1 | /////////////////////////////////////////////////////////// |
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2 | version = "$Id$"; |
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3 | category="Noncommutative"; |
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4 | info=" |
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5 | LIBRARY: ncfactor.lib Tools for factorization in some noncommutative algebras |
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6 | AUTHORS: Albert Heinle, albert.heinle@rwth-aachen.de |
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7 | @* Viktor Levandovskyy, levandov@math.rwth-aachen.de |
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8 | |
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9 | OVERVIEW: In this library, new methods for factorization on polynomials |
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10 | are implemented for two algebras, both generated by two generators (Weyl and |
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11 | shift algebras) over a field K. Recall, that the first Weyl algebra over K |
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12 | is generated by x,d obeying the relation d*x=x*d+1. |
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13 | @* The first shift algebra over K is generated by x,s obeying the relation s*x=x*s+s. |
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14 | @* More detailled description of the algorithms can be found at |
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15 | @url{http://www.math.rwth-aachen.de/\~Albert.Heinle}. |
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16 | |
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17 | Guide: We are interested in computing a tree of factorizations, that is at the moment |
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18 | a list of all found factorizations is returned. It may contain factorizations, which |
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19 | are further reducible. |
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20 | |
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21 | PROCEDURES: |
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22 | facFirstWeyl(h); factorization in the first Weyl algebra |
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23 | testNCfac(l[,h[,1]]); tests factorizations from a given list for correctness |
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24 | facSubWeyl(h,X,D); factorization in the first Weyl algebra as a subalgebra |
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25 | facFirstShift(h); factorization in the first shift algebra |
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26 | homogfacFirstQWeyl(h); [-1,1]-homogeneous factorization in the first Q-Weyl algebra |
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27 | homogfacFirstQWeyl_all(h); [-1,1] homogeneous factorization(complete) in the first Q-Weyl algebra |
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28 | "; |
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29 | |
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30 | LIB "general.lib"; |
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31 | LIB "nctools.lib"; |
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32 | LIB "involut.lib"; |
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33 | LIB "freegb.lib"; // for isVar |
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34 | |
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35 | proc tst_ncfactor() |
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36 | { |
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37 | example facFirstWeyl; |
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38 | example facFirstShift; |
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39 | example facSubWeyl; |
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40 | example testNCfac; |
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41 | example homogfacFirstQWeyl; |
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42 | example homogfacFirstQWeyl_all; |
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43 | } |
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44 | |
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45 | ///////////////////////////////////////////////////// |
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46 | //==================================================* |
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47 | //deletes double-entries in a list of factorization |
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48 | //without evaluating the product. |
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49 | static proc delete_dublicates_noteval(list l) |
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50 | {//proc delete_dublicates_noteval |
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51 | list result= l; |
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52 | int j; int k; int i; |
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53 | int deleted = 0; |
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54 | int is_equal; |
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55 | for (i = 1; i<= size(l); i++) |
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56 | {//Iterate over the different factorizations |
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57 | for (j = i+1; j<= size(l); j++) |
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58 | {//Compare the i'th factorization to the j'th |
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59 | if (size(l[i])!= size(l[j])) |
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60 | {//different sizes => not equal |
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61 | j++; |
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62 | continue; |
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63 | }//different sizes => not equal |
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64 | is_equal = 1; |
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65 | for (k = 1; k <= size(l[i]);k++) |
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66 | {//Compare every entry |
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67 | if (l[i][k]!=l[j][k]) |
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68 | { |
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69 | is_equal = 0; |
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70 | break; |
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71 | } |
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72 | }//Compare every entry |
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73 | if (is_equal == 1) |
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74 | {//Delete this entry, because there is another equal one int the list |
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75 | result = delete(result, i-deleted); |
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76 | deleted = deleted+1; |
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77 | break; |
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78 | }//Delete this entry, because there is another equal one int the list |
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79 | }//Compare the i'th factorization to the j'th |
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80 | }//Iterate over the different factorizations |
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81 | return(result); |
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82 | }//proc delete_dublicates_noteval |
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83 | |
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84 | //================================================== |
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85 | //deletes the double-entries in a list with |
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86 | //evaluating the products |
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87 | static proc delete_dublicates_eval(list l) |
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88 | {//proc delete_dublicates_eval |
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89 | list result=l; |
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90 | int j; int k; int i; |
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91 | int deleted = 0; |
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92 | int is_equal; |
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93 | for (i = 1; i<= size(result); i++) |
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94 | {//Iterating over all elements in result |
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95 | for (j = i+1; j<= size(result); j++) |
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96 | {//comparing with the other elements |
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97 | if (product(result[i]) == product(result[j])) |
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98 | {//There are two equal results; throw away that one with the smaller size |
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99 | if (size(result[i])>=size(result[j])) |
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100 | {//result[i] has more entries |
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101 | result = delete(result,j); |
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102 | continue; |
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103 | }//result[i] has more entries |
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104 | else |
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105 | {//result[j] has more entries |
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106 | result = delete(result,i); |
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107 | i--; |
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108 | break; |
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109 | }//result[j] has more entries |
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110 | }//There are two equal results; throw away that one with the smaller size |
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111 | }//comparing with the other elements |
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112 | }//Iterating over all elements in result |
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113 | return(result); |
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114 | }//proc delete_dublicates_eval |
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115 | |
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116 | |
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117 | //==================================================* |
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118 | //given a list of factors g and a desired size nof, the following |
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119 | //procedure combines the factors, such that we recieve a |
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120 | //list of the length nof. |
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121 | static proc combinekfinlf(list g, int nof, intvec limits) //nof stands for "number of factors" |
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122 | {//Procedure combinekfinlf |
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123 | list result; |
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124 | int i; int j; int k; //iteration variables |
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125 | list fc; //fc stands for "factors combined" |
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126 | list temp; //a temporary store for factors |
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127 | def nofgl = size(g); //nofgl stands for "number of factors of the given list" |
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128 | if (nofgl == 0) |
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129 | {//g was the empty list |
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130 | return(result); |
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131 | }//g was the empty list |
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132 | if (nof <= 0) |
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133 | {//The user wants to recieve a negative number or no element as a result |
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134 | return(result); |
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135 | }//The user wants to recieve a negative number or no element as a result |
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136 | if (nofgl == nof) |
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137 | {//There are no factors to combine |
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138 | if (limitcheck(g,limits)) |
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139 | { |
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140 | result = result + list(g); |
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141 | } |
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142 | return(result); |
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143 | }//There are no factors to combine |
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144 | if (nof == 1) |
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145 | {//User wants to get just one factor |
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146 | if (limitcheck(list(product(g)),limits)) |
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147 | { |
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148 | result = result + list(list(product(g))); |
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149 | } |
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150 | return(result); |
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151 | }//User wants to get just one factor |
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152 | for (i = nof; i > 1; i--) |
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153 | {//computing the possibilities that have at least one original factor from g |
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154 | for (j = i; j>=1; j--) |
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155 | {//shifting the window of combinable factors to the left |
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156 | //fc below stands for "factors combined" |
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157 | fc = combinekfinlf(list(g[(j)..(j+nofgl - i)]),nof - i + 1,limits); |
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158 | for (k = 1; k<=size(fc); k++) |
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159 | {//iterating over the different solutions of the smaller problem |
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160 | if (j>1) |
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161 | {//There are g_i before the combination |
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162 | if (j+nofgl -i < nofgl) |
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163 | {//There are g_i after the combination |
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164 | temp = list(g[1..(j-1)]) + fc[k] + list(g[(j+nofgl-i+1)..nofgl]); |
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165 | }//There are g_i after the combination |
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166 | else |
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167 | {//There are no g_i after the combination |
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168 | temp = list(g[1..(j-1)]) + fc[k]; |
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169 | }//There are no g_i after the combination |
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170 | }//There are g_i before the combination |
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171 | if (j==1) |
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172 | {//There are no g_i before the combination |
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173 | if (j+ nofgl -i <nofgl) |
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174 | {//There are g_i after the combination |
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175 | temp = fc[k]+ list(g[(j + nofgl - i +1)..nofgl]); |
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176 | }//There are g_i after the combination |
