\begin{lisp:documentation}{poly$-$scalar$-$composition}{FUNCTION}{f g {\sf \&optional} (order \#'lex$>$) }
Returns a polynomial obtained by substituting a list of polynomials
G=(G1,G2,...,GN) into a polynomial F(X1,X2,...,XN). All polynomials
are assumed to be in the internal form, so variables do not
explicitly apprear in the calculation. 
\end{lisp:documentation}

\begin{lisp:documentation}{poly$-$composition}{FUNCTION}{f g {\sf \&optional} (order \#'lex$>$) }
Return the composition of a polynomial map F with a polynomial map
G. Both maps are represented as lists of polynomials, and each
polynomial is in the internal alist representation. The restriction
is that the length of the list G must be the number of variables in
the list F. 
\end{lisp:documentation}

\begin{lisp:documentation}{poly$-$dynamic$-$power}{FUNCTION}{f n {\sf \&optional} (order \#'lex$>$) }
Calculate the composition FoFo...oF (n times), where
F is a polynomial map represented as a list of polynomials.
\end{lisp:documentation}

\begin{lisp:documentation}{poly$-$scalar$-$evaluate}{FUNCTION}{f x {\sf \&optional} (order \#'lex$>$) }
Evaluate a polynomial F at a point X. This operation is implemented
through POLY$-$SCALAR$-$COMPOSITION.
\end{lisp:documentation}

\begin{lisp:documentation}{poly$-$evaluate}{FUNCTION}{f x {\sf \&optional} (order \#'lex$>$) }
Evaluate a polynomial map F, represented as list of polynomials, at a
point X. 
\end{lisp:documentation}

\begin{lisp:documentation}{factorial}{FUNCTION}{n {\sf \&optional} (k n) {\sf \&aux} (result 1) }
Return N!/(N$-$K)!=N(N$-$1)(N$-$K+1).
\end{lisp:documentation}

\begin{lisp:documentation}{poly$-$scalar$-$diff}{FUNCTION}{f m }
Return the partial derivative of a polynomial F over multiple
 variables according to multiindex M.
\end{lisp:documentation}

\begin{lisp:documentation}{poly$-$diff}{FUNCTION}{f m }
Return the partial derivative of a polynomial map F, represented as a
list of polynomials, with respect to several variables, according to
multi$-$index M. 
\end{lisp:documentation}

\begin{lisp:documentation}{standard$-$vector}{FUNCTION}{n k {\sf \&optional} (coeff 1) {\sf \&aux} (v (make$-$list n :initial$-$element 0)) }
Returns vector (0 0 ... 1 ... 0 0) of length N, where 1 appears on
K$-$th place. 
\end{lisp:documentation}

\begin{lisp:documentation}{scalar$-$partial}{FUNCTION}{f k {\sf \&optional} (l 1) }
Returns the L$-$th partial derivative of a polynomial F over the
K$-$th variable. 
\end{lisp:documentation}

\begin{lisp:documentation}{partial}{FUNCTION}{f k {\sf \&optional} (l 1) }
Returns the L$-$th partial derivative over the K$-$th variable, of a
polynomial map F, represented as a list of polynomials.
\end{lisp:documentation}

\begin{lisp:documentation}{determinant}{FUNCTION}{f {\sf \&optional} (order \#'lex$>$) {\sf \&aux} (result nil) }
Returns the determinant of a polynomial matrix F, which is a list of
rows of the matrix, each row is a list of polynomials.  The algorithm
is recursive expansion along columns.
\end{lisp:documentation}

\begin{lisp:documentation}{minor}{FUNCTION}{f i j {\sf \&optional} (order \#'lex$>$) }
Calculate the minor of a polynomial matrix F with respect to entry
(I,J). 
\end{lisp:documentation}

\begin{lisp:documentation}{drop$-$row}{FUNCTION}{f i }
Discards the I$-$th row from a polynomial matrix F.
\end{lisp:documentation}

\begin{lisp:documentation}{drop$-$column}{FUNCTION}{f j }
Discards the J$-$th column from a polynomial matrix F.
\end{lisp:documentation}

\begin{lisp:documentation}{drop$-$elt}{FUNCTION}{lst j }
Discards the J$-$th element from a list LST.
\end{lisp:documentation}

\begin{lisp:documentation}{matrix$-$}{FUNCTION}{f g {\sf \&optional} (order \#'lex$>$) }
Returns difference of two polynomial matrices F and G.
\end{lisp:documentation}

