1 | \begin{lisp:documentation}{poly$-$scalar$-$composition}{FUNCTION}{f g {\sf \&optional} (order \#'lex$>$) }
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2 | Returns a polynomial obtained by substituting a list of polynomials
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3 | G=(G1,G2,...,GN) into a polynomial F(X1,X2,...,XN). All polynomials
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4 | are assumed to be in the internal form, so variables do not
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5 | explicitly apprear in the calculation.
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6 | \end{lisp:documentation}
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7 |
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8 | \begin{lisp:documentation}{poly$-$composition}{FUNCTION}{f g {\sf \&optional} (order \#'lex$>$) }
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9 | Return the composition of a polynomial map F with a polynomial map
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10 | G. Both maps are represented as lists of polynomials, and each
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11 | polynomial is in the internal alist representation. The restriction
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12 | is that the length of the list G must be the number of variables in
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13 | the list F.
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14 | \end{lisp:documentation}
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15 |
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16 | \begin{lisp:documentation}{poly$-$dynamic$-$power}{FUNCTION}{f n {\sf \&optional} (order \#'lex$>$) }
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17 | Calculate the composition FoFo...oF (n times), where
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18 | F is a polynomial map represented as a list of polynomials.
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19 | \end{lisp:documentation}
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20 |
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21 | \begin{lisp:documentation}{poly$-$scalar$-$evaluate}{FUNCTION}{f x {\sf \&optional} (order \#'lex$>$) }
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22 | Evaluate a polynomial F at a point X. This operation is implemented
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23 | through POLY$-$SCALAR$-$COMPOSITION.
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24 | \end{lisp:documentation}
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25 |
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26 | \begin{lisp:documentation}{poly$-$evaluate}{FUNCTION}{f x {\sf \&optional} (order \#'lex$>$) }
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27 | Evaluate a polynomial map F, represented as list of polynomials, at a
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28 | point X.
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29 | \end{lisp:documentation}
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30 |
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31 | \begin{lisp:documentation}{factorial}{FUNCTION}{n {\sf \&optional} (k n) {\sf \&aux} (result 1) }
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32 | Return N!/(N$-$K)!=N(N$-$1)(N$-$K+1).
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33 | \end{lisp:documentation}
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34 |
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35 | \begin{lisp:documentation}{poly$-$scalar$-$diff}{FUNCTION}{f m }
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36 | Return the partial derivative of a polynomial F over multiple
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37 | variables according to multiindex M.
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38 | \end{lisp:documentation}
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39 |
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40 | \begin{lisp:documentation}{poly$-$diff}{FUNCTION}{f m }
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41 | Return the partial derivative of a polynomial map F, represented as a
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42 | list of polynomials, with respect to several variables, according to
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43 | multi$-$index M.
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44 | \end{lisp:documentation}
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45 |
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46 | \begin{lisp:documentation}{standard$-$vector}{FUNCTION}{n k {\sf \&optional} (coeff 1) {\sf \&aux} (v (make$-$list n :initial$-$element 0)) }
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47 | Returns vector (0 0 ... 1 ... 0 0) of length N, where 1 appears on
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48 | K$-$th place.
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49 | \end{lisp:documentation}
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50 |
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51 | \begin{lisp:documentation}{scalar$-$partial}{FUNCTION}{f k {\sf \&optional} (l 1) }
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52 | Returns the L$-$th partial derivative of a polynomial F over the
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53 | K$-$th variable.
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54 | \end{lisp:documentation}
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55 |
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56 | \begin{lisp:documentation}{partial}{FUNCTION}{f k {\sf \&optional} (l 1) }
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57 | Returns the L$-$th partial derivative over the K$-$th variable, of a
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58 | polynomial map F, represented as a list of polynomials.
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59 | \end{lisp:documentation}
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60 |
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61 | \begin{lisp:documentation}{determinant}{FUNCTION}{f {\sf \&optional} (order \#'lex$>$) {\sf \&aux} (result nil) }
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62 | Returns the determinant of a polynomial matrix F, which is a list of
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63 | rows of the matrix, each row is a list of polynomials. The algorithm
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64 | is recursive expansion along columns.
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65 | \end{lisp:documentation}
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66 |
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67 | \begin{lisp:documentation}{minor}{FUNCTION}{f i j {\sf \&optional} (order \#'lex$>$) }
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68 | Calculate the minor of a polynomial matrix F with respect to entry
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69 | (I,J).
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70 | \end{lisp:documentation}
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71 |
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72 | \begin{lisp:documentation}{drop$-$row}{FUNCTION}{f i }
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73 | Discards the I$-$th row from a polynomial matrix F.
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74 | \end{lisp:documentation}
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75 |
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76 | \begin{lisp:documentation}{drop$-$column}{FUNCTION}{f j }
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77 | Discards the J$-$th column from a polynomial matrix F.
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78 | \end{lisp:documentation}
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79 |
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80 | \begin{lisp:documentation}{drop$-$elt}{FUNCTION}{lst j }
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81 | Discards the J$-$th element from a list LST.
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82 | \end{lisp:documentation}
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83 |
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84 | \begin{lisp:documentation}{matrix$-$}{FUNCTION}{f g {\sf \&optional} (order \#'lex$>$) }
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85 | Returns difference of two polynomial matrices F and G.
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86 | \end{lisp:documentation}
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87 |
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88 | \begin{lisp:documentation}{scalar$-$times$-$matrix}{FUNCTION}{s f }
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89 | Returns a product of a polynomial S by a polynomial matrix F.
