Polynomials of Bounded Tree Width (Extended Abstract)

Abstract

We introduce a new sparsity conditions on multivariate polynomials in n variables (over some ring R) and show that under this condition many otherwise intractable computational problems involving these polynomials become solvable in polynomial (even linear) time in n (in the BSS-model over R). To define our sparsity condition we associate with these polynomials a hypergraph and study classes of polynomials where this hypergraph has tree width at most k for some fixed k ∈ ℕ. We are interested in three cases: (1) The evaluation of multivariate polynomials where the number of monomials is O(2 n ). (2) The question whether a system of n polynomials p i (\(\overline x \)) ∈ R[\(\overline x \)] of fixed degree d in n variables has a root in R n . (3) For infinite ordered rings R ord , a polynomial of fixed degree d in n variables p (\(\overline x \)) ∈ R ord [\(\overline x \)] and a finite subset A ⊂ R ord we want to know whether p(ā) < 0 for all ā ∈ R ord n . Our method uses graph theoretic and model theoretic tools developed in the last 15 years and applies them to the algebraic setting. This work is an extension of work by B. Courcelle, J.A. Makowsky and U. Rotics.