The two-dimensional Prouhet–Tarry–Escott problem

Abstract

In this paper we generalize the Prouhet–Tarry–Escott problem (PTE) to any dimension. The one-dimensional PTE problem is the classical PTE problem. We concentrate on the two-dimensional version which asks, given parameters n , k ∈ N , for two different multi-sets { ( x 1 , y 1 ) , … , ( x n , y n ) } , { ( x 1 ′ , y 1 ′ ) , … , ( x n ′ , y n ′ ) } of points from Z 2 such that ∑ i = 1 n x i j y i d − j = ∑ i = 1 n x i ′ j y i ′ d − j for all d , j ∈ { 0 , … , k } with j ⩽ d . We present parametric solutions for n ∈ { 2 , 3 , 4 , 6 } with optimal size, i.e., with k = n − 1 . We show that these solutions come from convex 2 n-gons with all vertices in Z 2 such that every line parallel to a side contains an even number of vertices and prove that such convex 2 n-gons do not exist for other values of n. Furthermore we show that solutions to the two-dimensional PTE problem yield solutions to the one-dimensional PTE problem. Finally, we address the PTE problem over the Gaussian integers.