POLY - A MAPLE package for polynomials in Rⁿ

The MAPLEV4 package poly contains routines for type checking of polynomials in several variables, converting polynomials into vectors, solving linear equations between polynomials, determining if a polynomial belongs to the vector space spanned by other polynomials, and enumerating the monomials in a list (a "hash table").

• There is a global variable global_n which denotes the number of (algebraically independent) variables x1, x2, ... which the polynomials will be written in. This must be set in the beginning before running any of the procedures below.
• There are two MAPLE "types":
  • `type/MY_VARIABLES` returns `true` if the arguement is one of the variables x1, ..., x.global_n;
  • `type/MONOMIAL` returns `true` if the argument is a monomial in MY_VARIABLES
• There is a procedure monomial_part which takes a polynomial f in MY_VARIABLES and returns the coefficientless monomial factor if f has only one term. The procedure monomial_terms takes a polynomial f in MY_VARIABLES and returns the list of monomial terms (without coefficients) in f. The analogous procedure coeff_part takes a polynomial f in MY_VARIABLES and returns the list of coefficients in f.
• There is a procedure enum_of_monomial which takes in a monomial f and returns the integer k such that f = fk, where f1(=x1), f2, ... is a certain complete enumeration of all the monomials in MY_VARIABLES. We order the monomials first by their total degree then by their order of occurence in (x1+...+x.global_n)^k. We call this number the "enumeration" of the monomial. Incidently, the number of linearly independent terms in (x1+...+xn)^k is binomial(k+n-1,n-1). The procedures find_in_multi_exp and partial_binomial_sum are auxilliary.

  The procedure

  monomial_of_enum is the inverse of the procedure enum_of_monomial described above.
• The procedure poly2vector converts a polynomial in MY_VARIABLES into a vector associated to the monomial terms via the enumeration. The dimension of the vector (actually, represented as a list) depends on the inputted polynomial. The correspondence satisfies fk |----> ek where ek = [0, ..., 0, 1, 0, ...] is the vector with a 1 in the kth position, zeroes elsewhere, and where f1, f2, ... is the complete listing of the monomials mentioned earlier. n_vectors is auxilliary.

  The procedure

  vector2poly is the inverse to the poly2vector procedure.
• The procedure poly_basis takes a list of poynomials in MY_VARIABLES and returns a maximal sublist of linearly independent polynomials.
• is_in is a procedure which returns 1 if the 1st argument belongs to the vector space spanned by the vectors (represented as lists) in the 2nd argument. It returns 0 otherwise. It is implicitly assumed the 2nd argument is a list of basis vectors (All vectors are represented as lists.)
• is_in_poly is the analogous procedure using polynomials in MY_VARIABLES in place of vectors.
• The procedure solve_poly takes two arguments: the 1st a polynomial f and the 2nd a list of polynomials p1, p2, ..., pr. It tries to find a linear combination of the pi's which equals f. Returns 0 if no linear combination exists and a list of coefficients
• The procedure diff_inner computes the "differential inner product" of two polynomials in MY_VARIABLES: diff(f,g)=f(D1,...,Dn)(g(x1,...,xn), where Di denotes the derivative with respect to xi.

  The MAPLEV4 polynomials in several variables package has basic linear algebra commands which work with polynomials. There is a poly worksheet which may be downloaded as well. If you have MAPLEV4 and Windows 3.1 or better then here is how to load the poly package:

  • create a subdirectory "poly" of your maplev4/share/algebra directory,
  • copy poly.mpl into this poly subdirectory,
  • read this text file into maple (using read(`d:/maplev4/share/algebra/poly/poly.mpl`);) and save the resulting poly.m file in the "maplve4/share/algebra" subdirectory,
  • in a new session of maple type:

    with(share): readshare(poly,algebra);

  • To work with polynomials in 3 variables x1,x2,x3, for example, type

    global_n:=3;

    followed by whatever commands in poly.mpl you wish.

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  Last updated 5-9-97.