The Prouhet–Tarry–Escott problem, indecomposability of polynomials and Diophantine equations

Abstract

In this paper, we show how the subjects mentioned in the title are related. First we study the structure of partitions of A subseteq {1, dots , n} in k-sets such that the first k-1 symmetric polynomials of the elements of the k-sets coincide. Then we apply this result to derive a decomposability result for the polynomial f_A(x) := prod _{x in A} (x-a). Finally we prove two theorems on the structure of the solutions (x, y) of the Diophantine equation f_A(x)=P(y) where P(y)in mathbb {Q}[y] and on shifted power values of f_A(x).