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177 | }//There are no g_i before the combination |
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178 | if (limitcheck(temp,limits)) |
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179 | { |
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180 | result = result + list(temp); |
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181 | } |
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182 | }//iterating over the different solutions of the smaller problem |
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183 | }//shifting the window of combinable factors to the left |
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184 | }//computing the possibilities that have at least one original factor from g |
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185 | for (i = 2; i<=nofgl/nof;i++) |
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186 | {//getting the other possible results |
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187 | result = result + combinekfinlf(list(product(list(g[1..i])))+list(g[(i+1)..nofgl]),nof,limits); |
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188 | }//getting the other possible results |
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189 | result = delete_dublicates_noteval(result); |
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190 | return(result); |
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191 | }//Procedure combinekfinlf |
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192 | |
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193 | |
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194 | //==================================================* |
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195 | //merges two sets of factors ignoring common |
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196 | //factors |
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197 | static proc merge_icf(list l1, list l2, intvec limits) |
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198 | {//proc merge_icf |
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199 | list g; |
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200 | list f; |
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201 | int i; int j; |
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202 | if (size(l1)==0) |
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203 | { |
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204 | return(list()); |
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205 | } |
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206 | if (size(l2)==0) |
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207 | { |
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208 | return(list()); |
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209 | } |
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210 | if (size(l2)<=size(l1)) |
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211 | {//l1 will be our g, l2 our f |
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212 | g = l1; |
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213 | f = l2; |
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214 | }//l1 will be our g, l2 our f |
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215 | else |
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216 | {//l1 will be our f, l2 our g |
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217 | g = l2; |
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218 | f = l1; |
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219 | }//l1 will be our f, l2 our g |
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220 | def result = combinekfinlf(g,size(f),limits); |
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221 | for (i = 1 ; i<= size(result); i++) |
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222 | {//Adding the factors of f to every possibility listed in temp |
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223 | for (j = 1; j<= size(f); j++) |
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224 | { |
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225 | result[i][j] = result[i][j]+f[j]; |
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226 | } |
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227 | if(!limitcheck(result[i],limits)) |
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228 | { |
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229 | result = delete(result,i); |
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230 | i--; |
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231 | } |
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232 | }//Adding the factors of f to every possibility listed in temp |
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233 | return(result); |
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234 | }//proc merge_icf |
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235 | |
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236 | //==================================================* |
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237 | //merges two sets of factors with respect to the occurrence |
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238 | //of common factors |
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239 | static proc merge_cf(list l1, list l2, intvec limits) |
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240 | {//proc merge_cf |
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241 | list g; |
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242 | list f; |
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243 | int i; int j; |
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244 | list pre; |
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245 | list post; |
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246 | list candidate; |
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247 | list temp; |
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248 | int temppos; |
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249 | if (size(l1)==0) |
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250 | {//the first list is empty |
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251 | return(list()); |
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252 | }//the first list is empty |
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253 | if(size(l2)==0) |
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254 | {//the second list is empty |
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255 | return(list()); |
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256 | }//the second list is empty |
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257 | if (size(l2)<=size(l1)) |
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258 | {//l1 will be our g, l2 our f |
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259 | g = l1; |
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260 | f = l2; |
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261 | }//l1 will be our g, l2 our f |
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262 | else |
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263 | {//l1 will be our f, l2 our g |
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264 | g = l2; |
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265 | f = l1; |
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266 | }//l1 will be our f, l2 our g |
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267 | list M; |
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268 | for (i = 2; i<size(f); i++) |
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269 | {//finding common factors of f and g... |
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270 | for (j=2; j<size(g);j++) |
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271 | {//... with g |
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272 | if (f[i] == g[j]) |
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273 | {//we have an equal pair |
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274 | M = M + list(list(i,j)); |
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275 | }//we have an equal pair |
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276 | }//... with g |
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277 | }//finding common factors of f and g... |
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278 | if (g[1]==f[1]) |
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279 | {//Checking for the first elements to be equal |
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280 | M = M + list(list(1,1)); |
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281 | }//Checking for the first elements to be equal |
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282 | if (g[size(g)]==f[size(f)]) |
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283 | {//Checking for the last elements to be equal |
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284 | M = M + list(list(size(f),size(g))); |
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285 | }//Checking for the last elements to be equal |
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286 | list result;//= list(list()); |
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287 | while(size(M)>0) |
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288 | {//set of equal pairs is not empty |
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289 | temp = M[1]; |
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290 | temppos = 1; |
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291 | for (i = 2; i<=size(M); i++) |
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292 | {//finding the minimal element of M |
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293 | if (M[i][1]<=temp[1]) |
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294 | {//a possible candidate that is smaller than temp could have been found |
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295 | if (M[i][1]==temp[1]) |
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296 | {//In this case we must look at the second number |
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297 | if (M[i][2]< temp[2]) |
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298 | {//the candidate is smaller |
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299 | temp = M[i]; |
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300 | temppos = i; |
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301 | }//the candidate is smaller |
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302 | }//In this case we must look at the second number |
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303 | else |
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304 | {//The candidate is definately smaller |
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305 | temp = M[i]; |
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306 | temppos = i; |
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307 | }//The candidate is definately smaller |
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308 | }//a possible candidate that is smaller than temp could have been found |
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309 | }//finding the minimal element of M |
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310 | M = delete(M, temppos); |
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311 | if(temp[1]>1) |
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312 | {//There are factors to combine before the equal factor |
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313 | if (temp[1]<size(f)) |
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314 | {//The most common case |
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315 | //first the combinations ignoring common factors |
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316 | pre = merge_icf(list(f[1..(temp[1]-1)]),list(g[1..(temp[2]-1)]),limits); |
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317 | post = merge_icf(list(f[(temp[1]+1)..size(f)]),list(g[(temp[2]+1..size(g))]),limits); |
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318 | for (i = 1; i <= size(pre); i++) |
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319 | {//all possible pre's... |
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320 | for (j = 1; j<= size(post); j++) |
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321 | {//...combined with all possible post's |
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322 | candidate = pre[i]+list(f[temp[1]])+post[j]; |
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323 | if (limitcheck(candidate,limits)) |
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324 | { |
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325 | result = result + list(candidate); |
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326 | } |
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327 | }//...combined with all possible post's |
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328 | }//all possible pre's... |
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329 | //Now the combinations with respect to common factors |
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330 | post = merge_cf(list(f[(temp[1]+1)..size(f)]),list(g[(temp[2]+1..size(g))]),limits); |
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331 | if (size(post)>0) |
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332 | {//There are factors to combine |
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333 | for (i = 1; i <= size(pre); i++) |
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334 | {//all possible pre's... |
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335 | for (j = 1; j<= size(post); j++) |
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336 | {//...combined with all possible post's |
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337 | candidate= pre[i]+list(f[temp[1]])+post[j]; |
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338 | if (limitcheck(candidate,limits)) |
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339 | { |
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340 | result = result + list(candidate); |
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341 | } |
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342 | }//...combined with all possible post's |
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343 | }//all possible pre's... |
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344 | }//There are factors to combine |
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345 | }//The most common case |
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346 | else |
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347 | {//the last factor is the common one |
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348 | pre = merge_icf(list(f[1..(temp[1]-1)]),list(g[1..(temp[2]-1)]),limits); |
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349 | for (i = 1; i<= size(pre); i++) |
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350 | {//iterating over the possible pre-factors |
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351 | candidate = pre[i]+list(f[temp[1]]); |
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352 | if (limitcheck(candidate,limits)) |
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353 | { |
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354 | result = result + list(candidate); |
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355 | } |
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356 | }//iterating over the possible pre-factors |
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357 | }//the last factor is the common one |
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358 | }//There are factors to combine before the equal factor |
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359 | else |
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360 | {//There are no factors to combine before the equal factor |
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361 | if (temp[1]<size(f)) |
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362 | {//Just a check for security |
