Two Families of Algorithms for Symbolic Polynomials

We consider multivariate polynomials with exponents that are themselves integer-valued multivariate polynomials, and we present algorithms to compute their GCD and factorization. The algorithms fall into two families: algebraic extension methods and projection methods. The first family of algorithms uses the algebraic independence of x, x, x 2 , x, etc, to solve related problems with more indeterminates. Some subtlety is needed to avoid problems with fixed divisors of the exponent polynomials. The second family of algorithms uses evaluation and interpolation of the exponent polynomials. While these methods can run into unlucky evaluation points, in many cases they can be more appealing. Additionally, we also treat the case of symbolic exponents on rational coefficients (e.g. 4 2+n − 81) and show how to avoid integer factorization. ∗Supported by the NSERC Discovery Grant program.