# Identity Testing for Radical Expressions

We study the Radical Identity Testing problem (RIT): Given an algebraic circuit representing a polynomial f∈ℤ[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 radicals √(a_1), …,√(a_k), i.e., test whether f(√(a_1), …,√(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.

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