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363 | //first without common factors |
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364 | post=merge_icf(list(f[(temp[1]+1)..size(f)]),list(g[(temp[2]+1..size(g))]),limits); |
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365 | for (i = 1; i<=size(post); i++) |
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366 | { |
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367 | candidate = list(f[temp[1]])+post[i]; |
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368 | if (limitcheck(candidate,limits)) |
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369 | { |
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370 | result = result + list(candidate); |
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371 | } |
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372 | } |
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373 | //Now with common factors |
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374 | post = merge_cf(list(f[(temp[1]+1)..size(f)]),list(g[(temp[2]+1..size(g))]),limits); |
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375 | if(size(post)>0) |
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376 | {//we could find other combinations |
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377 | for (i = 1; i<=size(post); i++) |
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378 | { |
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379 | candidate = list(f[temp[1]])+post[i]; |
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380 | if (limitcheck(candidate,limits)) |
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381 | { |
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382 | result = result + list(candidate); |
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383 | } |
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384 | } |
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385 | }//we could find other combinations |
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386 | }//Just a check for security |
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387 | }//There are no factors to combine before the equal factor |
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388 | }//set of equal pairs is not empty |
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389 | return(result); |
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390 | }//proc merge_cf |
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391 | |
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392 | |
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393 | //==================================================* |
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394 | //merges two sets of factors |
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395 | |
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396 | static proc mergence(list l1, list l2, intvec limits) |
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397 | {//Procedure mergence |
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398 | list g; |
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399 | list f; |
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400 | int l; int k; |
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401 | list F; |
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402 | if (size(l2)<=size(l1)) |
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403 | {//l1 will be our g, l2 our f |
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404 | g = l1; |
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405 | f = l2; |
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406 | }//l1 will be our g, l2 our f |
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407 | else |
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408 | {//l1 will be our f, l2 our g |
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409 | g = l2; |
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410 | f = l1; |
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411 | }//l1 will be our f, l2 our g |
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412 | list result; |
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413 | for (l = size(f); l>=1; l--) |
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414 | {//all possibilities to combine the factors of f |
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415 | F = combinekfinlf(f,l,limits); |
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416 | for (k = 1; k<= size(F);k++) |
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417 | {//for all possibilities of combinations of the factors of f |
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418 | result = result + merge_cf(F[k],g,limits); |
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419 | result = result + merge_icf(F[k],g,limits); |
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420 | }//for all possibilities of combinations of the factors of f |
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421 | }//all possibilities to combine the factors of f |
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422 | return(result); |
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423 | }//Procedure mergence |
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424 | |
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425 | |
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426 | //================================================== |
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427 | //Checks, whether a list of factors doesn't exceed the given limits |
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428 | static proc limitcheck(list g, intvec limits) |
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429 | {//proc limitcheck |
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430 | int i; |
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431 | if (size(limits)!=3) |
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432 | {//check the input |
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433 | return(0); |
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434 | }//check the input |
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435 | if(size(g)==0) |
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436 | { |
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437 | return(0); |
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438 | } |
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439 | def prod = product(g); |
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440 | intvec iv11 = intvec(1,1); |
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441 | intvec iv10 = intvec(1,0); |
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442 | intvec iv01 = intvec(0,1); |
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443 | def limg = intvec(deg(prod,iv11) ,deg(prod,iv10),deg(prod,iv01)); |
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444 | for (i = 1; i<=size(limg);i++) |
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445 | {//the final check |
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446 | if(limg[i]>limits[i]) |
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447 | { |
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448 | return(0); |
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449 | } |
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450 | }//the final check |
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451 | return(1); |
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452 | }//proc limitcheck |
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453 | |
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454 | |
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455 | //==================================================* |
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456 | //one factorization of a homogeneous polynomial |
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457 | //in the first Weyl Algebra |
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458 | static proc homogfacFirstWeyl(poly h) |
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459 | "USAGE: homogfacFirstWeyl(h); h is a homogeneous polynomial in the |
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460 | first Weyl algebra with respect to the weight vector [-1,1] |
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461 | RETURN: list |
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462 | PURPOSE: Computes a factorization of a homogeneous polynomial h with |
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463 | respect to the weight vector [-1,1] in the first Weyl algebra |
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464 | THEORY: @code{homogfacFirstWeyl} returns a list with a factorization of the given, |
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465 | [-1,1]-homogeneous polynomial. If the degree of the polynomial is k with |
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466 | k positive, the last k entries in the output list are the second |
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467 | variable. If k is positive, the last k entries will be x. The other |
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468 | entries will be irreducible polynomials of degree zero or 1 resp. -1. |
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469 | SEE ALSO: homogfacFirstWeyl_all |
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470 | "{//proc homogfacFirstWeyl |
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471 | int p = printlevel-voice+2;//for dbprint |
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472 | def r = basering; |
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473 | poly hath; |
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474 | int i; int j; |
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475 | intvec ivm11 = intvec(-1,1); |
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476 | if (!homogwithorder(h,ivm11)) |
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477 | {//The given polynomial is not homogeneous |
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478 | ERROR("Given polynomial was not [-1,1]-homogeneous"); |
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479 | return(list()); |
---|
480 | }//The given polynomial is not homogeneous |
---|
481 | if (h==0) |
---|
482 | { |
---|
483 | return(list(0)); |
---|
484 | } |
---|
485 | list result; |
---|
486 | int m = deg(h,ivm11); |
---|
487 | dbprint(p,"==> Splitting the polynomial in A_0 and A_k-Part"); |
---|
488 | if (m!=0) |
---|
489 | {//The degree is not zero |
---|
490 | if (m <0) |
---|
491 | {//There are more x than y |
---|
492 | hath = lift(var(1)^(-m),h)[1,1]; |
---|
493 | for (i = 1; i<=-m; i++) |
---|
494 | { |
---|
495 | result = result + list(var(1)); |
---|
496 | } |
---|
497 | }//There are more x than y |
---|
498 | else |
---|
499 | {//There are more y than x |
---|
500 | hath = lift(var(2)^m,h)[1,1]; |
---|
501 | for (i = 1; i<=m;i++) |
---|
502 | { |
---|
503 | result = result + list(var(2)); |
---|
504 | } |
---|
505 | }//There are more y than x |
---|
506 | }//The degree is not zero |
---|
507 | else |
---|
508 | {//The degree is zero |
---|
509 | hath = h; |
---|
510 | }//The degree is zero |
---|
511 | dbprint(p,"==> Done"); |
---|
512 | //beginning to transform x^i*y^i in theta(theta-1)...(theta-i+1) |
---|
513 | list mons; |
---|
514 | dbprint(p,"==> Putting the monomials in the A_0-part in a list."); |
---|
515 | for(i = 1; i<=size(hath);i++) |
---|
516 | {//Putting the monomials in a list |
---|
517 | mons = mons+list(hath[i]); |
---|
518 | }//Putting the monomials in a list |
---|
519 | dbprint(p,"==> Done"); |
---|
520 | dbprint(p,"==> Mapping this monomials to K[theta]"); |
---|
521 | ring tempRing = 0,(x,y,theta),dp; |
---|
522 | setring tempRing; |
---|
523 | map thetamap = r,x,y; |
---|
524 | list mons = thetamap(mons); |
---|
525 | poly entry; |
---|
526 | for (i = 1; i<=size(mons);i++) |
---|
527 | {//transforming the monomials as monomials in theta |
---|
528 | entry = leadcoef(mons[i]); |
---|
529 | for (j = 0; j<leadexp(mons[i])[2];j++) |
---|
530 | { |
---|
531 | entry = entry * (theta-j); |
---|
532 | } |
---|
533 | mons[i] = entry; |
---|
534 | }//transforming the monomials as monomials in theta |
---|
535 | dbprint(p,"==> Done"); |
---|
536 | dbprint(p,"==> Factorize the A_0-Part in K[theta]"); |
---|
537 | list azeroresult = factorize(sum(mons)); |
---|
538 | dbprint(p,"==> Successful"); |
---|
539 | list azeroresult_return_form; |
---|
540 | for (i = 1; i<=size(azeroresult[1]);i++) |
---|
541 | {//rewrite the result of the commutative factorization |
---|
542 | for (j = 1; j <= azeroresult[2][i];j++) |
---|
543 | { |
---|
544 | azeroresult_return_form = azeroresult_return_form + list(azeroresult[1][i]); |
---|
545 | } |
---|
546 | }//rewrite the result of the commutative factorization |
---|
547 | dbprint(p,"==> Mapping back to A_0."); |
---|
548 | setring(r); |
---|
549 | map finalmap = tempRing,var(1),var(2),var(1)*var(2); |
---|
550 | list tempresult = finalmap(azeroresult_return_form); |
---|
551 | dbprint(p,"Successful."); |
---|
552 | for (i = 1; i<=size(tempresult);i++) |
---|
553 | {//factorizations of theta resp. theta +1 |
---|
554 | if(tempresult[i]==var(1)*var(2)) |
---|
555 | { |
---|
556 | tempresult = insert(tempresult,var(1),i-1); |
---|
557 | i++; |
---|
558 | tempresult[i]=var(2); |
---|
559 | } |
---|
560 | if(tempresult[i]==var(2)*var(1)) |
---|
561 | { |
---|
562 | tempresult = insert(tempresult,var(2),i-1); |
---|
563 | i++; |
---|
564 | tempresult[i]=var(1); |
---|
565 | } |
---|
566 | }//factorizations of theta resp. theta +1 |
---|
567 | result = tempresult+result; |
---|
568 | return(result); |
---|
569 | }//proc homogfacFirstWeyl |
---|
570 | /* example */ |
---|
571 | /* { */ |
---|
572 | /* "EXAMPLE:";echo=2; */ |
---|
573 | /* ring R = 0,(x,y),Ws(-1,1); */ |
---|
574 | /* def r = nc_algebra(1,1); */ |
---|
575 | /* setring(r); */ |
---|
576 | /* poly h = (x^2*y^2+1)*(x^4); */ |
---|
577 | /* homogfacFirstWeyl(h); */ |
---|
578 | /* } */ |
---|
579 | |
---|
580 | //================================================== |
---|
581 | //Computes all possible homogeneous factorizations |
---|
582 | static proc homogfacFirstWeyl_all(poly h) |
---|
583 | "USAGE: homogfacFirstWeyl_all(h); h is a homogeneous polynomial in the first Weyl algebra |
---|
584 | with respect to the weight vector [-1,1] |
---|
585 | RETURN: list |
---|
586 | PURPOSE: Computes all factorizations of a homogeneous polynomial h with respect |
---|
587 | to the weight vector [-1,1] in the first Weyl algebra |
---|
588 | THEORY: @code{homogfacFirstWeyl} returns a list with all factorization of the given, |
---|
589 | homogeneous polynomial. It uses the output of homogfacFirstWeyl and permutes |
---|
590 | its entries with respect to the commutation rule. Furthermore, if a |
---|
591 | factor of degree zero is irreducible in K[\theta], but reducible in |
---|
592 | the first Weyl algebra, the permutations of this element with the other |
---|
593 | entries will also be computed. |
---|
594 | SEE ALSO: homogfacFirstWeyl |
---|
595 | "{//proc HomogfacFirstWeylAll |
---|
596 | int p=printlevel-voice+2;//for dbprint |
---|
597 | intvec iv11= intvec(1,1); |
---|
598 | if (deg(h,iv11) <= 0 ) |
---|
599 | {//h is a constant |
---|
600 | dbprint(p,"Given polynomial was not homogeneous"); |
---|
601 | return(list(list(h))); |
---|
602 | }//h is a constant |
---|
603 | def r = basering; |
---|
604 | list one_hom_fac; //stands for one homogeneous factorization |
---|
605 | int i; int j; int k; |
---|
606 | intvec ivm11 = intvec(-1,1); |
---|
607 | dbprint(p,"==> Calculate one homogeneous factorization using homogfacFirstWeyl"); |
---|
608 | //Compute again a homogeneous factorization |
---|
609 | one_hom_fac = homogfacFirstWeyl(h); |
---|
610 | dbprint(p,"Successful"); |
---|
611 | if (size(one_hom_fac) == 0) |
---|
612 | {//there is no homogeneous factorization or the polynomial was not homogeneous |
---|
613 | return(list()); |
---|
614 | }//there is no homogeneous factorization or the polynomial was not homogeneous |
---|
615 | //divide list in A0-Part and a list of x's resp. y's |
---|
