Identity Testing for Radical Expressions

Abstract

We study the Radical Identity Testing problem (RIT): Given an algebraic circuit over integers representing a multivariate polynomial 𝑓 (𝑥 1 , . . . , 𝑥 𝑘 ) and nonnegative integers 𝑎 1 , . . . , 𝑎 𝑘 and 𝑑 1 , . . . , 𝑑 𝑘 , written in binary, test whether the polynomial vanishes at the real radicalsWe place the problem in coNP assuming the Generalised Riemann Hypothesis (GRH), improving on the straightforward PSPACE upper bound obtained by reduction to the existential theory of reals. Next we consider a restricted version, called 2-RIT, where the radicals are square roots of prime numbers, written in binary. It was known since the work of Chen and Kao [16] that 2-RIT is at least as hard as the polynomial identity testing problem, however no better upper bound than PSPACE was known prior to our work. We show that 2-RIT is in coRP assuming GRH and in coNP unconditionally. Our proof relies on theorems from algebraic and analytic number theory, such as the Chebotarev density theorem and quadratic reciprocity. CCS CONCEPTS• Mathematics of computing → Probabilistic algorithms; • Computing methodologies → Algebraic algorithms; Number theory algorithms.