1 | ////////////////////////////////////////////////////////// |
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2 | version="version ncfactor.lib 4.0.0.0 Jun_2013 "; // $Id$ |
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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, aheinle@uwaterloo.ca |
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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 | tst_ncfactor(); Runs the examples of all contained not static functions. Test thing. |
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29 | "; |
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30 | |
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31 | LIB "general.lib"; |
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32 | LIB "nctools.lib"; |
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33 | LIB "involut.lib"; |
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34 | LIB "freegb.lib"; // for isVar |
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35 | LIB "crypto.lib"; //for introot |
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36 | LIB "matrix.lib"; //for submatrix |
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37 | LIB "solve.lib"; //right now not needed |
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38 | LIB "poly.lib"; //for content |
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39 | |
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40 | proc tst_ncfactor() |
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41 | " |
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42 | A little test if the library works correct. |
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43 | Runs simply all examples of non-static functions. |
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44 | " |
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45 | { |
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46 | example facFirstWeyl; |
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47 | example facFirstShift; |
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48 | example facSubWeyl; |
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49 | example testNCfac; |
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50 | example homogfacFirstQWeyl; |
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51 | example homogfacFirstQWeyl_all; |
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52 | } |
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53 | example |
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54 | { |
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55 | "EXAMPLE:";echo=2; |
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56 | tst_ncfactor(); |
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57 | } |
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58 | |
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59 | ///////////////////////////////////////////////////// |
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60 | //==================================================* |
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61 | //deletes double-entries in a list of factorization |
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62 | //without evaluating the product. |
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63 | static proc delete_dublicates_noteval(list l) |
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64 | " |
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65 | INPUT: A list of lists; Output same as e.g. FacFirstWeyl. Containing different factorizations |
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66 | of a polynomial |
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67 | OUTPUT: If there are dublicates in this list, this procedure deletes them and returns the list |
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68 | without double entries |
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69 | " |
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70 | {//proc delete_dublicates_noteval |
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71 | list result= l; |
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72 | int j; int k; int i; |
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73 | int deleted = 0; |
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74 | int is_equal; |
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75 | for (i = 1; i<= size(l); i++) |
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76 | {//Iterate over the different factorizations |
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77 | for (j = i+1; j<= size(l); j++) |
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78 | {//Compare the i'th factorization to the j'th |
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79 | if (size(l[i])!= size(l[j])) |
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80 | {//different sizes => not equal |
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81 | j++; |
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82 | continue; |
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83 | }//different sizes => not equal |
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84 | is_equal = 1; |
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85 | for (k = 1; k <= size(l[i]);k++) |
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86 | {//Compare every entry |
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87 | if (l[i][k]!=l[j][k]) |
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88 | { |
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89 | is_equal = 0; |
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90 | break; |
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91 | } |
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92 | }//Compare every entry |
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93 | if (is_equal == 1) |
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94 | {//Delete this entry, because there is another equal one int the list |
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95 | result = delete(result, i-deleted); |
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96 | deleted = deleted+1; |
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97 | break; |
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98 | }//Delete this entry, because there is another equal one int the list |
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99 | }//Compare the i'th factorization to the j'th |
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100 | }//Iterate over the different factorizations |
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101 | return(result); |
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102 | }//proc delete_dublicates_noteval |
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103 | |
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104 | //================================================== |
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105 | //deletes the double-entries in a list with |
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106 | //evaluating the products |
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107 | static proc delete_dublicates_eval(list l) |
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108 | " |
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109 | DEPRECATED |
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110 | " |
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111 | {//proc delete_dublicates_eval |
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112 | list result=l; |
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113 | int j; int k; int i; |
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114 | int deleted = 0; |
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115 | int is_equal; |
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116 | for (i = 1; i<= size(result); i++) |
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117 | {//Iterating over all elements in result |
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118 | for (j = i+1; j<= size(result); j++) |
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119 | {//comparing with the other elements |
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120 | if (product(result[i]) == product(result[j])) |
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121 | {//There are two equal results; throw away that one with the smaller size |
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122 | if (size(result[i])>=size(result[j])) |
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123 | {//result[i] has more entries |
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124 | result = delete(result,j); |
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125 | continue; |
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126 | }//result[i] has more entries |
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127 | else |
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128 | {//result[j] has more entries |
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129 | result = delete(result,i); |
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130 | i--; |
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131 | break; |
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132 | }//result[j] has more entries |
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133 | }//There are two equal results; throw away that one with the smaller size |
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134 | }//comparing with the other elements |
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135 | }//Iterating over all elements in result |
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136 | return(result); |
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137 | }//proc delete_dublicates_eval |
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138 | |
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139 | |
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140 | //==================================================* |
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141 | |
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142 | static proc combinekfinlf(list g, int nof) //nof stands for "number of factors" |
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143 | " |
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144 | given a list of factors g and a desired size nof, this |
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145 | procedure combines the factors, such that we recieve a |
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146 | list of the length nof. |
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147 | INPUT: A list of containing polynomials or any type where the *-operator is existent |
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148 | OUTPUT: All possibilities (without permutation of the given list) to combine the polynomials |
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149 | into nof polynomials given by the user. |
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150 | " |
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151 | {//Procedure combinekfinlf |
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152 | list result; |
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153 | int i; int j; int k; //iteration variables |
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154 | list fc; //fc stands for "factors combined" |
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155 | list temp; //a temporary store for factors |
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156 | def nofgl = size(g); //nofgl stands for "number of factors of the given list" |
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157 | if (nofgl == 0) |
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158 | {//g was the empty list |
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159 | return(result); |
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160 | }//g was the empty list |
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161 | if (nof <= 0) |
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162 | {//The user wants to recieve a negative number or no element as a result |
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163 | return(result); |
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164 | }//The user wants to recieve a negative number or no element as a result |
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165 | if (nofgl == nof) |
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166 | {//There are no factors to combine |
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167 | result = result + list(g); |
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168 | return(result); |
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169 | }//There are no factors to combine |
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170 | if (nof == 1) |
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171 | {//User wants to get just one factor |
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172 | result = result + list(list(product(g))); |
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173 | return(result); |
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174 | }//User wants to get just one factor |
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175 | for (i = nof; i > 1; i--) |
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176 | {//computing the possibilities that have at least one original factor from g |
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177 | for (j = i; j>=1; j--) |
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178 | {//shifting the window of combinable factors to the left |
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179 | //fc below stands for "factors combined" |
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180 | fc = combinekfinlf(list(g[(j)..(j+nofgl - i)]),nof - i + 1); |
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181 | for (k = 1; k<=size(fc); k++) |
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182 | {//iterating over the different solutions of the smaller problem |
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183 | if (j>1) |
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184 | {//There are g_i before the combination |
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185 | if (j+nofgl -i < nofgl) |
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186 | {//There are g_i after the combination |
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187 | temp = list(g[1..(j-1)]) + fc[k] + list(g[(j+nofgl-i+1)..nofgl]); |
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188 | }//There are g_i after the combination |
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189 | else |
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190 | {//There are no g_i after the combination |
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191 | temp = list(g[1..(j-1)]) + fc[k]; |
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192 | }//There are no g_i after the combination |
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193 | }//There are g_i before the combination |
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194 | if (j==1) |
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195 | {//There are no g_i before the combination |
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196 | if (j+ nofgl -i <nofgl) |
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197 | {//There are g_i after the combination |
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198 | temp = fc[k]+ list(g[(j + nofgl - i +1)..nofgl]); |
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199 | }//There are g_i after the combination |
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200 | }//There are no g_i before the combination |
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201 | result = result + list(temp); |
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202 | }//iterating over the different solutions of the smaller problem |
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203 | }//shifting the window of combinable factors to the left |
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204 | }//computing the possibilities that have at least one original factor from g |
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205 | for (i = 2; i<=nofgl div nof;i++) |
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206 | {//getting the other possible results |
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207 | result = result + combinekfinlf(list(product(list(g[1..i])))+list(g[(i+1)..nofgl]),nof); |
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208 | }//getting the other possible results |
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209 | result = delete_dublicates_noteval(result); |
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210 | return(result); |
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211 | }//Procedure combinekfinlf |
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212 | |
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213 | |
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214 | //==================================================* |
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215 | //merges two sets of factors ignoring common |
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216 | //factors |
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217 | static proc merge_icf(list l1, list l2, intvec limits) |
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218 | " |
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219 | DEPRECATED |
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220 | " |
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221 | {//proc merge_icf |
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222 | list g; |
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223 | list f; |
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224 | int i; int j; |
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225 | if (size(l1)==0) |
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226 | { |
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227 | return(list()); |
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228 | } |
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229 | if (size(l2)==0) |
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230 | { |
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231 | return(list()); |
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232 | } |
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233 | if (size(l2)<=size(l1)) |
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234 | {//l1 will be our g, l2 our f |
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235 | g = l1; |
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236 | f = l2; |
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237 | }//l1 will be our g, l2 our f |
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238 | else |
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239 | {//l1 will be our f, l2 our g |
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240 | g = l2; |
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241 | f = l1; |
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242 | }//l1 will be our f, l2 our g |
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243 | def result = combinekfinlf(g,size(f),limits); |
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244 | for (i = 1 ; i<= size(result); i++) |
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245 | {//Adding the factors of f to every possibility listed in temp |
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246 | for (j = 1; j<= size(f); j++) |
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247 | { |
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248 | result[i][j] = result[i][j]+f[j]; |
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249 | } |
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250 | if(!limitcheck(result[i],limits)) |
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251 | { |
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252 | result = delete(result,i); |
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253 | continue; |
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254 | } |
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255 | for (j = 1; j<=size(f);j++) |
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256 | {//Delete entry if there is a zero or an integer as a factor |
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257 | if (deg(result[i][j]) <= 0) |
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258 | {//found one |
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259 | result = delete(result,i); |
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260 | i--; |
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261 | break; |
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262 | }//found one |
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263 | }//Delete entry if there is a zero as factor |
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264 | }//Adding the factors of f to every possibility listed in temp |
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265 | return(result); |
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266 | }//proc merge_icf |
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267 | |
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268 | //==================================================* |
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269 | //merges two sets of factors with respect to the occurrence |
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270 | //of common factors |
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271 | static proc merge_cf(list l1, list l2, intvec limits) |
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272 | " |
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273 | DEPRECATED |
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274 | " |
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275 | {//proc merge_cf |
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276 | list g; |
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277 | list f; |
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278 | int i; int j; |
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279 | list pre; |
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280 | list post; |
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281 | list candidate; |
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282 | list temp; |
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283 | int temppos; |
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284 | if (size(l1)==0) |
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285 | {//the first list is empty |
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286 | return(list()); |
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287 | }//the first list is empty |
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288 | if(size(l2)==0) |
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289 | {//the second list is empty |
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290 | return(list()); |
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291 | }//the second list is empty |
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292 | if (size(l2)<=size(l1)) |
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293 | {//l1 will be our g, l2 our f |
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294 | g = l1; |
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295 | f = l2; |
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296 | }//l1 will be our g, l2 our f |
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297 | else |
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298 | {//l1 will be our f, l2 our g |
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299 | g = l2; |
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300 | f = l1; |
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301 | }//l1 will be our f, l2 our g |
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302 | list M; |
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303 | for (i = 2; i<size(f); i++) |
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304 | {//finding common factors of f and g... |
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305 | for (j=2; j<size(g);j++) |
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306 | {//... with g |
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307 | if (f[i] == g[j]) |
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308 | {//we have an equal pair |
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309 | M = M + list(list(i,j)); |
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310 | }//we have an equal pair |
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311 | }//... with g |
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312 | }//finding common factors of f and g... |
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313 | if (g[1]==f[1]) |
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314 | {//Checking for the first elements to be equal |
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315 | M = M + list(list(1,1)); |
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316 | }//Checking for the first elements to be equal |
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317 | if (g[size(g)]==f[size(f)]) |
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318 | {//Checking for the last elements to be equal |
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319 | M = M + list(list(size(f),size(g))); |
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320 | }//Checking for the last elements to be equal |
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321 | list result;//= list(list()); |
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322 | while(size(M)>0) |
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323 | {//set of equal pairs is not empty |
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324 | temp = M[1]; |
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325 | temppos = 1; |
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326 | for (i = 2; i<=size(M); i++) |
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327 | {//finding the minimal element of M |
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328 | if (M[i][1]<=temp[1]) |
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329 | {//a possible candidate that is smaller than temp could have been found |
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330 | if (M[i][1]==temp[1]) |
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331 | {//In this case we must look at the second number |
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332 | if (M[i][2]< temp[2]) |
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333 | {//the candidate is smaller |
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334 | temp = M[i]; |
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335 | temppos = i; |
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336 | }//the candidate is smaller |
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337 | }//In this case we must look at the second number |
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338 | else |
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339 | {//The candidate is definately smaller |
