Identity Testing for Radical Expressions

Abstract

We study the Radical Identity Testing problem (RIT): Given an algebraic circuit representing a polynomial $f\in \mathbb{Z}[x_1, \ldots, x_k]$ and nonnegative integers $a_1, \ldots, a_k$ and $d_1, \ldots,$ $d_k$, written in binary, test whether the polynomial vanishes at the real radicals $\sqrt[d_1]{a_1}, \ldots,\sqrt[d_k]{a_k}$, i.e., test whether $f(\sqrt[d_1]{a_1}, \ldots,\sqrt[d_k]{a_k}) = 0$. We 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 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.