Membership in Plynomial Ideals over Q Is Exponential Space Complete

A polynomial ideal membership problem is an (n+1)-tuple P = 〈p, p1, p2,..., p n 〉 where p and the p i are multivariate polynomials over some ring, and the problem is to determine whether p is in the ideal generated by the p i . For polynomials over the integers or rationals, it is known that this problem is exponential space hard. Here, we show that the problem for multivariate polynomials over the rationals is solvable in exponential space, establishing its exponential space completeness.