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340 | temp = M[i]; |
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341 | temppos = i; |
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342 | }//The candidate is definately smaller |
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343 | }//a possible candidate that is smaller than temp could have been found |
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344 | }//finding the minimal element of M |
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345 | M = delete(M, temppos); |
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346 | if(temp[1]>1) |
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347 | {//There are factors to combine before the equal factor |
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348 | if (temp[1]<size(f)) |
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349 | {//The most common case |
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350 | //first the combinations ignoring common factors |
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351 | pre = merge_icf(list(f[1..(temp[1]-1)]),list(g[1..(temp[2]-1)]),limits); |
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352 | post = merge_icf(list(f[(temp[1]+1)..size(f)]),list(g[(temp[2]+1..size(g))]),limits); |
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353 | for (i = 1; i <= size(pre); i++) |
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354 | {//all possible pre's... |
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355 | for (j = 1; j<= size(post); j++) |
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356 | {//...combined with all possible post's |
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357 | candidate = pre[i]+list(f[temp[1]])+post[j]; |
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358 | if (limitcheck(candidate,limits)) |
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359 | { |
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360 | result = result + list(candidate); |
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361 | } |
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362 | }//...combined with all possible post's |
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363 | }//all possible pre's... |
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364 | //Now the combinations with respect to common factors |
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365 | post = merge_cf(list(f[(temp[1]+1)..size(f)]),list(g[(temp[2]+1..size(g))]),limits); |
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366 | if (size(post)>0) |
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367 | {//There are factors to combine |
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368 | for (i = 1; i <= size(pre); i++) |
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369 | {//all possible pre's... |
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370 | for (j = 1; j<= size(post); j++) |
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371 | {//...combined with all possible post's |
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372 | candidate= pre[i]+list(f[temp[1]])+post[j]; |
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373 | if (limitcheck(candidate,limits)) |
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374 | { |
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375 | result = result + list(candidate); |
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376 | } |
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377 | }//...combined with all possible post's |
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378 | }//all possible pre's... |
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379 | }//There are factors to combine |
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380 | }//The most common case |
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381 | else |
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382 | {//the last factor is the common one |
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383 | pre = merge_icf(list(f[1..(temp[1]-1)]),list(g[1..(temp[2]-1)]),limits); |
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384 | for (i = 1; i<= size(pre); i++) |
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385 | {//iterating over the possible pre-factors |
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386 | candidate = pre[i]+list(f[temp[1]]); |
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387 | if (limitcheck(candidate,limits)) |
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388 | { |
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389 | result = result + list(candidate); |
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390 | } |
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391 | }//iterating over the possible pre-factors |
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392 | }//the last factor is the common one |
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393 | }//There are factors to combine before the equal factor |
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394 | else |
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395 | {//There are no factors to combine before the equal factor |
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396 | if (temp[1]<size(f)) |
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397 | {//Just a check for security |
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398 | //first without common factors |
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399 | post=merge_icf(list(f[(temp[1]+1)..size(f)]),list(g[(temp[2]+1..size(g))]),limits); |
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400 | for (i = 1; i<=size(post); i++) |
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401 | { |
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402 | candidate = list(f[temp[1]])+post[i]; |
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403 | if (limitcheck(candidate,limits)) |
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404 | { |
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405 | result = result + list(candidate); |
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406 | } |
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407 | } |
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408 | //Now with common factors |
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409 | post = merge_cf(list(f[(temp[1]+1)..size(f)]),list(g[(temp[2]+1..size(g))]),limits); |
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410 | if(size(post)>0) |
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411 | {//we could find other combinations |
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412 | for (i = 1; i<=size(post); i++) |
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413 | { |
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414 | candidate = list(f[temp[1]])+post[i]; |
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415 | if (limitcheck(candidate,limits)) |
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416 | { |
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417 | result = result + list(candidate); |
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418 | } |
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419 | } |
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420 | }//we could find other combinations |
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421 | }//Just a check for security |
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422 | }//There are no factors to combine before the equal factor |
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423 | }//set of equal pairs is not empty |
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424 | for (i = 1; i <= size(result); i++) |
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425 | {//delete those combinations, who have an entry with degree less or equal 0 |
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426 | for (j = 1; j<=size(result[i]);j++) |
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427 | {//Delete entry if there is a zero or an integer as a factor |
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428 | if (deg(result[i][j]) <= 0) |
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429 | {//found one |
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430 | result = delete(result,i); |
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431 | i--; |
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432 | break; |
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433 | }//found one |
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434 | }//Delete entry if there is a zero as factor |
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435 | }//delete those combinations, who have an entry with degree less or equal 0 |
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436 | return(result); |
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437 | }//proc merge_cf |
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438 | |
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439 | |
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440 | //==================================================* |
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441 | //merges two sets of factors |
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442 | |
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443 | static proc mergence(list l1, list l2, intvec limits) |
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444 | " |
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445 | DEPRECATED |
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446 | " |
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447 | {//Procedure mergence |
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448 | list g; |
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449 | list f; |
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450 | int k; int i; int j; |
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451 | list F = list(); |
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452 | list G = list(); |
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453 | list tempEntry; |
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454 | list comb; |
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455 | if (size(l2)<=size(l1)) |
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456 | {//l1 will be our g, l2 our f |
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457 | g = l1; |
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458 | f = l2; |
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459 | }//l1 will be our g, l2 our f |
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460 | else |
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461 | {//l1 will be our f, l2 our g |
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462 | g = l2; |
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463 | f = l1; |
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464 | }//l1 will be our f, l2 our g |
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465 | if (size(f)==1 or size(g)==1) |
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466 | {//One of them just has one entry |
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467 | if (size(f)== 1) {f = list(1) + f;} |
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468 | if (size(g) == 1) {g = list(1) + g;} |
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469 | }//One of them just has one entry |
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470 | //first, we need to add some latent -1's to the list f and to the list g in order |
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471 | //to get really all possibilities of combinations later |
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472 | for (i=1;i<=size(f)-1;i++) |
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473 | {//first iterator |
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474 | for (j=i+1;j<=size(f);j++) |
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475 | {//second iterator |
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476 | tempEntry = f; |
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477 | tempEntry[i] = (-1)*tempEntry[i]; |
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478 | tempEntry[j] = (-1)*tempEntry[j]; |
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479 | F = F + list(tempEntry); |
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480 | }//secont iterator |
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481 | }//first iterator |
---|
482 | F = F + list(f); |
---|
483 | //And now same game with g |
---|
484 | for (i=1;i<=size(g)-1;i++) |
---|
485 | {//first iterator |
---|
486 | for (j=i+1;j<=size(g);j++) |
---|
487 | {//second iterator |
---|
488 | tempEntry = g; |
---|
489 | tempEntry[i] = (-1)*tempEntry[i]; |
---|
490 | tempEntry[j] = (-1)*tempEntry[j]; |
---|
491 | G = G + list(tempEntry); |
---|
492 | }//secont iterator |
---|
493 | }//first iterator |
---|
494 | G = G + list(g); |
---|
495 | //Done with that |
---|
496 | |
---|
497 | list result; |
---|
498 | for (i = 1; i<=size(F); i++) |
---|
499 | {//Iterate over all entries in F |
---|
500 | for (j = 1;j<=size(G);j++) |
---|
501 | {//Same with G |
---|
502 | comb = combinekfinlf(F[i],2,limits); |
---|
503 | for (k = 1; k<= size(comb);k++) |
---|
504 | {//for all possibilities of combinations of the factors of f |
---|
505 | result = result + merge_cf(comb[k],G[j],limits); |
---|
506 | result = result + merge_icf(comb[k],G[j],limits); |
---|
507 | result = delete_dublicates_noteval(result); |
---|
508 | }//for all possibilities of combinations of the factors of f |
---|
509 | }//Same with G |
---|
510 | }//Iterate over all entries in F |
---|
511 | return(result); |
---|
512 | }//Procedure mergence |
---|
513 | |
---|
514 | |
---|
515 | //================================================== |
---|
516 | //Checks, whether a list of factors doesn't exceed the given limits |
---|
517 | static proc limitcheck(list g, intvec limits) |
---|
518 | " |
---|
519 | DEPRECATED |
---|
520 | " |
---|
521 | {//proc limitcheck |
---|
522 | int i; |
---|
523 | if (size(limits)!=3) |
---|
524 | {//check the input |
---|
525 | return(0); |
---|
526 | }//check the input |
---|
527 | if(size(g)==0) |
---|
528 | { |
---|
529 | return(0); |
---|
530 | } |
---|
531 | def prod = product(g); |
---|
532 | intvec iv11 = intvec(1,1); |
---|
533 | intvec iv10 = intvec(1,0); |
---|
534 | intvec iv01 = intvec(0,1); |
---|
535 | def limg = intvec(deg(prod,iv11) ,deg(prod,iv10),deg(prod,iv01)); |
---|
536 | for (i = 1; i<=size(limg);i++) |
---|
537 | {//the final check |
---|
538 | if(limg[i]>limits[i]) |
---|
539 | { |
---|
540 | return(0); |
---|
541 | } |
---|
542 | }//the final check |
---|
543 | return(1); |
---|
544 | }//proc limitcheck |
---|
545 | |
---|
546 | |
---|
547 | //==================================================* |
---|
548 | //one factorization of a homogeneous polynomial |
---|
549 | //in the first Weyl Algebra |
---|
550 | static proc homogfacFirstWeyl(poly h) |
---|
551 | "USAGE: homogfacFirstWeyl(h); h is a homogeneous polynomial in the |
---|
552 | first Weyl algebra with respect to the weight vector [-1,1] |
---|
553 | RETURN: list |
---|
554 | PURPOSE: Computes a factorization of a homogeneous polynomial h with |
---|
555 | respect to the weight vector [-1,1] in the first Weyl algebra |
---|
556 | THEORY: @code{homogfacFirstWeyl} returns a list with a factorization of the given, |
---|
557 | [-1,1]-homogeneous polynomial. If the degree of the polynomial is k with |
---|
558 | k positive, the last k entries in the output list are the second |
---|
559 | variable. If k is positive, the last k entries will be x. The other |
---|
560 | entries will be irreducible polynomials of degree zero or 1 resp. -1. |
---|
561 | SEE ALSO: homogfacFirstWeyl_all |
---|
562 | "{//proc homogfacFirstWeyl |
---|
563 | int p = printlevel-voice+2;//for dbprint |
---|
564 | def r = basering; |
---|
565 | poly hath; |
---|
566 | int i; int j; |
---|
567 | string dbprintWhitespace = ""; |
---|
568 | for (i = 1; i<=voice;i++) |
---|
569 | {dbprintWhitespace = dbprintWhitespace + " ";} |
---|
570 | intvec ivm11 = intvec(-1,1); |
---|
571 | if (!homogwithorder(h,ivm11)) |
---|
572 | {//The given polynomial is not homogeneous |
---|
573 | ERROR("Given polynomial was not [-1,1]-homogeneous"); |
---|
574 | return(list()); |
---|
575 | }//The given polynomial is not homogeneous |
---|
576 | if (h==0) |
---|
577 | { |
---|
578 | return(list(0)); |
---|
579 | } |
---|
580 | list result; |
---|
581 | int m = deg(h,ivm11); |
---|
582 | dbprint(p,dbprintWhitespace +" Splitting the polynomial in A_0 and A_k-Part"); |
---|
583 | if (m!=0) |
---|
584 | {//The degree is not zero |
---|
585 | if (m <0) |
---|
586 | {//There are more x than y |
---|
587 | hath = lift(var(1)^(-m),h)[1,1]; |
---|
588 | for (i = 1; i<=-m; i++) |
---|
589 | { |
---|
590 | result = result + list(var(1)); |
---|
591 | } |
---|
592 | }//There are more x than y |
---|
593 | else |
---|
594 | {//There are more y than x |
---|
595 | hath = lift(var(2)^m,h)[1,1]; |
---|
596 | for (i = 1; i<=m;i++) |
---|
597 | { |
---|
598 | result = result + list(var(2)); |
---|
599 | } |
---|
600 | }//There are more y than x |
---|
601 | }//The degree is not zero |
---|
602 | else |
---|
603 | {//The degree is zero |
---|
604 | hath = h; |
---|
605 | }//The degree is zero |
---|
606 | dbprint(p,dbprintWhitespace+" Done"); |
---|
607 | //beginning to transform x^i*y^i in theta(theta-1)...(theta-i+1) |
---|
608 | list mons; |
---|
609 | dbprint(p,dbprintWhitespace+" Putting the monomials in the A_0-part in a list."); |
---|
610 | for(i = 1; i<=size(hath);i++) |
---|
611 | {//Putting the monomials in a list |
---|
612 | mons = mons+list(hath[i]); |
---|
613 | }//Putting the monomials in a list |
---|
614 | dbprint(p,dbprintWhitespace+" Done"); |
---|
615 | dbprint(p,dbprintWhitespace+" Mapping this monomials to K[theta]"); |
---|
616 | ring tempRing = 0,(x,y,theta),dp; |
---|
617 | setring tempRing; |
---|
618 | map thetamap = r,x,y; |
---|
619 | list mons = thetamap(mons); |
---|
620 | poly entry; |
---|
621 | for (i = 1; i<=size(mons);i++) |
---|
622 | {//transforming the monomials as monomials in theta |
---|
623 | entry = leadcoef(mons[i]); |
---|
624 | for (j = 0; j<leadexp(mons[i])[2];j++) |
---|
625 | { |
---|
626 | entry = entry * (theta-j); |
---|
627 | } |
---|
628 | mons[i] = entry; |
---|
629 | }//transforming the monomials as monomials in theta |
---|
630 | dbprint(p,dbprintWhitespace+" Done"); |
---|
631 | dbprint(p,dbprintWhitespace+" Factorize the A_0-Part in K[theta]"); |
---|
632 | list azeroresult = factorize(sum(mons)); |
---|
633 | dbprint(p,dbprintWhitespace+" Successful"); |
---|
634 | list azeroresult_return_form; |
---|
635 | for (i = 1; i<=size(azeroresult[1]);i++) |
---|
636 | {//rewrite the result of the commutative factorization |
---|
637 | for (j = 1; j <= azeroresult[2][i];j++) |
---|
638 | { |
---|
639 | azeroresult_return_form = azeroresult_return_form + list(azeroresult[1][i]); |
---|
640 | } |
---|
641 | }//rewrite the result of the commutative factorization |
---|
642 | dbprint(p,dbprintWhitespace+" Mapping back to A_0."); |
---|
643 | setring(r); |
---|
644 | map finalmap = tempRing,var(1),var(2),var(1)*var(2); |
---|
645 | list tempresult = finalmap(azeroresult_return_form); |
---|
646 | dbprint(p,dbprintWhitespace+"Successful."); |
---|
647 | for (i = 1; i<=size(tempresult);i++) |
---|
648 | {//factorizations of theta resp. theta +1 |
---|
649 | if(tempresult[i]==var(1)*var(2)) |
---|
650 | { |
---|
651 | tempresult = insert(tempresult,var(1),i-1); |
---|
652 | i++; |
---|
653 | tempresult[i]=var(2); |
---|
654 | } |
---|
655 | if(tempresult[i]==var(2)*var(1)) |
---|
656 | { |
---|
657 | tempresult = insert(tempresult,var(2),i-1); |
---|
658 | i++; |
---|
659 | tempresult[i]=var(1); |
---|
660 | } |
---|
661 | }//factorizations of theta resp. theta +1 |
---|
662 | result = tempresult+result; |
---|
663 | return(result); |
---|
664 | }//proc homogfacFirstWeyl |
---|
665 | /* example */ |
---|
666 | /* { */ |
---|
667 | /* "EXAMPLE:";echo=2; */ |
---|
668 | /* ring R = 0,(x,y),Ws(-1,1); */ |
---|
669 | /* def r = nc_algebra(1,1); */ |
---|
670 | /* setring(r); */ |
---|
671 | /* poly h = (x^2*y^2+1)*(x^4); */ |
---|
672 | /* homogfacFirstWeyl(h); */ |
---|
673 | /* } */ |
---|
674 | |
---|
675 | //================================================== |
---|
676 | //Computes all possible homogeneous factorizations |
---|
677 | static proc homogfacFirstWeyl_all(poly h) |
---|
678 | "USAGE: homogfacFirstWeyl_all(h); h is a homogeneous polynomial in the first Weyl algebra |
---|
679 | with respect to the weight vector [-1,1] |
---|
680 | RETURN: list |
---|
681 | PURPOSE: Computes all factorizations of a homogeneous polynomial h with respect |
---|
682 | to the weight vector [-1,1] in the first Weyl algebra |
---|
683 | THEORY: @code{homogfacFirstWeyl} returns a list with all factorization of the given, |
---|
684 | homogeneous polynomial. It uses the output of homogfacFirstWeyl and permutes |
---|
685 | its entries with respect to the commutation rule. Furthermore, if a |
---|
686 | factor of degree zero is irreducible in K[ heta], but reducible in |
---|
687 | the first Weyl algebra, the permutations of this element with the other |
---|
688 | entries will also be computed. |
---|
689 | SEE ALSO: homogfacFirstWeyl |
---|
690 | "{//proc HomogfacFirstWeylAll |
---|
691 | int p=printlevel-voice+2;//for dbprint |
---|
692 | intvec iv11= intvec(1,1); |
---|
693 | if (deg(h,iv11) <= 0 ) |
---|
694 | {//h is a constant |
---|
695 | dbprint(p,"Given polynomial was not homogeneous"); |
---|
696 | return(list(list(h))); |
---|
697 | }//h is a constant |
---|
698 | def r = basering; |
---|
699 | list one_hom_fac; //stands for one homogeneous factorization |
---|
700 | int i; int j; int k; |
---|
701 | string dbprintWhitespace = ""; |
---|
702 | for (i = 1; i<=voice;i++) |
---|
703 | {dbprintWhitespace = dbprintWhitespace + " ";} |
---|
704 | intvec ivm11 = intvec(-1,1); |
---|
705 | dbprint(p,dbprintWhitespace +" Calculate one homogeneous factorization using homogfacFirstWeyl"); |
---|
706 | //Compute again a homogeneous factorization |
---|
707 | one_hom_fac = homogfacFirstWeyl(h); |
---|
708 | dbprint(p,dbprintWhitespace +"Successful"); |
---|
709 | if (size(one_hom_fac) == 0) |
---|
710 | {//there is no homogeneous factorization or the polynomial was not homogeneous |
---|
711 | return(list()); |
---|
712 | }//there is no homogeneous factorization or the polynomial was not homogeneous |
---|
713 | //divide list in A0-Part and a list of x's resp. y's |
---|
714 | list list_not_azero = list(); |
---|
715 | list list_azero; |
---|
716 | list k_factor; |
---|
717 | int is_list_not_azero_empty = 1; |
---|
718 | int is_list_azero_empty = 1; |
---|
719 | k_factor = list(one_hom_fac[1]); |
---|
720 | if (absValue(deg(h,ivm11))<size(one_hom_fac)-1) |
---|
721 | {//There is a nontrivial A_0-part |
---|
722 | list_azero = one_hom_fac[2..(size(one_hom_fac)-absValue(deg(h,ivm11)))]; |
---|
723 | is_list_azero_empty = 0; |
---|
724 | }//There is a nontrivial A_0 part |
---|
725 | dbprint(p,dbprintWhitespace +" Combine x,y to xy in the factorization again."); |
---|
726 | for (i = 1; i<=size(list_azero)-1;i++) |
---|
727 | {//in homogfacFirstWeyl, we factorized theta, and this will be made undone |
---|
728 | if (list_azero[i] == var(1)) |
---|
729 | { |
---|
730 | if (list_azero[i+1]==var(2)) |
---|
731 | { |
---|
732 | list_azero[i] = var(1)*var(2); |
---|
733 | list_azero = delete(list_azero,i+1); |
---|
734 | } |
---|
735 | } |
---|
736 | if (list_azero[i] == var(2)) |
---|
737 | { |
---|
738 | if (list_azero[i+1]==var(1)) |
---|
739 | { |
---|
740 | list_azero[i] = var(2)*var(1); |
---|
741 | list_azero = delete(list_azero,i+1); |
---|
742 | } |
---|
743 | } |
---|
744 | }//in homogfacFirstWeyl, we factorized theta, and this will be made undone |
---|
745 | dbprint(p,dbprintWhitespace +" Done"); |
---|
746 | if(deg(h,ivm11)!=0) |
---|
747 | {//list_not_azero is not empty |
---|
748 | list_not_azero = |
---|
749 | one_hom_fac[(size(one_hom_fac)-absValue(deg(h,ivm11))+1)..size(one_hom_fac)]; |
---|
750 | is_list_not_azero_empty = 0; |
---|
751 | }//list_not_azero is not empty |
---|
752 | //Map list_azero in K[theta] |
---|
753 | dbprint(p,dbprintWhitespace +" Map list_azero to K[theta]"); |
---|
754 | ring tempRing = 0,(x,y,theta), dp; |
---|
755 | setring(tempRing); |
---|
756 | poly entry; |
---|
757 | map thetamap = r,x,y; |
---|
758 | if(!is_list_not_azero_empty) |
---|
759 | {//Mapping in Singular is only possible, if the list before |
---|
760 | //contained at least one element of the other ring |
---|
761 | list list_not_azero = thetamap(list_not_azero); |
---|
762 | }//Mapping in Singular is only possible, if the list before |
---|
763 | //contained at least one element of the other ring |
---|
764 | if(!is_list_azero_empty) |
---|
765 | {//Mapping in Singular is only possible, if the list before |
---|
766 | //contained at least one element of the other ring |
---|
767 | list list_azero= thetamap(list_azero); |
---|
768 | }//Mapping in Singular is only possible, if the list before |
---|
769 | //contained at least one element of the other ring |
---|
770 | list k_factor = thetamap(k_factor); |
---|
771 | list tempmons; |
---|
772 | dbprint(p,dbprintWhitespace +" Done"); |
---|
773 | for(i = 1; i<=size(list_azero);i++) |
---|
774 | {//rewrite the polynomials in A1 as polynomials in K[theta] |
---|
775 | tempmons = list(); |
---|
776 | for (j = 1; j<=size(list_azero[i]);j++) |
---|
777 | { |
---|
778 | tempmons = tempmons + list(list_azero[i][j]); |
---|
779 | } |
---|
780 | for (j = 1 ; j<=size(tempmons);j++) |
---|
781 | { |
---|
782 | entry = leadcoef(tempmons[j]); |
---|
783 | for (k = 0; k < leadexp(tempmons[j])[2];k++) |
---|
784 | { |
---|
785 | entry = entry*(theta-k); |
---|
786 | } |
---|
787 | tempmons[j] = entry; |
---|
788 | } |
---|
789 | list_azero[i] = sum(tempmons); |
---|
790 | }//rewrite the polynomials in A1 as polynomials in K[theta] |
---|
791 | //Compute all permutations of the A0-part |
---|
792 | dbprint(p,dbprintWhitespace +" Compute all permutations of the A_0-part with the first resp. |
---|
793 | the snd. variable"); |
---|
794 | list result; |
---|
795 | int shift_sign; |
---|
796 | int shift; |
---|
797 | poly shiftvar; |
---|
798 | if (size(list_not_azero)!=0) |
---|
799 | {//Compute all possibilities to permute the x's resp. the y's in the list |
---|
800 | if (list_not_azero[1] == x) |
---|
801 | {//h had a negative weighted degree |
---|
802 | shift_sign = 1; |
---|
803 | shiftvar = x; |
---|
804 | }//h had a negative weighted degree |
---|
805 | else |
---|
806 | {//h had a positive weighted degree |
---|
807 | shift_sign = -1; |
---|
808 | shiftvar = y; |
---|
809 | }//h had a positive weighted degree |
---|
810 | result = permpp(list_azero + list_not_azero); |
---|
811 | for (i = 1; i<= size(result); i++) |
---|
812 | {//adjust the a_0-parts |
---|
813 | shift = 0; |
---|
814 | for (j=1; j<=size(result[i]);j++) |
---|
815 | { |
---|
816 | if (result[i][j]==shiftvar) |
---|
817 | { |
---|
818 | shift = shift + shift_sign; |
---|
819 | } |
---|
820 | else |
---|
821 | { |
---|
822 | result[i][j] = subst(result[i][j],theta,theta + shift); |
---|
823 | } |
---|
824 | } |
---|
825 | }//adjust the a_0-parts |
---|
826 | }//Compute all possibilities to permute the x's resp. the y's in the list |
