The two-dimensional Prouhet–Tarry–Escott problem

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 {(x1,y1),…,(xn,yn)}, of points from Z2 such that 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 2n-gons with all vertices in Z2 such that every line parallel to a side contains an even number of vertices and prove that such convex 2n-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.