The Spectrum of a Rational Function

Abstract

Let k be an infinite field, [Formula: see text] be a finite set of k, and Q ∈ k[x] (with x = (x1, …, xn) and n ≥ 2) be a nonconstant polynomial. The main goal of this paper is to construct a polynomial P(x) ∈ k[x] with suitably large partial degrees in x1, …, xn−1 such that P and Q are coprime, and P − aQ is reducible for all a in [Formula: see text].