An extension of the irreducibility criteria of Ehrenfeucht and Tverberg

Abstract

In 1956, Ehrenfeucht proved that a polynomial f ₁(x ₁) + · + f _n (x _n ) with complex coefficients in the variables x ₁, ..., x _n is irreducible over the field of complex numbers provided the degrees of the polynomials f ₁(x ₁), ..., f _n (x _n ) have greatest common divisor one. In 1964, Tverberg extended this result by showing that when n = 3, then f ₁(x ₁) + · + f _n (x _n ) belonging to K[x ₁, ..., x _n ] is irreducible over any field K of characteristic zero provided the degree of each f _i is positive. Clearly a polynomial F = f ₁(x ₁) + · + f _n (x _n ) is reducible over a field K of characteristic p ≠ 0 if F can be written as F = (g ₁(x ₁)) _p + (g ₂(x ₂)) ^p + · + (g _n (x _n )) ^p + c[g ₁(x ₁) + g 2(x 2) + · + g _n (x _n )] where c is in K and each g _i (x _i ) is in K[x _i ]. In 1966, Tverberg proved that the converse of the above simple fact holds in the particular case when n = 3 and K is an algebraically closed field of characteristic p > 0. In this article, we prove an extension of Tverberg's result by showing that this converse holds for any n ≥ 3.