---|
827 | else |
---|
828 | {//The result is just all the permutations of the a_0-part |
---|
829 | result = permpp(list_azero); |
---|
830 | }//The result is just all the permutations of the a_0 part |
---|
831 | if (size(result)==0) |
---|
832 | { |
---|
833 | return(result); |
---|
834 | } |
---|
835 | dbprint(p,dbprintWhitespace +" Done"); |
---|
836 | dbprint(p,dbprintWhitespace +" Searching for theta resp. theta+1 in the list and fact. them"); |
---|
837 | //Now we are going deeper and search for theta resp. theta + 1, substitute |
---|
838 | //them by xy resp. yx and go on permuting |
---|
839 | int found_theta; |
---|
840 | int thetapos; |
---|
841 | list leftpart; |
---|
842 | list rightpart; |
---|
843 | list lparts; |
---|
844 | list rparts; |
---|
845 | list tempadd; |
---|
846 | for (i = 1; i<=size(result) ; i++) |
---|
847 | {//checking every entry of result for theta or theta +1 |
---|
848 | found_theta = 0; |
---|
849 | for(j=1;j<=size(result[i]);j++) |
---|
850 | { |
---|
851 | if (result[i][j]==theta) |
---|
852 | {//the jth entry is theta and can be written as x*y |
---|
853 | thetapos = j; |
---|
854 | result[i]= insert(result[i],x,j-1); |
---|
855 | j++; |
---|
856 | result[i][j] = y; |
---|
857 | found_theta = 1; |
---|
858 | break; |
---|
859 | }//the jth entry is theta and can be written as x*y |
---|
860 | if(result[i][j] == theta +1) |
---|
861 | { |
---|
862 | thetapos = j; |
---|
863 | result[i] = insert(result[i],y,j-1); |
---|
864 | j++; |
---|
865 | result[i][j] = x; |
---|
866 | found_theta = 1; |
---|
867 | break; |
---|
868 | } |
---|
869 | } |
---|
870 | if (found_theta) |
---|
871 | {//One entry was theta resp. theta +1 |
---|
872 | leftpart = result[i]; |
---|
873 | leftpart = leftpart[1..thetapos]; |
---|
874 | rightpart = result[i]; |
---|
875 | rightpart = rightpart[(thetapos+1)..size(rightpart)]; |
---|
876 | lparts = list(leftpart); |
---|
877 | rparts = list(rightpart); |
---|
878 | //first deal with the left part |
---|
879 | if (leftpart[thetapos] == x) |
---|
880 | { |
---|
881 | shift_sign = 1; |
---|
882 | shiftvar = x; |
---|
883 | } |
---|
884 | else |
---|
885 | { |
---|
886 | shift_sign = -1; |
---|
887 | shiftvar = y; |
---|
888 | } |
---|
889 | for (j = size(leftpart); j>1;j--) |
---|
890 | {//drip x resp. y |
---|
891 | if (leftpart[j-1]==shiftvar) |
---|
892 | {//commutative |
---|
893 | j--; |
---|
894 | continue; |
---|
895 | }//commutative |
---|
896 | if (deg(leftpart[j-1],intvec(-1,1,0))!=0) |
---|
897 | {//stop here |
---|
898 | break; |
---|
899 | }//stop here |
---|
900 | //Here, we can only have a a0- part |
---|
901 | leftpart[j] = subst(leftpart[j-1],theta, theta + shift_sign); |
---|
902 | leftpart[j-1] = shiftvar; |
---|
903 | lparts = lparts + list(leftpart); |
---|
904 | }//drip x resp. y |
---|
905 | //and now deal with the right part |
---|
906 | if (rightpart[1] == x) |
---|
907 | { |
---|
908 | shift_sign = 1; |
---|
909 | shiftvar = x; |
---|
910 | } |
---|
911 | else |
---|
912 | { |
---|
913 | shift_sign = -1; |
---|
914 | shiftvar = y; |
---|
915 | } |
---|
916 | for (j = 1 ; j < size(rightpart); j++) |
---|
917 | { |
---|
918 | if (rightpart[j+1] == shiftvar) |
---|
919 | { |
---|
920 | j++; |
---|
921 | continue; |
---|
922 | } |
---|
923 | if (deg(rightpart[j+1],intvec(-1,1,0))!=0) |
---|
924 | { |
---|
925 | break; |
---|
926 | } |
---|
927 | rightpart[j] = subst(rightpart[j+1], theta, theta - shift_sign); |
---|
928 | rightpart[j+1] = shiftvar; |
---|
929 | rparts = rparts + list(rightpart); |
---|
930 | } |
---|
931 | //And now, we put all possibilities together |
---|
932 | tempadd = list(); |
---|
933 | for (j = 1; j<=size(lparts); j++) |
---|
934 | { |
---|
935 | for (k = 1; k<=size(rparts);k++) |
---|
936 | { |
---|
937 | tempadd = tempadd + list(lparts[j]+rparts[k]); |
---|
938 | } |
---|
939 | } |
---|
940 | tempadd = delete(tempadd,1); // The first entry is already in the list |
---|
941 | result = result + tempadd; |
---|
942 | continue; //We can may be not be done already with the ith entry |
---|
943 | }//One entry was theta resp. theta +1 |
---|
944 | }//checking every entry of result for theta or theta +1 |
---|
945 | dbprint(p,dbprintWhitespace +" Done"); |
---|
946 | //map back to the basering |
---|
947 | dbprint(p,dbprintWhitespace +" Mapping back everything to the basering"); |
---|
948 | setring(r); |
---|
949 | map finalmap = tempRing, var(1), var(2),var(1)*var(2); |
---|
950 | list result = finalmap(result); |
---|
951 | for (i=1; i<=size(result);i++) |
---|
952 | {//adding the K factor |
---|
953 | result[i] = k_factor + result[i]; |
---|
954 | }//adding the k-factor |
---|
955 | dbprint(p,dbprintWhitespace +" Done"); |
---|
956 | dbprint(p,dbprintWhitespace +" Delete double entries in the list."); |
---|
957 | result = delete_dublicates_noteval(result); |
---|
958 | dbprint(p,dbprintWhitespace +" Done"); |
---|
959 | return(result); |
---|
960 | }//proc HomogfacFirstWeylAll |
---|
961 | /* example */ |
---|
962 | /* { */ |
---|
963 | /* "EXAMPLE:";echo=2; */ |
---|
964 | /* ring R = 0,(x,y),Ws(-1,1); */ |
---|
965 | /* def r = nc_algebra(1,1); */ |
---|
966 | /* setring(r); */ |
---|
967 | /* poly h = (x^2*y^2+1)*(x^4); */ |
---|
968 | /* homogfacFirstWeyl_all(h); */ |
---|
969 | /* } */ |
---|
970 | |
---|
971 | //==================================================* |
---|
972 | //Computes all permutations of a given list |
---|
973 | static proc perm(list l) |
---|
974 | " |
---|
975 | DEPRECATED |
---|
976 | " |
---|
977 | {//proc perm |
---|
978 | int i; int j; |
---|
979 | list tempresult; |
---|
980 | list result; |
---|
981 | if (size(l)==0) |
---|
982 | { |
---|
983 | return(list()); |
---|
984 | } |
---|
985 | if (size(l)==1) |
---|
986 | { |
---|
987 | return(list(l)); |
---|
988 | } |
---|
989 | for (i = 1; i<=size(l); i++ ) |
---|
990 | { |
---|
991 | tempresult = perm(delete(l,i)); |
---|
992 | for (j = 1; j<=size(tempresult);j++) |
---|
993 | { |
---|
994 | tempresult[j] = list(l[i])+tempresult[j]; |
---|
995 | } |
---|
996 | result = result+tempresult; |
---|
997 | } |
---|
998 | return(result); |
---|
999 | }//proc perm |
---|
1000 | |
---|
1001 | //================================================== |
---|
1002 | //computes all permutations of a given list by |
---|
1003 | //ignoring equal entries (faster than perm) |
---|
1004 | static proc permpp(list l) |
---|
1005 | " |
---|
1006 | INPUT: A list with entries of a type, where the ==-operator is defined |
---|
1007 | OUTPUT: A list with all permutations of this given list. |
---|
1008 | " |
---|
1009 | {//proc permpp |
---|
1010 | int i; int j; |
---|
1011 | list tempresult; |
---|
1012 | list l_without_double; |
---|
1013 | list l_without_double_pos; |
---|
1014 | int double_entry; |
---|
1015 | list result; |
---|
1016 | if (size(l)==0) |
---|
1017 | { |
---|
1018 | return(list()); |
---|
1019 | } |
---|
1020 | if (size(l)==1) |
---|
1021 | { |
---|
1022 | return(list(l)); |
---|
1023 | } |
---|
1024 | for (i = 1; i<=size(l);i++) |
---|
1025 | {//Filling the list with unique entries |
---|
1026 | double_entry = 0; |
---|
1027 | for (j = 1; j<=size(l_without_double);j++) |
---|
1028 | { |
---|
1029 | if (l_without_double[j] == l[i]) |
---|
1030 | { |
---|
1031 | double_entry = 1; |
---|
1032 | break; |
---|
1033 | } |
---|
1034 | } |
---|
1035 | if (!double_entry) |
---|
1036 | { |
---|
1037 | l_without_double = l_without_double + list(l[i]); |
---|
1038 | l_without_double_pos = l_without_double_pos + list(i); |
---|
1039 | } |
---|
1040 | }//Filling the list with unique entries |
---|
1041 | for (i = 1; i<=size(l_without_double); i++ ) |
---|
1042 | { |
---|
1043 | tempresult = permpp(delete(l,l_without_double_pos[i])); |
---|
1044 | for (j = 1; j<=size(tempresult);j++) |
---|
1045 | { |
---|
1046 | tempresult[j] = list(l_without_double[i])+tempresult[j]; |
---|
1047 | } |
---|
1048 | result = result+tempresult; |
---|
1049 | } |
---|
1050 | return(result); |
---|
1051 | }//proc permpp |
---|
1052 | |
---|
1053 | //================================================== |
---|
1054 | //factorization of the first Weyl Algebra |
---|
1055 | |
---|
1056 | //The following procedure just serves the purpose to |
---|
1057 | //transform the input into an appropriate input for |
---|
1058 | //the procedure sfacwa, where the ring must contain the |
---|
1059 | //variables in a certain order. |
---|
1060 | proc facFirstWeyl(poly h) |
---|
1061 | "USAGE: facFirstWeyl(h); h a polynomial in the first Weyl algebra |
---|
1062 | RETURN: list |
---|
1063 | PURPOSE: compute all factorizations of a polynomial in the first Weyl algebra |
---|
1064 | THEORY: Implements the new algorithm by A. Heinle and V. Levandovskyy, see the thesis of A. Heinle |
---|
1065 | ASSUME: basering is the first Weyl algebra |
---|
1066 | NOTE: Every entry of the output list is a list with factors for one possible factorization. |
---|
1067 | The first factor is always a constant (1, if no nontrivial constant could be excluded). |
---|
1068 | EXAMPLE: example facFirstWeyl; shows examples |
---|
1069 | SEE ALSO: facSubWeyl, testNCfac, facFirstShift |
---|
1070 | "{//proc facFirstWeyl |
---|
1071 | //Definition of printlevel variable |
---|
1072 | int p = printlevel-voice+2; |
---|
1073 | int i; |
---|
1074 | string dbprintWhitespace = ""; |
---|
1075 | for (i = 1; i<=voice;i++) |
---|
1076 | {dbprintWhitespace = dbprintWhitespace + " ";} |
---|
1077 | dbprint(p,dbprintWhitespace +" Checking if the given algebra is a Weyl algebra"); |
---|
1078 | //Redefine the ring in my standard form |
---|
1079 | if (!isWeyl()) |
---|
1080 | {//Our basering is not the Weyl algebra |
---|
1081 | ERROR("Ring was not the first Weyl algebra"); |
---|
1082 | return(list()); |
---|
1083 | }//Our basering is not the Weyl algebra |
---|
1084 | dbprint(p,dbprintWhitespace +" Successful"); |
---|
1085 | dbprint(p,dbprintWhitespace +" Checking, if the given ring is the first Weyl algebra"); |
---|
1086 | if(nvars(basering)!=2) |
---|
1087 | {//Our basering is the Weyl algebra, but not the first |
---|
1088 | ERROR("Ring is not the first Weyl algebra"); |
---|
1089 | return(list()); |
---|
1090 | }//Our basering is the Weyl algebra, but not the first |
---|
1091 | |
---|
1092 | //A last check before we start the real business: Is h already given as a polynomial just |
---|
1093 | //in one variable? |
---|
1094 | if (deg(h,intvec(1,0))== 0 or deg(h,intvec(0,1)) == 0) |
---|
1095 | {//h is in K[x] or in K[d] |
---|
1096 | if (deg(h,intvec(1,0))== 0 and deg(h,intvec(0,1)) == 0) |
---|
1097 | {//We just have a constant |
---|
1098 | return(list(list(h))); |
---|
1099 | }//We just have a constant |
---|
1100 | dbprint(p,dbprintWhitespace+"Polynomial was given in one variable. |
---|
1101 | Performing commutative factorization."); |
---|
1102 | int theCommVar; |
---|
1103 | if (deg(h,intvec(1,0)) == 0) |
---|
1104 | {//The second variable is the variable to factorize |
---|
1105 | theCommVar = 2; |
---|
1106 | }//The second variable is the variable to factorize |
---|
1107 | else{theCommVar = 1;} |
---|
1108 | def r = basering; |
---|
1109 | ring tempRing = 0,(var(theCommVar)),dp; |
---|
1110 | if (theCommVar == 1){map mapToCommutative = r,var(1),1;} |
---|
1111 | else {map mapToCommutative = r,1,var(1);} |
---|
1112 | poly h = mapToCommutative(h); |
---|
1113 | list tempResult = factorize(h); |
---|
1114 | list result = list(list()); |
---|
1115 | int j; |
---|
1116 | for (i = 1; i<=size(tempResult[1]); i++) |
---|
1117 | { |
---|
1118 | for (j = 1; j<=tempResult[2][i]; j++) |
---|
1119 | { |
---|
1120 | result[1] = result[1] + list(tempResult[1][i]); |
---|
1121 | } |
---|
1122 | } |
---|
1123 | //mapping back |
---|
1124 | setring(r); |
---|
1125 | map mapBackFromCommutative = tempRing,var(theCommVar); |
---|
1126 | def result = mapBackFromCommutative(result); |
---|
1127 | dbprint(p,dbprintWhitespace+"result:"); |
---|
1128 | dbprint(p,result); |
---|
1129 | dbprint(p,dbprintWhitespace+"Computing all permutations of this factorization"); |
---|
1130 | poly constantFactor = result[1][1]; |
---|
1131 | result[1] = delete(result[1],1);//Deleting the constant factor |
---|
1132 | result=permpp(result[1]); |
---|
1133 | for (i = 1; i<=size(result);i++) |
---|
1134 | {//Insert constant factor |
---|
1135 | result[i] = insert(result[i],constantFactor); |
---|
1136 | }//Insert constant factor |
---|
1137 | dbprint(p,dbprintWhitespace+"Done."); |
---|
1138 | return(result); |
---|
1139 | }//h is in K[x] or in K[d] |
---|
1140 | |
---|
1141 | |
---|
1142 | dbprint(p,dbprintWhitespace +" Successful"); |
---|
1143 | list result = list(); |
---|
1144 | int j; int k; int l; //counter |
---|
1145 | if (ringlist(basering)[6][1,2] == -1) //manual of ringlist will tell you why |
---|
1146 | { |
---|
1147 | dbprint(p,dbprintWhitespace +" positions of the variables have to be switched"); |
---|
1148 | def r = basering; |
---|
1149 | ring tempRing = ringlist(r)[1][1],(x,y),Ws(-1,1); // very strange: |
---|
1150 | // setting Wp(-1,1) leads to SegFault; to clarify why!!! |
---|
1151 | def NTR = nc_algebra(1,1); |
---|
1152 | setring NTR ; |
---|
1153 | map transf = r, var(2), var(1); |
---|
1154 | dbprint(p,dbprintWhitespace +" Successful"); |
---|
1155 | list resulttemp = sfacwa(h); |
---|
1156 | setring(r); |
---|
1157 | map transfback = NTR, var(2),var(1); |
---|
1158 | result = transfback(resulttemp); |
---|
1159 | } |
---|
1160 | else |
---|
1161 | { |
---|
1162 | dbprint(p, dbprintWhitespace +" factorization of the polynomial with the routine sfacwa"); |
---|
1163 | result = sfacwa(h); |
---|
1164 | dbprint(p,dbprintWhitespace +" Done"); |
---|
1165 | } |
---|
1166 | if (homogwithorder(h,intvec(-1,1))) |
---|
1167 | { |
---|
1168 | dbprint(p, dbprintWhitespace + " Polynomial was homogeneous, therefore we have |
---|
1169 | already a complete factorization and do not have to go through the factors recursively."); |
---|
1170 | return(result); |
---|
1171 | } |
---|
1172 | dbprint(p,dbprintWhitespace + "We have the following intermediate list of inhomogeneous |
---|
1173 | factorizations:"); |
---|
1174 | dbprint(p,result); |
---|
1175 | dbprint(p,dbprintWhitespace +" recursively check factors for irreducibility"); |
---|
1176 | list recursivetemp; |
---|
1177 | int changedSomething; |
---|
1178 | for(i = 1; i<=size(result);i++) |
---|
1179 | {//recursively factorize factors |
---|
1180 | if(size(result[i])>2) |
---|
1181 | {//Nontrivial factorization |
---|
1182 | for (j=2;j<=size(result[i]);j++) |
---|
1183 | {//Factorize every factor |
---|
1184 | recursivetemp = facFirstWeyl(result[i][j]); |
---|
1185 | //if(size(recursivetemp)>1) |
---|
1186 | //{//we have a nontrivial factorization |
---|
1187 | changedSomething = 0; |
---|
1188 | for(k=1; k<=size(recursivetemp);k++) |
---|
1189 | {//insert factorized factors |
---|
1190 | if(size(recursivetemp[k])>2) |
---|
1191 | {//nontrivial |
---|
1192 | changedSomething = 1; |
---|
1193 | result = insert(result,result[i],i); |
---|
1194 | for(l = size(recursivetemp[k]);l>=2;l--) |
---|
1195 | { |
---|
1196 | result[i+1] = insert(result[i+1],recursivetemp[k][l],j); |
---|
1197 | } |
---|
1198 | result[i+1] = delete(result[i+1],j); |
---|
1199 | }//nontrivial |
---|
1200 | }//insert factorized factors |
---|
1201 | if (changedSomething) |
---|
1202 | { |
---|
1203 | result = delete(result,i); |
---|
1204 | } |
---|
1205 | //}//we have a nontrivial factorization |
---|
1206 | }//Factorize every factor |
---|
1207 | }//Nontrivial factorization |
---|
1208 | }//recursively factorize factors |
---|
1209 | dbprint(p,dbprintWhitespace +" Done"); |
---|
1210 | if (size(result)==0) |
---|
1211 | {//only the trivial factorization could be found |
---|
1212 | result = list(list(1,h)); |
---|
1213 | }//only the trivial factorization could be found |
---|
1214 | list resultWithInterchanges; |
---|
1215 | dbprint(p,dbprintWhitespace+ "And the result without interchanges with homogeneous factors is:"); |
---|
1216 | dbprint(p,result); |
---|
1217 | for (i = 1; i <= size(result) ; i++) |
---|
1218 | {//applying the interchanges to result |
---|
1219 | resultWithInterchanges = resultWithInterchanges + |
---|
1220 | checkForHomogInhomogInterchangability(result[i],2,size(result[i])); |
---|
1221 | }//applying the interchanges to result |
---|
1222 | dbprint(p,dbprintWhitespace + "With interchanges, the result is:"); |
---|
1223 | dbprint(p,resultWithInterchanges); |
---|
1224 | //now, refine the possible redundant list |
---|
1225 | return( delete_dublicates_noteval(resultWithInterchanges) ); |
---|
1226 | }//proc facFirstWeyl |
---|
1227 | example |
---|
1228 | { |
---|
1229 | "EXAMPLE:";echo=2; |
---|
1230 | ring R = 0,(x,y),dp; |
---|
1231 | def r = nc_algebra(1,1); |
---|
1232 | setring(r); |
---|
1233 | poly h = (x^2*y^2+x)*(x+1); |
---|
1234 | facFirstWeyl(h); |
---|
1235 | } |
---|
1236 | |
---|
1237 | ////////////////////////////////////////////////// |
---|
1238 | /////BRANDNEW!!!!//////////////////// |
---|
1239 | ////////////////////////////////////////////////// |
---|
1240 | |
---|
1241 | static proc checkForHomogInhomogInterchangability(list factors, posLeft, posRight) |
---|
1242 | " |
---|
1243 | INPUT: A list consisting of factors of a certain polynomial in the first Weyl |
---|
1244 | algebra, factors, and a position from the left and the right, where the last swap was done. |
---|
1245 | OUTPUT: A list containing lists consisting of factors of a certain polynomial in the first Weyl |
---|
1246 | algebra. |
---|
1247 | The purpose of this function is to check whether we can interchange certain inhomogeneous factors |
---|
1248 | with homogeneous ones. If it is possible, this function returns a list of lists |
---|
1249 | of possible interchanges. |
---|
1250 | |
---|
1251 | The idea came because of an example, where we need an extra swap in the end, otherwise we would |
---|
1252 | not capture all factorizations. The example was |
---|
1253 | h = x4d7+11x3d6+x2d7+x2d6+x3d4+29x2d5+xd6+8xd5+d6+5x2d3+14xd4+13d4+5xd2+d3+d; |
---|
1254 | |
---|
1255 | ASSUMPTIONS: |
---|
1256 | |
---|
1257 | - All factors are irreducible |
---|
1258 | " |
---|
1259 | {//checkForHomogInhomogInterchangability |
---|
1260 | int p = printlevel-voice+2; |
---|
1261 | string dbprintWhitespace = ""; |
---|
1262 | int i; int j; int k; |
---|
1263 | for (i = 1; i<=voice;i++) |
---|
1264 | {dbprintWhitespace = dbprintWhitespace + " ";} |
---|
1265 | if (size(factors) <= 2 || posLeft >= posRight - 1) |
---|
1266 | {//easiest case: There is nothing to swap |
---|
1267 | return (list(factors)); |
---|
1268 | }//easiest case: There is nothing to swap |
---|
1269 | list result = list(factors); |
---|
1270 | list tempResultEntries; |
---|
1271 | list tempSwaps; |
---|
1272 | list tempSwapsTempEntry; |
---|
1273 | list attemptToSwap; |
---|
1274 | intvec ivm11 = intvec(-1,1); |
---|
1275 | dbprint(p, dbprintWhitespace+"We try to swap elements in the following list:"); |
---|
1276 | dbprint(p, factors); |
---|
1277 | for (i = posLeft; i < posRight; i++) |
---|
1278 | {//checking within the window posLeft <--> posRight, if there are interchanges possible |
---|
1279 | if (homogwithorder(factors[i],ivm11) && !homogwithorder(factors[i+1],ivm11)) |
---|
1280 | {//position i is homogeneous, position i+1 is not ==> trying to swap |
---|
1281 | attemptToSwap = extractHomogeneousDivisorsRight(factors[i]*factors[i+1]); |
---|
1282 | if (size(attemptToSwap[1])>1) |
---|
1283 | {//Bingo, we were able to swap this one element |
---|
1284 | dbprint(p,dbprintWhitespace+"We can swap entry "+string(i)+" and "+ string(i+1)); |
---|
1285 | dbprint(p,dbprintWhitespace+"The elements look like the following after the swap:"); |
---|
1286 | dbprint(p,attemptToSwap); |
---|
1287 | tempSwapsTempEntry = list(); |
---|
1288 | for (j = size(factors); j >=1; j--) |
---|
1289 | {//creating a new entry for the resulting list, replacing the swap in factors |
---|
1290 | if (j==i+1) |
---|
1291 | { |
---|
1292 | for (k = size(attemptToSwap[1]); k >=1 ; k--) |
---|
1293 | { |
---|
1294 | tempSwapsTempEntry = insert(tempSwapsTempEntry, attemptToSwap[1][k]); |
---|
1295 | } |
---|
1296 | j--; //Because we changed entry i+1 and i |
---|
1297 | } |
---|
1298 | else |
---|
1299 | { |
---|
1300 | tempSwapsTempEntry = insert(tempSwapsTempEntry,factors[j]); |
---|
1301 | } |
---|
1302 | }//creating a new entry for the resulting list, replacing the swap in factors |
---|
1303 | tempSwaps = insert(tempSwaps,list(list(i+1,posRight),tempSwapsTempEntry)); |
---|
1304 | }//Bingo, we were able to swap this one element |
---|
1305 | }//position i is homogeneous, position i+1 is not ==> trying to swap |
---|
1306 | else |
---|
1307 | { |
---|
1308 | if(!homogwithorder(factors[i],ivm11) && homogwithorder(factors[i+1],ivm11)) |
---|
1309 | {//position i+1 is homogeneous, position i is not ==> trying to swap |
---|
1310 | attemptToSwap = extractHomogeneousDivisorsLeft(factors[i]*factors[i+1]); |
---|
1311 | if (size(attemptToSwap[1])>1) |
---|
1312 | {//Bingo, we were able to swap this one element |
---|
1313 | dbprint(p,dbprintWhitespace+"We can swap entry "+string(i)+" and "+ string(i+1)); |
---|
1314 | dbprint(p,dbprintWhitespace+"The elements look like the following after the swap:"); |
---|
1315 | dbprint(p,attemptToSwap); |
---|
1316 | tempSwapsTempEntry = list(); |
---|
1317 | for (j = size(factors); j >=1; j--) |
---|
1318 | {//creating a new entry for the resulting list, replacing the swap in factors |
---|
1319 | if (j==i+1) |
---|
1320 | { |
---|
1321 | for (k = size(attemptToSwap[1]); k >=1 ; k--) |
---|
1322 | { |
---|
1323 | tempSwapsTempEntry = insert(tempSwapsTempEntry, attemptToSwap[1][k]); |
---|
1324 | } |
---|
1325 | j--; //Because we changed entry i+1 and i |
---|
1326 | } |
---|
1327 | else |
---|
1328 | { |
---|
1329 | tempSwapsTempEntry = insert(tempSwapsTempEntry,factors[j]); |
---|
1330 | } |
---|
1331 | }//creating a new entry for the resulting list, replacing the swap in factors |
---|
1332 | tempSwaps = insert(tempSwaps,list(list(posLeft,i),tempSwapsTempEntry)); |
---|
1333 | }//Bingo, we were able to swap this one element |
---|
1334 | }//position i+1 is homogeneous, position i is not ==> trying to swap |
---|
1335 | } |
---|
1336 | }//checking within the window posLeft <--> posRight, if there are interchanges possible |
---|
1337 | //Now we will recursively call the function for all swapped entries. |
---|
1338 | dbprint(p,dbprintWhitespace+ "Our list of different factorizations is now:"); |
---|
1339 | dbprint(p,tempSwaps); |
---|
1340 | for (i = 1; i<=size(tempSwaps);i++) |
---|
1341 | {//recursive call to all formerly attempted swaps. |
---|
1342 | tempResultEntries=checkForHomogInhomogInterchangability(tempSwaps[i][2], |
---|
1343 | tempSwaps[i][1][1],tempSwaps[i][1][2]); |
---|
1344 | result = result + tempResultEntries; |
---|
1345 | }//recursive call to all formerly attempted swaps. |
---|
1346 | result = delete_dublicates_noteval(result); |
---|
1347 | return(result); |
---|
1348 | }//checkForHomogInhomogInterchangability |
---|
1349 | |
---|
1350 | static proc sfacwa(poly h) |
---|
1351 | "INPUT: A polynomial h in the first Weyl algebra |
---|
1352 | OUTPUT: A list of factorizations, where the factors might still be reducible. |
---|
1353 | ASSUMPTIONS: |
---|
1354 | - Our basering is the first Weyl algebra; the x is the first variable, |
---|
1355 | the differential operator the second. |
---|
1356 | " |
---|
1357 | {//proc sfacwa |
---|
1358 | int i; int j; int k; |
---|
1359 | int p = printlevel-voice+2; |
---|
1360 | string dbprintWhitespace = ""; |
---|
1361 | number commonCoefficient = content(h); |
---|
1362 | for (i = 1; i<=voice;i++) |
---|
1363 | {dbprintWhitespace = dbprintWhitespace + " ";} |
---|
1364 | dbprint(p,dbprintWhitespace + " Extracting homogeneous left and right factors"); |
---|
1365 | if(homogwithorder(h,intvec(-1,1))) |
---|
1366 | {//we are already dealing with a -1,1 homogeneous poly |
---|
1367 | dbprint(p,dbprintWhitespace+" Given polynomial is -1,1 homogeneous. Start homog. |
---|
1368 | fac. and ret. its result"); |
---|
1369 | return(homogfacFirstWeyl_all(h)); |
---|
1370 | }//we are already dealing with a -1,1 homogeneous poly |
---|
1371 | list resulttemp = extractHomogeneousDivisors(h/commonCoefficient); |
---|
1372 | //resulttemp = resulttemp + list(list(h/commonCoefficient)); |
---|
1373 | list inhomogeneousFactorsToFactorize; |
---|
1374 | int isAlreadyInInhomogList; |
---|
1375 | dbprint(p,dbprintWhitespace +" Done"); |
---|
1376 | dbprint(p,dbprintWhitespace +" Making Set of inhomogeneous polynomials we have to factorize."); |
---|
1377 | for (i = 1; i<=size(resulttemp); i++) |
---|
1378 | {//Going through all different kinds of factorizations where we extracted homogeneous factors |
---|
1379 | for (j = 1;j<=size(resulttemp[i]);j++) |
---|
1380 | {//searching for the inhomogeneous factor |
---|
1381 | if (!homogwithorder(resulttemp[i][j],intvec(-1,1))) |
---|
1382 | {//We have found our candidate |
---|
1383 | isAlreadyInInhomogList = 0; |
---|
1384 | for (k = 1; k<=size(inhomogeneousFactorsToFactorize);k++) |
---|
1385 | {//Checking if our candidate is already in our tofactorize-list |
---|
1386 | if (inhomogeneousFactorsToFactorize[k]==resulttemp[i][j]) |
---|
1387 | {//The candidate was already in the list |
---|
1388 | isAlreadyInInhomogList = 1; |
---|
1389 | break; |
---|
1390 | }//The candidate was already in the list |
---|
1391 | }//Checking if our candidate is already in our tofactorize-list |
---|
1392 | if (!isAlreadyInInhomogList) |
---|
1393 | { |
---|
1394 | inhomogeneousFactorsToFactorize=inhomogeneousFactorsToFactorize + list(resulttemp[i][j]); |
---|
1395 | } |
---|
1396 | }//We have found our candidate |
---|
1397 | }//searching for the inhomogeneous factor |
---|
1398 | }//Going through all different kinds of factorizations where we extracted homogeneous factors |
---|
1399 | dbprint(p,dbprintWhitespace +" Done"); |
---|
1400 | dbprint(p,dbprintWhitespace + "The set is:"); |
---|
1401 | dbprint(p,inhomogeneousFactorsToFactorize); |
---|
1402 | dbprint(p,dbprintWhitespace+ "Factorizing the different occuring inhomogeneous factors"); |
---|
1403 | for (i = 1; i<= size(inhomogeneousFactorsToFactorize); i++) |
---|
1404 | {//Factorizing all kinds of inhomogeneous factors |
---|
1405 | inhomogeneousFactorsToFactorize[i] = sfacwa2(inhomogeneousFactorsToFactorize[i]); |
---|
1406 | for (j = 1; j<=size(inhomogeneousFactorsToFactorize[i]);j++) |
---|
1407 | {//Deleting the leading coefficient since we don't need him |
---|
1408 | if (deg(inhomogeneousFactorsToFactorize[i][j][1],intvec(1,1))==0) |
---|
1409 | { |
---|
1410 | inhomogeneousFactorsToFactorize[i][j] = delete(inhomogeneousFactorsToFactorize[i][j],1); |
---|
1411 | } |
---|
1412 | }//Deleting the leading coefficient since we don't need him |
---|
1413 | }//Factorizing all kinds of inhomogeneous factors |
---|
1414 | dbprint(p,dbprintWhitespace +" Done"); |
---|
1415 | dbprint(p,dbprintWhitespace +" Putting the factorizations in the lists"); |
---|
1416 | list result; |
---|
1417 | int posInhomogPoly; |
---|
1418 | int posInhomogFac; |
---|
1419 | for (i = 1; i<=size(resulttemp); i++) |
---|
1420 | {//going through all by now calculated factorizations |
---|
1421 | for (j = 1;j<=size(resulttemp[i]); j++) |
---|
1422 | {//Finding the inhomogeneous factor |
---|
1423 | if (!homogwithorder(resulttemp[i][j],intvec(-1,1))) |
---|
1424 | {//Found it |
---|
1425 | posInhomogPoly = j; |
---|
1426 | break; |
---|
1427 | }//Found it |
---|
1428 | }//Finding the inhomogeneous factor |
---|
1429 | for (k = 1; k<=size(inhomogeneousFactorsToFactorize);k++) |
---|
1430 | {//Finding the matching inhomogeneous factorization we already determined |
---|
