[1] | 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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