Irreducibility Testing of Sparse Polynomials over Integers is NP-hard and Probabilistic Polynomial Time

Abstract

In this paper, we prove that irreducibility testing for sparse polynomials in $\mathbb{Z}[X]$ is coNP-hard under deterministic polynomial-time many-one reductions. That is, there exists no polynomial-time irreducibility test algorithm unless $\mathrm P=\mathrm{NP}$. We also provide an upper bound for this problem by demonstrating that the same irreducibility testing problem lies in $\mathsf{PP}\subseteq\mathsf{PSPACE}$; in other words, it can be solved in probabilistic polynomial time and, moreover, within polynomial space. The fact that $\mathsf{SparseIrred}_{\mathbb{Z}}\in\mathsf{PP}$ relies on the quasi-Riemann hypothesis result of Cureton--Thorner and on the uniform effective prime-ideal bound by Frei--Sofos--Widmer just posted recently.