1431 | if(product(inhomogeneousFactorsToFactorize[k][1]) == resulttemp[i][j]) |
---|
1432 | {//found it |
---|
1433 | posInhomogFac = k; |
---|
1434 | break; |
---|
1435 | }//Found it |
---|
1436 | }//Finding the matching inhomogeneous factorization we already determined |
---|
1437 | for (j = 1; j <= size(inhomogeneousFactorsToFactorize[posInhomogFac]); j++) |
---|
1438 | { |
---|
1439 | result = insert(result, resulttemp[i]); |
---|
1440 | result[1] = delete(result[1],posInhomogPoly); |
---|
1441 | for (k =size(inhomogeneousFactorsToFactorize[posInhomogFac][j]);k>=1; k--) |
---|
1442 | {//Inserting factorizations |
---|
1443 | result[1] = insert(result[1],inhomogeneousFactorsToFactorize[posInhomogFac][j][k], |
---|
1444 | posInhomogPoly-1); |
---|
1445 | }//Inserting factorizations |
---|
1446 | dbprint(p,dbprintWhitespace + "Added a factorization to result, namely:"); |
---|
1447 | dbprint(p, result[1]); |
---|
1448 | } |
---|
1449 | }//going through all by now calculated factorizations |
---|
1450 | dbprint(p,dbprintWhitespace +" Done"); |
---|
1451 | result = delete_dublicates_noteval(result); |
---|
1452 | for (i = 1; i<=size(result);i++) |
---|
1453 | {//Putting the content everywhere |
---|
1454 | result[i] = insert(result[i],commonCoefficient); |
---|
1455 | }//Putting the content everywhere |
---|
1456 | return(result); |
---|
1457 | }//proc sfacwa |
---|
1458 | |
---|
1459 | static proc sfacwa2(poly h) |
---|
1460 | " |
---|
1461 | Subprocedure of sfacwa |
---|
1462 | Assumptions: |
---|
1463 | - h is not in K[x] or in K[d], or even in K. These cases are caught by the input |
---|
1464 | - The coefficients are integer values and the gcd of the coefficients is 1 |
---|
1465 | " |
---|
1466 | {//proc sfacwa2 |
---|
1467 | int p=printlevel-voice+2; // for dbprint |
---|
1468 | int i; |
---|
1469 | string dbprintWhitespace = ""; |
---|
1470 | for (i = 1; i<=voice;i++) |
---|
1471 | {dbprintWhitespace = dbprintWhitespace + " ";} |
---|
1472 | intvec ivm11 = intvec(-1,1); |
---|
1473 | intvec iv11 = intvec(1,1); |
---|
1474 | intvec iv10 = intvec(1,0); |
---|
1475 | intvec iv01 = intvec(0,1); |
---|
1476 | intvec iv1m1 = intvec(1,-1); |
---|
1477 | poly p_max; poly p_min; poly q_max; poly q_min; |
---|
1478 | map invo = basering,-var(1),var(2); |
---|
1479 | list calculatedRightFactors; |
---|
1480 | if(homogwithorder(h,ivm11)) |
---|
1481 | {//Unnecessary how we are using it, but if one wants to use it on its own, we are stating it here |
---|
1482 | dbprint(p,dbprintWhitespace+" Given polynomial is -1,1 homogeneous. |
---|
1483 | Start homog. fac. and ret. its result"); |
---|
1484 | return(homogfacFirstWeyl_all(h)); |
---|
1485 | }//Unnecessary how we are using it, but if one wants to use it on its own, we are stating it here |
---|
1486 | list result = list(); |
---|
1487 | int j; int k; int l; |
---|
1488 | dbprint(p,dbprintWhitespace+" Computing the degree-limits of the factorization"); |
---|
1489 | //end finding the limits |
---|
1490 | dbprint(p,dbprintWhitespace+" Computing the maximal and the minimal |
---|
1491 | homogeneous part of the given polynomial"); |
---|
1492 | list M = computeCombinationsMinMaxHomog(h); |
---|
1493 | dbprint(p,dbprintWhitespace+" Done."); |
---|
1494 | dbprint(p,dbprintWhitespace+" Filtering invalid combinations in M."); |
---|
1495 | for (i = 1 ; i<= size(M); i++) |
---|
1496 | {//filter valid combinations |
---|
1497 | if (product(M[i]) == h) |
---|
1498 | {//We have one factorization |
---|
1499 | result = result + divides(M[i][1],h,invo,1); |
---|
1500 | dbprint(p,dbprintWhitespace+"Result list updated:"); |
---|
1501 | dbprint(p,dbprintWhitespace+string(result)); |
---|
1502 | M = delete(M,i); |
---|
1503 | continue; |
---|
1504 | }//We have one factorization |
---|
1505 | }//filter valid combinations |
---|
1506 | dbprint(p,dbprintWhitespace+"Done."); |
---|
1507 | dbprint(p,dbprintWhitespace+"The size of M is "+string(size(M))); |
---|
1508 | for (i = 1; i<=size(M); i++) |
---|
1509 | {//Iterate over all first combinations (p_max + p_min)(q_max + q_min) |
---|
1510 | dbprint(p,dbprintWhitespace+" Combination No. "+string(i)+" in M:" ); |
---|
1511 | p_max = jet(M[i][1],deg(M[i][1],ivm11),ivm11)-jet(M[i][1],deg(M[i][1],ivm11)-1,ivm11); |
---|
1512 | p_min = jet(M[i][1],deg(M[i][1],iv1m1),iv1m1)-jet(M[i][1],deg(M[i][1],iv1m1)-1,iv1m1); |
---|
1513 | q_max = jet(M[i][2],deg(M[i][2],ivm11),ivm11)-jet(M[i][2],deg(M[i][2],ivm11)-1,ivm11); |
---|
1514 | q_min = jet(M[i][2],deg(M[i][2],iv1m1),iv1m1)-jet(M[i][2],deg(M[i][2],iv1m1)-1,iv1m1); |
---|
1515 | dbprint(p,dbprintWhitespace+" pmax = "+string(p_max)); |
---|
1516 | dbprint(p,dbprintWhitespace+" pmin = "+string(p_min)); |
---|
1517 | dbprint(p,dbprintWhitespace+" qmax = "+string(q_max)); |
---|
1518 | dbprint(p,dbprintWhitespace+" qmin = "+string(q_min)); |
---|
1519 | //Check, whether p_max + p_min or q_max and q_min are already left or right divisors. |
---|
1520 | if (divides(p_min + p_max,h,invo)) |
---|
1521 | { |
---|
1522 | dbprint(p,dbprintWhitespace+" Got one result."); |
---|
1523 | result = result + divides(p_min + p_max,h,invo,1); |
---|
1524 | } |
---|
1525 | else |
---|
1526 | { |
---|
1527 | if (divides(q_min + q_max,h,invo)) |
---|
1528 | { |
---|
1529 | dbprint(p,dbprintWhitespace+" Got one result."); |
---|
1530 | result = result + divides(q_min + q_max, h , invo, 1); |
---|
1531 | } |
---|
1532 | } |
---|
1533 | //Now the check, if deg(p_max) = deg(p_min)+1 (and the same with q_max and q_min) |
---|
1534 | |
---|
1535 | if (deg(p_max, ivm11) == deg(p_min, ivm11) +1 or deg(q_max, ivm11) == deg(q_min, ivm11) +1 ) |
---|
1536 | {//Therefore, p_max + p_min must be a left factor or we can dismiss the combination |
---|
1537 | dbprint(p,dbprintWhitespace+" There are no homogeneous parts we can put between |
---|
1538 | pmax and pmin resp. qmax and qmin."); |
---|
1539 | //TODO: Prove, that then also a valid right factor is not possible |
---|
1540 | M = delete(M,i); |
---|
1541 | continue; |
---|
1542 | }//Therefore, p_max + p_min must be a left factor or we can dismiss the combination |
---|
1543 | |
---|
1544 | //Done with the Check |
---|
1545 | |
---|
1546 | //If we come here, there are still homogeneous parts to be added to p_max + p_min |
---|
1547 | //AND to q_max and q_min in |
---|
1548 | //order to obtain a real factor |
---|
1549 | //We use the procedure determineRestOfHomogParts to find our q. |
---|
1550 | dbprint(p,dbprintWhitespace+" Solving for the other homogeneous parts in q"); |
---|
1551 | calculatedRightFactors = determineRestOfHomogParts(p_max,p_min,q_max,q_min,h); |
---|
1552 | dbprint(p,dbprintWhitespace+" Done with it. Found "+string(size(calculatedRightFactors)) |
---|
1553 | +" solutions."); |
---|
1554 | for (j = 1; j<=size(calculatedRightFactors);j++) |
---|
1555 | {//Check out whether we really have right factors of h in calculatedRightFactors |
---|
1556 | if (divides(calculatedRightFactors[j],h,invo)) |
---|
1557 | { |
---|
1558 | result = result + divides(calculatedRightFactors[j],h,invo,1); |
---|
1559 | } |
---|
1560 | else |
---|
1561 | { |
---|
1562 | dbprint(p,"Solution for max and min homog found, but not a divisor of h"); |
---|
1563 | //TODO: Proof, why this can happen. |
---|
1564 | } |
---|
1565 | }//Check out whether we really have right factors of h in calculatedRightFactors |
---|
1566 | }//Iterate over all first combinations (p_max + p_min)(q_max + q_min) |
---|
1567 | |
---|
1568 | |
---|
1569 | result = delete_dublicates_noteval(result); |
---|
1570 | //print(M); |
---|
1571 | if (size(result) == 0) |
---|
1572 | {//no factorization found |
---|
1573 | result = list(list(h)); |
---|
1574 | }//no factorization found |
---|
1575 | return(result); |
---|
1576 | }//proc sfacwa2 |
---|
1577 | |
---|
1578 | static proc determineRestOfHomogParts(poly pmax, poly pmin, poly qmax, poly qmin, poly h) |
---|
1579 | "INPUT: Polynomials p_max, p_min, q_max, q_min and h. The maximum homogeneous part h_max of h is |
---|
1580 | given by p_max*pmin, the minimum homogeneous part h_min of h is given by p_min*q_min. |
---|
1581 | OUTPUT: A list of right factors q of h that have q_max and q_min as their maximum respectively |
---|
1582 | minimum homogeneous part. Empty list, if those elements are not existent |
---|
1583 | ASSUMPTIONS: |
---|
1584 | - deg(p_max,intvec(-1,1))>deg(p_min,intvec(-1,1)) +1 |
---|
1585 | - deg(q_max,intvec(-1,1))>deg(q_min,intvec(-1,1)) +1 |
---|
1586 | - p_max*q_max = h_max |
---|
1587 | - p_min*q_min = h_min |
---|
1588 | - The basering is the first Weyl algebra |
---|
1589 | " |
---|
1590 | {//proc determineRestOfHomogParts |
---|
1591 | int p=printlevel-voice+2; // for dbprint |
---|
1592 | string dbprintWhitespace = ""; |
---|
1593 | int i; |
---|
1594 | for (i = 1; i<=voice;i++) |
---|
1595 | {dbprintWhitespace = dbprintWhitespace + " ";} |
---|
1596 | int kappa = Min(intvec(deg(h,intvec(1,0)), deg(h,intvec(0,1)))); |
---|
1597 | def R = basering; |
---|
1598 | int n1 = deg(pmax,intvec(-1,1)); |
---|
1599 | int nk = -deg(pmin,intvec(1,-1)); |
---|
1600 | int m1 = deg(qmax,intvec(-1,1)); |
---|
1601 | int ml = -deg(qmin,intvec(1,-1)); |
---|
1602 | int j; int k; |
---|
1603 | ideal mons; |
---|
1604 | |
---|
1605 | dbprint(p,dbprintWhitespace+" Extracting zero homog. parts of pmax, qmax, pmin, qmin and h."); |
---|
1606 | //Extracting the zero homogeneous part of the given polynomials |
---|
1607 | ideal pandqZero = pmax,pmin,qmax,qmin; |
---|
1608 | if (n1 > 0){pandqZero[1] = lift(var(2)^n1,pmax)[1,1];} |
---|
1609 | else{if (n1 < 0){pandqZero[1] = lift(var(1)^(-n1),pmax)[1,1];} |
---|
1610 | else{pandqZero[1] = pmax;}} |
---|
1611 | if (nk > 0){pandqZero[2] = lift(var(2)^nk,pmin)[1,1];} |
---|
1612 | else{if (nk < 0){pandqZero[2] = lift(var(1)^(-nk),pmin)[1,1];} |
---|
1613 | else{pandqZero[2] = pmin;}} |
---|
1614 | if (m1 > 0){pandqZero[3] = lift(var(2)^m1,qmax)[1,1];} |
---|
1615 | else{if (m1 < 0){pandqZero[3] = lift(var(1)^(-m1),qmax)[1,1];} |
---|
1616 | else{pandqZero[3] = qmax;}} |
---|
1617 | if (ml > 0){pandqZero[4] = lift(var(2)^ml,qmin)[1,1];} |
---|
1618 | else{if (ml < 0){pandqZero[4] = lift(var(1)^(-ml),qmin)[1,1];} |
---|
1619 | else{pandqZero[4] = qmin;}} |
---|
1620 | list hZeroinR = homogDistribution(h); |
---|
1621 | for (i = 1; i<=size(hZeroinR);i++) |
---|
1622 | {//Extracting the zero homogeneous parts of the homogeneous summands of h |
---|
1623 | if (hZeroinR[i][1] > 0){hZeroinR[i][2] = lift(var(2)^hZeroinR[i][1],hZeroinR[i][2])[1,1];} |
---|
1624 | if (hZeroinR[i][1] < 0){hZeroinR[i][2] = lift(var(1)^(-hZeroinR[i][1]),hZeroinR[i][2])[1,1];} |
---|
1625 | }//Extracting the zero homogeneous parts of the homogeneous summands of h |
---|
1626 | dbprint(p,dbprintWhitespace+" Done!"); |
---|
1627 | //Moving everything into the ring K[theta] |
---|
1628 | dbprint(p,dbprintWhitespace+" Moving everything into the ring K[theta]"); |
---|
1629 | ring KTheta = 0,(x,d,theta),dp; |
---|
1630 | map thetamap = R, x, d; |
---|
1631 | poly entry; |
---|
1632 | ideal mons; |
---|
1633 | ideal pandqZero; |
---|
1634 | list hZeroinKTheta; |
---|
1635 | setring(R); |
---|
1636 | |
---|
1637 | //Starting with p and q |
---|
1638 | for (k=1; k<=4; k++) |
---|
1639 | {//Transforming pmax(0),qmax(0),pmin(0),qmin(0) in theta-polys |
---|
1640 | mons = ideal(); |
---|
1641 | for(i = 1; i<=size(pandqZero[k]);i++) |
---|
1642 | {//Putting the monomials in a list |
---|
1643 | mons[size(mons)+1] = pandqZero[k][i]; |
---|
1644 | }//Putting the monomials in a list |
---|
1645 | setring(KTheta); |
---|
1646 | mons = thetamap(mons); |
---|
1647 | for (i = 1; i<=size(mons);i++) |
---|
1648 | {//transforming the monomials as monomials in theta |
---|
1649 | entry = leadcoef(mons[i]); |
---|
1650 | for (j = 0; j<leadexp(mons[i])[2];j++) |
---|
1651 | { |
---|
1652 | entry = entry * (theta-j); |
---|
1653 | } |
---|
1654 | mons[i] = entry; |
---|
1655 | }//transforming the monomials as monomials in theta |
---|
1656 | pandqZero[size(pandqZero)+1] = sum(mons); |
---|
1657 | setring(R); |
---|
1658 | }//Transforming pmax(0),qmax(0),pmin(0),qmin(0) in theta-polys |
---|
1659 | |
---|
1660 | //Now hZero |
---|
1661 | for (k = size(hZeroinR); k>= 1;k--) |
---|
1662 | {//Transforming the different homogeneous parts of h into polys in K[theta] |
---|
1663 | mons = ideal(); |
---|
1664 | for(i = 1; i<=size(hZeroinR[k][2]);i++) |
---|
1665 | {//Putting the monomials in a list |
---|
1666 | mons[size(mons)+1] = hZeroinR[k][2][i]; |
---|
1667 | }//Putting the monomials in a list |
---|
1668 | setring(KTheta); |
---|
1669 | mons = thetamap(mons); |
---|
1670 | for (i = 1; i<=size(mons);i++) |
---|
1671 | {//transforming the monomials as monomials in theta |
---|
1672 | entry = leadcoef(mons[i]); |
---|
1673 | for (j = 0; j<leadexp(mons[i])[2];j++) |
---|
1674 | { |
---|
1675 | entry = entry * (theta-j); |
---|
1676 | } |
---|
1677 | mons[i] = entry; |
---|
1678 | }//transforming the monomials as monomials in theta |
---|
1679 | hZeroinKTheta = hZeroinKTheta + list(sum(mons)); |
---|
1680 | setring(R); |
---|
1681 | }//Transforming the different homogeneous parts of h into polys in K[theta] |
---|
1682 | dbprint(p,dbprintWhitespace+" Done!"); |
---|
1683 | //Making the solutionRing |
---|
1684 | ring solutionRing = 0,(theta,q(0..(kappa+1)*(m1-ml-1)-1)),lp; |
---|
1685 | dbprint(p,dbprintWhitespace+" Our solution ring is given by "+ string(solutionRing)); |
---|
1686 | //mapping the different ps and qs and HZeros |
---|
1687 | dbprint(p,dbprintWhitespace+" Setting up our solution system."); |
---|
1688 | list ps; |
---|
1689 | ideal pandqZero = imap(KTheta,pandqZero); |
---|
1690 | ps[1] = list(n1,pandqZero[1]); |
---|
1691 | ps[n1-nk+1] = list(nk,pandqZero[2]); |
---|
1692 | for (i = 2; i<=n1-nk; i++) |
---|
1693 | { |
---|
1694 | ps[i] = list(n1-i+1,0); |
---|
1695 | } |
---|
1696 | list qs; |
---|
1697 | qs[1] = list(m1,pandqZero[3]); |
---|
1698 | qs[m1-ml+1] = list(ml,pandqZero[4]); |
---|
1699 | for (i = 2; i<=m1-ml; i++) |
---|
1700 | { |
---|
1701 | qs[i] = list(m1-i+1,0); |
---|
1702 | for (j = 0; j<=kappa; j++) |
---|
1703 | { |
---|
1704 | qs[i][2] = qs[i][2] + q((i-2)*(kappa+1)+j)*theta^j; |
---|
1705 | } |
---|
1706 | } |
---|
1707 | list hZero = imap(KTheta,hZeroinKTheta); |
---|
1708 | for (i = 1; i<=size(hZero); i++) |
---|
1709 | { |
---|
1710 | hZero[i] = list(n1+m1-i+1,hZero[i]); |
---|
1711 | } |
---|
1712 | |
---|
1713 | //writing and solving the system |
---|
1714 | list lhs; |
---|
1715 | lhs[1] = list(ps[1][2],1); |
---|
1716 | list rhs; |
---|
1717 | rhs[n1-nk+1] = list(ps[size(ps)][2],1); |
---|
1718 | for (i = 2; i<= n1-nk+1;i++) |
---|
1719 | { |
---|
1720 | lhs[i] = list(hZero[i][2],subst(qs[1][2],theta,theta+n1-i+1)*gammaForTheta(ps[i][1],qs[1][1])); |
---|
1721 | rhs[n1-nk-i+2] = list(hZero[size(hZero)-i+1][2],subst(qs[m1-ml+1][2],theta, |
---|
1722 | theta+nk+i-1)*gammaForTheta(ps[n1-nk-i+2][1],qs[m1-ml+1][1])); |
---|
1723 | for (j = 1; j<i; j++) |
---|
1724 | { |
---|
1725 | for (k = 1; k<=size(qs);k++) |
---|
1726 | { |
---|
1727 | if(ps[j][1]+qs[k][1] == hZero[i][1]) |
---|
1728 | { |
---|
1729 | lhs[i][1] = lhs[i][1]*lhs[j][2]; |
---|
1730 | lhs[i][2] = lhs[i][2]*lhs[j][2]; |
---|
1731 | lhs[i][1] = lhs[i][1]-lhs[j][1]*subst(qs[k][2],theta, theta + n1- j +1) |
---|
1732 | *gammaForTheta(ps[j][1],qs[k][1]); |
---|
1733 | } |
---|
1734 | if(ps[n1-nk+2-j][1] + qs[m1-ml+2-k][1] ==hZero[size(hZero)-i+1][1]) |
---|
1735 | { |
---|
1736 | rhs[n1-nk-i+2][1] = rhs[n1-nk-i+2][1]*rhs[n1-nk+2-j][2]; |
---|
1737 | rhs[n1-nk-i+2][2] = rhs[n1-nk-i+2][2]*rhs[n1-nk+2-j][2]; |
---|
1738 | rhs[n1-nk-i+2][1] = rhs[n1-nk-i+2][1]-rhs[n1-nk+2-j][1]*subst(qs[m1-ml+2-k][2],theta, |
---|
1739 | theta + nk -j+1)*gammaForTheta(ps[n1-nk+2-j][1],qs[m1-ml+2-k][1]); |
---|
1740 | } |
---|
1741 | } |
---|
1742 | } |
---|
1743 | } |
---|
1744 | list eqs; |
---|
1745 | poly tempgcd; |
---|
1746 | poly templhscoeff; |
---|
1747 | poly temprhscoeff; |
---|
1748 | for (i = 2; i<=n1-nk;i++) |
---|
1749 | { |
---|
1750 | if (gcd(rhs[i][2],lhs[i][2]) == 1) |
---|
1751 | { |
---|
1752 | eqs = eqs + list(lhs[i][1]*rhs[i][2] -rhs[i][1]*lhs[i][2]); |
---|
1753 | } |
---|
1754 | else |
---|
1755 | { |
---|
1756 | tempgcd = gcd(rhs[i][2],lhs[i][2]); |
---|
1757 | templhscoeff = quotient(rhs[i][2],tempgcd)[1]; |
---|
1758 | temprhscoeff = quotient(lhs[i][2],tempgcd)[1]; |
---|
1759 | eqs = eqs + list(lhs[i][1]*templhscoeff -rhs[i][1]*temprhscoeff); |
---|
1760 | } |
---|
1761 | } |
---|
1762 | matrix tempCoefMatrix; |
---|
1763 | ideal solutionSystemforqs; |
---|
1764 | for (i= 1; i<=size(eqs); i++) |
---|
1765 | { |
---|
1766 | tempCoefMatrix = coef(eqs[i],theta); |
---|
1767 | solutionSystemforqs = solutionSystemforqs + ideal(submat(tempCoefMatrix,intvec(2), |
---|
1768 | intvec(1..ncols(tempCoefMatrix)))); |
---|
1769 | } |
---|
1770 | dbprint(p,dbprintWhitespace+" Solution system for the coefficients of q is given by:"); |
---|
1771 | dbprint(p,solutionSystemforqs); |
---|
1772 | option(redSB); |
---|
1773 | dbprint(p,dbprintWhitespace+" Calculating reduced Groebner Basis of that system."); |
---|
1774 | solutionSystemforqs = slimgb(solutionSystemforqs); |
---|
1775 | dbprint(p,dbprintWhitespace+" Done!, the solution for the system is:"); |
---|
1776 | dbprint(p,dbprintWhitespace+string(solutionSystemforqs)); |
---|
1777 | if(vdim(slimgb(solutionSystemforqs+theta))==0) |
---|
1778 | {//No solution in this case. Return the empty list |
---|
1779 | dbprint(p,dbprintWhitespace+"The Groebner Basis of the solution system was <1>."); |
---|
1780 | setring(R); |
---|
1781 | return(list()); |
---|
1782 | }//No solution in this case. Return the empty list |
---|
1783 | if(vdim(slimgb(solutionSystemforqs+theta))==-1) |
---|
1784 | {//My conjecture is that this would never happen |
---|
1785 | //ERROR("This is an counterexample to your conjecture. We have infinitely many solutions"); |
---|
1786 | //TODO: See, what we would do here |
---|
1787 | dbprint(p,dbprintWhitespace+"There are infinitely many solution to this system. |
---|
1788 | We will return the empty list."); |
---|
1789 | setring(R); |
---|
1790 | return(list()); |
---|
1791 | }//My conjecture is that this would never happen |
---|
1792 | else |
---|
1793 | {//We have finitely many solutions |
---|
1794 | if(vdim(slimgb(solutionSystemforqs+theta))==1) |
---|
1795 | {//exactly one solution |
---|
1796 | for (i = 2; i<= size(qs)-1;i++) |
---|
1797 | { |
---|
1798 | qs[i][2] = NF(qs[i][2],solutionSystemforqs); |
---|
1799 | } |
---|
1800 | setring(R); |
---|
1801 | map backFromSolutionRing = solutionRing,var(1)*var(2); |
---|
1802 | list qs = backFromSolutionRing(qs); |
---|
1803 | list result = list(0); |
---|
1804 | for (i = 1; i<=size(qs); i++) |
---|
1805 | { |
---|
1806 | if (qs[i][1]>0){qs[i][2] = qs[i][2]*var(2)^qs[i][1];} |
---|
1807 | if (qs[i][1]<0){qs[i][2] = qs[i][2]*var(1)^(-qs[i][1]);} |
---|
1808 | result[1] = result[1] + qs[i][2]; |
---|
1809 | } |
---|
1810 | dbprint(p,dbprintWhitespace+"Found one unique solution. Returning the result."); |
---|
1811 | return(result); |
---|
1812 | }//exactly one solution |
---|
1813 | else |
---|
1814 | {//We have more than one solution, but finitely many |
---|
1815 | dbprint(p,dbprintWhitespace+"Finitely many, but more than one solution. |
---|
1816 | Right now the conjecture is that this cannot happen."); |
---|
1817 | dbprint(p,dbprintWhitespace+"Yay to counterexample :-) We can have more than one solution."); |
---|
1818 | //TODO: Some theoretical work why this can never happan |
---|
1819 | return(list()); |
---|
1820 | }//We have more than one solution, but finitely many |
---|
1821 | }//We have finitely many solutions |
---|
1822 | }//proc determineRestOfHomogParts |
---|
1823 | |
---|
1824 | static proc gammaForTheta(int j1,int j2) |
---|
1825 | " |
---|
1826 | INPUT: Two integers j1 and j2 |
---|
1827 | OUTPUT: A polynomial in the first variable of the given ring. It calculates the following function: |
---|
1828 | / 1, if j1,j2>0 or j1,j2 <= 0 |
---|
1829 | | prod_{kappa = 0}^{|j1|-1}(var(1)-kappa), if j1<0, j2>0, |j1|<=|j2| |
---|
1830 | gamma_{j1,j2}:= < prod_{kappa = 0}^{j2-1}(var(1)-kappa-|j1|+|j2|), if j1<0, j2>0, |j1|>|j2| |
---|
1831 | | prod_{kappa = 1}^{j1}(var(1)+kappa), if j1>0, j2<0, |j1|<=|j2| |
---|
1832 | \ prod_{kappa = 1}^{|j2|}(\var(1)+kappa+|j1|-|j2|), if j1>0, j2<0, |j1|>|j2| |
---|
1833 | ASSUMPTION: |
---|
1834 | - Ring has at least one variable |
---|
1835 | " |
---|
1836 | {//gammaForTheta |
---|
1837 | if (j1<=0 && j2 <=0) {return(1);} |
---|
1838 | if (j1>=0 && j2 >=0){return(1);} |
---|
1839 | poly result; |
---|
1840 | int i; |
---|
1841 | if (j1<0 && j2>0) |
---|
1842 | {//case 2 or 3 from description above |
---|
1843 | if (absValue(j1)<=absValue(j2)) |
---|
1844 | {//Case 2 holds here |
---|
1845 | result = 1; |
---|
1846 | for (i = 0;i<absValue(j1);i++) |
---|
1847 | { |
---|
1848 | result = result*(var(1)-i); |
---|
1849 | } |
---|
1850 | return(result); |
---|
1851 | }//Case 2 holds here |
---|
1852 | else |
---|
1853 | {//Case 3 holds here |
---|
1854 | result = 1; |
---|
1855 | for (i = 0; i<j2; i++) |
---|
1856 | { |
---|
1857 | result = result*(var(1)-i-absValue(j1)+absValue(j2)); |
---|
1858 | } |
---|
1859 | return(result); |
---|
1860 | }//Case 3 holds here |
---|
1861 | }//case 2 or 3 from description above |
---|
1862 | else |
---|
1863 | {//Case 4 or 5 from description above hold |
---|
1864 | if (absValue(j1)<=absValue(j2)) |
---|
1865 | {//Case 4 holds |
---|
1866 | result = 1; |
---|
1867 | for (i = 1; i<=j1; i++) |
---|
1868 | { |
---|
1869 | result = result*(var(1)+i); |
---|
1870 | } |
---|
1871 | return(result); |
---|
1872 | }//Case 4 holds |
---|
1873 | else |
---|
1874 | {//Case 5 holds |
---|
1875 | result = 1; |
---|
1876 | for (i = 1; i<=absValue(j2); i++) |
---|
1877 | { |
---|
1878 | result = result*(var(1)+i+absValue(j1)-absValue(j2)); |
---|
1879 | } |
---|
1880 | return(result); |
---|
1881 | }//Case 5 holds |
---|
1882 | }//Case 4 or 5 from description above hold |
---|
1883 | }//gammaForTheta |
---|
1884 | |
---|
1885 | static proc extractHomogeneousDivisors(poly h) |
---|
1886 | "INPUT: A polynomial h in the first Weyl algebra |
---|
1887 | OUTPUT: If h is homogeneous, then all factorizations of h are returned. |
---|
1888 | If h is inhomogeneous, then a list l is returned whose entries |
---|
1889 | are again lists k = [k_1,...,k_n], where k_1*...*k_n = h and there |
---|
1890 | exists an i in {1,...,n}, such that k_i is inhomogeneous and there |
---|
1891 | is no homogeneous polynomial that divides this k_i neither from the |
---|
1892 | left nor from the right. All the other entries in k are homogeneous |
---|
1893 | polynomials. |
---|
1894 | " |
---|
1895 | {//extractHomogeneousDivisors |
---|
1896 | int p=printlevel-voice+2; // for dbprint |
---|
1897 | string dbprintWhitespace = ""; |
---|
1898 | int i; int j; int k; int l; |
---|
1899 | list result; |
---|
1900 | for (i = 1; i<=voice;i++) |
---|
1901 | {dbprintWhitespace = dbprintWhitespace + " ";} |
---|
1902 | if (homogwithorder(h,intvec(-1,1))) |
---|
1903 | {//given polynomial was homogeneous already |
---|
1904 | dbprint(p,dbprintWhitespace+"Polynomial was homogeneous. Just returning all factorizations."); |
---|
1905 | result = homogfacFirstWeyl_all(h); |
---|
1906 | for (i = 1; i<=size(result);i++) |
---|
1907 | {//removing the first entry (coefficient) from the list result |
---|
1908 | result[i] = delete(result[i],1); |
---|
1909 | }//removing the first entry (coefficient) from the list result |
---|
1910 | return(result); |
---|
1911 | }//given polynomial was homogeneous already |
---|
1912 | dbprint(p,dbprintWhitespace+"Calculating list with all homogeneous left divisors extracted"); |
---|
1913 | list leftDivisionPossibilities = extractHomogeneousDivisorsLeft(h); |
---|
1914 | dbprint(p,dbprintWhitespace+"Done. The result is:"); |
---|
1915 | dbprint(p,leftDivisionPossibilities); |
---|
1916 | dbprint(p,dbprintWhitespace+"Calculating list with all homogeneous Right divisors extracted"); |
---|
1917 | list rightDivisionPossibilities = extractHomogeneousDivisorsRight(h); |
---|
1918 | dbprint(p,dbprintWhitespace+"Done. The result is:"); |
---|
1919 | dbprint(p,rightDivisionPossibilities); |
---|
1920 | list tempList; |
---|
1921 | dbprint(p,dbprintWhitespace+"Calculating remaining right and left homogeneous divisors"); |
---|
1922 | for (i = 1; i<=size(leftDivisionPossibilities); i++) |
---|
1923 | {//iterating through the list with extracted left divisors |
---|
1924 | tempList = extractHomogeneousDivisorsRight |
---|
1925 | (leftDivisionPossibilities[i][size(leftDivisionPossibilities[i])]); |
---|
1926 | leftDivisionPossibilities[i] = delete(leftDivisionPossibilities[i], |
---|
1927 | size(leftDivisionPossibilities[i])); |
---|
1928 | for (j=1;j<=size(tempList);j++) |
---|
1929 | {//Updating the list for Result |
---|
1930 | tempList[j] = leftDivisionPossibilities[i] + tempList[j]; |
---|
1931 | }//Updating the list for Result |
---|
1932 | result = result + tempList; |
---|
1933 | }//iterating through the list with extracted left divisors |
---|
1934 | for (i = 1; i<=size(rightDivisionPossibilities); i++) |
---|
1935 | {//iterating through the list with extracted left divisors |
---|
1936 | tempList = extractHomogeneousDivisorsLeft(rightDivisionPossibilities[i][1]); |
---|
1937 | rightDivisionPossibilities[i] = delete(rightDivisionPossibilities[i],1); |
---|
1938 | for (j=1;j<=size(tempList);j++) |
---|
1939 | {//Updating the list for Result |
---|
1940 | tempList[j] = tempList[j]+rightDivisionPossibilities[i]; |
---|
1941 | }//Updating the list for Result |
---|
1942 | result = result + tempList; |
---|
1943 | }//iterating through the list with extracted left divisors |
---|
1944 | dbprint(p,dbprintWhitespace+"Done"); |
---|
1945 | int posInhomog; |
---|
1946 | poly hath = 1; |
---|
1947 | list tempResult; |
---|
1948 | dbprint(p,dbprintWhitespace+"Checking if we can swap left resp. right |
---|
1949 | divisors and updating result."); |
---|
1950 | for (i = 1; i<= size(result); i++) |
---|
1951 | {//Checking if we can swap left resp. right divisors |
---|
1952 | for (j = 1; j<=size(result[i]);j++) |
---|
1953 | {//finding the position of the inhomogeneous element in the list |
---|
1954 | if(!homogwithorder(result[i][j],intvec(-1,1))) |
---|
1955 | { |
---|
1956 | posInhomog = j; |
---|
1957 | break; |
---|
1958 | } |
---|
1959 | }//finding the position of the inhomogeneous element in the list |
---|
1960 | hath = result[i][posInhomog]; |
---|
1961 | for(j=posInhomog-1;j>=1;j--) |
---|
1962 | { |
---|
1963 | hath = result[i][j]*hath; |
---|
1964 | tempList = extractHomogeneousDivisorsRight(hath); |
---|
1965 | if(size(tempList[1])==1) |
---|
1966 | {//We could not swap this element to the right |
---|
1967 | break; |
---|
1968 | }//We could not swap this element to the right |
---|
1969 | dbprint(p,dbprintWhitespace+"A swapping (left) of an element was possible"); |
---|
1970 | for(k = 1; k<=size(tempList);k++) |
---|
1971 | { |
---|
1972 | tempResult = insert(tempResult,result[i]); |
---|
1973 | for (l = j;l<=posInhomog;l++) |
---|
1974 | { |
---|
1975 | tempResult[1] = delete(tempResult[1],j); |
---|
1976 | } |
---|
1977 | for (l = size(tempList[k]);l>=1;l--) |
---|
1978 | { |
---|
1979 | tempResult[1] = insert(tempResult[1],tempList[k][l],j-1); |
---|
1980 | } |
---|
1981 | } |
---|
1982 | } |
---|
1983 | hath = result[i][posInhomog]; |
---|
1984 | for(j=posInhomog+1;j<=size(result[i]);j++) |
---|
1985 | { |
---|
1986 | hath = hath*result[i][j]; |
---|
1987 | tempList = extractHomogeneousDivisorsLeft(hath); |
---|
1988 | if(size(tempList[1])==1) |
---|
1989 | {//We could not swap this element to the right |
---|
1990 | break; |
---|
1991 | }//We could not swap this element to the right |
