p2); |
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315 | zeroReduce(p1-p3); |
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316 | zeroReduce(p2-p3); |
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317 | |
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318 | |
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319 | // 3) Anti-alias filter:(m=16, 1 Var) |
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320 | |
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321 | LIB "ringgb.lib"; |
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322 | ring r = (integer, 2, 16), c, dp; |
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323 | poly p1 = 156*c^6+62724*c^5+17968*c^4+18661*c^3+43593*c^2+40224*c+13281; |
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324 | poly p2 = 156*c^6+5380*c^5+1584*c^4+43237*c^3+27209*c^2+40224*c+13281; |
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325 | poly p3 = 156*c^6+5380*c^5+1584*c^4+10469*c^3+27209*c^2+7456*c+13281; |
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326 | zeroReduce(p1-p2); |
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327 | zeroReduce(p1-p3); |
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328 | zeroReduce(p2-p3); |
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329 | |
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330 | |
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331 | // 4) PSK:(m=16, 2 Var) |
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332 | |
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333 | LIB "ringgb.lib"; |
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334 | ring r = (integer, 2, 16), (x,y), dp; |
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335 | poly p1 = 4166*x^4+16666*x^3*y+25000*x^2*y^2+15536*x^2+16666*x*y^4+31072*x*y+4166*y^4+15536*y^2+34464; |
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336 | poly p2 = 4166*x^4+16666*x^3*y+8616*x^2*y^2+16384*x^2*y+15536*x^2+282*x*y^4+47456*x*y+53318*y^4+31920*y^2+34464; |
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337 | poly p3 = 4166*x^4+16666*x^3*y+8616*x^2*y^2+16384*x^2*y+15536*x^2+282*x*y^4+47456*x*y+4166*y^4+15536*y^2+34464; |
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338 | zeroReduce(p1-p2); |
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339 | zeroReduce(p1-p3); |
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340 | zeroReduce(p2-p3); |
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341 | |
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342 | // Ref: A. Peymandoust G. De Micheli, Application of Symbolic Computer Algebra in High-Level Data-Flow |
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343 | // Synthesis, IEEE Transactions on CAD/ICAS, Vol. 22, No. 9, September 2003, pp.1154-1165. |
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344 | |
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345 | */ |
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