616 | list list_not_azero = list(); |
---|
617 | list list_azero; |
---|
618 | list k_factor; |
---|
619 | int is_list_not_azero_empty = 1; |
---|
620 | int is_list_azero_empty = 1; |
---|
621 | k_factor = list(one_hom_fac[1]); |
---|
622 | if (absValue(deg(h,ivm11))<size(one_hom_fac)-1) |
---|
623 | {//There is a nontrivial A_0-part |
---|
624 | list_azero = one_hom_fac[2..(size(one_hom_fac)-absValue(deg(h,ivm11)))]; |
---|
625 | is_list_azero_empty = 0; |
---|
626 | }//There is a nontrivial A_0 part |
---|
627 | dbprint(p,"==> Combine x,y to xy in the factorization again."); |
---|
628 | for (i = 1; i<=size(list_azero)-1;i++) |
---|
629 | {//in homogfacFirstWeyl, we factorized theta, and this will be made undone |
---|
630 | if (list_azero[i] == var(1)) |
---|
631 | { |
---|
632 | if (list_azero[i+1]==var(2)) |
---|
633 | { |
---|
634 | list_azero[i] = var(1)*var(2); |
---|
635 | list_azero = delete(list_azero,i+1); |
---|
636 | } |
---|
637 | } |
---|
638 | if (list_azero[i] == var(2)) |
---|
639 | { |
---|
640 | if (list_azero[i+1]==var(1)) |
---|
641 | { |
---|
642 | list_azero[i] = var(2)*var(1); |
---|
643 | list_azero = delete(list_azero,i+1); |
---|
644 | } |
---|
645 | } |
---|
646 | }//in homogfacFirstWeyl, we factorized theta, and this will be made undone |
---|
647 | dbprint(p,"==> Done"); |
---|
648 | if(deg(h,ivm11)!=0) |
---|
649 | {//list_not_azero is not empty |
---|
650 | list_not_azero = |
---|
651 | one_hom_fac[(size(one_hom_fac)-absValue(deg(h,ivm11))+1)..size(one_hom_fac)]; |
---|
652 | is_list_not_azero_empty = 0; |
---|
653 | }//list_not_azero is not empty |
---|
654 | //Map list_azero in K[theta] |
---|
655 | dbprint(p,"==> Map list_azero to K[theta]"); |
---|
656 | ring tempRing = 0,(x,y,theta), dp; |
---|
657 | setring(tempRing); |
---|
658 | poly entry; |
---|
659 | map thetamap = r,x,y; |
---|
660 | if(!is_list_not_azero_empty) |
---|
661 | {//Mapping in Singular is only possible, if the list before |
---|
662 | //contained at least one element of the other ring |
---|
663 | list list_not_azero = thetamap(list_not_azero); |
---|
664 | }//Mapping in Singular is only possible, if the list before |
---|
665 | //contained at least one element of the other ring |
---|
666 | if(!is_list_azero_empty) |
---|
667 | {//Mapping in Singular is only possible, if the list before |
---|
668 | //contained at least one element of the other ring |
---|
669 | list list_azero= thetamap(list_azero); |
---|
670 | }//Mapping in Singular is only possible, if the list before |
---|
671 | //contained at least one element of the other ring |
---|
672 | list k_factor = thetamap(k_factor); |
---|
673 | list tempmons; |
---|
674 | dbprint(p,"==> Done"); |
---|
675 | for(i = 1; i<=size(list_azero);i++) |
---|
676 | {//rewrite the polynomials in A1 as polynomials in K[theta] |
---|
677 | tempmons = list(); |
---|
678 | for (j = 1; j<=size(list_azero[i]);j++) |
---|
679 | { |
---|
680 | tempmons = tempmons + list(list_azero[i][j]); |
---|
681 | } |
---|
682 | for (j = 1 ; j<=size(tempmons);j++) |
---|
683 | { |
---|
684 | entry = leadcoef(tempmons[j]); |
---|
685 | for (k = 0; k < leadexp(tempmons[j])[2];k++) |
---|
686 | { |
---|
687 | entry = entry*(theta-k); |
---|
688 | } |
---|
689 | tempmons[j] = entry; |
---|
690 | } |
---|
691 | list_azero[i] = sum(tempmons); |
---|
692 | }//rewrite the polynomials in A1 as polynomials in K[theta] |
---|
693 | //Compute all permutations of the A0-part |
---|
694 | dbprint(p,"==> Compute all permutations of the A_0-part with the first resp. the snd. variable"); |
---|
695 | list result; |
---|
696 | int shift_sign; |
---|
697 | int shift; |
---|
698 | poly shiftvar; |
---|
699 | if (size(list_not_azero)!=0) |
---|
700 | {//Compute all possibilities to permute the x's resp. the y's in the list |
---|
701 | if (list_not_azero[1] == x) |
---|
702 | {//h had a negative weighted degree |
---|
703 | shift_sign = 1; |
---|
704 | shiftvar = x; |
---|
705 | }//h had a negative weighted degree |
---|
706 | else |
---|
707 | {//h had a positive weighted degree |
---|
708 | shift_sign = -1; |
---|
709 | shiftvar = y; |
---|
710 | }//h had a positive weighted degree |
---|
711 | result = permpp(list_azero + list_not_azero); |
---|
712 | for (i = 1; i<= size(result); i++) |
---|
713 | {//adjust the a_0-parts |
---|
714 | shift = 0; |
---|
715 | for (j=1; j<=size(result[i]);j++) |
---|
716 | { |
---|
717 | if (result[i][j]==shiftvar) |
---|
718 | { |
---|
719 | shift = shift + shift_sign; |
---|
720 | } |
---|
721 | else |
---|
722 | { |
---|
723 | result[i][j] = subst(result[i][j],theta,theta + shift); |
---|
724 | } |
---|
725 | } |
---|
726 | }//adjust the a_0-parts |
---|
727 | }//Compute all possibilities to permute the x's resp. the y's in the list |
---|
728 | else |
---|
729 | {//The result is just all the permutations of the a_0-part |
---|
730 | result = permpp(list_azero); |
---|
731 | }//The result is just all the permutations of the a_0 part |
---|
732 | if (size(result)==0) |
---|
733 | { |
---|
734 | return(result); |
---|
735 | } |
---|
736 | dbprint(p,"==> Done"); |
---|
737 | dbprint(p,"==> Searching for theta resp. theta + 1 in the list and factorize them"); |
---|
738 | //Now we are going deeper and search for theta resp. theta + 1, substitute |
---|
739 | //them by xy resp. yx and go on permuting |
---|
740 | int found_theta; |
---|
741 | int thetapos; |
---|
742 | list leftpart; |
---|
743 | list rightpart; |
---|
744 | list lparts; |
---|
745 | list rparts; |
---|
746 | list tempadd; |
---|
747 | for (i = 1; i<=size(result) ; i++) |
---|
748 | {//checking every entry of result for theta or theta +1 |
---|
749 | found_theta = 0; |
---|
750 | for(j=1;j<=size(result[i]);j++) |
---|
751 | { |
---|
752 | if (result[i][j]==theta) |
---|
753 | {//the jth entry is theta and can be written as x*y |
---|
754 | thetapos = j; |
---|
755 | result[i]= insert(result[i],x,j-1); |
---|
756 | j++; |
---|
757 | result[i][j] = y; |
---|
758 | found_theta = 1; |
---|
759 | break; |
---|
760 | }//the jth entry is theta and can be written as x*y |
---|
761 | if(result[i][j] == theta +1) |
---|
762 | { |
---|
763 | thetapos = j; |
---|
764 | result[i] = insert(result[i],y,j-1); |
---|
765 | j++; |
---|
766 | result[i][j] = x; |
---|
767 | found_theta = 1; |
---|
768 | break; |
---|
769 | } |
---|
770 | } |
---|
771 | if (found_theta) |
---|
772 | {//One entry was theta resp. theta +1 |
---|
773 | leftpart = result[i]; |
---|
774 | leftpart = leftpart[1..thetapos]; |
---|
775 | rightpart = result[i]; |
---|
776 | rightpart = rightpart[(thetapos+1)..size(rightpart)]; |
---|
777 | lparts = list(leftpart); |
---|
778 | rparts = list(rightpart); |
---|
779 | //first deal with the left part |
---|
780 | if (leftpart[thetapos] == x) |
---|
781 | { |
---|
782 | shift_sign = 1; |
---|
783 | shiftvar = x; |
---|
784 | } |
---|
785 | else |
---|
786 | { |
---|
787 | shift_sign = -1; |
---|
788 | shiftvar = y; |
---|
789 | } |
---|
790 | for (j = size(leftpart); j>1;j--) |
---|
791 | {//drip x resp. y |
---|
792 | if (leftpart[j-1]==shiftvar) |
---|
793 | {//commutative |
---|
794 | j--; |
---|
795 | continue; |
---|
796 | }//commutative |
---|
797 | if (deg(leftpart[j-1],intvec(-1,1,0))!=0) |
---|
798 | {//stop here |
---|
799 | break; |
---|
800 | }//stop here |
---|
801 | //Here, we can only have a a0- part |
---|
802 | leftpart[j] = subst(leftpart[j-1],theta, theta + shift_sign); |
---|
803 | leftpart[j-1] = shiftvar; |
---|
804 | lparts = lparts + list(leftpart); |
---|
805 | }//drip x resp. y |
---|
806 | //and now deal with the right part |
---|
807 | if (rightpart[1] == x) |
---|
808 | { |
---|
809 | shift_sign = 1; |
---|
810 | shiftvar = x; |
---|
811 | } |
---|
812 | else |
---|
813 | { |
---|
814 | shift_sign = -1; |
---|
815 | shiftvar = y; |
---|
816 | } |
---|
817 | for (j = 1 ; j < size(rightpart); j++) |
---|
818 | { |
---|
819 | if (rightpart[j+1] == shiftvar) |
---|
820 | { |
---|
821 | j++; |
---|
822 | continue; |
---|
823 | } |
---|
824 | if (deg(rightpart[j+1],intvec(-1,1,0))!=0) |
---|
825 | { |
---|
826 | break; |
---|
827 | } |
---|
828 | rightpart[j] = subst(rightpart[j+1], theta, theta - shift_sign); |
---|
829 | rightpart[j+1] = shiftvar; |
---|
830 | rparts = rparts + list(rightpart); |
---|
831 | } |
---|
832 | //And now, we put all possibilities together |
---|
833 | tempadd = list(); |
---|
834 | for (j = 1; j<=size(lparts); j++) |
---|
835 | { |
---|
836 | for (k = 1; k<=size(rparts);k++) |
---|
837 | { |
---|
838 | tempadd = tempadd + list(lparts[j]+rparts[k]); |
---|
839 | } |
---|
840 | } |
---|
841 | tempadd = delete(tempadd,1); // The first entry is already in the list |
---|
842 | result = result + tempadd; |
---|
843 | continue; //We can may be not be done already with the ith entry |
---|
844 | }//One entry was theta resp. theta +1 |
---|
845 | }//checking every entry of result for theta or theta +1 |
---|
846 | dbprint(p,"==> Done"); |
---|
847 | //map back to the basering |
---|
848 | dbprint(p,"==> Mapping back everything to the basering"); |
---|
849 | setring(r); |
---|
850 | map finalmap = tempRing, var(1), var(2),var(1)*var(2); |
---|
851 | list result = finalmap(result); |
---|
852 | for (i=1; i<=size(result);i++) |
---|
853 | {//adding the K factor |
---|
854 | result[i] = k_factor + result[i]; |
---|
855 | }//adding the k-factor |
---|
856 | dbprint(p,"==> Done"); |
---|
857 | dbprint(p,"==> Delete double entries in the list."); |
---|
858 | result = delete_dublicates_noteval(result); |
---|
859 | dbprint(p,"==> Done"); |
---|
860 | return(result); |
---|
861 | }//proc HomogfacFirstWeylAll |
---|
862 | /* example */ |
---|
863 | /* { */ |
---|
864 | /* "EXAMPLE:";echo=2; */ |
---|
865 | /* ring R = 0,(x,y),Ws(-1,1); */ |
---|
866 | /* def r = nc_algebra(1,1); */ |
---|
867 | /* setring(r); */ |
---|
868 | /* poly h = (x^2*y^2+1)*(x^4); */ |
---|
869 | /* homogfacFirstWeyl_all(h); */ |
---|
870 | /* } */ |
---|
871 | |
---|
872 | //==================================================* |
---|
873 | //Computes all permutations of a given list |
---|
874 | static proc perm(list l) |
---|
875 | {//proc perm |
---|
876 | int i; int j; |
---|
877 | list tempresult; |
---|
878 | list result; |
---|
879 | if (size(l)==0) |
---|
880 | { |
---|
881 | return(list()); |
---|
882 | } |
---|
883 | if (size(l)==1) |
---|
884 | { |
---|
885 | return(list(l)); |
---|
886 | } |
---|
887 | for (i = 1; i<=size(l); i++ ) |
---|
888 | { |
---|
889 | tempresult = perm(delete(l,i)); |
---|
890 | for (j = 1; j<=size(tempresult);j++) |
---|
891 | { |
---|
892 | tempresult[j] = list(l[i])+tempresult[j]; |
---|
893 | } |
---|
894 | result = result+tempresult; |
---|
895 | } |
---|
896 | return(result); |
---|
897 | }//proc perm |
---|
898 | |
---|
899 | //================================================== |
---|
900 | //computes all permutations of a given list by |
---|
901 | //ignoring equal entries (faster than perm) |
---|
902 | static proc permpp(list l) |
---|
903 | {//proc permpp |
---|
904 | int i; int j; |
---|
905 | list tempresult; |
---|
906 | list l_without_double; |
---|
907 | list l_without_double_pos; |
---|
908 | int double_entry; |
---|
909 | list result; |
---|
910 | if (size(l)==0) |
---|
911 | { |
---|
912 | return(list()); |
---|
913 | } |
---|
914 | if (size(l)==1) |
---|
915 | { |
---|
916 | return(list(l)); |
---|
917 | } |
---|
918 | for (i = 1; i<=size(l);i++) |
---|
919 | {//Filling the list with unique entries |
---|
920 | double_entry = 0; |
---|
921 | for (j = 1; j<=size(l_without_double);j++) |
---|
922 | { |
---|
923 | if (l_without_double[j] == l[i]) |
---|
924 | { |
---|
925 | double_entry = 1; |
---|
926 | break; |
---|
927 | } |
---|
928 | } |
---|
929 | if (!double_entry) |
---|
930 | { |
---|
931 | l_without_double = l_without_double + list(l[i]); |
---|
932 | l_without_double_pos = l_without_double_pos + list(i); |
---|
933 | } |
---|
934 | }//Filling the list with unique entries |
---|
935 | for (i = 1; i<=size(l_without_double); i++ ) |
---|
936 | { |
---|
937 | tempresult = permpp(delete(l,l_without_double_pos[i])); |
---|
938 | for (j = 1; j<=size(tempresult);j++) |
---|
939 | { |
---|
940 | tempresult[j] = list(l_without_double[i])+tempresult[j]; |
---|
941 | } |
---|
942 | result = result+tempresult; |
---|
943 | } |
---|
944 | return(result); |
---|
945 | }//proc permpp |
---|
946 | |
---|
947 | //================================================== |
---|
948 | //factorization of the first Weyl Algebra |
---|
949 | |
---|
950 | //The following procedure just serves the purpose to |
---|
951 | //transform the input into an appropriate input for |
---|
952 | //the procedure sfacwa, where the ring must contain the |
---|
953 | //variables in a certain order. |
---|
954 | proc facFirstWeyl(poly h) |
---|
955 | "USAGE: facFirstWeyl(h); h a polynomial in the first Weyl algebra |
---|
956 | RETURN: list |
---|
957 | PURPOSE: compute all factorizations of a polynomial in the first Weyl algebra |
---|
958 | THEORY: Implements the new algorithm by A. Heinle and V. Levandovskyy, see the thesis of A. Heinle |
---|
959 | ASSUME: basering is the first Weyl algebra |
---|
960 | NOTE: Every entry of the output list is a list with factors for one possible factorization. |
---|
961 | The first factor is always a constant (1, if no nontrivial constant could be excluded). |
---|
962 | EXAMPLE: example facFirstWeyl; shows examples |
---|
963 | SEE ALSO: facSubWeyl, testNCfac, facFirstShift |
---|
964 | "{//proc facFirstWeyl |
---|
965 | //Definition of printlevel variable |
---|
966 | int p = printlevel-voice+2; |
---|
967 | dbprint(p,"==> Checking if the given algebra is a Weyl algebra"); |
---|
968 | //Redefine the ring in my standard form |
---|
969 | if (!isWeyl()) |
---|
970 | {//Our basering is not the Weyl algebra |
---|
971 | ERROR("Ring was not the first Weyl algebra"); |
---|
972 | return(list()); |
---|
973 | }//Our basering is not the Weyl algebra |
---|
974 | dbprint(p,"==> Successful"); |
---|
975 | dbprint(p,"==> Checking, if the given ring is the first Weyl algebra"); |
---|
976 | if(nvars(basering)!=2) |
---|
977 | {//Our basering is the Weyl algebra, but not the first |
---|
978 | ERROR("==>Ring is not the first Weyl algebra"); |
---|
979 | return(list()); |
---|
980 | }//Our basering is the Weyl algebra, but not the first |
---|
981 | dbprint(p,"==> Successful"); |
---|
982 | list result = list(); |
---|
983 | int i;int j; int k; int l; //counter |
---|
984 | if (ringlist(basering)[6][1,2] == -1) //manual of ringlist will tell you why |
---|
985 | { |
---|
986 | dbprint(p,"==> positions of the variables have to be switched"); |
---|
987 | def r = basering; |
---|
988 | ring tempRing = ringlist(r)[1][1],(x,y),Ws(-1,1); // very strange: |
---|
989 | // setting Wp(-1,1) leads to SegFault; to clarify why!!! |
---|
990 | def NTR = nc_algebra(1,1); |
---|
991 | setring NTR ; |
---|
992 | map transf = r, var(2), var(1); |
---|
993 | dbprint(p,"==> Successful"); |
---|
994 | dbprint(p, "==> factorization of the polynomial with the routine sfacwa"); |
---|
995 | list resulttemp = sfacwa(transf(h)); |
---|
996 | dbprint(p,"==> Done"); |
---|
997 | setring(r); |
---|
998 | map transfback = NTR, var(2),var(1); |
---|
999 | result = transfback(resulttemp); |
---|
1000 | } |
---|
1001 | else |
---|
1002 | { |
---|
1003 | dbprint(p, "==> factorization of the polynomial with the routine sfacwa"); |
---|
1004 | result = sfacwa(h); |
---|
1005 | dbprint(p,"==> Done"); |
---|
1006 | } |
---|
1007 | dbprint(p,"==> recursively check factors for irreducibility"); |
---|
1008 | list recursivetemp; |
---|
1009 | for(i = 1; i<=size(result);i++) |
---|
1010 | {//recursively factorize factors |
---|
1011 | if(size(result[i])>2) |
---|
1012 | {//Nontrivial factorization |
---|
1013 | for (j=2;j<=size(result[i]);j++) |
---|
1014 | {//Factorize every factor |
---|
1015 | recursivetemp = facFirstWeyl(result[i][j]); |
---|
1016 | if(size(recursivetemp)>1) |
---|
1017 | {//we have a nontrivial factorization |
---|
1018 | for(k=1; k<=size(recursivetemp);k++) |
---|
1019 | {//insert factorized factors |
---|
1020 | if(size(recursivetemp[k])>2) |
---|
1021 | {//nontrivial |
---|
1022 | result = insert(result,result[i],i); |
---|
1023 | for(l = size(recursivetemp[k]);l>=2;l--) |
---|
1024 | { |
---|
1025 | result[i+1] = insert(result[i+1],recursivetemp[k][l],j); |
---|
1026 | } |
---|
1027 | result[i+1] = delete(result[i+1],j); |