---|
1992 | dbprint(p,dbprintWhitespace+"A swapping (right) of an element was possible"); |
---|
1993 | for(k = 1; k<=size(tempList);k++) |
---|
1994 | { |
---|
1995 | tempResult = insert(tempResult,result[i]); |
---|
1996 | for (l=posInhomog; l<=j;l++) |
---|
1997 | { |
---|
1998 | tempResult[1] = delete(tempResult[1],posInhomog); |
---|
1999 | } |
---|
2000 | for (l = size(tempList[k]);l>=1;l--) |
---|
2001 | { |
---|
2002 | tempResult[1] = insert(tempResult[1],tempList[k][l],posInhomog-1); |
---|
2003 | } |
---|
2004 | } |
---|
2005 | } |
---|
2006 | }//Checking if we can swap left resp. right divisors |
---|
2007 | result = result + tempResult; |
---|
2008 | result = delete_dublicates_noteval(result); |
---|
2009 | return(result); |
---|
2010 | }//extractHomogeneousDivisors |
---|
2011 | |
---|
2012 | static proc extractHomogeneousDivisorsLeft(poly h) |
---|
2013 | "INPUT: A polynomial h in the first Weyl algebra |
---|
2014 | OUTPUT: If h is homogeneous, then all factorizations of h are returned. |
---|
2015 | If h is inhomogeneous, then a list l is returned whose entries |
---|
2016 | are again lists k = [k_1,...,k_n], where k_1*...*k_n = h. |
---|
2017 | The entry k_n is inhomogeneous and has no other homogeneous |
---|
2018 | left divisors any more. |
---|
2019 | All the other entries in k are homogeneous |
---|
2020 | polynomials. |
---|
2021 | " |
---|
2022 | {//extractHomogeneousDivisorsLeft |
---|
2023 | int p=printlevel-voice+2; // for dbprint |
---|
2024 | string dbprintWhitespace = ""; |
---|
2025 | int i;int j; int k; |
---|
2026 | list result; |
---|
2027 | poly hath; |
---|
2028 | list recResult; |
---|
2029 | map invo = basering,-var(1),var(2); |
---|
2030 | for (i = 1; i<=voice;i++) |
---|
2031 | {dbprintWhitespace = dbprintWhitespace + " ";} |
---|
2032 | if (homogwithorder(h,intvec(-1,1))) |
---|
2033 | {//given polynomial was homogeneous already |
---|
2034 | dbprint(p,dbprintWhitespace+"Polynomial was homogeneous. Just returning all factorizations."); |
---|
2035 | result = homogfacFirstWeyl_all(h); |
---|
2036 | for (i = 1; i<=size(result);i++) |
---|
2037 | {//removing the first entry (coefficient) from the list result |
---|
2038 | result[i] = delete(result[i],1); |
---|
2039 | }//removing the first entry (coefficient) from the list result |
---|
2040 | return(result); |
---|
2041 | }//given polynomial was homogeneous already |
---|
2042 | list hlist = homogDistribution(h); |
---|
2043 | dbprint(p,dbprintWhitespace+ " Computing factorizations of all homogeneous summands."); |
---|
2044 | for (i = 1; i<= size(hlist); i++) |
---|
2045 | { |
---|
2046 | hlist[i] = homogfacFirstWeyl_all(hlist[i][2]); |
---|
2047 | if (size(hlist[i][1])==1) |
---|
2048 | {//One homogeneous part just has a trivial factorization |
---|
2049 | if(hlist[i][1][1] == 0) |
---|
2050 | { |
---|
2051 | hlist = delete(hlist,i); |
---|
2052 | continue; |
---|
2053 | } |
---|
2054 | else |
---|
2055 | { |
---|
2056 | return(list(list(h))); |
---|
2057 | } |
---|
2058 | }//One homogeneous part just has a trivial factorization |
---|
2059 | } |
---|
2060 | dbprint(p,dbprintWhitespace+ " Done."); |
---|
2061 | dbprint(p,dbprintWhitespace+ " Trying to find Left divisors"); |
---|
2062 | list alreadyConsideredCandidates; |
---|
2063 | poly candidate; |
---|
2064 | int isCandidate; |
---|
2065 | for (i = 1; i<=size(hlist[1]);i++) |
---|
2066 | {//Finding candidates for homogeneous left divisors of h |
---|
2067 | candidate = hlist[1][i][2]; |
---|
2068 | isCandidate = 0; |
---|
2069 | for (j=1;j<=size(alreadyConsideredCandidates);j++) |
---|
2070 | { |
---|
2071 | if(alreadyConsideredCandidates[j] == candidate) |
---|
2072 | { |
---|
2073 | isCandidate =1; |
---|
2074 | break; |
---|
2075 | } |
---|
2076 | } |
---|
2077 | if(isCandidate) |
---|
2078 | { |
---|
2079 | i++; |
---|
2080 | continue; |
---|
2081 | } |
---|
2082 | else |
---|
2083 | { |
---|
2084 | alreadyConsideredCandidates = alreadyConsideredCandidates + list(candidate); |
---|
2085 | } |
---|
2086 | dbprint(p,dbprintWhitespace+"Checking if "+string(candidate)+" is a homogeneous left divisor"); |
---|
2087 | for (j = 2; j<=size(hlist);j++) |
---|
2088 | {//Iterating through the other homogeneous parts |
---|
2089 | isCandidate = 0; |
---|
2090 | for(k=1; k<=size(hlist[j]);k++) |
---|
2091 | { |
---|
2092 | if(hlist[j][k][2]==candidate) |
---|
2093 | { |
---|
2094 | isCandidate = 1; |
---|
2095 | break; |
---|
2096 | } |
---|
2097 | } |
---|
2098 | if(!isCandidate) |
---|
2099 | { |
---|
2100 | break; |
---|
2101 | } |
---|
2102 | }//Iterating through the other homogeneous parts |
---|
2103 | if(isCandidate) |
---|
2104 | {//candidate was really a left divisor |
---|
2105 | dbprint(p,dbprintWhitespace+string(candidate)+" is a homogeneous left divisor"); |
---|
2106 | hath = involution(lift(involution(candidate,invo),involution(h,invo))[1,1],invo); |
---|
2107 | recResult = extractHomogeneousDivisorsLeft(hath); |
---|
2108 | for (j = 1; j<=size(recResult); j++) |
---|
2109 | { |
---|
2110 | recResult[j] = insert(recResult[j],candidate); |
---|
2111 | } |
---|
2112 | result = result + recResult; |
---|
2113 | }//Candidate was really a left divisor |
---|
2114 | }//Finding candidates for homogeneous left divisors of h |
---|
2115 | if (size(result)==0) |
---|
2116 | { |
---|
2117 | return(list(list(h))); |
---|
2118 | } |
---|
2119 | return(result); |
---|
2120 | }//extractHomogeneousDivisorsLeft |
---|
2121 | |
---|
2122 | static proc extractHomogeneousDivisorsRight(poly h) |
---|
2123 | "INPUT: A polynomial h in the first Weyl algebra |
---|
2124 | OUTPUT: If h is homogeneous, then all factorizations of h are returned. |
---|
2125 | If h is inhomogeneous, then a list l is returned whose entries |
---|
2126 | are again lists k = [k_1,...,k_n], where k_1*...*k_n = h. |
---|
2127 | The entry k_1 is inhomogeneous and has no other homogeneous |
---|
2128 | right divisors any more. |
---|
2129 | All the other entries in k are homogeneous |
---|
2130 | polynomials. |
---|
2131 | " |
---|
2132 | {//extractHomogeneousDivisorsRight |
---|
2133 | int p=printlevel-voice+2; // for dbprint |
---|
2134 | string dbprintWhitespace = ""; |
---|
2135 | int i;int j; int k; |
---|
2136 | list result; |
---|
2137 | poly hath; |
---|
2138 | list recResult; |
---|
2139 | for (i = 1; i<=voice;i++) |
---|
2140 | {dbprintWhitespace = dbprintWhitespace + " ";} |
---|
2141 | if (homogwithorder(h,intvec(-1,1))) |
---|
2142 | {//given polynomial was homogeneous already |
---|
2143 | dbprint(p,dbprintWhitespace+"Polynomial was homogeneous. Just returning all factorizations."); |
---|
2144 | result = homogfacFirstWeyl_all(h); |
---|
2145 | for (i = 1; i<=size(result);i++) |
---|
2146 | {//removing the first entry (coefficient) from the list result |
---|
2147 | result[i] = delete(result[i],1); |
---|
2148 | }//removing the first entry (coefficient) from the list result |
---|
2149 | return(result); |
---|
2150 | }//given polynomial was homogeneous already |
---|
2151 | list hlist = homogDistribution(h); |
---|
2152 | dbprint(p,dbprintWhitespace+ " Computing factorizations of all homogeneous summands."); |
---|
2153 | for (i = 1; i<= size(hlist); i++) |
---|
2154 | { |
---|
2155 | hlist[i] = homogfacFirstWeyl_all(hlist[i][2]); |
---|
2156 | if (size(hlist[i][1])==1) |
---|
2157 | {//One homogeneous part just has a trivial factorization |
---|
2158 | if(hlist[i][1][1] == 0) |
---|
2159 | { |
---|
2160 | hlist = delete(hlist,i); |
---|
2161 | continue; |
---|
2162 | } |
---|
2163 | else |
---|
2164 | { |
---|
2165 | return(list(list(h))); |
---|
2166 | } |
---|
2167 | }//One homogeneous part just has a trivial factorization |
---|
2168 | } |
---|
2169 | dbprint(p,dbprintWhitespace+ " Done."); |
---|
2170 | dbprint(p,dbprintWhitespace+ " Trying to find right divisors"); |
---|
2171 | list alreadyConsideredCandidates; |
---|
2172 | poly candidate; |
---|
2173 | int isCandidate; |
---|
2174 | for (i = 1; i<=size(hlist[1]);i++) |
---|
2175 | {//Finding candidates for homogeneous left divisors of h |
---|
2176 | candidate = hlist[1][i][size(hlist[1][i])]; |
---|
2177 | isCandidate = 0; |
---|
2178 | for (j=1;j<=size(alreadyConsideredCandidates);j++) |
---|
2179 | { |
---|
2180 | if(alreadyConsideredCandidates[j] == candidate) |
---|
2181 | { |
---|
2182 | isCandidate =1; |
---|
2183 | break; |
---|
2184 | } |
---|
2185 | } |
---|
2186 | if(isCandidate) |
---|
2187 | { |
---|
2188 | i++; |
---|
2189 | continue; |
---|
2190 | } |
---|
2191 | else |
---|
2192 | { |
---|
2193 | alreadyConsideredCandidates = alreadyConsideredCandidates + list(candidate); |
---|
2194 | } |
---|
2195 | dbprint(p,dbprintWhitespace+"Checking if "+string(candidate)+" is a homogeneous r-divisor"); |
---|
2196 | for (j = 2; j<=size(hlist);j++) |
---|
2197 | {//Iterating through the other homogeneous parts |
---|
2198 | isCandidate = 0; |
---|
2199 | for(k=1; k<=size(hlist[j]);k++) |
---|
2200 | { |
---|
2201 | if(hlist[j][k][size(hlist[j][k])]==candidate) |
---|
2202 | { |
---|
2203 | isCandidate = 1; |
---|
2204 | break; |
---|
2205 | } |
---|
2206 | } |
---|
2207 | if(!isCandidate) |
---|
2208 | { |
---|
2209 | break; |
---|
2210 | } |
---|
2211 | }//Iterating through the other homogeneous parts |
---|
2212 | if(isCandidate) |
---|
2213 | {//candidate was really a left divisor |
---|
2214 | dbprint(p,dbprintWhitespace+string(candidate)+" is a homogeneous right divisor"); |
---|
2215 | hath = lift(candidate,h)[1,1]; |
---|
2216 | recResult = extractHomogeneousDivisorsRight(hath); |
---|
2217 | for (j = 1; j<=size(recResult); j++) |
---|
2218 | { |
---|
2219 | recResult[j] = insert(recResult[j],candidate,size(recResult[j])); |
---|
2220 | } |
---|
2221 | result = result + recResult; |
---|
2222 | }//Candidate was really a left divisor |
---|
2223 | }//Finding candidates for homogeneous left divisors of h |
---|
2224 | if (size(result)==0) |
---|
2225 | { |
---|
2226 | result = list(list(h)); |
---|
2227 | } |
---|
2228 | return(result); |
---|
2229 | }//extractHomogeneousDivisorsRight |
---|
2230 | |
---|
2231 | static proc fromZeroHomogToThetaPoly(poly h) |
---|
2232 | " |
---|
2233 | //DEPRECATED |
---|
2234 | INPUT: A polynomial h in the first Weyl algebra, homogeneous of degree 0 |
---|
2235 | OUTPUT: The ring Ktheta with a polynomial result representing h as polynomial in |
---|
2236 | theta |
---|
2237 | ASSUMPTIONS: |
---|
2238 | - h is homogeneous of degree 0 with respect to the [-1,1] weight vector |
---|
2239 | - The basering is the first Weyl algebra |
---|
2240 | " |
---|
2241 | {//proc fromZeroHomogToThetaPoly |
---|
2242 | int i; int j; |
---|
2243 | list mons; |
---|
2244 | if(!homogwithorder(h,intvec(-1,1))) |
---|
2245 | {//Input not a homogeneous polynomial causes an error |
---|
2246 | ERROR("The input was not a homogeneous polynomial"); |
---|
2247 | }//Input not a homogeneous polynomial causes an error |
---|
2248 | if(deg(h,intvec(-1,1))!=0) |
---|
2249 | {//Input does not have degree 0 |
---|
2250 | ERROR("The input did not have degree 0"); |
---|
2251 | }//Input does not have degree 0 |
---|
2252 | for(i = 1; i<=size(h);i++) |
---|
2253 | {//Putting the monomials in a list |
---|
2254 | mons = mons+list(h[i]); |
---|
2255 | }//Putting the monomials in a list |
---|
2256 | ring KTheta = 0,(x,y,theta),dp; |
---|
2257 | setring KTheta; |
---|
2258 | map thetamap = r,x,y; |
---|
2259 | list mons = thetamap(mons); |
---|
2260 | poly entry; |
---|
2261 | for (i = 1; i<=size(mons);i++) |
---|
2262 | {//transforming the monomials as monomials in theta |
---|
2263 | entry = leadcoef(mons[i]); |
---|
2264 | for (j = 0; j<leadexp(mons[i])[2];j++) |
---|
2265 | { |
---|
2266 | entry = entry * (theta-j); |
---|
2267 | } |
---|
2268 | mons[i] = entry; |
---|
2269 | }//transforming the monomials as monomials in theta |
---|
2270 | poly result = sum(mons); |
---|
2271 | keepring KTheta; |
---|
2272 | }//proc fromZeroHomogToThetaPoly |
---|
2273 | |
---|
2274 | static proc fromHomogDistributionListToPoly(list l) |
---|
2275 | "returns the corresponding polynomial to a given output of homogDistribution" |
---|
2276 | {//proc fromHomogDistributionListToPoly |
---|
2277 | poly result = 0; |
---|
2278 | int i; |
---|
2279 | for (i = 1; i<=size(l);i++) |
---|
2280 | { |
---|
2281 | result = result + l[i][2]; |
---|
2282 | } |
---|
2283 | return(result); |
---|
2284 | }//proc fromHomogDistributionListToPoly |
---|
2285 | |
---|
2286 | static proc divides(poly p1, poly p2,map invo, list #) |
---|
2287 | "Tests, whether p1 divides p2 either from left or from right. The involution invo is needed |
---|
2288 | for checking both sides. The optional argument is needed in order to also return the other factor. |
---|
2289 | RETURN: If no optional argument is given, it will just return 1 or 0. |
---|
2290 | Otherwise a list with at least one element |
---|
2291 | Case 1: p1 does not divide p2 from any side. Then the output will be the empty list. |
---|
2292 | Case 2: p2 does divide p2 from one side at least. |
---|
2293 | Then it returns a list with tuples p,q, such that p or q equals p1 and |
---|
2294 | pq = p2. |
---|
2295 | ASSUMPTIONS: - The map invo is an involution on the basering." |
---|
2296 | {//proc divides |
---|
2297 | list result = list(); |
---|
2298 | poly tempfactor; |
---|
2299 | if (involution(reduce(involution(p2,invo),std(involution(ideal(p1),invo))),invo)==0) |
---|
2300 | {//p1 is a left divisor |
---|
2301 | if(size(#)==0){return(1);} |
---|
2302 | tempfactor = involution(lift(involution(p1,invo),involution(p2,invo))[1,1],invo); |
---|
2303 | if (leadcoef(p1)<0 && leadcoef(tempfactor)<0) |
---|
2304 | {//both have a negative leading coefficient |
---|
2305 | result = result +list(list(1,-p1, -tempfactor)); |
---|
2306 | }//both have a negative leading coefficient |
---|
2307 | else |
---|
2308 | { |
---|
2309 | if (leadcoef(p1)*leadcoef(tempfactor)<0) |
---|
2310 | {//One of them has a negative leading coefficient |
---|
2311 | if (leadcoef(p1)<0) |
---|
2312 | { |
---|
2313 | result = result + list(list(-1,-p1,tempfactor)); |
---|
2314 | } |
---|
2315 | else |
---|
2316 | { |
---|
2317 | result = result + list(list(-1,p1,-tempfactor)); |
---|
2318 | } |
---|
2319 | }//One of them has a negative leading coefficient |
---|
2320 | else |
---|
2321 | {//no negative coefficient at all |
---|
2322 | result = result + list(list(1,p1,tempfactor)); |
---|
2323 | }//no negative coefficient at all |
---|
2324 | } |
---|
2325 | }//p1 is a left divisor |
---|
2326 | |
---|
2327 | if (reduce(p2,std(ideal(p1))) == 0) |
---|
2328 | {//p1 is a right divisor |
---|
2329 | if(size(#)==0){return(1);} |
---|
2330 | tempfactor = lift(p1, p2)[1,1]; |
---|
2331 | if (leadcoef(p1)<0 && leadcoef(tempfactor)<0) |
---|
2332 | {//both have a negative leading coefficient |
---|
2333 | result = result +list(list(1, -tempfactor,-p1)); |
---|
2334 | }//both have a negative leading coefficient |
---|
2335 | else |
---|
2336 | { |
---|
2337 | if (leadcoef(p1)*leadcoef(tempfactor)<0) |
---|
2338 | {//One of them has a negative leading coefficient |
---|
2339 | if (leadcoef(p1)<0) |
---|
2340 | { |
---|
2341 | result = result + list(list(-1,tempfactor,-p1)); |
---|
2342 | } |
---|
2343 | else |
---|
2344 | { |
---|
2345 | result = result + list(list(-1,-tempfactor,p1)); |
---|
2346 | } |
---|
2347 | }//One of them has a negative leading coefficient |
---|
2348 | else |
---|
2349 | {//no negative coefficient at all |
---|
2350 | result = result + list(list(1,tempfactor,p1)); |
---|
2351 | }//no negative coefficient at all |
---|
2352 | } |
---|
2353 | }//p1 is already a right divisor |
---|
2354 | if (size(#)==0){return(0);} |
---|
2355 | return(result); |
---|
2356 | }//proc divides |
---|
2357 | |
---|
2358 | static proc computeCombinationsMinMaxHomog(poly h) |
---|
2359 | "Input: A polynomial h in the first Weyl Algebra |
---|
2360 | Output: Combinations of the form (p_max + p_min)(q_max + q_min), such that p_max, p_min, |
---|
2361 | q_max and q_min are homogeneous and p_max*q_max equals the maximal homogeneous |
---|
2362 | part in h, and p_max * q_max equals the minimal homogeneous part in h. |
---|
2363 | h is not homogeneous. |
---|
2364 | "{//proc computeCombinationsMinMaxHomog |
---|
2365 | intvec ivm11 = intvec(-1,1); |
---|
2366 | intvec iv11 = intvec(1,1); |
---|
2367 | intvec iv10 = intvec(1,0); |
---|
2368 | intvec iv01 = intvec(0,1); |
---|
2369 | intvec iv1m1 = intvec(1,-1); |
---|
2370 | poly maxh = jet(h,deg(h,ivm11),ivm11)-jet(h,deg(h,ivm11)-1,ivm11); |
---|
2371 | poly minh = jet(h,deg(h,iv1m1),iv1m1)-jet(h,deg(h,iv1m1)-1,iv1m1); |
---|
2372 | list f1 = homogfacFirstWeyl_all(maxh); |
---|
2373 | list f2 = homogfacFirstWeyl_all(minh); |
---|
2374 | list result = list(); |
---|
2375 | int i; |
---|
2376 | int j; |
---|
2377 | //TODOs: Nur die kombinieren, die auch wirklich eine brauchbare Kombination sind. |
---|
2378 | //Also, wir duerfen nicht aus den Geraden herausfliegen |
---|
2379 | list pqmax = list(); |
---|
2380 | list pqmin = list(); |
---|
2381 | list tempList = list(); |
---|
2382 | |
---|
2383 | //First, we are going to deal with our most hated guys: The Coefficients. |
---|
2384 | // |
---|
2385 | list coeffTuplesMax = getAllCoeffTuplesComb(factorizeInt(number(f1[1][1]))); |
---|
2386 | //We can assume without loss of generality, that p_max has a |
---|
2387 | //nonnegative leading coefficient |
---|
2388 | for (i = 1; i<=size(coeffTuplesMax);i++) |
---|
2389 | {//Deleting all tuples with negative entries for p_max |
---|
2390 | if (coeffTuplesMax[i][1]<0) |
---|
2391 | { |
---|
2392 | coeffTuplesMax = delete(coeffTuplesMax,i); |
---|
2393 | continue; |
---|
2394 | } |
---|
2395 | }//Deleting all tuples with negative entries for p_max |
---|
2396 | list coeffTuplesMin = getAllCoeffTuplesComb(factorizeInt(number(f2[1][1]))); |
---|
2397 | |
---|
2398 | //Now, we will be actally dealing with the Combinations. |
---|
2399 | //Let's start with the pqmax |
---|
2400 | if (size(f1[1]) == 1) |
---|
2401 | {//the maximal homogeneous factor is a constant |
---|
2402 | pqmax = coeffTuplesMax; |
---|
2403 | }//the maximal homogeneous factor is a constant |
---|
2404 | else |
---|
2405 | {//the maximal homogeneous factor is not a constant |
---|
2406 | for (i = 1; i<=size(f1); i++) |
---|
2407 | {//We can forget about the first coefficient now. Therefore we will delete him from the list. |
---|
2408 | f1[i] = delete(f1[i],1); |
---|
2409 | if(size(f1[i])==1) |
---|
2410 | {//trivial thing |
---|
2411 | for (j = 1; j<=size(coeffTuplesMax); j++) |
---|
2412 | { |
---|
2413 | pqmax = pqmax + list(list(coeffTuplesMax[j][1],coeffTuplesMax[j][2]*f1[i][1])); |
---|
2414 | pqmax = pqmax + list(list(coeffTuplesMax[j][1]*f1[i][1],coeffTuplesMax[j][2])); |
---|
2415 | } |
---|
2416 | f1 = delete(f1,i); |
---|
2417 | continue; |
---|
2418 | }//trivial thing |
---|
2419 | }//We can forget about the first coefficient now. Therefore we will delete him from the list. |
---|
2420 | if (size(f1)>0) |
---|
2421 | { |
---|
2422 | tempList = getAllCombOfHomogFact(f1); |
---|
2423 | for (i = 1; i<=size(tempList); i++) |
---|
2424 | {//Every combination combined with the coefficient possibilities |
---|
2425 | for (j = 1; j<=size(coeffTuplesMax); j++) |
---|
2426 | {//iterating through the possible coefficient choices |
---|
2427 | pqmax = pqmax + list(list(coeffTuplesMax[j][1]*tempList[i][1], |
---|
2428 | coeffTuplesMax[j][2]*tempList[i][2])); |
---|
2429 | }//iterating through the possible coefficient choices |
---|
2430 | }//Every combination combined with the coefficient possibilities |
---|
2431 | for (i = 1; i<=size(coeffTuplesMax); i++) |
---|
2432 | { |
---|
2433 | pqmax = pqmax + list(list(coeffTuplesMax[i][1],maxh/coeffTuplesMax[i][1])); |
---|
2434 | pqmax = pqmax + list(list(maxh/coeffTuplesMax[i][2],coeffTuplesMax[i][2])); |
---|
2435 | } |
---|
2436 | } |
---|
2437 | }//the maximal homogeneous factor is not a constant |
---|
2438 | //Now we go to pqmin |
---|
2439 | if (size(f2[1]) == 1) |
---|
2440 | {//the minimal homogeneous factor is a constant |
---|
2441 | pqmin = coeffTuplesMin; |
---|
2442 | }//the minimal homogeneous factor is a constant |
---|
2443 | else |
---|
2444 | {//the minimal homogeneous factor is not a constant |
---|
2445 | for (i = 1; i<=size(f2); i++) |
---|
2446 | {//We can forget about the first coefficient now. Therefore we will delete him from the list. |
---|
2447 | f2[i] = delete(f2[i],1); |
---|
2448 | if(size(f2[i])==1) |
---|
2449 | {//trivial thing |
---|
2450 | for (j = 1; j<=size(coeffTuplesMin); j++) |
---|
2451 | { |
---|
2452 | pqmin = pqmin + list(list(coeffTuplesMin[j][1],coeffTuplesMin[j][2]*f2[i][1])); |
---|
2453 | pqmin = pqmin + list(list(coeffTuplesMin[j][1]*f2[i][1],coeffTuplesMin[j][2])); |
---|
2454 | } |
---|
2455 | f2 = delete(f2,i); |
---|
2456 | continue; |
---|
2457 | } |
---|
2458 | }//We can forget about the first coefficient now. Therefore we will delete him from the list. |
---|
2459 | if(size(f2)>0) |
---|
2460 | { |
---|
2461 | tempList = getAllCombOfHomogFact(f2); |
---|
2462 | for (i = 1; i<=size(tempList); i++) |
---|
2463 | {//Every combination combined with the coefficient possibilities |
---|
2464 | for (j = 1; j<=size(coeffTuplesMin); j++) |
---|
2465 | {//iterating through the possible coefficient choices |
---|
2466 | pqmin = pqmin + list(list(coeffTuplesMin[j][1]*tempList[i][1], |
---|
2467 | coeffTuplesMin[j][2]*tempList[i][2])); |
---|
2468 | }//iterating through the possible coefficient choices |
---|
2469 | }//Every combination combined with the coefficient possibilities |
---|
2470 | for (i = 1; i<=size(coeffTuplesMin); i++) |
---|
2471 | { |
---|
2472 | pqmin = pqmin + list(list(coeffTuplesMin[i][1],minh/coeffTuplesMin[i][1])); |
---|
2473 | pqmin = pqmin + list(list(minh/coeffTuplesMin[i][2],coeffTuplesMin[i][2])); |
---|
2474 | } |
---|
2475 | } |
---|
2476 | }//the minimal homogeneous factor is not a constant |
---|
2477 | |
---|
2478 | //and now we combine them together to obtain all possibilities. |
---|
2479 | for (i = 1; i<=size(pqmax); i++) |
---|
2480 | {//iterate over the maximal homogeneous combination possibilities |
---|
2481 | for (j = 1; j<=size(pqmin); j++) |
---|
2482 | {//iterate over the minimal homogeneous combiniation possibilities |
---|
2483 | if (deg(pqmax[i][1], ivm11)>=deg(pqmin[j][1],ivm11) and deg(pqmax[i][2], |
---|
2484 | ivm11)>=deg(pqmin[j][2],ivm11)) |
---|
2485 | { |
---|
2486 | if (pqmax[i][1]+pqmin[j][1]!=0 and pqmax[i][2]+pqmin[j][2]!=0) |
---|
2487 | { |
---|
2488 | |
---|
2489 | if (deg(h,ivm11)<=deg(h-(pqmax[i][1]+pqmin[j][1])*(pqmax[i][2]+pqmin[j][2]),ivm11)) |
---|
2490 | { |
---|
2491 | j++; |
---|
2492 | continue; |
---|
2493 | } |
---|
2494 | if (deg(h,iv1m1)<=deg(h-(pqmax[i][1]+pqmin[j][1])*(pqmax[i][2]+pqmin[j][2]),iv1m1)) |
---|
2495 | { |
---|
2496 | j++; |
---|
2497 | continue; |
---|
2498 | } |
---|
2499 | result = result +list(list(pqmax[i][1]+pqmin[j][1],pqmax[i][2]+pqmin[j][2])); |
---|
2500 | } |
---|
2501 | } |
---|
2502 | }//iterate over the minimal homogeneous combiniation possibilities |
---|
2503 | }//iterate over the maximal homogeneous combination possibilities |
---|
2504 | //Now deleting double entries |
---|
2505 | result = delete_dublicates_noteval(result); |
---|
2506 | return(result); |
---|
2507 | }//proc computeCombinationsMinMaxHomog |
---|
2508 | |
---|
2509 | static proc getAllCombOfHomogFact(list l) |
---|
2510 | "Gets called in computeCombinationsMinMaxHomog. It gets a list of different homogeneous |
---|
2511 | factorizations of |
---|
2512 | one homogeneous polynomial and returns the possibilities to combine them into two factors. |
---|
2513 | Assumptions: |
---|
2514 | - The list does not contain the first coefficient. |
---|
2515 | - The list contains at least one list with two elements." |
---|
2516 | {//proc getAllCombOfHomogFact |
---|
2517 | list result; |
---|
2518 | list leftAndRightHandSides; |
---|
2519 | int i; int j; |
---|
2520 | list tempset; |
---|
2521 | if (size(l)==1 and size(l[1])==2) |
---|
2522 | { |
---|
2523 | result = result + list(list(l[1][1],l[1][2])); |
---|
2524 | return(result); |
---|
2525 | } |
---|
2526 | leftAndRightHandSides = getPossibilitiesForRightSides(l); |
---|
2527 | for (i = 1; i<=size(leftAndRightHandSides); i++) |
---|
2528 | { |
---|
2529 | result =result+list(list(leftAndRightHandSides[i][1],product(leftAndRightHandSides[i][2][1]))); |
---|
2530 | //tidy up the right hand sides, because, if it is just one irreducible factor, we are done |
---|
2531 | for (j = 1; j<=size(leftAndRightHandSides[i][2]);j++) |
---|
2532 | {//Tidy up right hand sides |
---|
2533 | if (size(leftAndRightHandSides[i][2][j])<2) |
---|
2534 | {//Element can be dismissed |
---|
2535 | leftAndRightHandSides[i][2] = delete(leftAndRightHandSides[i][2],j); |
---|
2536 | continue; |
---|
2537 | }//Element can be dismissed |
---|
2538 | }//Tidy up right hand sides |
---|
2539 | if (size(leftAndRightHandSides[i][2])>0) |
---|
2540 | { |
---|
2541 | tempset = getAllCombOfHomogFact(leftAndRightHandSides[i][2]); |
---|
2542 | for (j = 1; j<=size(tempset);j++) |
---|
2543 | {//multiplying the first factor with the left hand side |
---|
2544 | result = result + list(list(leftAndRightHandSides[i][1]*tempset[j][1],tempset[j][2])); |
---|
2545 | }//multiplying the first factor with the left hand side |
---|
2546 | } |
---|
2547 | } |
---|
2548 | return(result); |
---|
2549 | }//proc getAllCombOfHomogFact |
---|
2550 | |
---|
2551 | static proc getPossibilitiesForRightSides(list l) |
---|
2552 | "Given a list of different factorizations l, this function returns a list of the form |
---|
2553 | (a,{(a_2,...,a_n)| (a,a_2,...,a_n) in A})" |
---|
2554 | {//getPossibilitiesForRightSide |
---|
2555 | list templ = l; |
---|
2556 | list result; |
---|
2557 | poly firstElement; |
---|
2558 | list rightSides; |
---|
2559 | list tempRightSide; |
---|
2560 | int i; int j; |
---|
2561 | while (size(templ)>0) |
---|
2562 | { |
---|
2563 | firstElement = templ[1][1]; |
---|
2564 | rightSides = list(); |
---|
2565 | for (i = 1; i<= size(templ); i++) |
---|
2566 | { |
---|
2567 | if (templ[i][1] == firstElement) |
---|
2568 | {//save the right sides |
---|
2569 | tempRightSide = list(); |
---|
2570 | for (j = 2; j<=size(templ[i]);j++) |
---|
2571 | { |
---|
2572 | tempRightSide = tempRightSide + list(templ[i][j]); |
---|
2573 | } |
---|
2574 | if (size(tempRightSide)!=0) |
---|
2575 | { |
---|
2576 | rightSides = rightSides + list(tempRightSide); |
---|
2577 | } |
---|
2578 | templ = delete(templ,i); |
---|
2579 | continue; |
---|
2580 | }//save the right sides |
---|
2581 | } |
---|
2582 | result = result + list(list(firstElement,rightSides)); |
---|
2583 | } |
---|
2584 | return(result); |
---|
2585 | }//getPossibilitiesForRightSide |
---|
2586 | |
---|
2587 | static proc getAllCoeffTuplesComb(list l)" |
---|
2588 | Given the output of factorizeInt ((a_1,...,a_n),(i_1,...,i_n)) , it returns all possible tuples |
---|
2589 | of the set {(a,b) | There exists an real N!=emptyset subset of {1,...,n}, such that |
---|
2590 | a = prod_{i \in N}a_i, b=prod_{i \not\in N} a_i} |
---|
2591 | Assumption: The list is sorted from smallest integer to highest. |
---|
2592 | - it is not the factorization of 0. |
---|
2593 | " |
---|
2594 | {//proc getAllCoeffTuplesComb |
---|
2595 | list result; |
---|
2596 | if (l[1][1] == 0) |
---|
2597 | { |
---|
2598 | ERROR("getAllCoeffTuplesComb: Zero Coefficients as leading and Tail Coeffs? |
---|
2599 | That is not possible. Something went wrong."); |
---|
2600 | } |
---|
2601 | if (size(l[1]) == 1) |
---|
2602 | {//Trivial Factorization, just 1 |
---|
2603 | if (l[1][1] == 1) |
---|
2604 | { |
---|
2605 | return(list(list(1,1),list(-1,-1))); |
---|
2606 | } |
---|
2607 | else |
---|
2608 | { |
---|
