Primes via Zeros: Interactive Proofs for Testing Primality of Natural Classes of Ideals

A central question in mathematics and computer science is the question of determining whether a given ideal $I$ is prime, which geometrically corresponds to the zero set of $I$, denoted $Z(I)$, being irreducible. The case of principal ideals (i.e., $m=1$) corresponds to the more familiar absolute irreducibility testing of polynomials, where the seminal work of (Kaltofen 1995) yields a randomized, polynomial time algorithm for this problem. However, when $m > 1$, the complexity of the primality testing problem seems much harder. The current best algorithms for this problem are only known to be in EXPSPACE. In this work, we significantly reduce the complexity-theoretic gap for the ideal primality testing problem for the important families of ideals $I$ (namely, radical ideals and equidimensional Cohen-Macaulay ideals). For these classes of ideals, assuming the Generalized Riemann Hypothesis, we show that primality testing lies in $\Sigma_3^p \cap \Pi_3^p$. This significantly improves the upper bound for these classes, approaching their lower bound, as the primality testing problem is coNP-hard for these classes of ideals. Another consequence of our results is that for equidimensional Cohen-Macaulay ideals, we get the first PSPACE algorithm for primality testing, exponentially improving the space and time complexity of prior known algorithms.