Some complexity results for polynomial ideals

In this paper, we survey some of our new results on the complexity of a number of problems related to polynomial ideals. We consider multivariate polynomials over some ring, like the integers or the rationals. For instance, a polynomial ideal membership problem is a (w + l)-tuple P = (f, g₁, g₂, ..., g_w) where f and the g_i are multivariate polynomials, and the problem is to determine whether f is in the ideal generated by the g_i. For polynomials over the integers or rationals, this problem is known to be exponential space complete. We discuss further complexity results for problems related to polynomial ideals, like the word and subword problems for commutative semigroups, a quantitative version of Hilbert's Nullstellensatz in a complexity theoretic version, and problems concerning the computation of reduced polynomials and Gröbner bases.