2609 | return(list(list(-1,1),list(1,-1))); |
---|
2610 | } |
---|
2611 | }//Trivial Factorization, just 1 |
---|
2612 | if (size(l[1]) == 2 and l[2][2]==1) |
---|
2613 | {//Just a prime number |
---|
2614 | if (l[1][1] == 1) |
---|
2615 | { |
---|
2616 | result = list(list(l[1][2],1),list(1,l[1][2])); |
---|
2617 | result = result + list(list(-l[1][2],-1),list(-1,-l[1][2])); |
---|
2618 | return(result); |
---|
2619 | } |
---|
2620 | else |
---|
2621 | { |
---|
2622 | result = list(list(l[1][2],-1),list(1,-l[1][2])); |
---|
2623 | result = result + list(list(-l[1][2],1),list(-1,l[1][2])); |
---|
2624 | return(result); |
---|
2625 | } |
---|
2626 | }//Just a prime number |
---|
2627 | //Now comes the interesting case: a product of primes |
---|
2628 | list tempPrimeFactors; |
---|
2629 | list tempPowersOfThem; |
---|
2630 | int i; |
---|
2631 | for (i = 2; i<=size(l[1]);i++) |
---|
2632 | {//Removing the starting 1 or -1 to get the N's |
---|
2633 | tempPrimeFactors[i-1] = l[1][i]; |
---|
2634 | tempPowersOfThem[i-1] = l[2][i]; |
---|
2635 | }//Removing the starting 1 or -1 to get the N's |
---|
2636 | list Ns = getAllSubsetsN(list(tempPrimeFactors,tempPowersOfThem)); |
---|
2637 | list tempTuples; |
---|
2638 | number productOfl = multiplyFactIntOutput(l); |
---|
2639 | if (productOfl<0){productOfl = -productOfl;} |
---|
2640 | tempTuples = tempTuples + list(list(1,productOfl),list(productOfl,1)); |
---|
2641 | for (i = 1; i<=size(Ns); i++) |
---|
2642 | { |
---|
2643 | if (productOfl/Ns[i]>Ns[i]) |
---|
2644 | {//TODO: BEWEISEN, dass das die einzigen Combos sind |
---|
2645 | tempTuples = tempTuples + list(list(Ns[i],productOfl/Ns[i]),list(productOfl/Ns[i],Ns[i])); |
---|
2646 | }//TODO: BEWEISEN, dass das die einzigen Combos sind |
---|
2647 | if (productOfl/Ns[i]==Ns[i]) |
---|
2648 | { |
---|
2649 | tempTuples = tempTuples + list(list(Ns[i],Ns[i])); |
---|
2650 | } |
---|
2651 | } |
---|
2652 | //And now, it just remains to get the -1s and 1-s correctly to the tuples |
---|
2653 | list tempEntry; |
---|
2654 | if (l[1][1] == 1) |
---|
2655 | { |
---|
2656 | for (i = 1; i<=size(tempTuples);i++) |
---|
2657 | {//Adding everything to result |
---|
2658 | tempEntry = tempTuples[i]; |
---|
2659 | result = result + list(tempEntry); |
---|
2660 | result = result + list(list(-tempEntry[1], -tempEntry[2])); |
---|
2661 | }//Adding everyThing to Result |
---|
2662 | } |
---|
2663 | else |
---|
2664 | { |
---|
2665 | for (i = 1; i<=size(tempTuples);i++) |
---|
2666 | {//Adding everything to result |
---|
2667 | tempEntry = tempTuples[i]; |
---|
2668 | result = result + list(list(tempEntry[1],-tempEntry[2])); |
---|
2669 | result = result + list(list(-tempEntry[1], tempEntry[2])); |
---|
2670 | }//Adding everyThing to Result |
---|
2671 | } |
---|
2672 | return(result); |
---|
2673 | }//proc getAllCoeffTuplesComb |
---|
2674 | |
---|
2675 | static proc contains(list l, int elem) |
---|
2676 | "Assumption: l is sorted" |
---|
2677 | {//Binary Search in list |
---|
2678 | if (size(l)<=1) |
---|
2679 | { |
---|
2680 | if(size(l) == 0){return(0);} |
---|
2681 | if (l[1]!=elem){return(0);} |
---|
2682 | else{return(1);} |
---|
2683 | } |
---|
2684 | int imax = size(l); |
---|
2685 | int imin = 1; |
---|
2686 | int imid; |
---|
2687 | while(imax >= imin) |
---|
2688 | { |
---|
2689 | imid = (imin + imax)/2; |
---|
2690 | if (l[imid] == elem){return(1);} |
---|
2691 | if (l[imid] <elem) {imin = imid +1;} |
---|
2692 | else{imax = imid -1;} |
---|
2693 | } |
---|
2694 | return(0) |
---|
2695 | }//Binary Search in list |
---|
2696 | |
---|
2697 | static proc getAllSubsetsN(list l) |
---|
2698 | " |
---|
2699 | Assumptions: |
---|
2700 | - The list is containing two lists. They can be assumed to be outputs of the function |
---|
2701 | factorizeInt. They have at least one entry. If it is exactly one entry, the second intvec should |
---|
2702 | contain a value at least 2. |
---|
2703 | " |
---|
2704 | { |
---|
2705 | list primeFactors=l[1]; |
---|
2706 | list powersOfThem = l[2]; |
---|
2707 | int i;int j; |
---|
2708 | //Casting the entries to be numbers |
---|
2709 | for (i=1; i<=size(primeFactors); i++) |
---|
2710 | { |
---|
2711 | primeFactors[i] = number(primeFactors[i]); |
---|
2712 | powersOfThem[i] = number(powersOfThem[i]); |
---|
2713 | } |
---|
2714 | |
---|
2715 | //Done |
---|
2716 | list result; |
---|
2717 | list tempPrimeFactors; |
---|
2718 | list tempPowersOfThem; |
---|
2719 | list tempset; |
---|
2720 | if (sum(powersOfThem) <=2) |
---|
2721 | {//Easy Case |
---|
2722 | return(list(primeFactors[1])); |
---|
2723 | }//Easy Case |
---|
2724 | if (size(primeFactors)==1) |
---|
2725 | {//Also Easy Case |
---|
2726 | for (j = 1; j<powersOfThem[1]; j++) |
---|
2727 | { |
---|
2728 | result = result + list(primeFactors[1]^j); |
---|
2729 | } |
---|
2730 | return(result); |
---|
2731 | }//Also Easy Case |
---|
2732 | for (i = 1; i<= size(primeFactors); i++) |
---|
2733 | {//Going through every entry |
---|
2734 | result = result + list(primeFactors[i]); |
---|
2735 | if (i == size(primeFactors)) |
---|
2736 | { |
---|
2737 | for (j = 1;j<powersOfThem[i];j++) |
---|
2738 | { |
---|
2739 | result = result + list (primeFactors[i]^j); |
---|
2740 | } |
---|
2741 | break; |
---|
2742 | } |
---|
2743 | if (powersOfThem[i]==1) |
---|
2744 | { |
---|
2745 | for (j = i+1;j<=size(primeFactors);j++) |
---|
2746 | { |
---|
2747 | tempPrimeFactors[j-i] = primeFactors[j]; |
---|
2748 | tempPowersOfThem[j-i] = powersOfThem[j]; |
---|
2749 | } |
---|
2750 | } |
---|
2751 | else |
---|
2752 | { |
---|
2753 | for (j = i; j<=size(primeFactors);j++) |
---|
2754 | { |
---|
2755 | tempPrimeFactors[j-i+1] = primeFactors[j]; |
---|
2756 | tempPowersOfThem[j-i+1] = powersOfThem[j]; |
---|
2757 | tempPowersOfThem[1] = tempPowersOfThem[1]-1; |
---|
2758 | } |
---|
2759 | } |
---|
2760 | tempset = getAllSubsetsN(list(tempPrimeFactors,tempPowersOfThem)); |
---|
2761 | for (j = 1; j<=size(tempset); j++) |
---|
2762 | { |
---|
2763 | result = result +list((tempset[j])*(primeFactors[i])); |
---|
2764 | } |
---|
2765 | }//Going through every entry |
---|
2766 | result = sort(result)[1]; |
---|
2767 | result = delete_dublicates_noteval(result); |
---|
2768 | return(result); |
---|
2769 | } |
---|
2770 | |
---|
2771 | static proc multiplyFactIntOutput(list l) |
---|
2772 | "Given the output of factorizeInt, this method computes the product of it." |
---|
2773 | {//proc multiplyFactIntOutput |
---|
2774 | int i; |
---|
2775 | number result = 1; |
---|
2776 | for (i = 1; i<=size(l[1]); i++) |
---|
2777 | { |
---|
2778 | result = result*(l[1][i])^(l[2][i]); |
---|
2779 | } |
---|
2780 | return(result); |
---|
2781 | }//proc multiplyFactIntOutput |
---|
2782 | |
---|
2783 | static proc fromListToIntvec(list l) |
---|
2784 | "Converter from List to intvec" |
---|
2785 | { |
---|
2786 | intvec result; int i; |
---|
2787 | for (i = 1; i<=size(l); i++) |
---|
2788 | { |
---|
2789 | result[i] = l[i]; |
---|
2790 | } |
---|
2791 | return(result); |
---|
2792 | } |
---|
2793 | |
---|
2794 | static proc fromIntvecToList(intvec l)" |
---|
2795 | Converter from intvec to list" |
---|
2796 | {//proc fromIntvecToList |
---|
2797 | list result = list(); |
---|
2798 | int i; |
---|
2799 | for (i = size(l); i>=1; i--) |
---|
2800 | { |
---|
2801 | result = insert(result, l[i]); |
---|
2802 | } |
---|
2803 | return(result); |
---|
2804 | }//proc fromIntvecToList |
---|
2805 | |
---|
2806 | |
---|
2807 | static proc factorizeInt(number n) |
---|
2808 | "Given an integer n, factorizeInt computes its factorization. The output is a list |
---|
2809 | containing two intvecs. The first contains the prime factors, the second its powers. |
---|
2810 | ASSUMPTIONS: |
---|
2811 | - n is given as integer number |
---|
2812 | "{ |
---|
2813 | if (n==0) |
---|
2814 | {return(list(list(0),list(1)));} |
---|
2815 | int i; |
---|
2816 | list temp = primefactors(n); |
---|
2817 | if (n<0) |
---|
2818 | {list result = list(list(-1),list(1));} |
---|
2819 | else |
---|
2820 | {list result = list(list(1),list(1));} |
---|
2821 | result[1] = result[1] + temp[1]; |
---|
2822 | result[2] = result[2] + temp[2]; |
---|
2823 | return(result); |
---|
2824 | } |
---|
2825 | |
---|
2826 | |
---|
2827 | static proc homogDistribution(poly h) |
---|
2828 | "Input: A polynomial in the first Weyl Algebra. |
---|
2829 | Output: A two-dimensional list of the following form. Every sublist contains exactly two entries. |
---|
2830 | One for the Z-degree of the corresponding homogeneous part (integer), and the homogeneous |
---|
2831 | polynomial itself, and those sublists are oredered by ascending degree. |
---|
2832 | For example a call of homogDistribution(x+d+1) would have the output |
---|
2833 | [1]: |
---|
2834 | [1]: |
---|
2835 | -1 |
---|
2836 | [2]: |
---|
2837 | x |
---|
2838 | [2]: |
---|
2839 | [1]: |
---|
2840 | 0 |
---|
2841 | [2]: |
---|
2842 | 1 |
---|
2843 | [3]: |
---|
2844 | [1]: |
---|
2845 | 1 |
---|
2846 | [2]: |
---|
2847 | d |
---|
2848 | "{//homogDistribution |
---|
2849 | if (h == 0) |
---|
2850 | {//trivial case where input is 0 |
---|
2851 | return(list(list(0,0))); |
---|
2852 | }//trivial case where input is 0 |
---|
2853 | if (!isWeyl()) |
---|
2854 | {//Our basering is not the Weyl algebra |
---|
2855 | ERROR("Ring was not the first Weyl algebra"); |
---|
2856 | return(list()); |
---|
2857 | }//Our basering is not the Weyl algebra |
---|
2858 | if(nvars(basering)!=2) |
---|
2859 | {//Our basering is the Weyl algebra, but not the first |
---|
2860 | ERROR("Ring is not the first Weyl algebra"); |
---|
2861 | return(list()); |
---|
2862 | }//Our basering is the Weyl algebra, but not the first |
---|
2863 | intvec ivm11 = intvec(-1,1); |
---|
2864 | intvec iv1m1 = intvec(1,-1); |
---|
2865 | poly tempH = h; |
---|
2866 | poly minh; |
---|
2867 | list result = list(); |
---|
2868 | int nextExpectedDegree = -deg(tempH,iv1m1); |
---|
2869 | while (tempH != 0) |
---|
2870 | { |
---|
2871 | minh = jet(tempH,deg(tempH,iv1m1),iv1m1)-jet(tempH,deg(tempH,iv1m1)-1,iv1m1); |
---|
2872 | while (deg(minh,ivm11)>nextExpectedDegree) |
---|
2873 | {//filling empty homogeneous spaces with 0 |
---|
2874 | result = result + list(list(nextExpectedDegree,0)); |
---|
2875 | nextExpectedDegree = nextExpectedDegree +1; |
---|
2876 | }//filling empty homogeneous spaces with 0 |
---|
2877 | result = result + list(list(deg(minh,ivm11),minh)); |
---|
2878 | tempH = tempH - minh; |
---|
2879 | nextExpectedDegree = nextExpectedDegree +1; |
---|
2880 | } |
---|
2881 | return(result); |
---|
2882 | }//homogDistribution |
---|
2883 | |
---|
2884 | static proc countHomogParts(poly h) |
---|
2885 | "Counts the homogeneous parts of a given polynomial h" |
---|
2886 | { |
---|
2887 | int i; |
---|
2888 | list outPutHD = homogDistribution(h); |
---|
2889 | int result = 0; |
---|
2890 | for (i = 1; i <=size(outPutHD); i++) |
---|
2891 | { |
---|
2892 | if (outPutHD[i][2] != 0){result++;} |
---|
2893 | } |
---|
2894 | return(result); |
---|
2895 | } |
---|
2896 | |
---|
2897 | ////////////////////////////////////////////////// |
---|
2898 | /////BRANDNEW!!!!//////////////////// |
---|
2899 | ////////////////////////////////////////////////// |
---|
2900 | |
---|
2901 | //================================================== |
---|
2902 | /*Singular has no way implemented to test polynomials |
---|
2903 | for homogenity with respect to a weight vector. |
---|
2904 | The following procedure does exactly this*/ |
---|
2905 | static proc homogwithorder(poly h, intvec weights) |
---|
2906 | {//proc homogwithorder |
---|
2907 | if(size(weights) != nvars(basering)) |
---|
2908 | {//The user does not know how many variables the current ring has |
---|
2909 | return(0); |
---|
2910 | }//The user does not know how many variables the current ring has |
---|
2911 | int i; |
---|
2912 | int dofp = deg(h,weights); //degree of polynomial |
---|
2913 | for (i = 1; i<=size(h);i++) |
---|
2914 | { |
---|
2915 | if (deg(h[i],weights)!=dofp) |
---|
2916 | { |
---|
2917 | return(0); |
---|
2918 | } |
---|
2919 | } |
---|
2920 | return(1); |
---|
2921 | }//proc homogwithorder |
---|
2922 | |
---|
2923 | //================================================== |
---|
2924 | //Testfac: Given a list with different factorizations of |
---|
2925 | // one polynomial, the following procedure checks |
---|
2926 | // whether they all refer to the same polynomial. |
---|
2927 | // If they do, the output will be a list, that contains |
---|
2928 | // the product of each factorization. If not, the empty |
---|
2929 | // list will be returned. |
---|
2930 | // If the optional argument # is given (i.e. the polynomial |
---|
2931 | // which is factorized by the elements of the given list), |
---|
2932 | // then we look, if the entries are factorizations of p |
---|
2933 | // and if not, a list with the products subtracted by p |
---|
2934 | // will be returned |
---|
2935 | proc testNCfac(list l, list #) |
---|
2936 | "USAGE: testNCfac(l[,p,b]); l is a list, p is an optional poly, b is 1 or 0 |
---|
2937 | RETURN: Case 1: No optional argument. In this case the output is 1, if the |
---|
2938 | entries in the given list represent the same polynomial or 0 |
---|
2939 | otherwise. |
---|
2940 | Case 2: One optional argument p is given. In this case it returns 1, |
---|
2941 | if all the entries in l are factorizations of p, otherwise 0. |
---|
2942 | Case 3: Second optional b is given. In this case a list is returned |
---|
2943 | containing the difference between the product of each entry in |
---|
2944 | l and p. |
---|
2945 | ASSUME: basering is the first Weyl algebra, the entries of l are polynomials |
---|
2946 | PURPOSE: Checks whether a list of factorizations contains factorizations of |
---|
2947 | the same element in the first Weyl algebra |
---|
2948 | THEORY: @code{testNCfac} multiplies out each factorization and checks whether |
---|
2949 | each factorization was a factorization of the same element. |
---|
2950 | @* - if there is only a list given, the output will be 0, if it |
---|
2951 | does not contain factorizations of the same element. Otherwise the output |
---|
2952 | will be 1. |
---|
2953 | @* - if there is a polynomial in the second argument, then the procedure checks |
---|
2954 | whether the given list contains factorizations of this polynomial. If it |
---|
2955 | does, then the output depends on the third argument. If it is not given, |
---|
2956 | the procedure will check whether the factorizations in the list |
---|
2957 | l are associated to this polynomial and return either 1 or 0, respectively. |
---|
2958 | If the third argument is given, the output will be a list with |
---|
2959 | the length of the given one and in each entry is the product of one |
---|
2960 | entry in l subtracted by the polynomial. |
---|
2961 | EXAMPLE: example testNCfac; shows examples |
---|
2962 | SEE ALSO: facFirstWeyl, facSubWeyl, facFirstShift |
---|
2963 | "{//proc testfac |
---|
2964 | int p = printlevel - voice + 2; |
---|
2965 | int i; |
---|
2966 | string dbprintWhitespace = ""; |
---|
2967 | for (i = 1; i<=voice;i++) |
---|
2968 | {dbprintWhitespace = dbprintWhitespace + " ";} |
---|
2969 | dbprint(p,dbprintWhitespace + " Checking the input"); |
---|
2970 | if (size(l)==0) |
---|
2971 | {//The empty list is given |
---|
2972 | dbprint(p,dbprintWhitespace + " Given list was empty"); |
---|
2973 | return(list()); |
---|
2974 | }//The empty list is given |
---|
2975 | if (size(#)>2) |
---|
2976 | {//We want max. two optional arguments |
---|
2977 | dbprint(p,dbprintWhitespace + " More than two optional arguments"); |
---|
2978 | return(list()); |
---|
2979 | }//We want max. two optional arguments |
---|
2980 | dbprint(p,dbprintWhitespace + " Done"); |
---|
2981 | list result; |
---|
2982 | int j; |
---|
2983 | if (size(#)==0) |
---|
2984 | {//No optional argument is given |
---|
2985 | dbprint(p,dbprintWhitespace + " No optional arguments"); |
---|
2986 | int valid = 1; |
---|
2987 | for (i = size(l);i>=1;i--) |
---|
2988 | {//iterate over the elements of the given list |
---|
2989 | if (size(result)>0) |
---|
2990 | { |
---|
2991 | if (product(l[i])!=result[size(l)-i]) |
---|
2992 | { |
---|
2993 | valid = 0; |
---|
2994 | break; |
---|
2995 | } |
---|
2996 | } |
---|
2997 | result = insert(result, product(l[i])); |
---|
2998 | }//iterate over the elements of the given list |
---|
2999 | return(valid); |
---|
3000 | }//No optional argument is given |
---|
3001 | else |
---|
3002 | { |
---|
3003 | dbprint(p,dbprintWhitespace + " Optional arguments are given."); |
---|
3004 | int valid = 1; |
---|
3005 | for (i = size(l);i>=1;i--) |
---|
3006 | {//iterate over the elements of the given list |
---|
3007 | if (product(l[i])!=#[1]) |
---|
3008 | { |
---|
3009 | valid = 0; |
---|
3010 | } |
---|
3011 | result = insert(result, product(l[i])-#[1]); |
---|
3012 | }//iterate over the elements of the given list |
---|
3013 | if(size(#)==2) |
---|
3014 | { |
---|
3015 | dbprint(p,dbprintWhitespace + " A third argument is given. Output is a list now."); |
---|
3016 | return(result); |
---|
3017 | } |
---|
3018 | return(valid); |
---|
3019 | } |
---|
3020 | }//proc testfac |
---|
3021 | example |
---|
3022 | { |
---|
3023 | "EXAMPLE:";echo=2; |
---|
3024 | ring r = 0,(x,y),dp; |
---|
3025 | def R = nc_algebra(1,1); |
---|
3026 | setring R; |
---|
3027 | poly h = (x^2*y^2+1)*(x^2); |
---|
3028 | def t1 = facFirstWeyl(h); |
---|
3029 | //fist a correct list |
---|
3030 | testNCfac(t1); |
---|
3031 | //now a correct list with the factorized polynomial |
---|
3032 | testNCfac(t1,h); |
---|
3033 | //now we put in an incorrect list without a polynomial |
---|
3034 | t1[3][3] = y; |
---|
3035 | testNCfac(t1); |
---|
3036 | // take h as additional input |
---|
3037 | testNCfac(t1,h); |
---|
3038 | // take h as additional input and output list of differences |
---|
3039 | testNCfac(t1,h,1); |
---|
3040 | } |
---|
3041 | //================================================== |
---|
3042 | //Procedure facSubWeyl: |
---|
3043 | //This procedure serves the purpose to compute a |
---|
3044 | //factorization of a given polynomial in a ring, whose subring |
---|
3045 | //is the first Weyl algebra. The polynomial must only contain |
---|
3046 | //the two arguments, which are also given by the user. |
---|
3047 | |
---|
3048 | proc facSubWeyl(poly h, X, D) |
---|
3049 | "USAGE: facSubWeyl(h,x,y); h, X, D polynomials |
---|
3050 | RETURN: list |
---|
3051 | ASSUME: X and D are variables of a basering, which satisfy DX = XD +1. |
---|
3052 | @* That is, they generate the copy of the first Weyl algebra in a basering. |
---|
3053 | @* Moreover, h is a polynomial in X and D only. |
---|
3054 | PURPOSE: compute factorizations of the polynomial, which depends on X and D. |
---|
3055 | EXAMPLE: example facSubWeyl; shows examples |
---|
3056 | SEE ALSO: facFirstWeyl, testNCfac, facFirstShift |
---|
3057 | "{ |
---|
3058 | int p = printlevel - voice + 2; |
---|
3059 | dbprint(p," Start initial Checks of the input."); |
---|
3060 | // basering can be anything having a Weyl algebra as subalgebra |
---|
3061 | def @r = basering; |
---|
3062 | //We begin to check the input for assumptions |
---|
3063 | // which are: X,D are vars of the basering, |
---|
3064 | if ( (isVar(X)!=1) || (isVar(D)!=1) || (size(X)>1) || (size(D)>1) || |
---|
3065 | (leadcoef(X) != number(1)) || (leadcoef(D) != number(1)) ) |
---|
3066 | { |
---|
3067 | ERROR("expected pure variables as generators of a subalgebra"); |
---|
3068 | } |
---|
3069 | // Weyl algebra: |
---|
3070 | poly w = D*X-X*D-1; // [D,X]=1 |
---|
3071 | poly u = D*X-X*D+1; // [X,D]=1 |
---|
3072 | if (u*w!=0) |
---|
3073 | { |
---|
3074 | // that is no combination gives Weyl |
---|
3075 | ERROR("2nd and 3rd argument do not generate a Weyl algebra"); |
---|
3076 | } |
---|
3077 | // one of two is correct |
---|
3078 | int isReverted = 0; // Reverted Weyl if dx=xd-1 holds |
---|
3079 | if (u==0) |
---|
3080 | { |
---|
3081 | isReverted = 1; |
---|
3082 | } |
---|
3083 | // else: do nothing |
---|
3084 | // DONE with assumptions, Input successfully checked |
---|
3085 | dbprint(p," Successful"); |
---|
3086 | intvec lexpofX = leadexp(X); |
---|
3087 | intvec lexpofD = leadexp(D); |
---|
3088 | int varnumX=1; |
---|
3089 | int varnumD=1; |
---|
3090 | while(lexpofX[varnumX] != 1) |
---|
3091 | { |
---|
3092 | varnumX++; |
---|
3093 | } |
---|
3094 | while(lexpofD[varnumD] != 1) |
---|
3095 | { |
---|
3096 | varnumD++; |
---|
3097 | } |
---|
3098 | /* VL : to add printlevel stuff */ |
---|
3099 | dbprint(p," Change positions of the two variables in the list, if needed"); |
---|
3100 | if (isReverted) |
---|
3101 | { |
---|
3102 | ring firstweyl = 0,(var(varnumD),var(varnumX)),dp; |
---|
3103 | def Firstweyl = nc_algebra(1,1); |
---|
3104 | setring Firstweyl; |
---|
3105 | ideal M = 0:nvars(@r); |
---|
3106 | M[varnumX]=var(2); |
---|
3107 | M[varnumD]=var(1); |
---|
3108 | map Q = @r,M; |
---|
3109 | poly h= Q(h); |
---|
3110 | } |
---|
3111 | else |
---|
3112 | { // that is unReverted |
---|
3113 | ring firstweyl = 0,(var(varnumX),var(varnumD)),dp; |
---|
3114 | def Firstweyl = nc_algebra(1,1); |
---|
3115 | setring Firstweyl; |
---|
3116 | poly h= imap(@r,h); |
---|
3117 | } |
---|
3118 | dbprint(p," Done!"); |
---|
3119 | list result = facFirstWeyl(h); |
---|
3120 | setring @r; |
---|
3121 | list result; |
---|
3122 | if (isReverted) |
---|
3123 | { |
---|
3124 | // map swap back |
---|
3125 | ideal M; M[1] = var(varnumD); M[2] = var(varnumX); |
---|
3126 | map S = Firstweyl, M; |
---|
3127 | result = S(result); |
---|
3128 | } |
---|
3129 | else |
---|
3130 | { |
---|
3131 | // that is unReverted |
---|
3132 | result = imap(Firstweyl,result); |
---|
3133 | } |
---|
3134 | return(result); |
---|
3135 | }//proc facSubWeyl |
---|
3136 | example |
---|
3137 | { |
---|
3138 | "EXAMPLE:";echo=2; |
---|
3139 | ring r = 0,(x,y,z),dp; |
---|
3140 | matrix D[3][3]; D[1,3]=-1; |
---|
3141 | def R = nc_algebra(1,D); // x,z generate Weyl subalgebra |
---|
3142 | setring R; |
---|
3143 | poly h = (x^2*z^2+x)*x; |
---|
3144 | list fact1 = facSubWeyl(h,x,z); |
---|
3145 | // compare with facFirstWeyl: |
---|
3146 | ring s = 0,(z,x),dp; |
---|
3147 | def S = nc_algebra(1,1); setring S; |
---|
3148 | poly h = (x^2*z^2+x)*x; |
---|
3149 | list fact2 = facFirstWeyl(h); |
---|
3150 | map F = R,x,0,z; |
---|
3151 | list fact1 = F(fact1); // it is identical to list fact2 |
---|
3152 | testNCfac(fact1); // check the correctness again |
---|
3153 | } |
---|
3154 | //================================================== |
---|
3155 | |
---|
3156 | //================================================== |
---|
3157 | //************From here: Shift-Algebra************** |
---|
3158 | //================================================== |
---|
3159 | //==================================================* |
---|
3160 | //one factorization of a homogeneous polynomial |
---|
3161 | //in the first Shift Algebra |
---|
3162 | static proc homogfacFirstShift(poly h) |
---|
3163 | {//proc homogfacFirstShift |
---|
3164 | int p=printlevel-voice+2; //for dbprint |
---|
3165 | def r = basering; |
---|
3166 | poly hath; |
---|
3167 | intvec iv01 = intvec(0,1); |
---|
3168 | int i; int j; |
---|
3169 | string dbprintWhitespace = ""; |
---|
3170 | for (i = 1; i<=voice;i++) |
---|
3171 | {dbprintWhitespace = dbprintWhitespace + " ";} |
---|
3172 | if (!homogwithorder(h,iv01)) |
---|
3173 | {//The given polynomial is not homogeneous |
---|
3174 | ERROR("The given polynomial is not homogeneous."); |
---|
3175 | return(list()); |
---|
3176 | }//The given polynomial is not homogeneous |
---|
3177 | if (h==0) |
---|
3178 | { |
---|
3179 | return(list(0)); |
---|
3180 | } |
---|
3181 | list result; |
---|
3182 | int m = deg(h,iv01); |
---|
3183 | dbprint(p,dbprintWhitespace+" exclude the homogeneous part of deg. 0"); |
---|
3184 | if (m>0) |
---|
3185 | {//The degree is not zero |
---|
3186 | hath = lift(var(2)^m,h)[1,1]; |
---|
3187 | for (i = 1; i<=m;i++) |
---|
3188 | { |
---|
3189 | result = result + list(var(2)); |
---|
3190 | } |
---|
3191 | }//The degree is not zero |
---|
3192 | else |
---|
3193 | {//The degree is zero |
---|
3194 | hath = h; |
---|
3195 | }//The degree is zero |
---|
3196 | ring tempRing = 0,(x),dp; |
---|
3197 | setring tempRing; |
---|
3198 | map thetamap = r,x,1; |
---|
3199 | poly hath = thetamap(hath); |
---|
3200 | dbprint(p,dbprintWhitespace+" Factorize it using commutative factorization."); |
---|
3201 | list azeroresult = factorize(hath); |
---|
3202 | list azeroresult_return_form; |
---|
3203 | for (i = 1; i<=size(azeroresult[1]);i++) |
---|
3204 | {//rewrite the result of the commutative factorization |
---|
3205 | for (j = 1; j <= azeroresult[2][i];j++) |
---|
3206 | { |
---|
3207 | azeroresult_return_form = azeroresult_return_form + list(azeroresult[1][i]); |
---|
3208 | } |
---|
3209 | }//rewrite the result of the commutative factorization |
---|
3210 | setring(r); |
---|
3211 | map finalmap = tempRing,var(1); |
---|
3212 | list tempresult = finalmap(azeroresult_return_form); |
---|
3213 | result = tempresult+result; |
---|
3214 | return(result); |
---|
3215 | }//proc homogfacFirstShift |
---|
3216 | |
---|
3217 | //================================================== |
---|
3218 | //Computes all possible homogeneous factorizations |
---|
3219 | static proc homogfacFirstShift_all(poly h) |
---|
3220 | {//proc HomogfacFirstShiftAll |
---|
3221 | int p=printlevel-voice+2; //for dbprint |
---|
3222 | intvec iv11 = intvec(1,1); |
---|
3223 | if (deg(h,iv11) <= 0 ) |
---|
3224 | {//h is a constant |
---|
3225 | return(list(list(h))); |
---|
3226 | }//h is a constant |
---|
3227 | def r = basering; |
---|
3228 | list one_hom_fac; //stands for one homogeneous factorization |
---|
3229 | int i; int j; int k; |
---|
3230 | string dbprintWhitespace = ""; |
---|
3231 | for (i = 1; i<=voice;i++) |
---|
3232 | {dbprintWhitespace = dbprintWhitespace + " ";} |
---|
3233 | int shiftcounter; |
---|
3234 | //Compute again a homogeneous factorization |
---|
3235 | dbprint(p,dbprintWhitespace+" Computing one homog. factorization of the polynomial"); |
---|
3236 | one_hom_fac = homogfacFirstShift(h); |
---|
3237 | one_hom_fac = delete(one_hom_fac,1); |
---|
3238 | if (size(one_hom_fac) == 0) |
---|
3239 | {//there is no homogeneous factorization or the polynomial was not homogeneous |
---|
3240 | return(list()); |
---|
3241 | }//there is no homogeneous factorization or the polynomial was not homogeneous |
---|
3242 | dbprint(p,dbprintWhitespace+" Permuting the 0-homogeneous part with the s"); |
---|
3243 | list result = permpp(one_hom_fac); |
---|
3244 | for (i = 1; i<=size(result);i++) |
---|
3245 | { |
---|
3246 | shiftcounter = 0; |
---|
3247 | for (j = 1; j<=size(result[i]); j++) |
---|
3248 | { |
---|
3249 | if (result[i][j]==var(2)) |
---|
3250 | { |
---|
3251 | shiftcounter++; |
---|
3252 | } |
---|
3253 | else |
---|
3254 | { |
---|
3255 | result[i][j] = subst(result[i][j], var(1), var(1)-shiftcounter); |
---|
3256 | } |
---|
3257 | } |
---|
3258 | result[i] = insert(result[i],1); |
---|
3259 | } |
---|
3260 | dbprint(p,dbprintWhitespace+" Deleting double entries in the resulting list"); |
---|
3261 | result = delete_dublicates_noteval(result); |
---|
3262 | return(result); |
---|
3263 | }//proc HomogfacFirstShiftAll |
---|
3264 | |
---|
3265 | //================================================== |
---|
3266 | //factorization of the first Shift Algebra |
---|
3267 | proc facFirstShift(poly h) |
---|
3268 | "USAGE: facFirstShift(h); h a polynomial in the first shift algebra |
---|
3269 | RETURN: list |
---|
3270 | PURPOSE: compute all factorizations of a polynomial in the first shift algebra |
---|
