Ellipsoidal designs and the Prouhet--Tarry--Escott problem
Hideki Matsumura, Masanori Sawa
TL;DR
The paper develops a unified framework linking ellipsoidal designs on $C_D(r)$ with the two-dimensional PTE problem $PTE_2$, establishing a generalized Sobolev theory and a Stroud-type bound that yields equality classifications for tight designs. It provides a combinatorial criterion showing how pairs of ellipsoidal designs generate $PTE_2$-solutions, and constructs a parametric ideal degree-5 $PTE_2$ solution, proving its equivalence to a two-dimensional Borwein solution and to the Alpers--Tijdeman construction over $\oldsymbol{Q}$, with a new family of ellipsoidal $5$-designs arising from this link. This work forges deep connections between design theory, lattice shells, and Diophantine problems, and demonstrates that rational ellipsoidal designs can produce richer parametric families than rational spherical designs. It also highlights open questions on rational tight designs and potential extensions to other orthogonal polynomial families, offering a new platform for exploring PTE-type problems through geometric designs.
Abstract
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.