Identity Testing for Radical Expressions

TL;DR

2-RIT is shown to be in coRP assuming GRH and in coNP unconditionally, improving on the straightforward PSPACE upper bound obtained by reduction to the existential theory of reals.

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.

Figures (5)

• Figure 1: An algebraic circuit computing the polynomial $p(x)=(1-2x)^{2^s}$, with the highest degree monomial $x^{2^s}$ and the coefficients double exponential $2^{2^s}$ in its size $s+3$.
• Figure 2: Our nondeterministic algorithm for the complement of RIT.
• Figure 3: Our randomised polynomial time algorithm for the complement of $2$-RIT.
• Figure 4: The scheme of the reduction from RIT to its variant where the input radicands are pairwise coprime and the exponents are all equal. In this simple example, $a_1$ is factored to $m_1m_2^2m_{\ell}$.
• Figure 5: Procedure to solve the congruence $x^2\equiv a \mod p$ for prime $p \equiv 5 \mod 8$.

Theorems & Definitions (34)

• Theorem 1
• Theorem 2
• Proposition 1
• proof : Proof sketch
• Example 1
• Lemma 2
• proof
• Lemma 3
• proof
• Example 2