Hard to Detect Factors of Univariate Integer Polynomials

We investigate the computational complexity of deciding whether a given univariate integer polynomial (Formula presented.) has a factor (Formula presented.) satisfying specific additional constraints. When the only constraint imposed on (Formula presented.) is to have a degree smaller than the degree of (Formula presented.) and greater than zero, the problem is equivalent to testing the irreducibility of (Formula presented.) and then it is solvable in polynomial time. We prove that deciding whether a given monic univariate integer polynomial has factors satisfying additional properties is NP-complete in the strong sense. In particular, given any constant value (Formula presented.), 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 (Formula presented.) (Theorem 1) 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 (Formula presented.) (Theorem 2). The list of all the properties we have investigated in this paper is reported at the end of Section Introduction.