Lower bounds for some decision problems over C

Lower bounds for some explicit decision problems over the complex numbers are given. The decision problems considered are certain zero-dimensional subsets of N x C, and can be assimilated to a countable family of polynomials g(i). More precisely, one should decide for input (i,x) if g(i)(x)=0. A lower bound for deciding if a polynomial g(i) vanishes at some x can be derived from a uniform lower bound for the evaluation of all f is an element of (g(i)). That bound is obtained by means of an arithmetic invariant of the roots of g(i), the Newton diagram of f and other known techniques. (C) 2002 Elsevier Science B.V. All rights reserved.