---|
1028 | }//nontrivial |
---|
1029 | }//insert factorized factors |
---|
1030 | }//we have a nontrivial factorization |
---|
1031 | }//Factorize every factor |
---|
1032 | }//Nontrivial factorization |
---|
1033 | }//recursively factorize factors |
---|
1034 | dbprint(p,"==> Done"); |
---|
1035 | if (size(result)==0) |
---|
1036 | {//only the trivial factorization could be found |
---|
1037 | result = list(list(1,h)); |
---|
1038 | }//only the trivial factorization could be found |
---|
1039 | //now, refine the possible redundant list |
---|
1040 | return( delete_dublicates_noteval(result) ); |
---|
1041 | }//proc facFirstWeyl |
---|
1042 | example |
---|
1043 | { |
---|
1044 | "EXAMPLE:";echo=2; |
---|
1045 | ring R = 0,(x,y),dp; |
---|
1046 | def r = nc_algebra(1,1); |
---|
1047 | setring(r); |
---|
1048 | poly h = (x^2*y^2+x)*(x+1); |
---|
1049 | facFirstWeyl(h); |
---|
1050 | } |
---|
1051 | |
---|
1052 | //This is the main program |
---|
1053 | static proc sfacwa(poly h) |
---|
1054 | "USAGE: sfacwa(h); h is a polynomial in the first Weyl algebra |
---|
1055 | RETURN: list |
---|
1056 | PURPOSE: Computes a factorization of a polynomial h in the first Weyl algebra |
---|
1057 | THEORY: @code{sfacwa} returns a list with some factorizations of the given |
---|
1058 | polynomial. The possibilities of the factorization of the highest |
---|
1059 | homogeneous part and those of the lowest will be merged. If during this |
---|
1060 | procedure a factorization of the polynomial occurs, it will be added to |
---|
1061 | the output list. For a more detailed description visit |
---|
1062 | @url{http://www.math.rwth-aachen.de/\~Albert.Heinle} |
---|
1063 | SEE ALSO: homogfacFirstWeyl_all, homogfacFirstWeyl |
---|
1064 | "{//proc sfacwa |
---|
1065 | int p=printlevel-voice+2; // for dbprint |
---|
1066 | intvec ivm11 = intvec(-1,1); |
---|
1067 | intvec iv11 = intvec(1,1); |
---|
1068 | intvec iv10 = intvec(1,0); |
---|
1069 | intvec iv01 = intvec(0,1); |
---|
1070 | intvec iv1m1 = intvec(1,-1); |
---|
1071 | if(homogwithorder(h,ivm11)) |
---|
1072 | { |
---|
1073 | dbprint(p,"==> Given polynomial is -1,1 homogeneous. Start homog. fac. and ret. its result"); |
---|
1074 | return(homogfacFirstWeyl_all(h)); |
---|
1075 | } |
---|
1076 | def r = basering; |
---|
1077 | map invo = basering,-var(1),var(2); |
---|
1078 | int i; int j; int k; |
---|
1079 | dbprint(p,"==> Computing the degree-limits of the factorization"); |
---|
1080 | intvec limits = deg(h,iv11) ,deg(h,iv10),deg(h,iv01); |
---|
1081 | def prod; |
---|
1082 | //end finding the limits |
---|
1083 | dbprint(p,"==> Computing the maximal and the minimal homogeneous part of the give polynomial"); |
---|
1084 | poly maxh = jet(h,deg(h,ivm11),ivm11)-jet(h,deg(h,ivm11)-1,ivm11); |
---|
1085 | poly minh = jet(h,deg(h,iv1m1),iv1m1)-jet(h,deg(h,iv1m1)-1,iv1m1); |
---|
1086 | list result; |
---|
1087 | list temp; |
---|
1088 | list homogtemp; |
---|
1089 | def invhath; |
---|
1090 | def invideal; |
---|
1091 | list M; list hatM; |
---|
1092 | dbprint(p,"==> Factorize the maximal and the minimal homog. part of the given polynomial"); |
---|
1093 | list f1 = homogfacFirstWeyl_all(maxh); |
---|
1094 | list f2 = homogfacFirstWeyl_all(minh); |
---|
1095 | int is_equal; |
---|
1096 | poly hath; |
---|
1097 | dbprint(p,"==> Merging the factorizations"); |
---|
1098 | for (i = 1; i<=size(f1);i++) |
---|
1099 | {//Merge all combinations |
---|
1100 | for (j = 1; j<=size(f2); j++) |
---|
1101 | { |
---|
1102 | M = M + mergence(f1[i],f2[j],limits); |
---|
1103 | } |
---|
1104 | }//Merge all combinations |
---|
1105 | dbprint(p,"==> Filtering invalid combinations"); |
---|
1106 | for (i = 1 ; i<= size(M); i++) |
---|
1107 | {//filter valid combinations |
---|
1108 | if (product(M[i]) == h) |
---|
1109 | {//We have one factorization |
---|
1110 | result = result + list(M[i]); |
---|
1111 | dbprint(p,"Result list updated:"); |
---|
1112 | dbprint(p,result); |
---|
1113 | M = delete(M,i); |
---|
1114 | continue; |
---|
1115 | }//We have one factorization |
---|
1116 | else |
---|
1117 | { |
---|
1118 | if (deg(h,ivm11)<=deg(h-product(M[i]),ivm11)) |
---|
1119 | { |
---|
1120 | M = delete(M,i); |
---|
1121 | continue; |
---|
1122 | } |
---|
1123 | if (deg(h,iv1m1)<=deg(h-product(M[i]),iv1m1)) |
---|
1124 | { |
---|
1125 | M = delete(M,i); |
---|
1126 | continue; |
---|
1127 | } |
---|
1128 | } |
---|
1129 | }//filter valid combinations |
---|
1130 | dbprint(p,"==> Deleting doublicates in the resulting list"); |
---|
1131 | M = delete_dublicates_eval(M); |
---|
1132 | dbprint(p,"==> Iterating over all possible Combinations"); |
---|
1133 | while(size(M)>0) |
---|
1134 | {//work on the elements of M |
---|
1135 | hatM = list(); |
---|
1136 | for(i = 1; i<=size(M); i++) |
---|
1137 | {//iterate over all elements of M |
---|
1138 | hath = h-product(M[i]); |
---|
1139 | temp = list(); |
---|
1140 | //First check for common inhomogeneous factors between hath and h |
---|
1141 | invhath = involution(hath,invo); |
---|
1142 | invideal = std(involution(ideal(M[i][1]),invo)); |
---|
1143 | if (involution(NF(invhath,invideal) ,invo)==0) |
---|
1144 | {//hath and h have a common factor on the left |
---|
1145 | j = 1; |
---|
1146 | f1 = M[i]; |
---|
1147 | if (j+1<=size(f1)) |
---|
1148 | {//Checking for more than one common factor |
---|
1149 | invideal = std(involution(ideal(product(f1[1..(j+1)])),invo)); |
---|
1150 | while(involution(NF(invhath,invideal),invo)==0) |
---|
1151 | { |
---|
1152 | if (j+1<size(f1)) |
---|
1153 | { |
---|
1154 | j++; |
---|
1155 | } |
---|
1156 | else |
---|
1157 | { |
---|
1158 | break; |
---|
1159 | } |
---|
1160 | } |
---|
1161 | }//Checking for more than one common factor |
---|
1162 | invideal = lift(involution(product(f1[1..j]),invo),invhath); |
---|
1163 | f2 = list(f1[1..j])+list(involution(invideal[1,1],invo)); |
---|
1164 | temp = temp + merge_cf(f2,f1,limits); |
---|
1165 | }//hath and h have a common factor on the left |
---|
1166 | if (reduce(hath, std(ideal(M[i][size(M[i])])))==0) |
---|
1167 | {//hath and h have a common factor on the right |
---|
1168 | j = size(M[i]); |
---|
1169 | f1 = M[i]; |
---|
1170 | if (j-1>0) |
---|
1171 | {//Checking for more than one factor |
---|
1172 | while(reduce(hath,std(ideal(product(f1[(j-1)..size(f1)]))))==0) |
---|
1173 | { |
---|
1174 | if (j-1>1) |
---|
1175 | { |
---|
1176 | j--; |
---|
1177 | } |
---|
1178 | else |
---|
1179 | { |
---|
1180 | break; |
---|
1181 | } |
---|
1182 | } |
---|
1183 | }//Checking for more than one factor |
---|
1184 | f2 = list(lift(product(f1[j..size(f1)]),hath)[1,1])+list(f1[j..size(f1)]); |
---|
1185 | temp = temp + merge_cf(f2,M[i],limits); |
---|
1186 | }//hath and h have a common factor on the right |
---|
1187 | //and now the homogeneous |
---|
1188 | maxh = jet(hath,deg(hath,ivm11),ivm11)-jet(hath,deg(hath,ivm11)-1,ivm11); |
---|
1189 | minh = jet(hath,deg(hath,iv1m1),iv1m1)-jet(hath,deg(hath,iv1m1)-1,iv1m1); |
---|
1190 | f1 = homogfacFirstWeyl_all(maxh); |
---|
1191 | f2 = homogfacFirstWeyl_all(minh); |
---|
1192 | for (j = 1; j<=size(f1);j++) |
---|
1193 | { |
---|
1194 | for (k=1; k<=size(f2);k++) |
---|
1195 | { |
---|
1196 | homogtemp = mergence(f1[j],f2[k],limits); |
---|
1197 | } |
---|
1198 | } |
---|
1199 | for (j = 1; j<= size(homogtemp); j++) |
---|
1200 | { |
---|
1201 | temp = temp + mergence(homogtemp[j],M[i],limits); |
---|
1202 | } |
---|
1203 | for (j = 1; j<=size(temp); j++) |
---|
1204 | {//filtering invalid entries in temp |
---|
1205 | if(product(temp[j])==h) |
---|
1206 | {//This is already a result |
---|
1207 | result = result + list(temp[j]); |
---|
1208 | temp = delete(temp,j); |
---|
1209 | continue; |
---|
1210 | }//This is already a result |
---|
1211 | if (deg(hath,ivm11)<=deg(hath-product(temp[j]),ivm11)) |
---|
1212 | { |
---|
1213 | temp = delete(temp,j); |
---|
1214 | continue; |
---|
1215 | } |
---|
1216 | }//filtering invalid entries in temp |
---|
1217 | hatM = hatM + temp; |
---|
1218 | }//iterate over all elements of M |
---|
1219 | M = hatM; |
---|
1220 | for (i = 1; i<=size(M);i++) |
---|
1221 | {//checking for complete factorizations |
---|
1222 | if (h == product(M[i])) |
---|
1223 | { |
---|
1224 | result = result + list(M[i]); |
---|
1225 | dbprint(p,"Result list updated:"); |
---|
1226 | dbprint(p,result); |
---|
1227 | M = delete(M,i); |
---|
1228 | continue; |
---|
1229 | } |
---|
1230 | }//checking for complete factorizations |
---|
1231 | M = delete_dublicates_eval(M); |
---|
1232 | }//work on the elements of M |
---|
1233 | //In the case, that there is none, write a constant factor before the factor of interest. |
---|
1234 | dbprint(p,"==> Done"); |
---|
1235 | for (i = 1 ; i<=size(result);i++) |
---|
1236 | {//add a constant factor |
---|
1237 | if (deg(result[i][1],iv11)!=0) |
---|
1238 | { |
---|
1239 | result[i] = insert(result[i],1); |
---|
1240 | } |
---|
1241 | }//add a constant factor |
---|
1242 | dbprint(p,"==> Deleting doublicates in the output list"); |
---|
1243 | result = delete_dublicates_noteval(result); |
---|
1244 | return(result); |
---|
1245 | }//proc sfacwa |
---|
1246 | |
---|
1247 | |
---|
1248 | //================================================== |
---|
1249 | /*Singular has no way implemented to test polynomials |
---|
1250 | for homogenity with respect to a weight vector. |
---|
1251 | The following procedure does exactly this*/ |
---|
1252 | static proc homogwithorder(poly h, intvec weights) |
---|
1253 | {//proc homogwithorder |
---|
1254 | if(size(weights) != nvars(basering)) |
---|
1255 | {//The user does not know how many variables the current ring has |
---|
1256 | return(0); |
---|
1257 | }//The user does not know how many variables the current ring has |
---|
1258 | int i; |
---|
1259 | int dofp = deg(h,weights); //degree of polynomial |
---|
1260 | for (i = 1; i<=size(h);i++) |
---|
1261 | { |
---|
1262 | if (deg(h[i],weights)!=dofp) |
---|
1263 | { |
---|
1264 | return(0); |
---|
1265 | } |
---|
1266 | } |
---|
1267 | return(1); |
---|
1268 | }//proc homogwithorder |
---|
1269 | |
---|
1270 | //================================================== |
---|
1271 | //Testfac: Given a list with different factorizations of |
---|
1272 | // one polynomial, the following procedure checks |
---|
1273 | // whether they all refer to the same polynomial. |
---|
1274 | // If they do, the output will be a list, that contains |
---|
1275 | // the product of each factorization. If not, the empty |
---|
1276 | // list will be returned. |
---|
1277 | // If the optional argument # is given (i.e. the polynomial |
---|
1278 | // which is factorized by the elements of the given list), |
---|
1279 | // then we look, if the entries are factorizations of p |
---|
1280 | // and if not, a list with the products subtracted by p |
---|
1281 | // will be returned |
---|
1282 | proc testNCfac(list l, list #) |
---|
1283 | "USAGE: testNCfac(l[,p,b]); l is a list, p is an optional poly, b is 1 or 0 |
---|
1284 | RETURN: Case 1: No optional argument. In this case the output is 1, if the |
---|
1285 | entries in the given list represent the same polynomial or 0 |
---|
1286 | otherwise. |
---|
1287 | Case 2: One optional argument p is given. In this case it returns 1, |
---|
1288 | if all the entries in l are factorizations of p, otherwise 0. |
---|
1289 | Case 3: Second optional b is given. In this case a list is returned |
---|
1290 | containing the difference between the product of each entry in |
---|
1291 | l and p. |
---|
1292 | ASSUME: basering is the first Weyl algebra, the entries of l are polynomials |
---|
1293 | PURPOSE: Checks whether a list of factorizations contains factorizations of |
---|
1294 | the same element in the first Weyl algebra |
---|
1295 | THEORY: @code{testNCfac} multiplies out each factorization and checks whether |
---|
1296 | each factorization was a factorization of the same element. |
---|
1297 | @* - if there is only a list given, the output will be 0, if it |
---|
1298 | does not contain factorizations of the same element. Otherwise the output |
---|
1299 | will be 1. |
---|
1300 | @* - if there is a polynomial in the second argument, then the procedure checks |
---|
1301 | whether the given list contains factorizations of this polynomial. If it |
---|
1302 | does, then the output depends on the third argument. If it is not given, |
---|
1303 | the procedure will check whether the factorizations in the list |
---|
1304 | l are associated to this polynomial and return either 1 or 0, respectively. |
---|
1305 | If the third argument is given, the output will be a list with |
---|
1306 | the length of the given one and in each entry is the product of one |
---|
1307 | entry in l subtracted by the polynomial. |
---|
1308 | EXAMPLE: example testNCfac; shows examples |
---|
1309 | SEE ALSO: facFirstWeyl, facSubWeyl, facFirstShift |
---|
1310 | "{//proc testfac |
---|
1311 | int p = printlevel - voice + 2; |
---|
1312 | dbprint(p,"==> Checking the input"); |
---|
1313 | if (size(l)==0) |
---|
1314 | {//The empty list is given |
---|
1315 | dbprint(p,"==> Given list was empty"); |
---|
1316 | return(list()); |
---|
1317 | }//The empty list is given |
---|
1318 | if (size(#)>2) |
---|
1319 | {//We want max. two optional arguments |
---|
1320 | dbprint(p,"==> More than two optional arguments"); |
---|
1321 | return(list()); |
---|
1322 | }//We want max. two optional arguments |
---|
1323 | dbprint(p,"==> Done"); |
---|
1324 | list result; |
---|
1325 | int i; int j; |
---|
1326 | if (size(#)==0) |
---|
1327 | {//No optional argument is given |
---|
1328 | dbprint(p,"==> No optional arguments"); |
---|
1329 | int valid = 1; |
---|
1330 | for (i = size(l);i>=1;i--) |
---|
1331 | {//iterate over the elements of the given list |
---|
1332 | if (size(result)>0) |
---|
1333 | { |
---|
1334 | if (product(l[i])!=result[size(l)-i]) |
---|
1335 | { |
---|
1336 | valid = 0; |
---|
1337 | break; |
---|
1338 | } |
---|
1339 | } |
---|
1340 | result = insert(result, product(l[i])); |
---|
1341 | }//iterate over the elements of the given list |
---|
1342 | return(valid); |
---|
1343 | }//No optional argument is given |
---|
1344 | else |
---|
1345 | { |
---|
1346 | dbprint(p,"==> Optional arguments are given."); |
---|
1347 | int valid = 1; |
---|
1348 | for (i = size(l);i>=1;i--) |
---|
1349 | {//iterate over the elements of the given list |
---|
1350 | if (product(l[i])!=#[1]) |
---|
1351 | { |
---|
1352 | valid = 0; |
---|
1353 | } |
---|
1354 | result = insert(result, product(l[i])-#[1]); |
---|
1355 | }//iterate over the elements of the given list |
---|
1356 | if(size(#)==2) |
---|
1357 | { |
---|
1358 | dbprint(p,"==> A third argument is given. Output is a list now."); |
---|
1359 | return(result); |
---|
1360 | } |
---|
1361 | return(valid); |
---|
1362 | } |
---|
1363 | }//proc testfac |
---|
1364 | example |
---|
1365 | { |
---|
1366 | "EXAMPLE:";echo=2; |
---|
1367 | ring r = 0,(x,y),dp; |
---|
1368 | def R = nc_algebra(1,1); |
---|
1369 | setring R; |
---|
1370 | poly h = (x^2*y^2+1)*(x^2); |
---|
1371 | def t1 = facFirstWeyl(h); |
---|
1372 | //fist a correct list |
---|
1373 | testNCfac(t1); |
---|
1374 | //now a correct list with the factorized polynomial |
---|
1375 | testNCfac(t1,h); |
---|
1376 | //now we put in an incorrect list without a polynomial |
---|
1377 | t1[3][3] = y; |
---|
1378 | testNCfac(t1); |
---|
1379 | // take h as additional input |
---|
1380 | testNCfac(t1,h); |
---|
1381 | // take h as additional input and output list of differences |
---|
1382 | testNCfac(t1,h,1); |
---|
1383 | } |
---|
1384 | //================================================== |
---|
1385 | //Procedure facSubWeyl: |
---|
1386 | //This procedure serves the purpose to compute a |
---|
1387 | //factorization of a given polynomial in a ring, whose subring |
---|
1388 | //is the first Weyl algebra. The polynomial must only contain |
---|
1389 | //the two arguments, which are also given by the user. |
---|
1390 | |
---|
1391 | proc facSubWeyl(poly h, X, D) |
---|
1392 | "USAGE: facSubWeyl(h,x,y); h, X, D polynomials |
---|
1393 | RETURN: list |
---|
1394 | ASSUME: X and D are variables of a basering, which satisfy DX = XD +1. |
---|
1395 | @* That is, they generate the copy of the first Weyl algebra in a basering. |
---|
1396 | @* Moreover, h is a polynomial in X and D only. |
---|
1397 | PURPOSE: compute factorizations of the polynomial, which depends on X and D. |