3271 | THEORY: Implements the new algorithm by A. Heinle and V. Levandovskyy, see the thesis of A. Heinle |
---|
3272 | ASSUME: basering is the first shift algebra |
---|
3273 | NOTE: Every entry of the output list is a list with factors for one possible factorization. |
---|
3274 | EXAMPLE: example facFirstShift; shows examples |
---|
3275 | SEE ALSO: testNCfac, facFirstWeyl, facSubWeyl |
---|
3276 | "{//facFirstShift |
---|
3277 | int p = printlevel - voice + 2; |
---|
3278 | int i; |
---|
3279 | string dbprintWhitespace = ""; |
---|
3280 | for (i = 1; i<=voice;i++) |
---|
3281 | {dbprintWhitespace = dbprintWhitespace + " ";} |
---|
3282 | dbprint(p,dbprintWhitespace +" Checking the input."); |
---|
3283 | if(nvars(basering)!=2) |
---|
3284 | {//Our basering is the Shift algebra, but not the first |
---|
3285 | ERROR("Basering is not the first shift algebra"); |
---|
3286 | return(list()); |
---|
3287 | }//Our basering is the Shift algebra, but not the first |
---|
3288 | def r = basering; |
---|
3289 | setring r; |
---|
3290 | list LR = ringlist(r); |
---|
3291 | number @n = leadcoef(LR[5][1,2]); |
---|
3292 | poly @p = LR[6][1,2]; |
---|
3293 | if ( @n!=number(1) ) |
---|
3294 | { |
---|
3295 | ERROR("Basering is not the first shift algebra"); |
---|
3296 | return(list()); |
---|
3297 | } |
---|
3298 | dbprint(p,dbprintWhitespace +" Done"); |
---|
3299 | list result = list(); |
---|
3300 | int j; int k; int l; //counter |
---|
3301 | // create a ring with the ordering which makes shift algebra |
---|
3302 | // graded |
---|
3303 | // def r = basering; // done before |
---|
3304 | ring tempRing = LR[1][1],(x,s),(a(0,1),Dp); |
---|
3305 | def tempRingnc = nc_algebra(1,s); |
---|
3306 | setring r; |
---|
3307 | // information on relations |
---|
3308 | if (@p == -var(1)) // reverted shift algebra |
---|
3309 | { |
---|
3310 | dbprint(p,dbprintWhitespace +" Reverted shift algebra. Swaping variables in Ringlist"); |
---|
3311 | setring(tempRingnc); |
---|
3312 | map transf = r, var(2), var(1); |
---|
3313 | setring(r); |
---|
3314 | map transfback = tempRingnc, var(2),var(1); |
---|
3315 | // result = transfback(resulttemp); |
---|
3316 | } |
---|
3317 | else |
---|
3318 | { |
---|
3319 | if ( @p == var(2)) // usual shift algebra |
---|
3320 | { |
---|
3321 | setring(tempRingnc); |
---|
3322 | map transf = r, var(1), var(2); |
---|
3323 | // result = facshift(h); |
---|
3324 | setring(r); |
---|
3325 | map transfback = tempRingnc, var(1),var(2); |
---|
3326 | } |
---|
3327 | else |
---|
3328 | { |
---|
3329 | ERROR("Basering is not the first shift algebra"); |
---|
3330 | return(list()); |
---|
3331 | } |
---|
3332 | } |
---|
3333 | // main calls |
---|
3334 | setring(tempRingnc); |
---|
3335 | dbprint(p,dbprintWhitespace +" Factorize the given polynomial with the subroutine sFacShift"); |
---|
3336 | list resulttemp = sFacShift(transf(h)); |
---|
3337 | dbprint(p,dbprintWhitespace +" Successful"); |
---|
3338 | setring(r); |
---|
3339 | result = transfback(resulttemp); |
---|
3340 | return( delete_dublicates_noteval(result) ); |
---|
3341 | }//facFirstShift |
---|
3342 | example |
---|
3343 | { |
---|
3344 | "EXAMPLE:";echo=2; |
---|
3345 | ring R = 0,(x,s),dp; |
---|
3346 | def r = nc_algebra(1,s); |
---|
3347 | setring(r); |
---|
3348 | poly h = (s^2*x+x)*s; |
---|
3349 | facFirstShift(h); |
---|
3350 | } |
---|
3351 | |
---|
3352 | static proc sFacShift(poly h) |
---|
3353 | " |
---|
3354 | USAGE: A static procedure to factorize a polynomial in the first Shift algebra, where all the |
---|
3355 | validity checks were made in advance. |
---|
3356 | INPUT: A polynomial h in the first Shift Algebra. |
---|
3357 | OUTPUT: A list of different factorizations of h, where the factors are irreducible |
---|
3358 | ASSUMPTIONS: |
---|
3359 | - The basering is the first Shift algebra and has n as first, and s as second variable, i.e. we |
---|
3360 | have var(2)*var(1) = var(1)*var(2)+1 |
---|
3361 | THEORY: If the given polynomial h is [0,1]-homogeneous, the routines for homogeneous factorizations |
---|
3362 | are called. Otherwise we map the polynomial into the first Weyl algebra (the first shift |
---|
3363 | algebra is a subring of the first Weyl algebra), and use facFirstWeyl to factorize it. Later |
---|
3364 | we map the factors back, if possible. |
---|
3365 | " |
---|
3366 | {//proc sFacShift |
---|
3367 | int p = printlevel - voice + 2; |
---|
3368 | int i; int j ; |
---|
3369 | string dbprintWhitespace = ""; |
---|
3370 | number commonCoefficient = content(h); |
---|
3371 | for (i = 1; i<=voice;i++) |
---|
3372 | {dbprintWhitespace = dbprintWhitespace + " ";} |
---|
3373 | //Checking if given polynomial is homogeneous |
---|
3374 | if(homogwithorder(h,intvec(0,1))) |
---|
3375 | {//The given polynomial is [0,1]-homogeneous |
---|
3376 | dbprint(p,dbprintWhitespace+"The polynomial is [0,1]-homogeneous. Returning the |
---|
3377 | homogeneous factorization"); |
---|
3378 | return(homogfacFirstShift_all(h)); |
---|
3379 | }//The given polynomial is [0,1]-homogeneous |
---|
3380 | |
---|
3381 | //---------- Start of interesting part ---------- |
---|
3382 | |
---|
3383 | dbprint(p,dbprintWhitespace+"Mapping the polynomial h into the first Weyl algebra."); |
---|
3384 | poly temph = h/commonCoefficient; |
---|
3385 | def ourBaseRing = basering; |
---|
3386 | ring tempWeylAlgebraComm = 0,(x,d),dp; |
---|
3387 | def tempWeylAlgebra = nc_algebra(1,1); |
---|
3388 | setring(tempWeylAlgebra); |
---|
3389 | map shiftMap = ourBaseRing, x*d, d; |
---|
3390 | poly h = shiftMap(temph); |
---|
3391 | dbprint(p,dbprintWhitespace+"Successful! The polynomial in the Weyl algebra is "+string(h)); |
---|
3392 | dbprint(p,dbprintWhitespace+"Factorizing the polynomial in the first Weyl algebra"); |
---|
3393 | list factorizationInWeyl = facFirstWeyl(h); |
---|
3394 | dbprint(p,dbprintWhitespace+"Successful! The factorization is given by:"); |
---|
3395 | dbprint(p,factorizationInWeyl); |
---|
3396 | list validCombinations; |
---|
3397 | |
---|
3398 | dbprint(p,dbprintWhitespace+"Now we will map this back to the shift algebra and filter |
---|
3399 | valid results"); |
---|
3400 | //-Now we map the results back to the shift algebra. But first, we need to combine them properly. |
---|
3401 | for (i = 1; i<=size(factorizationInWeyl); i++) |
---|
3402 | {//Deleting the first Coefficient factor |
---|
3403 | factorizationInWeyl[i] = delete(factorizationInWeyl[i],1); |
---|
3404 | validCombinations = validCombinations + combineNonnegative(factorizationInWeyl[i]); |
---|
3405 | }//Deleting the first Coefficient factor |
---|
3406 | if (size(validCombinations) == 0) |
---|
3407 | {//There are no valid combinations, therefore we can directly say, that h is irreducible |
---|
3408 | setring(ourBaseRing); |
---|
3409 | return(list(list(commonCoefficient, h/commonCoefficient))); |
---|
3410 | }//There are no valid combinations, therefore we can directly say, that h is irreducible |
---|
3411 | validCombinations = delete_dublicates_noteval(validCombinations); |
---|
3412 | setring(ourBaseRing); |
---|
3413 | map backFromWeyl = tempWeylAlgebra, var(1),var(2); |
---|
3414 | list validCombinations = backFromWeyl(validCombinations); |
---|
3415 | for (i = 1; i<=size(validCombinations); i++) |
---|
3416 | { |
---|
3417 | for (j = 1; j<=size(validCombinations[i]);j++) |
---|
3418 | { |
---|
3419 | setring(tempWeylAlgebra); |
---|
3420 | fromWeylToShiftPoly(validCombinations[i][j],ourBaseRing); |
---|
3421 | validCombinations[i][j] = result; |
---|
3422 | kill result; |
---|
3423 | kill tempResult; |
---|
3424 | kill zeroPoly; |
---|
3425 | kill fromWeyl; |
---|
3426 | } |
---|
3427 | } |
---|
3428 | for (i = 1; i<=size(validCombinations); i++) |
---|
3429 | {//Adding the common factor in the first position of the list |
---|
3430 | validCombinations[i] = insert(validCombinations[i],commonCoefficient); |
---|
3431 | }//Adding the common factor in the first position of the list |
---|
3432 | dbprint(dbprintWhitespace+"Done."); |
---|
3433 | //mapping |
---|
3434 | return(validCombinations); |
---|
3435 | }//proc sFacShift |
---|
3436 | |
---|
3437 | static proc combineNonnegative(list l) |
---|
3438 | " |
---|
3439 | USAGE: In sFacShift, when we want to map back the results of the factorization of the polynomial in |
---|
3440 | the first Weyl algebra to the shift algebra. We need to recombine the factors such that |
---|
3441 | we can map it back to the shift algebra without any problems. |
---|
3442 | INPUT: A list l containing one factorization of a polynomial in the first Weyl algebra. For example |
---|
3443 | for the polynomial (1+x)*(1+x+d) we would have the list [1,x+1,x+d+1]. |
---|
3444 | OUTPUT:If we can map every factor without a problem back to the shift algebra (i.e. if the smallest |
---|
3445 | homogeneous summand of every factor is of nonnegative degree), a list containing the same |
---|
3446 | list as given in the input is returned. |
---|
3447 | If otherwise some factors cause problems, we consider every possible combination (i.e. |
---|
3448 | products of the factors) and extract those where all factors have a smallest homogeneous |
---|
3449 | summand of nonnegative degree. |
---|
3450 | ASSUMPTIONS: |
---|
3451 | - Weyl algebra is given, and we have var(2)*var(1)=var(1)*var(2) +1 |
---|
3452 | " |
---|
3453 | {//combineNonnegative |
---|
3454 | int p = printlevel - voice + 2; |
---|
3455 | int i; |
---|
3456 | string dbprintWhitespace = ""; |
---|
3457 | for (i = 1; i<=voice;i++) |
---|
3458 | {dbprintWhitespace = dbprintWhitespace + " ";} |
---|
3459 | //First the easy case: all of the factors fulfill the condition of mapping to shift: |
---|
3460 | dbprint(p,dbprintWhitespace+"Checking, if the given factors |
---|
3461 | can already be mapped without a problem."); |
---|
3462 | int isValid = 1; |
---|
3463 | for (i = 1; i<=size(l);i++) |
---|
3464 | {//Checking for every entry if the condition is fulfilled. |
---|
3465 | if (deg(l[i],intvec(1,-1))>0) |
---|
3466 | {//Found one, where it is not fulfilled |
---|
3467 | isValid = 0; |
---|
3468 | break; |
---|
3469 | }//Found one, where it is not fulfilled |
---|
3470 | }//Checking for every entry if the condition is fulfilled. |
---|
3471 | dbprint(p,dbprintWhitespace+"Done."); |
---|
3472 | if (isValid) |
---|
3473 | {//We can map every factor to the shift algebra and do not need to combine anything |
---|
3474 | dbprint(p,dbprintWhitespace+"They can be mapped. Therefore we return them directly."); |
---|
3475 | return(list(l)); |
---|
3476 | }//We can map every factor to the shift algebra and do not need to combine anything |
---|
3477 | dbprint(p,dbprintWhitespace+"They cannot be mapped. Looking for valid combinations."); |
---|
3478 | //Starting with the case, where l only consists of 1 or two elements. |
---|
3479 | if(size(l)<=2) |
---|
3480 | {//The case where we won't call the function a second time |
---|
3481 | if (deg(product(l),intvec(1,-1))>0) |
---|
3482 | {//No way of a valid combination |
---|
3483 | return(list()); |
---|
3484 | }//No way of a valid combination |
---|
3485 | else |
---|
3486 | {//The product is the only possible and valid combination |
---|
3487 | return(list(list(product(l)))); |
---|
3488 | }//The product is the only possible and valid combination |
---|
3489 | }//The case where we won't call the function a second time |
---|
3490 | //---------- Easy pre-stuff done. now we combine the factors.---------- |
---|
3491 | int pos; |
---|
3492 | int j; int k; |
---|
3493 | dbprint(p,dbprintWhitespace+"Making combinations of two."); |
---|
3494 | list combinationsOfTwo = combinekfinlf(l,2); |
---|
3495 | dbprint(p,dbprintWhitespace+"Done. Now checking, if there are valid ones in between."); |
---|
3496 | list result; |
---|
3497 | list validLHS; |
---|
3498 | list validRHS; |
---|
3499 | for (i = 1; i<=size(combinationsOfTwo); i++) |
---|
3500 | {//go through all combinations and detect the valid ones |
---|
3501 | if(deg(combinationsOfTwo[i][1],intvec(1,-1))>0 or deg(combinationsOfTwo[i][2],intvec(1,-1))>0) |
---|
3502 | {//No chance, so no further treatment needed |
---|
3503 | i++; |
---|
3504 | continue; |
---|
3505 | }//No chance, so no further treatment needed |
---|
3506 | for (pos = 1; pos<=size(l);pos++) |
---|
3507 | {//find the position where the combination splits |
---|
3508 | if (product(l[1..pos]) == combinationsOfTwo[i][1]) |
---|
3509 | {//Found the position |
---|
3510 | break; |
---|
3511 | }//Found the position |
---|
3512 | }//find the position where the combination splits |
---|
3513 | dbprint(p,dbprintWhitespace+"Calling combineNonnegative recursively with argument " + |
---|
3514 | string(list(l[1..pos]))); |
---|
3515 | validLHS = combineNonnegative(list(l[1..pos])); |
---|
3516 | dbprint(p,dbprintWhitespace+"Calling combineNonnegative recursively with argument " + |
---|
3517 | string(list(l[pos+1..size(l)]))); |
---|
3518 | validRHS = combineNonnegative(list(l[pos+1..size(l)])); |
---|
3519 | for (j = 1; j<=size(validLHS); j++) |
---|
3520 | {//Combining the left hand side valid combnations... |
---|
3521 | for (k = 1; k<=size(validRHS); k++) |
---|
3522 | {//... with the right hand side valid combinations |
---|
3523 | result = insert(result, validLHS[j]+validRHS[k]); |
---|
3524 | }//... with the right hand side valid combinations |
---|
3525 | }//Combining the left hand side valid combnations... |
---|
3526 | }//go through all combinations and detect the valid ones |
---|
3527 | result = delete_dublicates_noteval(result); |
---|
3528 | dbprint(p,dbprintWhitespace+"Done."); |
---|
3529 | return(result); |
---|
3530 | }//combineNonnegative |
---|
3531 | |
---|
3532 | static proc fromWeylToShiftPoly(poly h, sAlgebra) |
---|
3533 | " |
---|
3534 | USAGE: Given a polynomial in the first Weyl algebra, this method returns it -- if possible -- |
---|
3535 | as an element in the first shift algebra, which is given in the method header. |
---|
3536 | INPUT: A polynomial h, and the first shift algebra as a ring |
---|
3537 | OUTPUT: The correct mapping in the shift Algebra |
---|
3538 | ASSUMPTIONS: |
---|
3539 | - The lowest [-1,1]-homogeneous summand of h is of nonnegative degree |
---|
3540 | - The shift algebra is given in the way that var(2)*var(1) = (var(1)+1)*var(2) |
---|
3541 | " |
---|
3542 | {//fromWeylToShiftPoly |
---|
3543 | int p = printlevel - voice + 2; |
---|
3544 | int i; |
---|
3545 | string dbprintWhitespace = ""; |
---|
3546 | for (i = 1; i<=voice;i++) |
---|
3547 | {dbprintWhitespace = dbprintWhitespace + " ";} |
---|
3548 | if (deg(h,intvec(1,-1))>0) |
---|
3549 | {//Wrong input polynomial |
---|
3550 | ERROR("The lowest [-1,1] homogeneous summand of "+string(h)+" is of negative degree."); |
---|
3551 | }//Wrong input polynomial |
---|
3552 | def ourHomeBase = basering; |
---|
3553 | list hDist = homogDistribution(h); |
---|
3554 | setring(sAlgebra); |
---|
3555 | poly result = 0; |
---|
3556 | poly tempResult; |
---|
3557 | poly zeroPoly; |
---|
3558 | map fromWeyl = ourHomeBase, var(1), var(2); |
---|
3559 | setring(ourHomeBase); |
---|
3560 | poly zeroPoly; |
---|
3561 | poly tempZeroPoly; |
---|
3562 | int j; int k; |
---|
3563 | int derDeg; |
---|
3564 | for (i = 1; i<=size(hDist);i++) |
---|
3565 | { |
---|
3566 | derDeg = hDist[i][1]; |
---|
3567 | setring(sAlgebra); |
---|
3568 | tempResult = 1; |
---|
3569 | setring(ourHomeBase); |
---|
3570 | zeroPoly = lift(d^derDeg, hDist[i][2])[1,1]; |
---|
3571 | for (j = 1; j<=size(zeroPoly); j++) |
---|
3572 | { |
---|
3573 | tempZeroPoly = zeroPoly[j]; |
---|
3574 | setring(sAlgebra); |
---|
3575 | zeroPoly = fromWeyl(tempZeroPoly); |
---|
3576 | tempResult = tempResult * leadcoef(zeroPoly); |
---|
3577 | setring(ourHomeBase); |
---|
3578 | for (k = 1; k<=deg(zeroPoly[j],intvec(0,1));k++) |
---|
3579 | { |
---|
3580 | setring(sAlgebra); |
---|
3581 | tempResult = tempResult*(var(1)-(k-1)); |
---|
3582 | setring(ourHomeBase); |
---|
3583 | } |
---|
3584 | setring(sAlgebra); |
---|
3585 | result = result + tempResult*var(2)^derDeg; |
---|
3586 | tempResult = 1; |
---|
3587 | setring(ourHomeBase); |
---|
3588 | } |
---|
3589 | } |
---|
3590 | setring(sAlgebra); |
---|
3591 | keepring(sAlgebra); |
---|
3592 | }//fromWeylToShiftPoly |
---|
3593 | |
---|
3594 | static proc refineFactList(list L) |
---|
3595 | { |
---|
3596 | // assume: list L is an output of factorization proc |
---|
3597 | // doing: remove doubled entries |
---|
3598 | int s = size(L); int sm; |
---|
3599 | int i,j,k,cnt; |
---|
3600 | list M, U, A, B; |
---|
3601 | A = L; |
---|
3602 | k = 0; |
---|
3603 | cnt = 1; |
---|
3604 | for (i=1; i<=s; i++) |
---|
3605 | { |
---|
3606 | if (size(A[i]) != 0) |
---|
3607 | { |
---|
3608 | M = A[i]; |
---|
3609 | // "probing with"; M; i; |
---|
3610 | B[cnt] = M; cnt++; |
---|
3611 | for (j=i+1; j<=s; j++) |
---|
3612 | { |
---|
3613 | if ( isEqualList(M,A[j]) ) |
---|
3614 | { |
---|
3615 | k++; |
---|
3616 | // U consists of intvecs with equal pairs |
---|
3617 | U[k] = intvec(i,j); |
---|
3618 | A[j] = 0; |
---|
3619 | } |
---|
3620 | } |
---|
3621 | } |
---|
3622 | } |
---|
3623 | kill A,U,M; |
---|
3624 | return(B); |
---|
3625 | } |
---|
3626 | example |
---|
3627 | { |
---|
3628 | "EXAMPLE:";echo=2; |
---|
3629 | ring R = 0,(x,s),dp; |
---|
3630 | def r = nc_algebra(1,1); |
---|
3631 | setring(r); |
---|
3632 | list l,m; |
---|
3633 | l = list(1,s2+1,x,s,x+s); |
---|
3634 | m = l,list(1,s,x,s,x),l; |
---|
3635 | refineFactList(m); |
---|
3636 | } |
---|
3637 | |
---|
3638 | static proc isEqualList(list L, list M) |
---|
3639 | { |
---|
3640 | // int boolean: 1=yes, 0 =no : test whether two lists are identical |
---|
3641 | int s = size(L); |
---|
3642 | if (size(M)!=s) { return(0); } |
---|
3643 | int j=1; |
---|
3644 | while ( (L[j]==M[j]) && (j<s) ) |
---|
3645 | { |
---|
3646 | j++; |
---|
3647 | } |
---|
3648 | if (L[j]==M[j]) |
---|
3649 | { |
---|
3650 | return(1); |
---|
3651 | } |
---|
3652 | return(0); |
---|
3653 | } |
---|
3654 | example |
---|
3655 | { |
---|
3656 | "EXAMPLE:";echo=2; |
---|
3657 | ring R = 0,(x,s),dp; |
---|
3658 | def r = nc_algebra(1,1); |
---|
3659 | setring(r); |
---|
3660 | list l,m; |
---|
3661 | l = list(1,s2+1,x,s,x+s); |
---|
3662 | m = l; |
---|
3663 | isEqualList(m,l); |
---|
3664 | } |
---|
3665 | |
---|
3666 | |
---|
3667 | ////////////////////////////////////////////////// |
---|
3668 | // Q-WEYL-SECTION |
---|
3669 | ////////////////////////////////////////////////// |
---|
3670 | |
---|
3671 | //================================================== |
---|
3672 | //A function to get the i'th triangular number |
---|
3673 | static proc triangNum(int n) |
---|
3674 | { |
---|
3675 | if (n == 0) |
---|
3676 | { |
---|
3677 | return(0); |
---|
3678 | } |
---|
3679 | return (n*(n+1) div 2); |
---|
3680 | } |
---|
3681 | |
---|
3682 | //==================================================* |
---|
3683 | //one factorization of a homogeneous polynomial |
---|
3684 | //in the first Q Weyl Algebra |
---|
3685 | proc homogfacFirstQWeyl(poly h) |
---|
3686 | "USAGE: homogfacFirstQWeyl(h); h is a homogeneous polynomial in the |
---|
3687 | first q-Weyl algebra with respect to the weight vector [-1,1] |
---|
3688 | RETURN: list |
---|
3689 | PURPOSE: Computes a factorization of a homogeneous polynomial h with |
---|
3690 | respect to the weight vector [-1,1] in the first q-Weyl algebra |
---|
3691 | THEORY: @code{homogfacFirstQWeyl} returns a list with a factorization of the given, |
---|
3692 | [-1,1]-homogeneous polynomial. If the degree of the polynomial is k with |
---|
3693 | k positive, the last k entries in the output list are the second |
---|
3694 | variable. If k is positive, the last k entries will be x. The other |
---|
3695 | entries will be irreducible polynomials of degree zero or 1 resp. -1. |
---|
3696 | SEE ALSO: homogfacFirstQWeyl_all |
---|
3697 | "{//proc homogfacFirstQWeyl |
---|
3698 | int p = printlevel-voice+2;//for dbprint |
---|
3699 | def r = basering; |
---|
3700 | poly hath; |
---|
3701 | int i; int j; |
---|
3702 | string dbprintWhitespace = ""; |
---|
3703 | for (i = 1; i<=voice;i++) |
---|
3704 | {dbprintWhitespace = dbprintWhitespace + " ";} |
---|
3705 | intvec ivm11 = intvec(-1,1); |
---|
3706 | if (!homogwithorder(h,ivm11)) |
---|
3707 | {//The given polynomial is not homogeneous |
---|
3708 | ERROR("Given polynomial was not [-1,1]-homogeneous"); |
---|
3709 | return(list()); |
---|
3710 | }//The given polynomial is not homogeneous |
---|
3711 | if (h==0) |
---|
3712 | { |
---|
3713 | return(list(0)); |
---|
3714 | } |
---|
3715 | list result; |
---|
3716 | int m = deg(h,ivm11); |
---|
3717 | dbprint(p,dbprintWhitespace+" Splitting the polynomial in A_0 and A_k-Part"); |
---|
3718 | if (m!=0) |
---|
3719 | {//The degree is not zero |
---|
3720 | if (m <0) |
---|
3721 | {//There are more x than y |
---|
3722 | hath = lift(var(1)^(-m),h)[1,1]; |
---|
3723 | for (i = 1; i<=-m; i++) |
---|
3724 | { |
---|
3725 | result = result + list(var(1)); |
---|
3726 | } |
---|
3727 | }//There are more x than y |
---|
3728 | else |
---|
3729 | {//There are more y than x |
---|
3730 | hath = lift(var(2)^m,h)[1,1]; |
---|
3731 | for (i = 1; i<=m;i++) |
---|
3732 | { |
---|
3733 | result = result + list(var(2)); |
---|
3734 | } |
---|
3735 | }//There are more y than x |
---|
3736 | }//The degree is not zero |
---|
3737 | else |
---|
3738 | {//The degree is zero |
---|
3739 | hath = h; |
---|
3740 | }//The degree is zero |
---|
3741 | dbprint(p," Done"); |
---|
3742 | //beginning to transform x^i*y^i in theta(theta-1)...(theta-i+1) |
---|
3743 | list mons; |
---|
3744 | dbprint(p," Putting the monomials in the A_0-part in a list."); |
---|
3745 | for(i = 1; i<=size(hath);i++) |
---|
3746 | {//Putting the monomials in a list |
---|
3747 | mons = mons+list(hath[i]); |
---|
3748 | }//Putting the monomials in a list |
---|
3749 | dbprint(p," Done"); |
---|
3750 | dbprint(p," Mapping this monomials to K(q)[theta]"); |
---|
3751 | //Now, map to the commutative ring with theta: |
---|
3752 | list tempRingList = ringlist(r); |
---|
3753 | tempRingList[2] = insert(tempRingList[2],"theta",2); //New variable theta = x*d |
---|
3754 | tempRingList = delete(tempRingList,5); |
---|
3755 | tempRingList = delete(tempRingList,5); //The ring should now be commutative |
---|
3756 | def tempRing = ring(tempRingList); |
---|
3757 | setring tempRing; |
---|
3758 | map thetamap = r,var(1),var(2); |
---|
3759 | list mons = thetamap(mons); |
---|
3760 | poly entry; |
---|
3761 | poly tempSummand; |
---|
3762 | for (i = 1; i<=size(mons);i++) |
---|
3763 | {//transforming the monomials as monomials in theta |
---|
3764 | entry = 1;//leadcoef(mons[i]) * q^(-triangNum(leadexp(mons[i])[2]-1)); |
---|
3765 | for (j = 0; j<leadexp(mons[i])[2];j++) |
---|
3766 | { |
---|
3767 | tempSummand = (par(1)^j-1)/(par(1)-1); |
---|
3768 | entry = entry * theta-tempSummand*entry; |
---|
3769 | } |
---|
3770 | //entry; |
---|
3771 | //leadcoef(mons[i]) * q^(-triangNum(leadexp(mons[i])[2]-1)); |
---|
3772 | mons[i] = entry*leadcoef(mons[i]) * par(1)^(-triangNum(leadexp(mons[i])[2]-1)); |
---|
3773 | }//transforming the monomials as monomials in theta |
---|
3774 | dbprint(p," Done"); |
---|
3775 | dbprint(p," Factorize the A_0-Part in K[theta]"); |
---|
3776 | list azeroresult = factorize(sum(mons)); |
---|
3777 | dbprint(p," Successful"); |
---|
3778 | list azeroresult_return_form; |
---|
3779 | for (i = 1; i<=size(azeroresult[1]);i++) |
---|
3780 | {//rewrite the result of the commutative factorization |
---|
3781 | for (j = 1; j <= azeroresult[2][i];j++) |
---|
3782 | { |
---|
3783 | azeroresult_return_form = azeroresult_return_form + list(azeroresult[1][i]); |
---|
3784 | } |
---|
3785 | }//rewrite the result of the commutative factorization |
---|
3786 | dbprint(p," Mapping back to A_0."); |
---|
3787 | setring(r); |
---|
3788 | map finalmap = tempRing,var(1),var(2),var(1)*var(2); |
---|
3789 | list tempresult = finalmap(azeroresult_return_form); |
---|
3790 | dbprint(p,"Successful."); |
---|
3791 | for (i = 1; i<=size(tempresult);i++) |
---|
3792 | {//factorizations of theta resp. theta +1 |
---|
3793 | if(tempresult[i]==var(1)*var(2)) |
---|
3794 | { |
---|
3795 | tempresult = insert(tempresult,var(1),i-1); |
---|
3796 | i++; |
---|
3797 | tempresult[i]=var(2); |
---|
3798 | } |
---|
3799 | if(tempresult[i]==var(2)*var(1)) |
---|
3800 | { |
---|
3801 | tempresult = insert(tempresult,var(2),i-1); |
---|
3802 | i++; |
---|
3803 | tempresult[i]=var(1); |
---|
3804 | } |
---|
3805 | }//factorizations of theta resp. theta +1 |
---|
3806 | result = tempresult+result; |
---|
3807 | //Correction of the result in the special q-Case: |
---|
3808 | for (j = 2 ; j<= size(result);j++) |
---|
3809 | {//Div the whole Term by the leading coefficient and multiply it to the first entry in result[i] |
---|
3810 | result[1] = result[1] * leadcoef(result[j]); |
---|
3811 | result[j] = 1/leadcoef(result[j]) * result[j]; |
---|
3812 | }//Div the whole Term by the leading coefficient and multiply it to the first entry in result[i] |
---|
3813 | return(result); |
---|
3814 | }//proc homogfacFirstQWeyl |
---|
3815 | example |
---|
3816 | { |
---|
3817 | "EXAMPLE:";echo=2; |
---|
3818 | ring R = (0,q),(x,d),dp; |
---|
3819 | def r = nc_algebra (q,1); |
---|
3820 | setring(r); |
---|
3821 | poly h = q^25*x^10*d^10+q^16*(q^4+q^3+q^2+q+1)^2*x^9*d^9+ |
---|
3822 | q^9*(q^13+3*q^12+7*q^11+13*q^10+20*q^9+26*q^8+30*q^7+ |
---|
3823 | 31*q^6+26*q^5+20*q^4+13*q^3+7*q^2+3*q+1)*x^8*d^8+ |
---|
3824 | q^4*(q^9+2*q^8+4*q^7+6*q^6+7*q^5+8*q^4+6*q^3+ |
---|
3825 | 4*q^2+2q+1)*(q^4+q^3+q^2+q+1)*(q^2+q+1)*x^7*d^7+ |
---|
3826 | 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+ |
---|
3827 | (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+ |
---|
3828 | 6*x^3*d^3+24; |
---|
3829 | homogfacFirstQWeyl(h); |
---|
3830 | } |
---|
3831 | |
---|
3832 | //================================================== |
---|
3833 | //Computes all possible homogeneous factorizations for an element in the first Q-Weyl Algebra |
---|
3834 | proc homogfacFirstQWeyl_all(poly h) |
---|
3835 | "USAGE: homogfacFirstQWeyl_all(h); h is a homogeneous polynomial in the first q-Weyl algebra |
---|
3836 | with respect to the weight vector [-1,1] |
---|
3837 | RETURN: list |
---|
3838 | PURPOSE: Computes all factorizations of a homogeneous polynomial h with respect |
---|
3839 | to the weight vector [-1,1] in the first q-Weyl algebra |
---|
3840 | THEORY: @code{homogfacFirstQWeyl} returns a list with all factorization of the given, |
---|
3841 | homogeneous polynomial. It uses the output of homogfacFirstQWeyl and permutes |
---|
3842 | its entries with respect to the commutation rule. Furthermore, if a |
---|
3843 | factor of degree zero is irreducible in K[ heta], but reducible in |
---|
3844 | the first q-Weyl algebra, the permutations of this element with the other |
---|
3845 | entries will also be computed. |
---|
3846 | SEE ALSO: homogfacFirstQWeyl |
---|
3847 | "{//proc HomogfacFirstQWeylAll |
---|
3848 | int p=printlevel-voice+2;//for dbprint |
---|
3849 | intvec iv11= intvec(1,1); |
---|
3850 | if (deg(h,iv11) <= 0 ) |
---|
3851 | {//h is a constant |
---|
3852 | dbprint(p,"Given polynomial was not homogeneous"); |
---|
3853 | return(list(list(h))); |
---|
3854 | }//h is a constant |
---|
3855 | def r = basering; |
---|
3856 | list one_hom_fac; //stands for one homogeneous factorization |
---|
3857 | int i; int j; int k; |
---|
3858 | intvec ivm11 = intvec(-1,1); |
---|
3859 | dbprint(p," Calculate one homogeneous factorization using homogfacFirstQWeyl"); |
---|
3860 | //Compute again a homogeneous factorization |
---|
3861 | one_hom_fac = homogfacFirstQWeyl(h); |
---|
3862 | dbprint(p,"Successful"); |
---|
3863 | if (size(one_hom_fac) == 0) |
---|
