Identity Testing for Radical Expressions

We study the Radical Identity Testing problem (RIT): Given an algebraic circuit representing a polynomial $f\in \mathbb{Z}[x1, \ldots, xk]$ and nonnegative integers $a1, \ldots, ak$ and $d1, \ldots,$ $dk$, written in binary, test whether the polynomial vanishes at the real radicals $\sqrt[d1]{a1}, \ldots,\sqrt[dk]{ak}$, i.e., test whether $f(\sqrt[d1]{a1}, \ldots,\sqrt[dk]{ak}) = 0$. 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.