Ellipsoidal designs and the Prouhet--Tarry--Escott problem

The notion of ellipsoidal design was first introduced by Pandey (2022) as a full generalization of spherical designs on the unit circle $S^1$. In this paper, we elucidate the advantages of examining the connections between ellipsoidal design and the two-dimensional Prouhet--Tarry--Escott problem, say ${\mathrm PTE}_2$, originally introduced by Alpers and Tijdeman (2007) as a natural generalization of the classical one-dimensional PTE problem (${\mathrm PTE}_1$). We first provide a combinatorial criterion for the construction of solutions of ${\mathrm PTE}_2$ from a pair of ellipsoidal designs. We also give an arithmetic proof of the Stroud-type bound for ellipsoidal designs, and then establish a classification theorem for designs with equality. Such a classification result is closely related to an open question on the existence of rational spherical $4$-designs on $S^1$, discussed in Cui, Xia and Xiang (2019). As far as the authors know, a solution found by Alpers and Tijdeman is the first and the only known parametric ideal solution of degree $5$ for ${\mathrm PTE}_2$. Moreover, as one of our main theorems, we prove that the Alpers--Tijdeman solution is equivalent to a certain two-dimensional extension of the famous Borwein solution for ${\mathrm PTE}_1$. As a by-product of this theorem, we discover a family of ellipsoidal $5$-designs among the Alpers--Tijdeman solution.