---|
1398 | EXAMPLE: example facSubWeyl; shows examples |
---|
1399 | SEE ALSO: facFirstWeyl, testNCfac, facFirstShift |
---|
1400 | "{ |
---|
1401 | int p = printlevel - voice + 2; |
---|
1402 | dbprint(p,"==> Start initial Checks of the input."); |
---|
1403 | // basering can be anything having a Weyl algebra as subalgebra |
---|
1404 | def @r = basering; |
---|
1405 | //We begin to check the input for assumptions |
---|
1406 | // which are: X,D are vars of the basering, |
---|
1407 | if ( (isVar(X)!=1) || (isVar(D)!=1) || (size(X)>1) || (size(D)>1) || |
---|
1408 | (leadcoef(X) != number(1)) || (leadcoef(D) != number(1)) ) |
---|
1409 | { |
---|
1410 | ERROR("expected pure variables as generators of a subalgebra"); |
---|
1411 | } |
---|
1412 | // Weyl algebra: |
---|
1413 | poly w = D*X-X*D-1; // [D,X]=1 |
---|
1414 | poly u = D*X-X*D+1; // [X,D]=1 |
---|
1415 | if (u*w!=0) |
---|
1416 | { |
---|
1417 | // that is no combination gives Weyl |
---|
1418 | ERROR("2nd and 3rd argument do not generate a Weyl algebra"); |
---|
1419 | } |
---|
1420 | // one of two is correct |
---|
1421 | int isReverted = 0; // Reverted Weyl if dx=xd-1 holds |
---|
1422 | if (u==0) |
---|
1423 | { |
---|
1424 | isReverted = 1; |
---|
1425 | } |
---|
1426 | // else: do nothing |
---|
1427 | // DONE with assumptions, Input successfully checked |
---|
1428 | dbprint(p,"==> Successful"); |
---|
1429 | intvec lexpofX = leadexp(X); |
---|
1430 | intvec lexpofD = leadexp(D); |
---|
1431 | int varnumX=1; |
---|
1432 | int varnumD=1; |
---|
1433 | while(lexpofX[varnumX] != 1) |
---|
1434 | { |
---|
1435 | varnumX++; |
---|
1436 | } |
---|
1437 | while(lexpofD[varnumD] != 1) |
---|
1438 | { |
---|
1439 | varnumD++; |
---|
1440 | } |
---|
1441 | /* VL : to add printlevel stuff */ |
---|
1442 | dbprint(p,"==> Change positions of the two variables in the list, if needed"); |
---|
1443 | if (isReverted) |
---|
1444 | { |
---|
1445 | ring firstweyl = 0,(var(varnumD),var(varnumX)),dp; |
---|
1446 | def Firstweyl = nc_algebra(1,1); |
---|
1447 | setring Firstweyl; |
---|
1448 | ideal M = 0:nvars(@r); |
---|
1449 | M[varnumX]=var(2); |
---|
1450 | M[varnumD]=var(1); |
---|
1451 | map Q = @r,M; |
---|
1452 | poly h= Q(h); |
---|
1453 | } |
---|
1454 | else |
---|
1455 | { // that is unReverted |
---|
1456 | ring firstweyl = 0,(var(varnumX),var(varnumD)),dp; |
---|
1457 | def Firstweyl = nc_algebra(1,1); |
---|
1458 | setring Firstweyl; |
---|
1459 | poly h= imap(@r,h); |
---|
1460 | } |
---|
1461 | dbprint(p,"==> Done!"); |
---|
1462 | list result = facFirstWeyl(h); |
---|
1463 | setring @r; |
---|
1464 | list result; |
---|
1465 | if (isReverted) |
---|
1466 | { |
---|
1467 | // map swap back |
---|
1468 | ideal M; M[1] = var(varnumD); M[2] = var(varnumX); |
---|
1469 | map S = Firstweyl, M; |
---|
1470 | result = S(result); |
---|
1471 | } |
---|
1472 | else |
---|
1473 | { |
---|
1474 | // that is unReverted |
---|
1475 | result = imap(Firstweyl,result); |
---|
1476 | } |
---|
1477 | return(result); |
---|
1478 | }//proc facSubWeyl |
---|
1479 | example |
---|
1480 | { |
---|
1481 | "EXAMPLE:";echo=2; |
---|
1482 | ring r = 0,(x,y,z),dp; |
---|
1483 | matrix D[3][3]; D[1,3]=-1; |
---|
1484 | def R = nc_algebra(1,D); // x,z generate Weyl subalgebra |
---|
1485 | setring R; |
---|
1486 | poly h = (x^2*z^2+x)*x; |
---|
1487 | list fact1 = facSubWeyl(h,x,z); |
---|
1488 | // compare with facFirstWeyl: |
---|
1489 | ring s = 0,(z,x),dp; |
---|
1490 | def S = nc_algebra(1,1); setring S; |
---|
1491 | poly h = (x^2*z^2+x)*x; |
---|
1492 | list fact2 = facFirstWeyl(h); |
---|
1493 | map F = R,x,0,z; |
---|
1494 | list fact1 = F(fact1); // it is identical to list fact2 |
---|
1495 | testNCfac(fact1); // check the correctness again |
---|
1496 | } |
---|
1497 | //================================================== |
---|
1498 | |
---|
1499 | //================================================== |
---|
1500 | //************From here: Shift-Algebra************** |
---|
1501 | //================================================== |
---|
1502 | //==================================================* |
---|
1503 | //one factorization of a homogeneous polynomial |
---|
1504 | //in the first Shift Algebra |
---|
1505 | static proc homogfacFirstShift(poly h) |
---|
1506 | {//proc homogfacFirstShift |
---|
1507 | int p=printlevel-voice+2; //for dbprint |
---|
1508 | def r = basering; |
---|
1509 | poly hath; |
---|
1510 | intvec iv01 = intvec(0,1); |
---|
1511 | int i; int j; |
---|
1512 | if (!homogwithorder(h,iv01)) |
---|
1513 | {//The given polynomial is not homogeneous |
---|
1514 | ERROR("The given polynomial is not homogeneous."); |
---|
1515 | return(list()); |
---|
1516 | }//The given polynomial is not homogeneous |
---|
1517 | if (h==0) |
---|
1518 | { |
---|
1519 | return(list(0)); |
---|
1520 | } |
---|
1521 | list result; |
---|
1522 | int m = deg(h,iv01); |
---|
1523 | dbprint(p,"==> exclude the homogeneous part of deg. 0"); |
---|
1524 | if (m>0) |
---|
1525 | {//The degree is not zero |
---|
1526 | hath = lift(var(2)^m,h)[1,1]; |
---|
1527 | for (i = 1; i<=m;i++) |
---|
1528 | { |
---|
1529 | result = result + list(var(2)); |
---|
1530 | } |
---|
1531 | }//The degree is not zero |
---|
1532 | else |
---|
1533 | {//The degree is zero |
---|
1534 | hath = h; |
---|
1535 | }//The degree is zero |
---|
1536 | ring tempRing = 0,(x),dp; |
---|
1537 | setring tempRing; |
---|
1538 | map thetamap = r,x,1; |
---|
1539 | poly hath = thetamap(hath); |
---|
1540 | dbprint(p,"==> Factorize it using commutative factorization."); |
---|
1541 | list azeroresult = factorize(hath); |
---|
1542 | list azeroresult_return_form; |
---|
1543 | for (i = 1; i<=size(azeroresult[1]);i++) |
---|
1544 | {//rewrite the result of the commutative factorization |
---|
1545 | for (j = 1; j <= azeroresult[2][i];j++) |
---|
1546 | { |
---|
1547 | azeroresult_return_form = azeroresult_return_form + list(azeroresult[1][i]); |
---|
1548 | } |
---|
1549 | }//rewrite the result of the commutative factorization |
---|
1550 | setring(r); |
---|
1551 | map finalmap = tempRing,var(1); |
---|
1552 | list tempresult = finalmap(azeroresult_return_form); |
---|
1553 | result = tempresult+result; |
---|
1554 | return(result); |
---|
1555 | }//proc homogfacFirstShift |
---|
1556 | |
---|
1557 | //================================================== |
---|
1558 | //Computes all possible homogeneous factorizations |
---|
1559 | static proc homogfacFirstShift_all(poly h) |
---|
1560 | {//proc HomogfacFirstShiftAll |
---|
1561 | int p=printlevel-voice+2; //for dbprint |
---|
1562 | intvec iv11 = intvec(1,1); |
---|
1563 | if (deg(h,iv11) <= 0 ) |
---|
1564 | {//h is a constant |
---|
1565 | return(list(list(h))); |
---|
1566 | }//h is a constant |
---|
1567 | def r = basering; |
---|
1568 | list one_hom_fac; //stands for one homogeneous factorization |
---|
1569 | int i; int j; int k; |
---|
1570 | int shiftcounter; |
---|
1571 | //Compute again a homogeneous factorization |
---|
1572 | dbprint(p,"==> Computing one homog. factorization of the polynomial"); |
---|
1573 | one_hom_fac = homogfacFirstShift(h); |
---|
1574 | one_hom_fac = delete(one_hom_fac,1); |
---|
1575 | if (size(one_hom_fac) == 0) |
---|
1576 | {//there is no homogeneous factorization or the polynomial was not homogeneous |
---|
1577 | return(list()); |
---|
1578 | }//there is no homogeneous factorization or the polynomial was not homogeneous |
---|
1579 | dbprint(p,"==> Permuting the 0-homogeneous part with the s"); |
---|
1580 | list result = permpp(one_hom_fac); |
---|
1581 | for (i = 1; i<=size(result);i++) |
---|
1582 | { |
---|
1583 | shiftcounter = 0; |
---|
1584 | for (j = 1; j<=size(result[i]); j++) |
---|
1585 | { |
---|
1586 | if (result[i][j]==var(2)) |
---|
1587 | { |
---|
1588 | shiftcounter++; |
---|
1589 | } |
---|
1590 | else |
---|
1591 | { |
---|
1592 | result[i][j] = subst(result[i][j], var(1), var(1)-shiftcounter); |
---|
1593 | } |
---|
1594 | } |
---|
1595 | result[i] = insert(result[i],1); |
---|
1596 | } |
---|
1597 | dbprint(p,"==> Deleting double entries in the resulting list"); |
---|
1598 | result = delete_dublicates_noteval(result); |
---|
1599 | return(result); |
---|
1600 | }//proc HomogfacFirstShiftAll |
---|
1601 | |
---|
1602 | //================================================== |
---|
1603 | //factorization of the first Shift Algebra |
---|
1604 | proc facFirstShift(poly h) |
---|
1605 | "USAGE: facFirstShift(h); h a polynomial in the first shift algebra |
---|
1606 | RETURN: list |
---|
1607 | PURPOSE: compute all factorizations of a polynomial in the first shift algebra |
---|
1608 | THEORY: Implements the new algorithm by A. Heinle and V. Levandovskyy, see the thesis of A. Heinle |
---|
1609 | ASSUME: basering is the first shift algebra |
---|
1610 | NOTE: Every entry of the output list is a list with factors for one possible factorization. |
---|
1611 | EXAMPLE: example facFirstShift; shows examples |
---|
1612 | SEE ALSO: testNCfac, facFirstWeyl, facSubWeyl |
---|
1613 | "{//facFirstShift |
---|
1614 | int p = printlevel - voice + 2; |
---|
1615 | dbprint(p,"==> Checking the input."); |
---|
1616 | if(nvars(basering)!=2) |
---|
1617 | {//Our basering is the Shift algebra, but not the first |
---|
1618 | ERROR("Basering is not the first shift algebra"); |
---|
1619 | return(list()); |
---|
1620 | }//Our basering is the Shift algebra, but not the first |
---|
1621 | def r = basering; |
---|
1622 | setring r; |
---|
1623 | list LR = ringlist(r); |
---|
1624 | number @n = leadcoef(LR[5][1,2]); |
---|
1625 | poly @p = LR[6][1,2]; |
---|
1626 | if ( @n!=number(1) ) |
---|
1627 | { |
---|
1628 | ERROR("Basering is not the first shift algebra"); |
---|
1629 | return(list()); |
---|
1630 | } |
---|
1631 | dbprint(p,"==> Done"); |
---|
1632 | list result = list(); |
---|
1633 | int i;int j; int k; int l; //counter |
---|
1634 | // create a ring with the ordering which makes shift algebra |
---|
1635 | // graded |
---|
1636 | // def r = basering; // done before |
---|
1637 | ring tempRing = LR[1][1],(x,s),(a(0,1),Dp); |
---|
1638 | def tempRingnc = nc_algebra(1,s); |
---|
1639 | setring r; |
---|
1640 | // information on relations |
---|
1641 | if (@p == -var(1)) // reverted shift algebra |
---|
1642 | { |
---|
1643 | dbprint(p,"==> Reverted shift algebra. Swaping variables in Ringlist"); |
---|
1644 | setring(tempRingnc); |
---|
1645 | map transf = r, var(2), var(1); |
---|
1646 | setring(r); |
---|
1647 | map transfback = tempRingnc, var(2),var(1); |
---|
1648 | // result = transfback(resulttemp); |
---|
1649 | } |
---|
1650 | else |
---|
1651 | { |
---|
1652 | if ( @p == var(2)) // usual shift algebra |
---|
1653 | { |
---|
1654 | setring(tempRingnc); |
---|
1655 | map transf = r, var(1), var(2); |
---|
1656 | // result = facshift(h); |
---|
1657 | setring(r); |
---|
1658 | map transfback = tempRingnc, var(1),var(2); |
---|
1659 | } |
---|
1660 | else |
---|
1661 | { |
---|
1662 | ERROR("Basering is not the first shift algebra"); |
---|
1663 | return(list()); |
---|
1664 | } |
---|
1665 | } |
---|
1666 | // main calls |
---|
1667 | setring(tempRingnc); |
---|
1668 | dbprint(p,"==> Factorize the given polynomial with the subroutine facshift"); |
---|
1669 | list resulttemp = facshift(transf(h)); |
---|
1670 | dbprint(p,"==> Successful"); |
---|
1671 | setring(r); |
---|
1672 | result = transfback(resulttemp); |
---|
1673 | dbprint(p,"==> Recursively check the found factors for reducibility."); |
---|
1674 | list recursivetemp; |
---|
1675 | for(i = 1; i<=size(result);i++) |
---|
1676 | {//recursively factorize factors |
---|
1677 | if(size(result[i])>2) |
---|
1678 | {//Nontrivial factorization |
---|
1679 | for (j=2;j<=size(result[i]);j++) |
---|
1680 | {//Factorize every factor |
---|
1681 | recursivetemp = facFirstShift(result[i][j]); |
---|
1682 | if(size(recursivetemp)>1) |
---|
1683 | {//we have a nontrivial factorization |
---|
1684 | for(k=1; k<=size(recursivetemp);k++) |
---|
1685 | {//insert factorized factors |
---|
1686 | if(size(recursivetemp[k])>2) |
---|
1687 | {//nontrivial |
---|
1688 | result = insert(result,result[i],i); |
---|
1689 | for(l = size(recursivetemp[k]);l>=2;l--) |
---|
1690 | { |
---|
1691 | result[i+1] = insert(result[i+1],recursivetemp[k][l],j); |
---|
1692 | } |
---|
1693 | result[i+1] = delete(result[i+1],j); |
---|
1694 | }//nontrivial |
---|
1695 | }//insert factorized factors |
---|
1696 | }//we have a nontrivial factorization |
---|
1697 | }//Factorize every factor |
---|
1698 | }//Nontrivial factorization |
---|
1699 | }//recursively factorize factors |
---|
1700 | //now, refine the possible redundant list |
---|
1701 | dbprint(p,"==> Done"); |
---|
1702 | return( delete_dublicates_noteval(result) ); |
---|
1703 | }//facFirstShift |
---|
1704 | example |
---|
1705 | { |
---|
1706 | "EXAMPLE:";echo=2; |
---|
1707 | ring R = 0,(x,s),dp; |
---|
1708 | def r = nc_algebra(1,s); |
---|
1709 | setring(r); |
---|
1710 | poly h = (s^2*x+x)*s; |
---|
1711 | facFirstShift(h); |
---|
1712 | } |
---|
1713 | |
---|
1714 | static proc facshift(poly h) |
---|
1715 | "USAGE: facshift(h); h is a polynomial in the first Shift algebra |
---|
1716 | RETURN: list |
---|
1717 | PURPOSE: Computes a factorization of a polynomial h in the first Shift algebra |
---|
1718 | THEORY: @code{facshift} returns a list with some factorizations of the given |
---|
1719 | polynomial. The possibilities of the factorization of the highest |
---|
1720 | homogeneous part and those of the lowest will be merged. If during this |
---|
1721 | procedure a factorization of the polynomial occurs, it will be added to |
---|
1722 | the output list. For a more detailled description visit |
---|
1723 | @url{http://www.math.rwth-aachen.de/\~Albert.Heinle} |
---|
1724 | SEE ALSO: homogfacFirstShift_all, homogfacFirstShift |
---|
1725 | "{//proc facshift |
---|
1726 | int p=printlevel-voice+2; // for dbprint |
---|
1727 | intvec iv01 = intvec(0,1); |
---|
1728 | intvec iv11 = intvec(1,1); |
---|
1729 | intvec iv10 = intvec(1,0); |
---|
1730 | intvec iv0m1 = intvec(0,-1); |
---|
1731 | if(homogwithorder(h,iv01)) |
---|
1732 | { |
---|
1733 | dbprint(p,"==> Given polynomial was 1,0-homogeneous. Calling subroutine for homog. polys."); |
---|
1734 | return(homogfacFirstShift_all(h)); |
---|
1735 | } |
---|
1736 | def r = basering; |
---|
1737 | map invo = basering,-var(1),-var(2); |
---|
1738 | int i; int j; int k; |
---|
1739 | intvec limits = deg(h,iv11) ,deg(h,iv10),deg(h,iv01); |
---|
1740 | def prod; |
---|
1741 | //end finding the limits |
---|
1742 | poly maxh = jet(h,deg(h,iv01),iv01)-jet(h,deg(h,iv01)-1,iv01); |
---|
1743 | poly minh = jet(h,deg(h,iv0m1),iv0m1)-jet(h,deg(h,iv0m1)-1,iv0m1); |
---|
1744 | list result; |
---|
1745 | list temp; |
---|
1746 | list homogtemp; |
---|
1747 | list M; list hatM; |
---|
1748 | def invhath; |
---|
1749 | def invideal; |
---|
1750 | dbprint(p,"==> Factorizing the highest and the lowest homogeneous part"); |
---|
1751 | list f1 = homogfacFirstShift_all(maxh); |
---|
1752 | list f2 = homogfacFirstShift_all(minh); |
---|
1753 | int is_equal; |
---|
1754 | poly hath; |
---|
1755 | dbprint(p,"==> Merging their factors"); |
---|
1756 | for (i = 1; i<=size(f1);i++) |
---|
1757 | {//Merge all combinations |
---|
1758 | for (j = 1; j<=size(f2); j++) |
---|
1759 | { |
---|
1760 | M = M + mergence(f1[i],f2[j],limits); |
---|
1761 | } |
---|
1762 | }//Merge all combinations |
---|
1763 | dbprint(p,"==> Filtering invalid combinations"); |
---|
1764 | for (i = 1 ; i<= size(M); i++) |
---|
1765 | {//filter valid combinations |
---|
1766 | if (product(M[i]) == h) |
---|
1767 | {//We have one factorization |
---|
1768 | result = result + list(M[i]); |
---|
1769 | dbprint(p,"Result list updated:"); |
---|
1770 | dbprint(p,result); |
---|
1771 | M = delete(M,i); |
---|
1772 | continue; |
---|
1773 | }//We have one factorization |
---|
1774 | else |
---|
1775 | { |
---|
1776 | if (deg(h,iv01)<=deg(h-product(M[i]),iv01)) |
---|
1777 | { |
---|
1778 | M = delete(M,i); |
---|
1779 | continue; |
---|
1780 | } |
---|
1781 | if (deg(h,iv0m1)<=deg(h-product(M[i]),iv0m1)) |
---|
1782 | { |
---|
1783 | M = delete(M,i); |
---|
1784 | continue; |
---|
1785 | } |
---|
1786 | } |
---|
1787 | }//filter valid combinations |
---|
1788 | dbprint(p,"==> Iterating over all elements in M"); |
---|
1789 | M = delete_dublicates_eval(M); |