3864 | {//there is no homogeneous factorization or the polynomial was not homogeneous |
---|
3865 | return(list()); |
---|
3866 | }//there is no homogeneous factorization or the polynomial was not homogeneous |
---|
3867 | //divide list in A0-Part and a list of x's resp. y's |
---|
3868 | list list_not_azero = list(); |
---|
3869 | list list_azero; |
---|
3870 | list k_factor; |
---|
3871 | int is_list_not_azero_empty = 1; |
---|
3872 | int is_list_azero_empty = 1; |
---|
3873 | k_factor = list(one_hom_fac[1]); |
---|
3874 | if (absValue(deg(h,ivm11))<size(one_hom_fac)-1) |
---|
3875 | {//There is a nontrivial A_0-part |
---|
3876 | list_azero = one_hom_fac[2..(size(one_hom_fac)-absValue(deg(h,ivm11)))]; |
---|
3877 | is_list_azero_empty = 0; |
---|
3878 | }//There is a nontrivial A_0 part |
---|
3879 | dbprint(p," Combine x,y to xy in the factorization again."); |
---|
3880 | for (i = 1; i<=size(list_azero)-1;i++) |
---|
3881 | {//in homogfacFirstQWeyl, we factorized theta, and this will be made undone |
---|
3882 | if (list_azero[i] == var(1)) |
---|
3883 | { |
---|
3884 | if (list_azero[i+1]==var(2)) |
---|
3885 | { |
---|
3886 | list_azero[i] = var(1)*var(2); |
---|
3887 | list_azero = delete(list_azero,i+1); |
---|
3888 | } |
---|
3889 | } |
---|
3890 | if (list_azero[i] == var(2)) |
---|
3891 | { |
---|
3892 | if (list_azero[i+1]==var(1)) |
---|
3893 | { |
---|
3894 | list_azero[i] = var(2)*var(1); |
---|
3895 | list_azero = delete(list_azero,i+1); |
---|
3896 | } |
---|
3897 | } |
---|
3898 | }//in homogfacFirstQWeyl, we factorized theta, and this will be made undone |
---|
3899 | dbprint(p," Done"); |
---|
3900 | if(deg(h,ivm11)!=0) |
---|
3901 | {//list_not_azero is not empty |
---|
3902 | list_not_azero = |
---|
3903 | one_hom_fac[(size(one_hom_fac)-absValue(deg(h,ivm11))+1)..size(one_hom_fac)]; |
---|
3904 | is_list_not_azero_empty = 0; |
---|
3905 | }//list_not_azero is not empty |
---|
3906 | //Map list_azero in K[theta] |
---|
3907 | dbprint(p," Map list_azero to K[theta]"); |
---|
3908 | //Now, map to the commutative ring with theta: |
---|
3909 | list tempRingList = ringlist(r); |
---|
3910 | tempRingList[2] = insert(tempRingList[2],"theta",2); //New variable theta = x*d |
---|
3911 | tempRingList = delete(tempRingList,5); |
---|
3912 | tempRingList = delete(tempRingList,5); //The ring should now be commutative |
---|
3913 | def tempRing = ring(tempRingList); |
---|
3914 | setring(tempRing); |
---|
3915 | poly entry; |
---|
3916 | map thetamap = r,var(1),var(2); |
---|
3917 | if(!is_list_not_azero_empty) |
---|
3918 | {//Mapping in Singular is only possible, if the list before |
---|
3919 | //contained at least one element of the other ring |
---|
3920 | list list_not_azero = thetamap(list_not_azero); |
---|
3921 | }//Mapping in Singular is only possible, if the list before |
---|
3922 | //contained at least one element of the other ring |
---|
3923 | if(!is_list_azero_empty) |
---|
3924 | {//Mapping in Singular is only possible, if the list before |
---|
3925 | //contained at least one element of the other ring |
---|
3926 | list list_azero= thetamap(list_azero); |
---|
3927 | }//Mapping in Singular is only possible, if the list before |
---|
3928 | //contained at least one element of the other ring |
---|
3929 | list k_factor = thetamap(k_factor); |
---|
3930 | list tempmons; |
---|
3931 | dbprint(p," Done"); |
---|
3932 | for(i = 1; i<=size(list_azero);i++) |
---|
3933 | {//rewrite the polynomials in A1 as polynomials in K[theta] |
---|
3934 | tempmons = list(); |
---|
3935 | for (j = 1; j<=size(list_azero[i]);j++) |
---|
3936 | { |
---|
3937 | tempmons = tempmons + list(list_azero[i][j]); |
---|
3938 | } |
---|
3939 | for (j = 1 ; j<=size(tempmons);j++) |
---|
3940 | { |
---|
3941 | //entry = leadcoef(tempmons[j]); |
---|
3942 | entry = leadcoef(tempmons[j]) * par(1)^(-triangNum(leadexp(tempmons[j])[2]-1)); |
---|
3943 | for (k = 0; k < leadexp(tempmons[j])[2];k++) |
---|
3944 | { |
---|
3945 | entry = entry*(theta-(par(1)^k-1)/(par(1)-1)); |
---|
3946 | } |
---|
3947 | tempmons[j] = entry; |
---|
3948 | } |
---|
3949 | list_azero[i] = sum(tempmons); |
---|
3950 | }//rewrite the polynomials in A1 as polynomials in K[theta] |
---|
3951 | //Compute all permutations of the A0-part |
---|
3952 | dbprint(p," Compute all permutations of the A_0-part with the first resp. the snd. variable"); |
---|
3953 | list result; |
---|
3954 | int shift_sign; |
---|
3955 | int shift; |
---|
3956 | poly shiftvar; |
---|
3957 | if (size(list_not_azero)!=0) |
---|
3958 | {//Compute all possibilities to permute the x's resp. the y's in the list |
---|
3959 | if (list_not_azero[1] == var(1)) |
---|
3960 | {//h had a negative weighted degree |
---|
3961 | shift_sign = 1; |
---|
3962 | shiftvar = var(1); |
---|
3963 | }//h had a negative weighted degree |
---|
3964 | else |
---|
3965 | {//h had a positive weighted degree |
---|
3966 | shift_sign = -1; |
---|
3967 | shiftvar = var(2); |
---|
3968 | }//h had a positive weighted degree |
---|
3969 | result = permpp(list_azero + list_not_azero); |
---|
3970 | for (i = 1; i<= size(result); i++) |
---|
3971 | {//adjust the a_0-parts |
---|
3972 | shift = 0; |
---|
3973 | for (j=1; j<=size(result[i]);j++) |
---|
3974 | { |
---|
3975 | if (result[i][j]==shiftvar) |
---|
3976 | { |
---|
3977 | shift = shift + shift_sign; |
---|
3978 | } |
---|
3979 | else |
---|
3980 | { |
---|
3981 | if (shift < 0) |
---|
3982 | {//We have two distict formulas for x and y. In this case use formula for y |
---|
3983 | if (shift == -1) |
---|
3984 | { |
---|
3985 | result[i][j] = subst(result[i][j],theta,1/par(1)*(theta - 1)); |
---|
3986 | } |
---|
3987 | else |
---|
3988 | { |
---|
3989 | result[i][j] = |
---|
3990 | subst(result[i][j], |
---|
3991 | theta, |
---|
3992 | 1/par(1)*((theta - 1)/par(1)^(absValue(shift)-1) |
---|
3993 | - (par(1)^(shift +2)-par(1))/(1-par(1)))); |
---|
3994 | } |
---|
3995 | }//We have two distict formulas for x and y. In this case use formula for y |
---|
3996 | if (shift > 0) |
---|
3997 | {//We have two distict formulas for x and y. In this case use formula for x |
---|
3998 | if (shift == 1) |
---|
3999 | { |
---|
4000 | result[i][j] = subst(result[i][j],theta,par(1)*theta + 1); |
---|
4001 | } |
---|
4002 | else |
---|
4003 | { |
---|
4004 | result[i][j] = |
---|
4005 | subst(result[i][j], |
---|
4006 | theta,par(1)^shift*theta+(par(1)^shift-1)/(par(1)-1)); |
---|
4007 | } |
---|
4008 | }//We have two distict formulas for x and y. In this case use formula for x |
---|
4009 | } |
---|
4010 | } |
---|
4011 | }//adjust the a_0-parts |
---|
4012 | }//Compute all possibilities to permute the x's resp. the y's in the list |
---|
4013 | else |
---|
4014 | {//The result is just all the permutations of the a_0-part |
---|
4015 | result = permpp(list_azero); |
---|
4016 | }//The result is just all the permutations of the a_0 part |
---|
4017 | if (size(result)==0) |
---|
4018 | { |
---|
4019 | return(result); |
---|
4020 | } |
---|
4021 | dbprint(p," Done"); |
---|
4022 | dbprint(p," Searching for theta resp. theta + 1 in the list and factorize them"); |
---|
4023 | //Now we are going deeper and search for theta resp. theta + 1, substitute |
---|
4024 | //them by xy resp. yx and go on permuting |
---|
4025 | int found_theta; |
---|
4026 | int thetapos; |
---|
4027 | list leftpart; |
---|
4028 | list rightpart; |
---|
4029 | list lparts; |
---|
4030 | list rparts; |
---|
4031 | list tempadd; |
---|
4032 | for (i = 1; i<=size(result) ; i++) |
---|
4033 | {//checking every entry of result for theta or theta +1 |
---|
4034 | found_theta = 0; |
---|
4035 | for(j=1;j<=size(result[i]);j++) |
---|
4036 | { |
---|
4037 | if (result[i][j]==theta) |
---|
4038 | {//the jth entry is theta and can be written as x*y |
---|
4039 | thetapos = j; |
---|
4040 | result[i]= insert(result[i],var(1),j-1); |
---|
4041 | j++; |
---|
4042 | result[i][j] = var(2); |
---|
4043 | found_theta = 1; |
---|
4044 | break; |
---|
4045 | }//the jth entry is theta and can be written as x*y |
---|
4046 | if(result[i][j] == par(1)*theta +1) |
---|
4047 | { |
---|
4048 | thetapos = j; |
---|
4049 | result[i] = insert(result[i],var(2),j-1); |
---|
4050 | j++; |
---|
4051 | result[i][j] = var(1); |
---|
4052 | found_theta = 1; |
---|
4053 | break; |
---|
4054 | } |
---|
4055 | } |
---|
4056 | if (found_theta) |
---|
4057 | {//One entry was theta resp. theta +1 |
---|
4058 | leftpart = result[i]; |
---|
4059 | leftpart = leftpart[1..thetapos]; |
---|
4060 | rightpart = result[i]; |
---|
4061 | rightpart = rightpart[(thetapos+1)..size(rightpart)]; |
---|
4062 | lparts = list(leftpart); |
---|
4063 | rparts = list(rightpart); |
---|
4064 | //first deal with the left part |
---|
4065 | if (leftpart[thetapos] == var(1)) |
---|
4066 | { |
---|
4067 | shift_sign = 1; |
---|
4068 | shiftvar = var(1); |
---|
4069 | } |
---|
4070 | else |
---|
4071 | { |
---|
4072 | shift_sign = -1; |
---|
4073 | shiftvar = var(2); |
---|
4074 | } |
---|
4075 | for (j = size(leftpart); j>1;j--) |
---|
4076 | {//drip x resp. y |
---|
4077 | if (leftpart[j-1]==shiftvar) |
---|
4078 | {//commutative |
---|
4079 | j--; |
---|
4080 | continue; |
---|
4081 | }//commutative |
---|
4082 | if (deg(leftpart[j-1],intvec(-1,1,0))!=0) |
---|
4083 | {//stop here |
---|
4084 | break; |
---|
4085 | }//stop here |
---|
4086 | //Here, we can only have a a0- part |
---|
4087 | if (shift_sign<0) |
---|
4088 | { |
---|
4089 | leftpart[j] = subst(leftpart[j-1],theta, 1/par(1)*(theta +shift_sign)); |
---|
4090 | } |
---|
4091 | if (shift_sign>0) |
---|
4092 | { |
---|
4093 | leftpart[j] = subst(leftpart[j-1],theta, par(1)*theta + shift_sign); |
---|
4094 | } |
---|
4095 | leftpart[j-1] = shiftvar; |
---|
4096 | lparts = lparts + list(leftpart); |
---|
4097 | }//drip x resp. y |
---|
4098 | //and now deal with the right part |
---|
4099 | if (rightpart[1] == var(1)) |
---|
4100 | { |
---|
4101 | shift_sign = 1; |
---|
4102 | shiftvar = var(1); |
---|
4103 | } |
---|
4104 | else |
---|
4105 | { |
---|
4106 | shift_sign = -1; |
---|
4107 | shiftvar = var(2); |
---|
4108 | } |
---|
4109 | for (j = 1 ; j < size(rightpart); j++) |
---|
4110 | { |
---|
4111 | if (rightpart[j+1] == shiftvar) |
---|
4112 | { |
---|
4113 | j++; |
---|
4114 | continue; |
---|
4115 | } |
---|
4116 | if (deg(rightpart[j+1],intvec(-1,1,0))!=0) |
---|
4117 | { |
---|
4118 | break; |
---|
4119 | } |
---|
4120 | if (shift_sign<0) |
---|
4121 | { |
---|
4122 | rightpart[j] = subst(rightpart[j+1], theta, par(1)*theta - shift_sign); |
---|
4123 | } |
---|
4124 | if (shift_sign>0) |
---|
4125 | { |
---|
4126 | rightpart[j] = subst(rightpart[j+1], theta, 1/par(1)*(theta - shift_sign)); |
---|
4127 | } |
---|
4128 | rightpart[j+1] = shiftvar; |
---|
4129 | rparts = rparts + list(rightpart); |
---|
4130 | } |
---|
4131 | //And now, we put all possibilities together |
---|
4132 | tempadd = list(); |
---|
4133 | for (j = 1; j<=size(lparts); j++) |
---|
4134 | { |
---|
4135 | for (k = 1; k<=size(rparts);k++) |
---|
4136 | { |
---|
4137 | tempadd = tempadd + list(lparts[j]+rparts[k]); |
---|
4138 | } |
---|
4139 | } |
---|
4140 | tempadd = delete(tempadd,1); // The first entry is already in the list |
---|
4141 | result = result + tempadd; |
---|
4142 | continue; //We can may be not be done already with the ith entry |
---|
4143 | }//One entry was theta resp. theta +1 |
---|
4144 | }//checking every entry of result for theta or theta +1 |
---|
4145 | dbprint(p," Done"); |
---|
4146 | //map back to the basering |
---|
4147 | dbprint(p," Mapping back everything to the basering"); |
---|
4148 | setring(r); |
---|
4149 | map finalmap = tempRing, var(1), var(2),var(1)*var(2); |
---|
4150 | list result = finalmap(result); |
---|
4151 | for (i=1; i<=size(result);i++) |
---|
4152 | {//adding the K factor |
---|
4153 | result[i] = k_factor + result[i]; |
---|
4154 | }//adding the k-factor |
---|
4155 | dbprint(p," Done"); |
---|
4156 | dbprint(p," Delete double entries in the list."); |
---|
4157 | result = delete_dublicates_noteval(result); |
---|
4158 | dbprint(p," Done"); |
---|
4159 | return(result); |
---|
4160 | }//proc HomogfacFirstQWeylAll |
---|
4161 | example |
---|
4162 | { |
---|
4163 | "EXAMPLE:";echo=2; |
---|
4164 | ring R = (0,q),(x,d),dp; |
---|
4165 | def r = nc_algebra (q,1); |
---|
4166 | setring(r); |
---|
4167 | poly h = q^25*x^10*d^10+q^16*(q^4+q^3+q^2+q+1)^2*x^9*d^9+ |
---|
4168 | q^9*(q^13+3*q^12+7*q^11+13*q^10+20*q^9+26*q^8+30*q^7+ |
---|
4169 | 31*q^6+26*q^5+20*q^4+13*q^3+7*q^2+3*q+1)*x^8*d^8+ |
---|
4170 | q^4*(q^9+2*q^8+4*q^7+6*q^6+7*q^5+8*q^4+6*q^3+ |
---|
4171 | 4*q^2+2q+1)*(q^4+q^3+q^2+q+1)*(q^2+q+1)*x^7*d^7+ |
---|
4172 | 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+ |
---|
4173 | (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+ |
---|
4174 | 6*x^3*d^3+24; |
---|
4175 | homogfacFirstQWeyl_all(h); |
---|
4176 | } |
---|
4177 | |
---|
4178 | //TODO: FirstQWeyl check the parameters... |
---|
4179 | |
---|
4180 | //================================================== |
---|
4181 | // EASY EXAMPLES FOR WEYL ALGEBRA |
---|
4182 | //================================================== |
---|
4183 | /* |
---|
4184 | Easy and fast example polynomials where one can find factorizations: K<x,d |dx=xd+1> |
---|
4185 | (x^2+d)*(x^2+d); |
---|
4186 | (x^2+x)*(x^2+d); |
---|
4187 | (x^3+x+1)*(x^4+d*x+2); |
---|
4188 | (x^2*d+d)*(d+x*d); |
---|
4189 | d^3+x*d^3+2*d^2+2*(x+1)*d^2+d+(x+2)*d; //Example 5 Grigoriev-Schwarz. |
---|
4190 | (d+1)*(d+1)*(d+x*d); //Landau Example projected to the first dimension. |
---|
4191 | */ |
---|
4192 | |
---|
4193 | //================================================== |
---|
4194 | //Some Bugs(fixed)/hard examples from Martin Lee: |
---|
4195 | //================================================== |
---|
4196 | // ex1, ex2 |
---|
4197 | /* |
---|
4198 | ring s = 0,(x,d),Ws(-1,1); |
---|
4199 | def S = nc_algebra(1,1); setring S; |
---|
4200 | poly a = 10x5d4+26x4d5+47x5d2-97x4d3; //Not so hard any more... Done in around 4 minutes |
---|
4201 | def l= facFirstWeyl (a); l; |
---|
4202 | kill l; |
---|
4203 | poly b = -5328x8d5-5328x7d6+720x9d2+720x8d3-16976x7d4-38880x6d5 |
---|
4204 | -5184x7d3-5184x6d4-3774x5d5+2080x8d+5760x7d2-6144x6d3-59616x5d4 |
---|
4205 | +3108x3d6-4098x6d2-25704x5d3-21186x4d4+8640x6d-17916x4d3+22680x2d5 |
---|
4206 | +2040x5d-4848x4d2-9792x3d3+3024x2d4-10704x3d2-3519x2d3+34776xd4 |
---|
4207 | +12096xd3+2898d4-5040x2d+8064d3+6048d2; //Still very hard... But it seems to be only because of the |
---|
4208 | //combinatorial explosion |
---|
4209 | def l= facFirstWeyl (b); l; |
---|
4210 | |
---|
4211 | // ex3: there was difference in answers => fixed |
---|
4212 | LIB "ncfactor.lib"; |
---|
4213 | ring r = 0,(x,y,z),dp; |
---|
4214 | matrix D[3][3]; D[1,3]=-1; |
---|
4215 | def R = nc_algebra(1,D); |
---|
4216 | setring R; |
---|
4217 | poly g= 7*z4*x+62*z3+26*z; |
---|
4218 | def l1= facSubWeyl (g, x, z); |
---|
4219 | l1; |
---|
4220 | //---- other ring |
---|
4221 | ring s = 0,(x,z),dp; |
---|
4222 | def S = nc_algebra(1,-1); setring S; |
---|
4223 | poly g= 7*z4*x+62*z3+26*z; |
---|
4224 | def l2= facFirstWeyl (g); |
---|
4225 | l2; |
---|
4226 | map F = R,x,0,z; |
---|
4227 | list l1 = F(l1); |
---|
4228 | l1; |
---|
4229 | //---- so the answers look different, check them! |
---|
4230 | testNCfac(l2); // ok |
---|
4231 | testNCfac(l1); // was not ok, but now it's been fixed!!! |
---|
4232 | |
---|
4233 | // selbst D und X so vertauschen dass sie erfuellt ist : ist gemacht |
---|
4234 | |
---|
4235 | */ |
---|
4236 | |
---|
4237 | /* |
---|
4238 | // bug from M Lee |
---|
4239 | LIB "ncfactor.lib"; |
---|
4240 | ring s = 0,(z,x),dp; |
---|
4241 | def S = nc_algebra(1,1); setring S; |
---|
4242 | poly f= -60z4x2-54z4-56zx3-59z2x-64; |
---|
4243 | def l= facFirstWeyl (f); |
---|
4244 | l; // before: empty list; after fix: 1 entry, f is irreducible |
---|
4245 | poly g = 75z3x2+92z3+24; |
---|
4246 | def l= facFirstWeyl (g); |
---|
4247 | l; //before: empty list, now: correct |
---|
4248 | */ |
---|
4249 | |
---|
4250 | /* more things from Martin Lee; fixed |
---|
4251 | ring R = 0,(x,s),dp; |
---|
4252 | def r = nc_algebra(1,s); |
---|
4253 | setring(r); |
---|
4254 | poly h = (s2*x+x)*s; |
---|
4255 | h= h* (x+s); |
---|
4256 | def l= facFirstShift(h); |
---|
4257 | l; // contained doubled entries: not anymore, fixed! |
---|
4258 | |
---|
4259 | ring R = 0,(x,s),dp; |
---|
4260 | def r = nc_algebra(1,-1); |
---|
4261 | setring(r); |
---|
4262 | poly h = (s2*x+x)*s; |
---|
4263 | h= h* (x+s); |
---|
4264 | def l= facFirstWeyl(h); |
---|
4265 | l; // contained doubled entries: not anymore, fixed! |
---|
4266 | |
---|
4267 | */ |
---|
4268 | |
---|
4269 | //====================================================================== |
---|
4270 | //Examples from TestSuite that are terminating in a reasonable time. |
---|
4271 | //====================================================================== |
---|
4272 | |
---|
4273 | //Counter example for old Algorithm, but now working: |
---|
4274 | /* |
---|
4275 | ring R = 0,(x,d),dp; |
---|
4276 | def r = nc_algebra(1,1); |
---|
4277 | setring(r); |
---|
4278 | LIB "ncfactor.lib"; |
---|
4279 | poly h = (1+x^2*d)^4; |
---|
4280 | list lsng = facFirstWeyl(h); |
---|
4281 | print(lsng); |
---|
4282 | */ |
---|
4283 | |
---|
4284 | //Example 2.7. from Master thesis |
---|
4285 | /* |
---|
4286 | ring R = 0,(x,d),dp; |
---|
4287 | def r = nc_algebra(1,1); |
---|
4288 | setring(r); |
---|
4289 | LIB "ncfactor.lib"; |
---|
4290 | poly h = (xdd + xd+1+ (xd+5)*x)*(((x*d)^2+1)*d + xd+3+ (xd+7)*x); |
---|
4291 | list lsng = facFirstWeyl(h); |
---|
4292 | print(lsng); |
---|
4293 | */ |
---|
4294 | |
---|
4295 | //Example with high combinatorial income |
---|
4296 | /* |
---|
4297 | ring R = 0,(x,d),dp; |
---|
4298 | def r = nc_algebra(1,1); |
---|
4299 | setring(r); |
---|
4300 | LIB "ncfactor.lib"; |
---|
4301 | poly h = (xdddd + (xd+1)*d*d+ (xd+5)*x*d*d)*(((x*d)^2+1)*d*x*x + (xd+3)*x*x+ (xd+7)*x*x*x); |
---|
4302 | list lsng = facFirstWeyl(h); |
---|
4303 | print(lsng); |
---|
4304 | */ |
---|
4305 | |
---|
4306 | //Once a bug, now working |
---|
4307 | /* |
---|
4308 | ring R = 0,(x,d),dp; |
---|
4309 | def r = nc_algebra(1,1); |
---|
4310 | setring(r); |
---|
4311 | LIB "ncfactor.lib"; |
---|
4312 | poly h = (x^2*d^2+x)*(x+1); |
---|
4313 | list lsng = facFirstWeyl(h); |
---|
4314 | print(lsng); |
---|
4315 | */ |
---|
4316 | |
---|
4317 | //Another one of that kind |
---|
4318 | /* |
---|
4319 | ring R = 0,(x,d),dp; |
---|
4320 | def r = nc_algebra(1,1); |
---|
4321 | setring(r); |
---|
4322 | LIB "ncfactor.lib"; |
---|
4323 | poly h = (x*d*d + (x*d)^5 +x)*((x*d+1)*d-(x*d-1)^5+x); |
---|
4324 | list lsng = facFirstWeyl(h); |
---|
4325 | print(lsng); |
---|
4326 | */ |
---|
4327 | |
---|
4328 | //Example of Victor for Shift Algebra |
---|
4329 | /* |
---|
4330 | ring s = 0,(n,Sn),dp; |
---|
4331 | def S = nc_algebra(1,Sn); setring S; |
---|
4332 | LIB "ncfactor.lib"; |
---|
4333 | list lsng = facFirstShift(n^2*Sn^2+3*n*Sn^2-n^2+2*Sn^2-3*n-2); |
---|
4334 | print(lsng); |
---|
4335 | */ |
---|
4336 | |
---|
4337 | //Interesting example, as there are actually also some complex solutions to it: |
---|
4338 | /* |
---|
4339 | ring R = 0,(x,d),dp; |
---|
4340 | def r = nc_algebra(1,1); |
---|
4341 | setring(r); |
---|
4342 | LIB "/Users/albertheinle/Studium/forschung/ncfactor/versionen/ncfactor.lib"; |
---|
4343 | poly h =(x^3+x+1)*(x^4+d*x+2);//Example for finitely many, but more than one solution in between. |
---|
4344 | list lsng = facFirstWeyl(h); |
---|
4345 | print(lsng); |
---|
4346 | */ |
---|
4347 | |
---|
4348 | //Another one of that kind: |
---|
4349 | /* |
---|
4350 | ring R = 0,(x,d),dp; |
---|
4351 | def r = nc_algebra(1,1); |
---|
4352 | setring(r); |
---|
4353 | LIB "ncfactor.lib"; |
---|
4354 | poly h =(x^2+d)*(x^2+d);//Example for finitely many, but more than one solution in between. |
---|
4355 | list lsng = facFirstWeyl(h); |
---|
4356 | print(lsng); |
---|
4357 | */ |
---|
4358 | |
---|
4359 | //Example by W. Koepf: |
---|
4360 | /* |
---|
4361 | ring R = 0,(x,d),dp; |
---|
4362 | def r = nc_algebra(1,1); |
---|
4363 | setring(r); |
---|
4364 | LIB "ncfactor.lib"; |
---|
4365 | poly h = (x^4-1)*x*d^2+(1+7*x^4)*d+8*x^3; |
---|
4366 | list lsng = facFirstWeyl(h); |
---|
4367 | print(lsng); |
---|
4368 | */ |
---|
4369 | |
---|
4370 | //Shift Example from W. Koepf |
---|
4371 | /* |
---|
4372 | ring R = 0,(n,s),dp; |
---|
4373 | def r = nc_algebra(1,s); |
---|
4374 | setring(r); |
---|
4375 | LIB "ncfactor.lib"; |
---|
4376 | poly h = n*(n+1)*s^2-2*n*(n+100)*s+(n+99)*(n+100); |
---|
4377 | list lsng = facFirstShift(h); |
---|
4378 | print(lsng); |
---|
4379 | */ |
---|
4380 | |
---|
4381 | //Tsai Example... Once hard, now easy... |
---|
4382 | /* |
---|
4383 | ring R = 0,(x,d),dp; |
---|
4384 | def r = nc_algebra(1,1); |
---|
4385 | setring(r); |
---|
4386 | LIB "ncfactor.lib"; |
---|
4387 | poly h = (x^6+2*x^4-3*x^2)*d^2-(4*x^5-4*x^4-12*x^2-12*x)*d + (6*x^4-12*x^3-6*x^2-24*x-12); |
---|
4388 | list lsng =facFirstWeyl(h); |
---|
4389 | print(lsng); |
---|
4390 | */ |
---|
4391 | |
---|
4392 | //====================================================================== |
---|
4393 | // Hard examples not yet calculatable in feasible amount of time |
---|
4394 | //====================================================================== |
---|
4395 | |
---|
4396 | //Also a counterexample for REDUCE. Very long Groebner basis computation in between. |
---|
4397 | /* |
---|
4398 | ring R = 0,(x,d),dp; |
---|
4399 | def r = nc_algebra(1,1); |
---|
4400 | setring(r); |
---|
4401 | LIB "ncfactor.lib"; |
---|
4402 | poly h = (d^4+x^2+dx+x)*(d^2+x^4+xd+d); |
---|
4403 | list lsng = facFirstWeyl(h); |
---|
4404 | print(lsng); |
---|
4405 | */ |
---|
4406 | |
---|
4407 | //Example from the Mainz-Group |
---|
4408 | /* |
---|
4409 | ring R = 0,(x,d),dp; |
---|
4410 | def r = nc_algebra(1,1); |
---|
4411 | setring(r); |
---|
4412 | poly dop6 = 1/35*x^4*(27-70*x+35*x^2)+ 1/35*x*(32+152*x+100*x^2-59*x^3+210*x^4+105*x^5)*d+ |
---|
4413 | (-10368/35-67056/35*x-35512/7*x^2-50328/7*x^3-40240/7*x^4-2400*x^5-400*x^6)*d^2+ |
---|
4414 | (-144/35*(x+1)*(1225*x^5+11025*x^4+37485*x^3+61335*x^2+50138*x+16584)-6912/35*(x+2)* |
---|
4415 | (x+1)*(105*x^4+1155*x^3+4456*x^2+7150*x+4212) -27648/35*(x+3)*(x+1)*(35*x^2+350*x+867)* |
---|
4416 | (x+2)^2)*d^3; |
---|
4417 | LIB "ncfactor.lib"; |
---|
4418 | printlevel = 5; |
---|
4419 | facFirstWeyl(dop6); |
---|
4420 | $;*/ |
---|
4421 | |
---|
4422 | //Another Mainz Example: |
---|
4423 | /* |
---|
4424 | LIB "ncfactor.lib"; |
---|
4425 | ring R = 0,(x,d),dp; |
---|
4426 | def r = nc_algebra(1,1); |
---|
4427 | setring(r); |
---|
4428 | poly dopp = 82547*x^4*d^4+60237*x^3*d^3+26772*x^5*d^5+2231*x^6*d^6+x*(1140138* |
---|
4429 | x^2*d^2-55872*x*d-3959658*x^3*d^3-8381805*x^4*d^4-3089576*x^5*d^5-274786* |
---|
4430 | x^6*d^6)+x^2*(-16658622*x*d-83427714*x^2*d^2-19715033*x^3*d^3+78915395*x^4 |
---|
4431 | *d^4+35337930*x^5*d^5+3354194*x^6*d^6)+x^3*(-99752472-1164881352*x*d+ |
---|
4432 | 4408536996*x^2*d^2+11774185985*x^3*d^3+5262196786*x^4*d^4+1046030561/2*x^5* |
---|
4433 | d^5-10564451/2*x^6*d^6)+x^4*(-1925782272+21995375398*x*d+123415803356*x^2* |
---|
4434 | d^2+302465300831/2*x^3*d^3+34140803907/2*x^4*d^4-15535653409*x^5*d^5-\ |
---|
4435 | 2277687768*x^6*d^6)+x^5*(71273525520+691398212366*x*d+901772633569*x^2*d^2+ |
---|
4436 | 2281275427069*x^3*d^3+2944352819911/2*x^4*d^4+836872370039/4*x^5*d^5+ |
---|
4437 | 9066399237/4*x^6*d^6)+x^6*(2365174430376+9596715855542*x*d+29459572469704*x^ |
---|
4438 | 2*d^2+92502197003786*x^3*d^3+65712473180525*x^4*d^4+13829360193674*x^5*d^5 |
---|
4439 | +3231449477251/4*x^6*d^6)+x^7*(26771079436836+117709870166226*x*d+ |
---|
4440 | 821686455179082*x^2*d^2+1803972139232179*x^3*d^3+1083654460691481*x^4*d^4+ |
---|
4441 | 858903621851785/4*x^5*d^5+50096565802957/4*x^6*d^6)+x^8*(179341727601960+ |
---|
4442 | 2144653944040630*x*d+13123246960284302*x^2*d^2+41138357917778169/2*x^3*d^3+ |
---|
4443 | 20605819587976401/2*x^4*d^4+3677396642905423/2*x^5*d^5+402688260229369/4*x^6 |
---|
4444 | *d^6)+x^9*(2579190935961288+43587063726809764*x*d+157045086382352387*x^2*d^ |
---|
4445 | 2+172175668477370223*x^3*d^3+138636285385875407/2*x^4*d^4+10707836398626232* |
---|
4446 | x^5*d^5+529435530567584*x^6*d^6)+x^10*(41501953525903392+558336731465626084* |
---|
4447 | x*d+1407267553543222268*x^2*d^2+1153046693323226808*x^3*d^3+ |
---|
4448 | 372331468563656085*x^4*d^4+48654019090240214*x^5*d^5+2114661191282167*x^6*d |
---|
4449 | ^6)+x^11*(364526077273381884+4158060401095928464*x*d+8646807662899324262*x^2* |
---|
4450 | d^2+5914675753405705400*x^3*d^3+1631934058875116005*x^4*d^4+ |
---|
4451 | 187371894330537204*x^5*d^5+7366806367019734*x^6*d^6)+x^12*( |
---|
4452 | 1759850321214603648+18265471270535733520*x*d+34201910114871110912*x^2*d^2+ |
---|
4453 | 21265221434709398152*x^3*d^3+5437363546219595036*x^4*d^4+594029113431041060* |
---|
4454 | x^5*d^5+22881659624561644*x^6*d^6)+x^13*(4648382639403200688+ |
---|
4455 | 45699084277107816096*x*d+81049061578449009384*x^2*d^2+48858488665016574368*x |
---|
4456 | ^3*d^3+12515362110098721444*x^4*d^4+1412152747420021048*x^5*d^5+ |
---|
4457 | 57196947123984972*x^6*d^6)+x^14*(5459369397960020544+55837825300341621824*x* |
---|
4458 | d+105671876924055409696*x^2*d^2+71551727420848766624*x^3*d^3+ |
---|
4459 | 21094786205096577808*x^4*d^4+2695663190297032192*x^5*d^5+118791751565613264* |
---|
4460 | x^6*d^6)+x^15*(1023333653580043776+47171127937488813824*x*d+ |
---|
4461 | 157258351906685700352*x^2*d^2+145765192195300531840*x^3*d^3+ |
---|
4462 | 49876215785510342176*x^4*d^4+6647374188802036864*x^5*d^5+287310278455067312* |
---|
4463 | x^6*d^6)+x^16*(11960091747366236160+250326608568269289472*x*d+ |
---|
4464 | 677587171115580981248*x^2*d^2+538246374825683603456*x^3*d^3+ |
---|
4465 | 161380433451548754048*x^4*d^4+19149099315354950144*x^5*d^5+ |
---|
4466 | 746433247985092544*x^6*d^6)+x^17*(42246252365448668160+657220532737851248640* |
---|
4467 | x*d+1531751689216283911680*x^2*d^2+1090829514212206064640*x^3*d^3+ |
---|
4468 | 299280728709430851840*x^4*d^4+32932767387222323200*x^5*d^5+ |
---|
4469 | 1202281367574179840*x^6*d^6)+x^18*(6239106101942784000+320638742839606579200* |
---|
4470 | x*d+873857213570556364800*x^2*d^2+645649080101933721600*x^3*d^3+ |
---|
4471 | 177008238160627276800*x^4*d^4+19165088507111475200*x^5*d^5+ |
---|
4472 | 683600826675660800*x^6*d^6)+x^19*(-60440251454613504000-476055211197689856000 |
---|
4473 | *x*d-733497382597635072000*x^2*d^2-386038662982742016000*x^3*d^3-\ |
---|
4474 | 83361486778142976000*x^4*d^4-7524999543181824000*x^5*d^5-232189492987008000* |
---|
4475 | x^6*d^6)+x^20*(1578562930483200000+12628503443865600000*x*d+ |
---|
4476 | 19732036631040000000*x^2*d^2+10523752869888000000*x^3*d^3+ |
---|
4477 | 2302070940288000000*x^4*d^4+210475057397760000*x^5*d^5+6577345543680000*x^6* |
---|
4478 | d^6); |
---|
4479 | printlevel = 3; |
---|
4480 | facFirstWeyl(dopp); |
---|
4481 | */ |
---|
4482 | |
---|
4483 | |
---|
4484 | |
---|
4485 | //Hard Example by Viktor: |
---|
4486 | /* |
---|
4487 | ring r = 0,(x,d), (dp); |
---|
4488 | def R = nc_algebra(1,1); |
---|
4489 | setring R; |
---|
4490 | LIB "/Users/albertheinle/Studium/forschung/ncfactor/versionen/ncfactor.lib"; |
---|
4491 | poly t = x; poly D =d; |
---|
4492 | poly p = 2*t^2*D^8-6*t*D^8+2*t^2*D^7+8*t*D^7+12*D^7-2*t^4*D^6+6*t^3*D^6+12*t*D^6-20*D^6 |
---|
4493 | -2*t^4*D^5-8*t^3*D^5-4*t^2*D^5+12*t*D^5-28*D^5-12*t^3*D^4-4*t^2*D^4-4*t*D^4-24*D^4+4*t^4*D^3 |
---|
4494 | -12*t^3*D^3+2*t^2*D^3-18*t*D^3+16*D^3+6*t^4*D^2-2*t^3*D^2+2*t^2*D^2-2*t*D^2+44*D^2+2*t^4*D |
---|
4495 | +12*t^3*D+2*t*D+4*t^3-8; |
---|
4496 | list lsng = facFirstWeyl(p); |
---|
4497 | print(lsng); |
---|
4498 | */ |
---|