Ellipsoidal designs and the Prouhet–Tarry–Escott problem

Abstract The notion of ellipsoidal design was first introduced by Pandey (Ramanujan J, 58(4):1245–1257, 2022) as a full generalization of spherical designs on the unit circle $$S^1$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>S</mml:mi> <mml:mn>1</mml:mn> </mml:msup> </mml:math> . In this paper, we elucidate the advantages of examining the connections between ellipsoidal design and the two-dimensional Prouhet–Tarry–Escott problem, say $${{\,\textrm{PTE}\,}}_2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mrow> <mml:mspace/> <mml:mtext>PTE</mml:mtext> <mml:mspace/> </mml:mrow> <mml:mn>2</mml:mn> </mml:msub> </mml:math> , originally introduced by Alpers and Tijdeman (J Number Theory, 123(2):403–412, 2007) as a natural generalization of the classical one-dimensional PTE problem ( $${{\,\textrm{PTE}\,}}_1$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mrow> <mml:mspace/> <mml:mtext>PTE</mml:mtext> <mml:mspace/> </mml:mrow> <mml:mn>1</mml:mn> </mml:msub> </mml:math> ). We first provide a combinatorial criterion for the construction of solutions of the $${{\,\textrm{PTE}\,}}_2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mrow> <mml:mspace/> <mml:mtext>PTE</mml:mtext> <mml:mspace/> </mml:mrow> <mml:mn>2</mml:mn> </mml:msub> </mml:math> 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 $$S^1$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>S</mml:mi> <mml:mn>1</mml:mn> </mml:msup> </mml:math> , discussed in Cui et al. (Adv Math, 352:541–571, 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 the $${{\,\textrm{PTE}\,}}_2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mrow> <mml:mspace/> <mml:mtext>PTE</mml:mtext> <mml:mspace/> </mml:mrow> <mml:mn>2</mml:mn> </mml:msub> </mml:math> . 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 the $${{\,\textrm{PTE}\,}}_1$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mrow> <mml:mspace/> <mml:mtext>PTE</mml:mtext> <mml:mspace/> </mml:mrow> <mml:mn>1</mml:mn> </mml:msub> </mml:math> . As a by-product of this theorem, we discover a family of ellipsoidal 5-designs among the Alpers–Tijdeman solution.