---|
1790 | while(size(M)>0) |
---|
1791 | {//work on the elements of M |
---|
1792 | hatM = list(); |
---|
1793 | for(i = 1; i<=size(M); i++) |
---|
1794 | {//iterate over all elements of M |
---|
1795 | hath = h-product(M[i]); |
---|
1796 | temp = list(); |
---|
1797 | //First check for common inhomogeneous factors between hath and h |
---|
1798 | invhath = involution(hath,invo); |
---|
1799 | invideal = std(involution(ideal(M[i][1]),invo)); |
---|
1800 | if (involution(NF(invhath, invideal ),invo)==0) |
---|
1801 | {//hath and h have a common factor on the left |
---|
1802 | j = 1; |
---|
1803 | f1 = M[i]; |
---|
1804 | if (j+1<=size(f1)) |
---|
1805 | {//Checking for more than one common factor |
---|
1806 | invideal = std(involution(ideal(product(f1[1..(j+1)])),invo)); |
---|
1807 | while(involution(NF(invhath,invideal),invo)==0) |
---|
1808 | { |
---|
1809 | if (j+1<size(f1)) |
---|
1810 | { |
---|
1811 | j++; |
---|
1812 | } |
---|
1813 | else |
---|
1814 | { |
---|
1815 | break; |
---|
1816 | } |
---|
1817 | } |
---|
1818 | }//Checking for more than one common factor |
---|
1819 | if (deg(product(f1[1..j]),iv11)!=0) |
---|
1820 | { |
---|
1821 | invideal = lift(involution(product(f1[1..j]),invo),invhath); |
---|
1822 | f2 = list(f1[1..j])+list(involution(invideal[1,1],invo)); |
---|
1823 | } |
---|
1824 | else |
---|
1825 | { |
---|
1826 | invideal = lift(product(f1[1..j]),involution(hath,invo)); |
---|
1827 | f2 = list(f1[1..j])+list(involution(invideal[1,1],invo)); |
---|
1828 | } |
---|
1829 | temp = temp + merge_cf(f2,f1,limits); |
---|
1830 | }//hath and h have a common factor on the left |
---|
1831 | if (reduce(hath, std(ideal(M[i][size(M[i])])))==0) |
---|
1832 | {//hath and h have a common factor on the right |
---|
1833 | j = size(M[i]); |
---|
1834 | f1 = M[i]; |
---|
1835 | if (j-1>0) |
---|
1836 | {//Checking for more than one factor |
---|
1837 | while(reduce(hath,std(ideal(product(f1[(j-1)..size(f1)]))))==0) |
---|
1838 | { |
---|
1839 | if (j-1>1) |
---|
1840 | { |
---|
1841 | j--; |
---|
1842 | } |
---|
1843 | else |
---|
1844 | { |
---|
1845 | break; |
---|
1846 | } |
---|
1847 | } |
---|
1848 | }//Checking for more than one factor |
---|
1849 | f2 = list(lift(product(f1[j..size(f1)]),hath)[1,1])+list(f1[j..size(f1)]); |
---|
1850 | temp = temp + merge_cf(f2,M[i],limits); |
---|
1851 | }//hath and h have a common factor on the right |
---|
1852 | //and now the homogeneous |
---|
1853 | maxh = jet(hath,deg(hath,iv01),iv01)-jet(hath,deg(hath,iv01)-1,iv01); |
---|
1854 | minh = jet(hath,deg(hath,iv0m1),iv0m1)-jet(hath,deg(hath,iv0m1)-1,iv0m1); |
---|
1855 | f1 = homogfacFirstShift_all(maxh); |
---|
1856 | f2 = homogfacFirstShift_all(minh); |
---|
1857 | for (j = 1; j<=size(f1);j++) |
---|
1858 | { |
---|
1859 | for (k=1; k<=size(f2);k++) |
---|
1860 | { |
---|
1861 | homogtemp = mergence(f1[j],f2[k],limits); |
---|
1862 | } |
---|
1863 | } |
---|
1864 | for (j = 1; j<= size(homogtemp); j++) |
---|
1865 | { |
---|
1866 | temp = temp + mergence(homogtemp[j],M[i],limits); |
---|
1867 | } |
---|
1868 | for (j = 1; j<=size(temp); j++) |
---|
1869 | {//filtering invalid entries in temp |
---|
1870 | if(product(temp[j])==h) |
---|
1871 | {//This is already a result |
---|
1872 | result = result + list(temp[j]); |
---|
1873 | temp = delete(temp,j); |
---|
1874 | continue; |
---|
1875 | }//This is already a result |
---|
1876 | if (deg(hath,iv01)<=deg(hath-product(temp[j]),iv01)) |
---|
1877 | { |
---|
1878 | temp = delete(temp,j); |
---|
1879 | continue; |
---|
1880 | } |
---|
1881 | }//filtering invalid entries in temp |
---|
1882 | hatM = hatM + temp; |
---|
1883 | }//iterate over all elements of M |
---|
1884 | M = hatM; |
---|
1885 | for (i = 1; i<=size(M);i++) |
---|
1886 | {//checking for complete factorizations |
---|
1887 | if (h == product(M[i])) |
---|
1888 | { |
---|
1889 | result = result + list(M[i]); |
---|
1890 | dbprint(p,"Result list updated:"); |
---|
1891 | dbprint(p,result); |
---|
1892 | M = delete(M,i); |
---|
1893 | continue; |
---|
1894 | } |
---|
1895 | }//checking for complete factorizations |
---|
1896 | M = delete_dublicates_eval(M); |
---|
1897 | }//work on the elements of M |
---|
1898 | //In the case, that there is none, write a constant factor before the factor of interest. |
---|
1899 | for (i = 1 ; i<=size(result);i++) |
---|
1900 | {//add a constant factor |
---|
1901 | if (deg(result[i][1],iv11)!=0) |
---|
1902 | { |
---|
1903 | result[i] = insert(result[i],1); |
---|
1904 | } |
---|
1905 | }//add a constant factor |
---|
1906 | dbprint(p,"==> Deleting double entries in the resulting list."); |
---|
1907 | result = delete_dublicates_noteval(result); |
---|
1908 | if (size(result)==0) |
---|
1909 | {//only the trivial factorization could be found |
---|
1910 | result = list(list(1,h)); |
---|
1911 | }//only the trivial factorization could be found |
---|
1912 | dbprint(p,"==> done"); |
---|
1913 | return(result); |
---|
1914 | }//proc facshift |
---|
1915 | |
---|
1916 | static proc refineFactList(list L) |
---|
1917 | { |
---|
1918 | // assume: list L is an output of factorization proc |
---|
1919 | // doing: remove doubled entries |
---|
1920 | int s = size(L); int sm; |
---|
1921 | int i,j,k,cnt; |
---|
1922 | list M, U, A, B; |
---|
1923 | A = L; |
---|
1924 | k = 0; |
---|
1925 | cnt = 1; |
---|
1926 | for (i=1; i<=s; i++) |
---|
1927 | { |
---|
1928 | if (size(A[i]) != 0) |
---|
1929 | { |
---|
1930 | M = A[i]; |
---|
1931 | // "probing with"; M; i; |
---|
1932 | B[cnt] = M; cnt++; |
---|
1933 | for (j=i+1; j<=s; j++) |
---|
1934 | { |
---|
1935 | if ( isEqualList(M,A[j]) ) |
---|
1936 | { |
---|
1937 | k++; |
---|
1938 | // U consists of intvecs with equal pairs |
---|
1939 | U[k] = intvec(i,j); |
---|
1940 | A[j] = 0; |
---|
1941 | } |
---|
1942 | } |
---|
1943 | } |
---|
1944 | } |
---|
1945 | kill A,U,M; |
---|
1946 | return(B); |
---|
1947 | } |
---|
1948 | example |
---|
1949 | { |
---|
1950 | "EXAMPLE:";echo=2; |
---|
1951 | ring R = 0,(x,s),dp; |
---|
1952 | def r = nc_algebra(1,1); |
---|
1953 | setring(r); |
---|
1954 | list l,m; |
---|
1955 | l = list(1,s2+1,x,s,x+s); |
---|
1956 | m = l,list(1,s,x,s,x),l; |
---|
1957 | refineFactList(m); |
---|
1958 | } |
---|
1959 | |
---|
1960 | static proc isEqualList(list L, list M) |
---|
1961 | { |
---|
1962 | // int boolean: 1=yes, 0 =no : test whether two lists are identical |
---|
1963 | int s = size(L); |
---|
1964 | if (size(M)!=s) { return(0); } |
---|
1965 | int j=1; |
---|
1966 | while ( (L[j]==M[j]) && (j<s) ) |
---|
1967 | { |
---|
1968 | j++; |
---|
1969 | } |
---|
1970 | if (L[j]==M[j]) |
---|
1971 | { |
---|
1972 | return(1); |
---|
1973 | } |
---|
1974 | return(0); |
---|
1975 | } |
---|
1976 | example |
---|
1977 | { |
---|
1978 | "EXAMPLE:";echo=2; |
---|
1979 | ring R = 0,(x,s),dp; |
---|
1980 | def r = nc_algebra(1,1); |
---|
1981 | setring(r); |
---|
1982 | list l,m; |
---|
1983 | l = list(1,s2+1,x,s,x+s); |
---|
1984 | m = l; |
---|
1985 | isEqualList(m,l); |
---|
1986 | } |
---|
1987 | |
---|
1988 | |
---|
1989 | ////////////////////////////////////////////////// |
---|
1990 | // Q-WEYL-SECTION |
---|
1991 | ////////////////////////////////////////////////// |
---|
1992 | |
---|
1993 | //================================================== |
---|
1994 | //A function to get the i'th triangular number |
---|
1995 | static proc triangNum(int n) |
---|
1996 | { |
---|
1997 | if (n == 0) |
---|
1998 | { |
---|
1999 | return(0); |
---|
2000 | } |
---|
2001 | return (n*(n+1)/2); |
---|
2002 | } |
---|
2003 | |
---|
2004 | //==================================================* |
---|
2005 | //one factorization of a homogeneous polynomial |
---|
2006 | //in the first Q Weyl Algebra |
---|
2007 | proc homogfacFirstQWeyl(poly h) |
---|
2008 | "USAGE: homogfacFirstQWeyl(h); h is a homogeneous polynomial in the |
---|
2009 | first q-Weyl algebra with respect to the weight vector [-1,1] |
---|
2010 | RETURN: list |
---|
2011 | PURPOSE: Computes a factorization of a homogeneous polynomial h with |
---|
2012 | respect to the weight vector [-1,1] in the first q-Weyl algebra |
---|
2013 | THEORY: @code{homogfacFirstQWeyl} returns a list with a factorization of the given, |
---|
2014 | [-1,1]-homogeneous polynomial. If the degree of the polynomial is k with |
---|
2015 | k positive, the last k entries in the output list are the second |
---|
2016 | variable. If k is positive, the last k entries will be x. The other |
---|
2017 | entries will be irreducible polynomials of degree zero or 1 resp. -1. |
---|
2018 | SEE ALSO: homogfacFirstWeyl, homogfacFirstQWeyl_all |
---|
2019 | "{//proc homogfacFirstQWeyl |
---|
2020 | int p = printlevel-voice+2;//for dbprint |
---|
2021 | def r = basering; |
---|
2022 | poly hath; |
---|
2023 | int i; int j; |
---|
2024 | intvec ivm11 = intvec(-1,1); |
---|
2025 | if (!homogwithorder(h,ivm11)) |
---|
2026 | {//The given polynomial is not homogeneous |
---|
2027 | ERROR("Given polynomial was not [-1,1]-homogeneous"); |
---|
2028 | return(list()); |
---|
2029 | }//The given polynomial is not homogeneous |
---|
2030 | if (h==0) |
---|
2031 | { |
---|
2032 | return(list(0)); |
---|
2033 | } |
---|
2034 | list result; |
---|
2035 | int m = deg(h,ivm11); |
---|
2036 | dbprint(p,"==> Splitting the polynomial in A_0 and A_k-Part"); |
---|
2037 | if (m!=0) |
---|
2038 | {//The degree is not zero |
---|
2039 | if (m <0) |
---|
2040 | {//There are more x than y |
---|
2041 | hath = lift(var(1)^(-m),h)[1,1]; |
---|
2042 | for (i = 1; i<=-m; i++) |
---|
2043 | { |
---|
2044 | result = result + list(var(1)); |
---|
2045 | } |
---|
2046 | }//There are more x than y |
---|
2047 | else |
---|
2048 | {//There are more y than x |
---|
2049 | hath = lift(var(2)^m,h)[1,1]; |
---|
2050 | for (i = 1; i<=m;i++) |
---|
2051 | { |
---|
2052 | result = result + list(var(2)); |
---|
2053 | } |
---|
2054 | }//There are more y than x |
---|
2055 | }//The degree is not zero |
---|
2056 | else |
---|
2057 | {//The degree is zero |
---|
2058 | hath = h; |
---|
2059 | }//The degree is zero |
---|
2060 | dbprint(p,"==> Done"); |
---|
2061 | //beginning to transform x^i*y^i in theta(theta-1)...(theta-i+1) |
---|
2062 | list mons; |
---|
2063 | dbprint(p,"==> Putting the monomials in the A_0-part in a list."); |
---|
2064 | for(i = 1; i<=size(hath);i++) |
---|
2065 | {//Putting the monomials in a list |
---|
2066 | mons = mons+list(hath[i]); |
---|
2067 | }//Putting the monomials in a list |
---|
2068 | dbprint(p,"==> Done"); |
---|
2069 | dbprint(p,"==> Mapping this monomials to K(q)[theta]"); |
---|
2070 | //Now, map to the commutative ring with theta: |
---|
2071 | list tempRingList = ringlist(r); |
---|
2072 | tempRingList[2] = insert(tempRingList[2],"theta",2); //New variable theta = x*d |
---|
2073 | tempRingList = delete(tempRingList,5); |
---|
2074 | tempRingList = delete(tempRingList,5); //The ring should now be commutative |
---|
2075 | def tempRing = ring(tempRingList); |
---|
2076 | setring tempRing; |
---|
2077 | map thetamap = r,var(1),var(2); |
---|
2078 | list mons = thetamap(mons); |
---|
2079 | poly entry; |
---|
2080 | poly tempSummand; |
---|
2081 | for (i = 1; i<=size(mons);i++) |
---|
2082 | {//transforming the monomials as monomials in theta |
---|
2083 | entry = 1;//leadcoef(mons[i]) * q^(-triangNum(leadexp(mons[i])[2]-1)); |
---|
2084 | for (j = 0; j<leadexp(mons[i])[2];j++) |
---|
2085 | { |
---|
2086 | tempSummand = (par(1)^j-1)/(par(1)-1); |
---|
2087 | entry = entry * theta-tempSummand*entry; |
---|
2088 | } |
---|
2089 | //entry; |
---|
2090 | //leadcoef(mons[i]) * q^(-triangNum(leadexp(mons[i])[2]-1)); |
---|
2091 | mons[i] = entry*leadcoef(mons[i]) * par(1)^(-triangNum(leadexp(mons[i])[2]-1)); |
---|
2092 | }//transforming the monomials as monomials in theta |
---|
2093 | dbprint(p,"==> Done"); |
---|
2094 | dbprint(p,"==> Factorize the A_0-Part in K[theta]"); |
---|
2095 | list azeroresult = factorize(sum(mons)); |
---|
2096 | dbprint(p,"==> Successful"); |
---|
2097 | list azeroresult_return_form; |
---|
2098 | for (i = 1; i<=size(azeroresult[1]);i++) |
---|
2099 | {//rewrite the result of the commutative factorization |
---|
2100 | for (j = 1; j <= azeroresult[2][i];j++) |
---|
2101 | { |
---|
2102 | azeroresult_return_form = azeroresult_return_form + list(azeroresult[1][i]); |
---|
2103 | } |
---|
2104 | }//rewrite the result of the commutative factorization |
---|
2105 | dbprint(p,"==> Mapping back to A_0."); |
---|
2106 | setring(r); |
---|
2107 | map finalmap = tempRing,var(1),var(2),var(1)*var(2); |
---|
2108 | list tempresult = finalmap(azeroresult_return_form); |
---|
2109 | dbprint(p,"Successful."); |
---|
2110 | for (i = 1; i<=size(tempresult);i++) |
---|
2111 | {//factorizations of theta resp. theta +1 |
---|
2112 | if(tempresult[i]==var(1)*var(2)) |
---|
2113 | { |
---|
2114 | tempresult = insert(tempresult,var(1),i-1); |
---|
2115 | i++; |
---|
2116 | tempresult[i]=var(2); |
---|
2117 | } |
---|
2118 | if(tempresult[i]==var(2)*var(1)) |
---|
2119 | { |
---|
2120 | tempresult = insert(tempresult,var(2),i-1); |
---|
2121 | i++; |
---|
2122 | tempresult[i]=var(1); |
---|
2123 | } |
---|
2124 | }//factorizations of theta resp. theta +1 |
---|
2125 | result = tempresult+result; |
---|
2126 | //Correction of the result in the special q-Case: |
---|
2127 | for (j = 2 ; j<= size(result);j++) |
---|
2128 | {//Div the whole Term by the leading coefficient and multiply it to the first entry in result[i] |
---|
2129 | result[1] = result[1] * leadcoef(result[j]); |
---|
2130 | result[j] = 1/leadcoef(result[j]) * result[j]; |
---|
2131 | }//Div the whole Term by the leading coefficient and multiply it to the first entry in result[i] |
---|
2132 | return(result); |
---|
2133 | }//proc homogfacFirstQWeyl |
---|
2134 | example |
---|
2135 | { |
---|
2136 | "EXAMPLE:";echo=2; |
---|
2137 | ring R = (0,q),(x,d),dp; |
---|
2138 | def r = nc_algebra (q,1); |
---|
2139 | setring(r); |
---|
2140 | poly h = q^25*x^10*d^10+q^16*(q^4+q^3+q^2+q+1)^2*x^9*d^9+ |
---|
2141 | q^9*(q^13+3*q^12+7*q^11+13*q^10+20*q^9+26*q^8+30*q^7+ |
---|
2142 | 31*q^6+26*q^5+20*q^4+13*q^3+7*q^2+3*q+1)*x^8*d^8+ |
---|
2143 | q^4*(q^9+2*q^8+4*q^7+6*q^6+7*q^5+8*q^4+6*q^3+ |
---|
2144 | 4*q^2+2q+1)*(q^4+q^3+q^2+q+1)*(q^2+q+1)*x^7*d^7+ |
---|
2145 | q*(q^2+q+1)*(q^5+2*q^4+2*q^3+3*q^2+2*q+1)*(q^4+q^3+q^2+q+1)*(q^2+1)*(q+1)*x^6*d^6+ |
---|
2146 | (q^10+5*q^9+12*q^8+21*q^7+29*q^6+33*q^5+31*q^4+24*q^3+15*q^2+7*q+12)*x^5*d^5+ |
---|
2147 | 6*x^3*d^3+24; |
---|
2148 | homogfacFirstQWeyl(h); |
---|
2149 | } |
---|
2150 | |
---|
2151 | |
---|
2152 | |
---|
2153 | //================================================== |
---|
2154 | //Computes all possible homogeneous factorizations for an element in the first Q-Weyl Algebra |
---|
2155 | proc homogfacFirstQWeyl_all(poly h) |
---|
2156 | "USAGE: homogfacFirstQWeyl_all(h); h is a homogeneous polynomial in the first q-Weyl algebra |
---|
2157 | with respect to the weight vector [-1,1] |
---|
2158 | RETURN: list |
---|
2159 | PURPOSE: Computes all factorizations of a homogeneous polynomial h with respect |
---|
2160 | to the weight vector [-1,1] in the first q-Weyl algebra |
---|
2161 | THEORY: @code{homogfacFirstQWeyl} returns a list with all factorization of the given, |
---|
2162 | homogeneous polynomial. It uses the output of homogfacFirstQWeyl and permutes |
---|
2163 | its entries with respect to the commutation rule. Furthermore, if a |
---|
2164 | factor of degree zero is irreducible in K[\theta], but reducible in |
---|
2165 | the first q-Weyl algebra, the permutations of this element with the other |
---|
2166 | entries will also be computed. |
---|
2167 | SEE ALSO: homogfacFirstQWeyl |
---|
2168 | "{//proc HomogfacFirstQWeylAll |
---|
2169 | int p=printlevel-voice+2;//for dbprint |
---|
2170 | intvec iv11= intvec(1,1); |
---|
2171 | if (deg(h,iv11) <= 0 ) |
---|
2172 | {//h is a constant |
---|
2173 | dbprint(p,"Given polynomial was not homogeneous"); |
---|
2174 | return(list(list(h))); |
---|
2175 | }//h is a constant |
---|
2176 | def r = basering; |
---|
2177 | list one_hom_fac; //stands for one homogeneous factorization |
---|
2178 | int i; int j; int k; |
---|
2179 | intvec ivm11 = intvec(-1,1); |
---|
2180 | dbprint(p,"==> Calculate one homogeneous factorization using homogfacFirstQWeyl"); |
---|
2181 | //Compute again a homogeneous factorization |
---|
2182 | one_hom_fac = homogfacFirstQWeyl(h); |