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90 | \end{lisp:documentation}
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91 |
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92 | \begin{lisp:documentation}{monom$-$times$-$matrix}{FUNCTION}{m f }
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93 | Returns a product of a monomial M by a polynomial matrix F.
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94 | \end{lisp:documentation}
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95 |
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96 | \begin{lisp:documentation}{term$-$times$-$matrix}{FUNCTION}{term f }
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97 | Returns a product of a term TERM by a polynomial matrix F.
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98 | \end{lisp:documentation}
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99 |
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100 | \begin{lisp:documentation}{poly$-$list$-$}{FUNCTION}{f g {\sf \&optional} (order \#'lex$>$) }
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101 | Returns the list of differences of two lists of polynomials
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102 | F and G (polynomial maps).
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103 | \end{lisp:documentation}
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104 |
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105 | \begin{lisp:documentation}{scalar$-$times$-$poly$-$list}{FUNCTION}{s f }
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106 | Returns the list of products of a polynomial S by the
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107 | list of polynomials F.
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108 | \end{lisp:documentation}
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109 |
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110 | \begin{lisp:documentation}{monom$-$times$-$poly$-$list}{FUNCTION}{m f }
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111 | Returns the list of products of a monomial M by the
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112 | list of polynomials F.
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113 | \end{lisp:documentation}
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114 |
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115 | \begin{lisp:documentation}{term$-$times$-$poly$-$list}{FUNCTION}{term f }
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116 | Returns the list of products of a term TERM by the
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117 | list of polynomials F.
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118 | \end{lisp:documentation}
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119 |
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120 | \begin{lisp:documentation}{characteristic$-$combination}{FUNCTION}{a b {\sf \&optional} (order \#'lex$>$) {\sf \&aux} (n (length b)) }
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121 | Returns A $-$ U1 * B1 $-$ U2 * B2 $-$ ... $-$ UM * BM where A is a
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122 | polynomial and B=(B1,B2,...,BM) is a polynomial list, where U1, U2,
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123 | ... , UM are new variables. These variables will be added to every
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124 | polynomial A and BI as the last M variables.
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125 | \end{lisp:documentation}
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126 |
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127 | \begin{lisp:documentation}{characteristic$-$combination$-$poly$-$list}{FUNCTION}{a b {\sf \&optional} (order \#'lex$>$) }
|
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128 | Returns A $-$ U1 * B1 $-$ U2 * B2 $-$ ... $-$ UM * BM where A is a
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129 | polynomial list and B=(B1, B2, ... , BM) is a list of polynomial
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130 | lists, where U1, U2, ... ,UM are new variables. These variables will
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131 | be added to every polynomial A and BI as the last M variables. Se
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132 | also CHARACTERISTIC$-$COMBINATION.
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133 | \end{lisp:documentation}
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134 |
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135 | \begin{lisp:documentation}{characteristic$-$matrix}{FUNCTION}{a {\sf \&optional} (order \#'lex$>$) (b (list (identity$-$matrix (length a) (length (caaaar a))))) }
|
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136 | Returns A $-$ U1*B1 $-$ U2*B2 $-$ ... $-$ UM * BM where A is a
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137 | polynomial matrix and B=(B1,B2,...,BM) is a list of polynomial
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138 | matrices, where U1, U2, .., UM are new variables. These variables
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139 | will be added to every polynomial A and BI as the last M variables.
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140 | Se also CHARACTERISTIC$-$COMBINATION.
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141 | \end{lisp:documentation}
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142 |
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143 | \begin{lisp:documentation}{characteristic$-$polynomial}{FUNCTION}{a {\sf \&optional} (order \#'lex$>$) (b (list (identity$-$matrix (length a) (length (caaaar a))))) }
|
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144 | Returns the generalized characteristic polynomial, i.e. the
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145 | determinant DET(A $-$ U1 * B1 $-$ U2 * B2 $-$ ... $-$ UM * BM), where
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146 | A and BI are square polynomial matrices in N variables. The resulting
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147 | polynomial will have N+M variables, with U1, U2, ..., UM added as the
|
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148 | last M variables.
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149 | \end{lisp:documentation}
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150 |
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151 | \begin{lisp:documentation}{identity$-$matrix}{FUNCTION}{dim nvars }
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152 | Return the polynomial matrix which is the identity matrix. DIM is the
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153 | requested dimension and NVARS is the number of variables of each
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154 | entry.
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155 | \end{lisp:documentation}
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156 |
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157 | \begin{lisp:documentation}{print$-$matrix}{FUNCTION}{f vars }
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158 | Prints a polynomial matrix F, using a list of symbols VARS as
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159 | variable names.
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160 | \end{lisp:documentation}
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161 |
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162 | \begin{lisp:documentation}{jacobi$-$matrix}{FUNCTION}{f {\sf \&optional} (m (length f)) (n (length (caaaar f))) }
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163 | Returns the Jacobi matrix of a polynomial list F over the first N
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164 | variables.
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165 | \end{lisp:documentation}
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166 |
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167 | \begin{lisp:documentation}{jacobian}{FUNCTION}{f {\sf \&optional} (order \#'lex$>$) (m (length f)) (n (length (caaaar f))) }
|
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168 | Returns the Jacobian (determinant) of a polynomial list F over the
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169 | first N variables.
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170 | \end{lisp:documentation}
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171 |
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