On the complexity of computing with planar algebraic curves

Abstract

In this paper, we give improved bounds for the computational complexity of computing with planar algebraic curves. More specifically, for arbitrary coprime polynomials f, g∈Z[x,y] and an arbitrary polynomial h∈Z[x,y], each of total degree less than n and with integer coefficients of absolute value less than 2τ, we show that each of the following problems can be solved in a deterministic way with a number of bit operations bounded by Õ(n6+n5τ), where we ignore polylogarithmic factors in n and τ:•The computation of isolating regions inC2for all complex solutions of the system f=g=0,•the computation of a separating form for the solutions of f=g=0,•the computation of the sign ofh at all real valued solutions of f=g=0, and•the computation of the topology of the planar algebraic curve C defined as the real valued vanishing set of the polynomial f. Our bound improves upon the best currently known bounds for the first three problems by a factor of n2 or more and closes the gap to the state-of-the-art randomized complexity for the last problem.