---|
2183 | dbprint(p,"Successful"); |
---|
2184 | if (size(one_hom_fac) == 0) |
---|
2185 | {//there is no homogeneous factorization or the polynomial was not homogeneous |
---|
2186 | return(list()); |
---|
2187 | }//there is no homogeneous factorization or the polynomial was not homogeneous |
---|
2188 | //divide list in A0-Part and a list of x's resp. y's |
---|
2189 | list list_not_azero = list(); |
---|
2190 | list list_azero; |
---|
2191 | list k_factor; |
---|
2192 | int is_list_not_azero_empty = 1; |
---|
2193 | int is_list_azero_empty = 1; |
---|
2194 | k_factor = list(one_hom_fac[1]); |
---|
2195 | if (absValue(deg(h,ivm11))<size(one_hom_fac)-1) |
---|
2196 | {//There is a nontrivial A_0-part |
---|
2197 | list_azero = one_hom_fac[2..(size(one_hom_fac)-absValue(deg(h,ivm11)))]; |
---|
2198 | is_list_azero_empty = 0; |
---|
2199 | }//There is a nontrivial A_0 part |
---|
2200 | dbprint(p,"==> Combine x,y to xy in the factorization again."); |
---|
2201 | for (i = 1; i<=size(list_azero)-1;i++) |
---|
2202 | {//in homogfacFirstQWeyl, we factorized theta, and this will be made undone |
---|
2203 | if (list_azero[i] == var(1)) |
---|
2204 | { |
---|
2205 | if (list_azero[i+1]==var(2)) |
---|
2206 | { |
---|
2207 | list_azero[i] = var(1)*var(2); |
---|
2208 | list_azero = delete(list_azero,i+1); |
---|
2209 | } |
---|
2210 | } |
---|
2211 | if (list_azero[i] == var(2)) |
---|
2212 | { |
---|
2213 | if (list_azero[i+1]==var(1)) |
---|
2214 | { |
---|
2215 | list_azero[i] = var(2)*var(1); |
---|
2216 | list_azero = delete(list_azero,i+1); |
---|
2217 | } |
---|
2218 | } |
---|
2219 | }//in homogfacFirstQWeyl, we factorized theta, and this will be made undone |
---|
2220 | dbprint(p,"==> Done"); |
---|
2221 | if(deg(h,ivm11)!=0) |
---|
2222 | {//list_not_azero is not empty |
---|
2223 | list_not_azero = |
---|
2224 | one_hom_fac[(size(one_hom_fac)-absValue(deg(h,ivm11))+1)..size(one_hom_fac)]; |
---|
2225 | is_list_not_azero_empty = 0; |
---|
2226 | }//list_not_azero is not empty |
---|
2227 | //Map list_azero in K[theta] |
---|
2228 | dbprint(p,"==> Map list_azero to K[theta]"); |
---|
2229 | //Now, map to the commutative ring with theta: |
---|
2230 | list tempRingList = ringlist(r); |
---|
2231 | tempRingList[2] = insert(tempRingList[2],"theta",2); //New variable theta = x*d |
---|
2232 | tempRingList = delete(tempRingList,5); |
---|
2233 | tempRingList = delete(tempRingList,5); //The ring should now be commutative |
---|
2234 | def tempRing = ring(tempRingList); |
---|
2235 | setring(tempRing); |
---|
2236 | poly entry; |
---|
2237 | map thetamap = r,var(1),var(2); |
---|
2238 | if(!is_list_not_azero_empty) |
---|
2239 | {//Mapping in Singular is only possible, if the list before |
---|
2240 | //contained at least one element of the other ring |
---|
2241 | list list_not_azero = thetamap(list_not_azero); |
---|
2242 | }//Mapping in Singular is only possible, if the list before |
---|
2243 | //contained at least one element of the other ring |
---|
2244 | if(!is_list_azero_empty) |
---|
2245 | {//Mapping in Singular is only possible, if the list before |
---|
2246 | //contained at least one element of the other ring |
---|
2247 | list list_azero= thetamap(list_azero); |
---|
2248 | }//Mapping in Singular is only possible, if the list before |
---|
2249 | //contained at least one element of the other ring |
---|
2250 | list k_factor = thetamap(k_factor); |
---|
2251 | list tempmons; |
---|
2252 | dbprint(p,"==> Done"); |
---|
2253 | for(i = 1; i<=size(list_azero);i++) |
---|
2254 | {//rewrite the polynomials in A1 as polynomials in K[theta] |
---|
2255 | tempmons = list(); |
---|
2256 | for (j = 1; j<=size(list_azero[i]);j++) |
---|
2257 | { |
---|
2258 | tempmons = tempmons + list(list_azero[i][j]); |
---|
2259 | } |
---|
2260 | for (j = 1 ; j<=size(tempmons);j++) |
---|
2261 | { |
---|
2262 | //entry = leadcoef(tempmons[j]); |
---|
2263 | entry = leadcoef(tempmons[j]) * par(1)^(-triangNum(leadexp(tempmons[j])[2]-1)); |
---|
2264 | for (k = 0; k < leadexp(tempmons[j])[2];k++) |
---|
2265 | { |
---|
2266 | entry = entry*(theta-(par(1)^k-1)/(par(1)-1)); |
---|
2267 | } |
---|
2268 | tempmons[j] = entry; |
---|
2269 | } |
---|
2270 | list_azero[i] = sum(tempmons); |
---|
2271 | }//rewrite the polynomials in A1 as polynomials in K[theta] |
---|
2272 | //Compute all permutations of the A0-part |
---|
2273 | dbprint(p,"==> Compute all permutations of the A_0-part with the first resp. the snd. variable"); |
---|
2274 | list result; |
---|
2275 | int shift_sign; |
---|
2276 | int shift; |
---|
2277 | poly shiftvar; |
---|
2278 | if (size(list_not_azero)!=0) |
---|
2279 | {//Compute all possibilities to permute the x's resp. the y's in the list |
---|
2280 | if (list_not_azero[1] == x) |
---|
2281 | {//h had a negative weighted degree |
---|
2282 | shift_sign = 1; |
---|
2283 | shiftvar = x; |
---|
2284 | }//h had a negative weighted degree |
---|
2285 | else |
---|
2286 | {//h had a positive weighted degree |
---|
2287 | shift_sign = -1; |
---|
2288 | shiftvar = y; |
---|
2289 | }//h had a positive weighted degree |
---|
2290 | result = permpp(list_azero + list_not_azero); |
---|
2291 | for (i = 1; i<= size(result); i++) |
---|
2292 | {//adjust the a_0-parts |
---|
2293 | shift = 0; |
---|
2294 | for (j=1; j<=size(result[i]);j++) |
---|
2295 | { |
---|
2296 | if (result[i][j]==shiftvar) |
---|
2297 | { |
---|
2298 | shift = shift + shift_sign; |
---|
2299 | } |
---|
2300 | else |
---|
2301 | { |
---|
2302 | if (shift < 0) |
---|
2303 | {//We have two distict formulas for x and y. In this case use formula for y |
---|
2304 | if (shift == -1) |
---|
2305 | { |
---|
2306 | result[i][j] = subst(result[i][j],theta,1/par(1)*(theta - 1)); |
---|
2307 | } |
---|
2308 | else |
---|
2309 | { |
---|
2310 | result[i][j] = |
---|
2311 | subst(result[i][j], |
---|
2312 | theta, |
---|
2313 | 1/par(1)*((theta - 1)/par(1)^(absValue(shift)-1) |
---|
2314 | - (par(1)^(shift +2)-par(1))/(1-par(1)))); |
---|
2315 | } |
---|
2316 | }//We have two distict formulas for x and y. In this case use formula for y |
---|
2317 | if (shift > 0) |
---|
2318 | {//We have two distict formulas for x and y. In this case use formula for x |
---|
2319 | if (shift == 1) |
---|
2320 | { |
---|
2321 | result[i][j] = subst(result[i][j],theta,par(1)*theta + 1); |
---|
2322 | } |
---|
2323 | else |
---|
2324 | { |
---|
2325 | result[i][j] = |
---|
2326 | subst(result[i][j], |
---|
2327 | theta,par(1)^shift*theta+(par(1)^shift-1)/(par(1)-1)); |
---|
2328 | } |
---|
2329 | }//We have two distict formulas for x and y. In this case use formula for x |
---|
2330 | } |
---|
2331 | } |
---|
2332 | }//adjust the a_0-parts |
---|
2333 | }//Compute all possibilities to permute the x's resp. the y's in the list |
---|
2334 | else |
---|
2335 | {//The result is just all the permutations of the a_0-part |
---|
2336 | result = permpp(list_azero); |
---|
2337 | }//The result is just all the permutations of the a_0 part |
---|
2338 | if (size(result)==0) |
---|
2339 | { |
---|
2340 | return(result); |
---|
2341 | } |
---|
2342 | dbprint(p,"==> Done"); |
---|
2343 | dbprint(p,"==> Searching for theta resp. theta + 1 in the list and factorize them"); |
---|
2344 | //Now we are going deeper and search for theta resp. theta + 1, substitute |
---|
2345 | //them by xy resp. yx and go on permuting |
---|
2346 | int found_theta; |
---|
2347 | int thetapos; |
---|
2348 | list leftpart; |
---|
2349 | list rightpart; |
---|
2350 | list lparts; |
---|
2351 | list rparts; |
---|
2352 | list tempadd; |
---|
2353 | for (i = 1; i<=size(result) ; i++) |
---|
2354 | {//checking every entry of result for theta or theta +1 |
---|
2355 | found_theta = 0; |
---|
2356 | for(j=1;j<=size(result[i]);j++) |
---|
2357 | { |
---|
2358 | if (result[i][j]==theta) |
---|
2359 | {//the jth entry is theta and can be written as x*y |
---|
2360 | thetapos = j; |
---|
2361 | result[i]= insert(result[i],x,j-1); |
---|
2362 | j++; |
---|
2363 | result[i][j] = y; |
---|
2364 | found_theta = 1; |
---|
2365 | break; |
---|
2366 | }//the jth entry is theta and can be written as x*y |
---|
2367 | if(result[i][j] == par(1)*theta +1) |
---|
2368 | { |
---|
2369 | thetapos = j; |
---|
2370 | result[i] = insert(result[i],y,j-1); |
---|
2371 | j++; |
---|
2372 | result[i][j] = x; |
---|
2373 | found_theta = 1; |
---|
2374 | break; |
---|
2375 | } |
---|
2376 | } |
---|
2377 | if (found_theta) |
---|
2378 | {//One entry was theta resp. theta +1 |
---|
2379 | leftpart = result[i]; |
---|
2380 | leftpart = leftpart[1..thetapos]; |
---|
2381 | rightpart = result[i]; |
---|
2382 | rightpart = rightpart[(thetapos+1)..size(rightpart)]; |
---|
2383 | lparts = list(leftpart); |
---|
2384 | rparts = list(rightpart); |
---|
2385 | //first deal with the left part |
---|
2386 | if (leftpart[thetapos] == x) |
---|
2387 | { |
---|
2388 | shift_sign = 1; |
---|
2389 | shiftvar = x; |
---|
2390 | } |
---|
2391 | else |
---|
2392 | { |
---|
2393 | shift_sign = -1; |
---|
2394 | shiftvar = y; |
---|
2395 | } |
---|
2396 | for (j = size(leftpart); j>1;j--) |
---|
2397 | {//drip x resp. y |
---|
2398 | if (leftpart[j-1]==shiftvar) |
---|
2399 | {//commutative |
---|
2400 | j--; |
---|
2401 | continue; |
---|
2402 | }//commutative |
---|
2403 | if (deg(leftpart[j-1],intvec(-1,1,0))!=0) |
---|
2404 | {//stop here |
---|
2405 | break; |
---|
2406 | }//stop here |
---|
2407 | //Here, we can only have a a0- part |
---|
2408 | if (shift_sign<0) |
---|
2409 | { |
---|
2410 | leftpart[j] = subst(leftpart[j-1],theta, 1/par(1)*(theta +shift_sign)); |
---|
2411 | } |
---|
2412 | if (shift_sign>0) |
---|
2413 | { |
---|
2414 | leftpart[j] = subst(leftpart[j-1],theta, par(1)*theta + shift_sign); |
---|
2415 | } |
---|
2416 | leftpart[j-1] = shiftvar; |
---|
2417 | lparts = lparts + list(leftpart); |
---|
2418 | }//drip x resp. y |
---|
2419 | //and now deal with the right part |
---|
2420 | if (rightpart[1] == x) |
---|
2421 | { |
---|
2422 | shift_sign = 1; |
---|
2423 | shiftvar = x; |
---|
2424 | } |
---|
2425 | else |
---|
2426 | { |
---|
2427 | shift_sign = -1; |
---|
2428 | shiftvar = y; |
---|
2429 | } |
---|
2430 | for (j = 1 ; j < size(rightpart); j++) |
---|
2431 | { |
---|
2432 | if (rightpart[j+1] == shiftvar) |
---|
2433 | { |
---|
2434 | j++; |
---|
2435 | continue; |
---|
2436 | } |
---|
2437 | if (deg(rightpart[j+1],intvec(-1,1,0))!=0) |
---|
2438 | { |
---|
2439 | break; |
---|
2440 | } |
---|
2441 | if (shift_sign<0) |
---|
2442 | { |
---|
2443 | rightpart[j] = subst(rightpart[j+1], theta, par(1)*theta - shift_sign); |
---|
2444 | } |
---|
2445 | if (shift_sign>0) |
---|
2446 | { |
---|
2447 | rightpart[j] = subst(rightpart[j+1], theta, 1/par(1)*(theta - shift_sign)); |
---|
2448 | } |
---|
2449 | rightpart[j+1] = shiftvar; |
---|
2450 | rparts = rparts + list(rightpart); |
---|
2451 | } |
---|
2452 | //And now, we put all possibilities together |
---|
2453 | tempadd = list(); |
---|
2454 | for (j = 1; j<=size(lparts); j++) |
---|
2455 | { |
---|
2456 | for (k = 1; k<=size(rparts);k++) |
---|
2457 | { |
---|
2458 | tempadd = tempadd + list(lparts[j]+rparts[k]); |
---|
2459 | } |
---|
2460 | } |
---|
2461 | tempadd = delete(tempadd,1); // The first entry is already in the list |
---|
2462 | result = result + tempadd; |
---|
2463 | continue; //We can may be not be done already with the ith entry |
---|
2464 | }//One entry was theta resp. theta +1 |
---|
2465 | }//checking every entry of result for theta or theta +1 |
---|
2466 | dbprint(p,"==> Done"); |
---|
2467 | //map back to the basering |
---|
2468 | dbprint(p,"==> Mapping back everything to the basering"); |
---|
2469 | setring(r); |
---|
2470 | map finalmap = tempRing, var(1), var(2),var(1)*var(2); |
---|
2471 | list result = finalmap(result); |
---|
2472 | for (i=1; i<=size(result);i++) |
---|
2473 | {//adding the K factor |
---|
2474 | result[i] = k_factor + result[i]; |
---|
2475 | }//adding the k-factor |
---|
2476 | dbprint(p,"==> Done"); |
---|
2477 | dbprint(p,"==> Delete double entries in the list."); |
---|
2478 | result = delete_dublicates_noteval(result); |
---|
2479 | dbprint(p,"==> Done"); |
---|
2480 | return(result); |
---|
2481 | }//proc HomogfacFirstQWeylAll |
---|
2482 | example |
---|
2483 | { |
---|
2484 | "EXAMPLE:";echo=2; |
---|
2485 | ring R = (0,q),(x,d),dp; |
---|
2486 | def r = nc_algebra (q,1); |
---|
2487 | setring(r); |
---|
2488 | poly h = q^25*x^10*d^10+q^16*(q^4+q^3+q^2+q+1)^2*x^9*d^9+ |
---|
2489 | q^9*(q^13+3*q^12+7*q^11+13*q^10+20*q^9+26*q^8+30*q^7+ |
---|
2490 | 31*q^6+26*q^5+20*q^4+13*q^3+7*q^2+3*q+1)*x^8*d^8+ |
---|
2491 | q^4*(q^9+2*q^8+4*q^7+6*q^6+7*q^5+8*q^4+6*q^3+ |
---|
2492 | 4*q^2+2q+1)*(q^4+q^3+q^2+q+1)*(q^2+q+1)*x^7*d^7+ |
---|
2493 | q*(q^2+q+1)*(q^5+2*q^4+2*q^3+3*q^2+2*q+1)*(q^4+q^3+q^2+q+1)*(q^2+1)*(q+1)*x^6*d^6+ |
---|
2494 | (q^10+5*q^9+12*q^8+21*q^7+29*q^6+33*q^5+31*q^4+24*q^3+15*q^2+7*q+12)*x^5*d^5+ |
---|
2495 | 6*x^3*d^3+24; |
---|
2496 | homogfacFirstQWeyl_all(h); |
---|
2497 | } |
---|
2498 | |
---|
2499 | //TODO: FirstQWeyl check the parameters... |
---|
2500 | |
---|
2501 | /* |
---|
2502 | Example polynomials where one can find factorizations: K<x,y |yx=xy+1> |
---|
2503 | (x^2+y)*(x^2+y); |
---|
2504 | (x^2+x)*(x^2+y); |
---|
2505 | (x^3+x+1)*(x^4+y*x+2); |
---|
2506 | (x^2*y+y)*(y+x*y); |
---|
2507 | y^3+x*y^3+2*y^2+2*(x+1)*y^2+y+(x+2)*y; //Example 5 Grigoriev-Schwarz. |
---|
2508 | (y+1)*(y+1)*(y+x*y); //Landau Example projected to the first dimension. |
---|
2509 | */ |
---|
2510 | |
---|
2511 | |
---|
2512 | /* very hard things from Martin Lee: |
---|
2513 | // ex1, ex2 |
---|
2514 | ring s = 0,(z,x),Ws(-1,1); |
---|
2515 | def S = nc_algebra(1,1); setring S; |
---|
2516 | poly a = 10z5x4+26z4x5+47z5x2-97z4x3; //Abgebrochen nach einer Stunde; yes, it takes long |
---|
2517 | def l= facFirstWeyl (a); l; |
---|
2518 | kill l; |
---|
2519 | poly b = -5328z8x5-5328z7x6+720z9x2+720z8x3-16976z7x4-38880z6x5 |
---|
2520 | -5184z7x3-5184z6x4-3774z5x5+2080z8x+5760z7x2-6144z6x3-59616z5x4 |
---|
2521 | +3108z3x6-4098z6x2-25704z5x3-21186z4x4+8640z6x-17916z4x3+22680z2x5 |
---|
2522 | +2040z5x-4848z4x2-9792z3x3+3024z2x4-10704z3x2-3519z2x3+34776zx4 |
---|
2523 | +12096zx3+2898x4-5040z2x+8064x3+6048x2; //Abgebrochen nach 1.5 Stunden;seems to be very complicated |
---|
2524 | def l= facFirstWeyl (b); l; |
---|
2525 | |
---|
2526 | // ex3: there was difference in answers => fixed |
---|
2527 | LIB "ncfactor.lib"; |
---|
2528 | ring r = 0,(x,y,z),dp; |
---|
2529 | matrix D[3][3]; D[1,3]=-1; |
---|
2530 | def R = nc_algebra(1,D); |
---|
2531 | setring R; |
---|
2532 | poly g= 7*z4*x+62*z3+26*z; |
---|
2533 | def l1= facSubWeyl (g, x, z); |
---|
2534 | l1; |
---|
2535 | //---- other ring |
---|
2536 | ring s = 0,(x,z),dp; |
---|
2537 | def S = nc_algebra(1,-1); setring S; |
---|
2538 | poly g= 7*z4*x+62*z3+26*z; |
---|
2539 | def l2= facFirstWeyl (g); |
---|
2540 | l2; |
---|
2541 | map F = R,x,0,z; |
---|
2542 | list l1 = F(l1); |
---|
2543 | l1; |
---|
2544 | //---- so the answers look different, check them! |
---|
2545 | testNCfac(l2); // ok |
---|
2546 | testNCfac(l1); // was not ok, but now it's been fixed!!! |
---|
2547 | |
---|
2548 | // selbst D und X so vertauschen dass sie erfuellt ist : ist gemacht |
---|
2549 | |
---|
2550 | */ |
---|
2551 | |
---|
2552 | /* |
---|
2553 | // bug from M Lee |
---|
2554 | LIB "ncfactor.lib"; |
---|
2555 | ring s = 0,(z,x),dp; |
---|
2556 | def S = nc_algebra(1,1); setring S; |
---|
2557 | poly f= -60z4x2-54z4-56zx3-59z2x-64; |
---|
2558 | def l= facFirstWeyl (f); |
---|
2559 | l; // before: empty list; after fix: 1 entry, f is irreducible |
---|
2560 | poly g = 75z3x2+92z3+24; |
---|
2561 | def l= facFirstWeyl (g); |
---|
2562 | l; //before: empty list, now: correct |
---|
2563 | */ |
---|
2564 | |
---|
2565 | /* more things from Martin Lee; fixed |
---|
2566 | ring R = 0,(x,s),dp; |
---|
2567 | def r = nc_algebra(1,s); |
---|
2568 | setring(r); |
---|
2569 | poly h = (s2*x+x)*s; |
---|
2570 | h= h* (x+s); |
---|
2571 | def l= facFirstShift(h); |
---|
2572 | l; // contained doubled entries: not anymore, fixed! |
---|
2573 | |
---|
2574 | ring R = 0,(x,s),dp; |
---|
2575 | def r = nc_algebra(1,-1); |
---|
2576 | setring(r); |
---|
2577 | poly h = (s2*x+x)*s; |
---|
2578 | h= h* (x+s); |
---|
2579 | def l= facFirstWeyl(h); |
---|
2580 | l; // contained doubled entries: not anymore, fixed! |
---|
2581 | |
---|
2582 | */ |
---|