;;; POLY-SCALAR-COMPOSITION (f g &optional (order #'lex>))           [FUNCTION]
;;;    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. 
;;;
;;; POLY-COMPOSITION (f g &optional (order #'lex>))                  [FUNCTION]
;;;    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. 
;;;
;;; POLY-DYNAMIC-POWER (f n &optional (order #'lex>))                [FUNCTION]
;;;    Calculate the composition FoFo...oF (n times), where
;;;    F is a polynomial map represented as a list of polynomials.
;;;
;;; POLY-SCALAR-EVALUATE (f x &optional (order #'lex>))              [FUNCTION]
;;;    Evaluate a polynomial F at a point X. This operation is implemented
;;;    through POLY-SCALAR-COMPOSITION.
;;;
;;; POLY-EVALUATE (f x &optional (order #'lex>))                     [FUNCTION]
;;;    Evaluate a polynomial map F, represented as list of polynomials, at a
;;;    point X. 
;;;
;;; FACTORIAL (n &optional (k n) &aux (result 1))                    [FUNCTION]
;;;    Return N!/(N-K)!=N(N-1)(N-K+1).
;;;
;;; POLY-SCALAR-DIFF (f m)                                           [FUNCTION]
;;;    Return the partial derivative of a polynomial F over multiple
;;;     variables according to multiindex M.
;;;
;;; POLY-DIFF (f m)                                                  [FUNCTION]
;;;    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. 
;;;
;;; STANDARD-VECTOR (n k &optional (coeff 1)                         [FUNCTION]
;;;                  &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. 
;;;
;;; SCALAR-PARTIAL (f k &optional (l 1))                             [FUNCTION]
;;;    Returns the L-th partial derivative of a polynomial F over the K-th
;;;    variable. 
;;;
;;; PARTIAL (f k &optional (l 1))                                    [FUNCTION]
;;;    Returns the L-th partial derivative over the K-th variable, of a
;;;    polynomial map F, represented as a list of polynomials.
;;;
;;; DETERMINANT (f &optional (order #'lex>) &aux (result nil))       [FUNCTION]
;;;    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.
;;;
;;; MINOR (f i j &optional (order #'lex>))                           [FUNCTION]
;;;    Calculate the minor of a polynomial matrix F with respect to entry
;;;    (I,J). 
;;;
;;; DROP-ROW (f i)                                                   [FUNCTION]
;;;    Discards the I-th row from a polynomial matrix F.
;;;
;;; DROP-COLUMN (f j)                                                [FUNCTION]
;;;    Discards the J-th column from a polynomial matrix F.
;;;
;;; DROP-ELT (lst j)                                                 [FUNCTION]
;;;    Discards the J-th element from a list LST.
;;;
;;; MATRIX- (f g &optional (order #'lex>))                           [FUNCTION]
;;;    Returns difference of two polynomial matrices F and G.
;;;
;;; SCALAR-TIMES-MATRIX (s f)                                        [FUNCTION]
;;;    Returns a product of a polynomial S by a polynomial matrix F.
;;;
;;; MONOM-TIMES-MATRIX (m f)                                         [FUNCTION]
;;;    Returns a product of a monomial M by a polynomial matrix F.
;;;
;;; TERM-TIMES-MATRIX (term f)                                       [FUNCTION]
;;;    Returns a product of a term TERM by a polynomial matrix F.
;;;
;;; POLY-LIST- (f g &optional (order #'lex>))                        [FUNCTION]
;;;    Returns the list of differences of two lists of polynomials
;;;    F and G (polynomial maps).
;;;
;;; SCALAR-TIMES-POLY-LIST (s f)                                     [FUNCTION]
;;;    Returns the list of products of a polynomial S by the
;;;    list of polynomials F.
;;;
;;; MONOM-TIMES-POLY-LIST (m f)                                      [FUNCTION]
;;;    Returns the list of products of a monomial M by the
;;;    list of polynomials F.
;;;
;;; TERM-TIMES-POLY-LIST (term f)                                    [FUNCTION]
;;;    Returns the list of products of a term TERM by the
;;;    list of polynomials F.
;;;
;;; CHARACTERISTIC-COMBINATION (a b &optional (order #'lex>)         [FUNCTION]
;;;                             &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.
;;;
;;; CHARACTERISTIC-COMBINATION-POLY-LIST (a b                        [FUNCTION]
;;;                                       &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. 
;;;
;;; CHARACTERISTIC-MATRIX (a &optional (order #'lex>)                [FUNCTION]
;;;                        (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. 
;;;
;;; CHARACTERISTIC-POLYNOMIAL (a &optional (order #'lex>)            [FUNCTION]
;;;                            (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. 
;;;
;;; IDENTITY-MATRIX (dim nvars)                                      [FUNCTION]
;;;    Return the polynomial matrix which is the identity matrix. DIM is the
;;;    requested dimension and NVARS is the number of variables of each
;;;    entry. 
;;;
;;; PRINT-MATRIX (f vars)                                            [FUNCTION]
;;;    Prints a polynomial matrix F, using a list of symbols VARS as
;;;    variable names. 
;;;
;;; JACOBI-MATRIX (f &optional (m (length f)) (n                     [FUNCTION]
;;;                (length (caaaar f))))
;;;    Returns the Jacobi matrix of a polynomial list F over the first N
;;;    variables. 
;;;
;;; JACOBIAN (f &optional (order #'lex>) (m (length f))              [FUNCTION]
;;;           (n (length (caaaar f))))
;;;    Returns the Jacobian (determinant) of a polynomial list F over the
;;;    first N variables. 
;;;