The complexity of root-finding in orders

TL;DR

This paper investigates the computational complexity of determining roots of polynomials over orders, revealing NP-completeness in most cases, full classification for degree ≤3, and undecidability for specific polynomials.

Contribution

It provides a comprehensive complexity classification for root-finding in orders, including NP-completeness results, polynomial-time cases, and undecidability proofs for certain polynomials.

Findings

NP-complete for almost all polynomials with probability 1

Polynomial-time algorithms for some degree ≤3 polynomials

Undecidable for specific polynomial (X^2+1)^2, assuming Hilbert's Tenth Problem

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 $3$.