Difference Polynomials and Their Generalizations

Abstract

this paper, we study irreducibility conditions of more general polynomials given by F (X; Y ) = cX ; c 2 k ; e 1; such that there exists t; 1 t e satisfying deg Y F (X; Y ) = degP t (Y ) = d and degP i (Y ) < di=t for i 6= t; 1 i e: Such a polynomial F (X; Y ) will be referred to as a quasi-dierence polynomial of the type (d; t) with respect to X: In 1990, L. Panaitopol and D. Stefanescu proved that a quasi-dierence polynomial of the type (d; t) with d and t coprime is irreducible over k[X] (cf. [5, Theorem 6]). In this direction, we go further and give an irreducibility criterion for a quasi-dierence polynomial of the type (d; t) with d and t not necessarily coprime, of which the criterion of Panaitopol and Stefanescu is a special case. Our method of proof is dierent from the one employed in [2] and [5], and is based on the idea of the proof of the Generalized Eisenstein's Irreducibility Criterion given in [4