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The two-dimensional Prouhet–Tarry–Escott problem

✍ Scribed by Andreas Alpers; Rob Tijdeman

Publisher
Elsevier Science
Year
2007
Tongue
English
Weight
147 KB
Volume
123
Category
Article
ISSN
0022-314X

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✦ Synopsis

In this paper we generalize the Prouhet-Tarry-Escott problem (PTE) to any dimension. The onedimensional 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

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 Z 2 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.

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