\begin{lisp:documentation}{scalar$-$times$-$matrix}{FUNCTION}{s f }
Returns a product of a polynomial S by a polynomial matrix F.
\end{lisp:documentation}

\begin{lisp:documentation}{monom$-$times$-$matrix}{FUNCTION}{m f }
Returns a product of a monomial M by a polynomial matrix F.
\end{lisp:documentation}

\begin{lisp:documentation}{term$-$times$-$matrix}{FUNCTION}{term f }
Returns a product of a term TERM by a polynomial matrix F.
\end{lisp:documentation}

\begin{lisp:documentation}{poly$-$list$-$}{FUNCTION}{f g {\sf \&optional} (order \#'lex$>$) }
Returns the list of differences of two lists of polynomials
F and G (polynomial maps).
\end{lisp:documentation}

\begin{lisp:documentation}{scalar$-$times$-$poly$-$list}{FUNCTION}{s f }
Returns the list of products of a polynomial S by the
list of polynomials F.
\end{lisp:documentation}

\begin{lisp:documentation}{monom$-$times$-$poly$-$list}{FUNCTION}{m f }
Returns the list of products of a monomial M by the
list of polynomials F.
\end{lisp:documentation}

\begin{lisp:documentation}{term$-$times$-$poly$-$list}{FUNCTION}{term f }
Returns the list of products of a term TERM by the
list of polynomials F.
\end{lisp:documentation}

\begin{lisp:documentation}{characteristic$-$combination}{FUNCTION}{a b {\sf \&optional} (order \#'lex$>$) {\sf \&aux} (n (length b)) }
Returns A $-$ U1 * B1 $-$ U2 * B2 $-$ ... $-$ UM * BM where A is a
polynomial and B=(B1,B2,...,BM) is a polynomial list, where U1, U2,
... , UM are new variables. These variables will be added to every
polynomial A and BI as the last M variables.
\end{lisp:documentation}

\begin{lisp:documentation}{characteristic$-$combination$-$poly$-$list}{FUNCTION}{a b {\sf \&optional} (order \#'lex$>$) }
Returns A $-$ U1 * B1 $-$ U2 * B2 $-$ ... $-$ UM * BM where A is a
polynomial list and B=(B1, B2, ... , BM) is a list of polynomial
lists, where U1, U2, ... ,UM are new variables. These variables will
be added to every polynomial A and BI as the last M variables. Se
also CHARACTERISTIC$-$COMBINATION. 
\end{lisp:documentation}

\begin{lisp:documentation}{characteristic$-$matrix}{FUNCTION}{a {\sf \&optional} (order \#'lex$>$) (b (list (identity$-$matrix (length a) (length (caaaar a))))) }
Returns A $-$ U1*B1 $-$ U2*B2 $-$ ... $-$ UM * BM where A is a
polynomial matrix and B=(B1,B2,...,BM) is a list of polynomial
matrices, where U1, U2, .., UM are new variables. These variables
will be added to every polynomial A and BI as the last M variables.
Se also CHARACTERISTIC$-$COMBINATION. 
\end{lisp:documentation}

\begin{lisp:documentation}{characteristic$-$polynomial}{FUNCTION}{a {\sf \&optional} (order \#'lex$>$) (b (list (identity$-$matrix (length a) (length (caaaar a))))) }
Returns the generalized characteristic polynomial, i.e. the
determinant DET(A $-$ U1 * B1 $-$ U2 * B2 $-$ ... $-$ UM * BM), where
A and BI are square polynomial matrices in N variables. The resulting
polynomial will have N+M variables, with U1, U2, ..., UM added as the
last M variables. 
\end{lisp:documentation}

\begin{lisp:documentation}{identity$-$matrix}{FUNCTION}{dim nvars }
Return the polynomial matrix which is the identity matrix. DIM is the
requested dimension and NVARS is the number of variables of each
entry. 
\end{lisp:documentation}

\begin{lisp:documentation}{print$-$matrix}{FUNCTION}{f vars }
Prints a polynomial matrix F, using a list of symbols VARS as
variable names. 
\end{lisp:documentation}

\begin{lisp:documentation}{jacobi$-$matrix}{FUNCTION}{f {\sf \&optional} (m (length f)) (n (length (caaaar f))) }
Returns the Jacobi matrix of a polynomial list F over the first N
variables. 
\end{lisp:documentation}

\begin{lisp:documentation}{jacobian}{FUNCTION}{f {\sf \&optional} (order \#'lex$>$) (m (length f)) (n (length (caaaar f))) }
Returns the Jacobian (determinant) of a polynomial list F over the
first N variables. 
\end{lisp:documentation}

