Hard to Detect Factors of Univariate Integer Polynomials

TL;DR

This paper explores the computational complexity of factoring univariate integer polynomials under various constraints, revealing that certain factor detection problems are NP-complete, unlike the polynomial-time solvable irreducibility test.

Contribution

It establishes the NP-completeness of detecting factors with specific properties in monic univariate integer polynomials, extending understanding of polynomial factorization complexity.

Findings

Irreducibility testing is polynomial-time solvable.

Detecting factors with prescribed evaluation values is NP-complete.

Finding pairs of factors with equal evaluations at a point is NP-complete.

Abstract

We investigate the computational complexity of deciding whether a given univariate integer polynomial p(x) has a factor q(x) satisfying specific additional constraints. When the only constraint imposed on q(x) is to have a degree smaller than the degree of p(x) and greater than zero, the problem is equivalent to testing the irreducibility of p(x) and then it is solvable in polynomial time. We prove that deciding whether a given monic univariate integer polynomial has factors satisfying additional properties may lead to NP-complete problems in the strong sense. In particular, given any constant value k in Z, we prove that it is NP-complete in the strong sense to detect the existence of a factor that returns a prescribed value when evaluated at x=k or to detect the existence of a pair of factors - whose product is equal to the original polynomial - that return the same value when evaluated at x=k.