The complexity of root-finding in orders

Abstract

Given an order, a commutative ring whose additive group is free of finite rank, a natural computational question is whether a fixed univariate polynomial $f \in \mathbb{Z}[X]$ has a root in this ring. In this paper, we show that the computational difficulty of this depends strongly on the arithmetic properties of $f$. We show that with probability 1, determining whether $f$ has a root is NP-complete. For $\text{deg } f \leq 3$ we give a full classification of the computational complexity: some special $f$ admit a polynomial-time algorithm, and for all other $f$ the problem is NP-complete. Additionally, we prove the problem is undecidable for $f = (X^2+1)^2$, conditional on Hilberts Tenth Problem for $\mathbb{Q}(i)$. The key ingredients for proving NP-completeness are a new source of NP-complete group-theoretic problems developed in previous work, and a full classification of cubic polynomials with discriminant divisible only by $3$.