Solutions to the multi-dimensional Prouhet-Tarry-Escott problem resulting from composition of balanced morphisms (Q515680)

In this very interesting paper, the author considers the following problem: Let \(\mathbb{S}\) be a semiring. For given integers \(d \geq 1\), \(k \geq 1\) and \(n \geq 2\), find an \(n\)-tuple of multisets \(\{A_z\}_{z \in \{0, 1, \ldots, n-1\}}\) of \(d\)-tuples of elements of \(\mathbb{S}\) such that, for all non-negative integers \(r_0, \ldots, r_{d-1}\) such that \(r_0 + \cdots + r_{d-1} < k\), the value \(\sum_{\langle a_0, \ldots, a_{d-1}\rangle \in A_z} A_z(\langle a_0, \ldots, a_{d-1}\rangle) a_{0}^{r_0} \cdots a_{d-1}^{r_{d-1}}\) does not depend on \(z\), where \(d\) is the dimension of the problem, \(k\) is the degree of the problem, and \(A_z(\langle a_0, \ldots, a_{d-1}\rangle)\) is the number of times the tuple \(\langle a_0, \ldots, a_{d-1}\rangle\) is contained in the multiset \(A_z\). This is an extended multi-dimensional version of the Prouhet-Tarry-Escott problem, a problem well-known in combinatorics on words. The author first looks for solutions when \(\mathbb{S}\) is the semiring of natural numbers. Then he shows how to construct solutions over an arbitrary semiring. The author's construction for the \(d=2\) case, which can be easily generalized to the \(d>2\) case, is based on the structure of (rectangular) array words obtained by composition of balanced array morphisms.