Geometric designs and Hilbert-Kamke equations of degree five for classical orthogonal polynomials

Abstract In this paper we elucidate the advantage of examining the connections between Hilbert-Kamke equations and geometric designs, or Chebyshev-type quadrature, for classical orthogonal polynomials. We first establish that if a 5-design with 6 rational points for a symmetric classical measure is parametrized by rational functions, then the corresponding measure should be the Chebyshev measure $$(1-t^2)^{-1/2}dt/\pi $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msup> <mml:mrow> <mml:mo>(</mml:mo> <mml:mn>1</mml:mn> <mml:mo>-</mml:mo> <mml:msup> <mml:mi>t</mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:mo>)</mml:mo> </mml:mrow> <mml:mrow> <mml:mo>-</mml:mo> <mml:mn>1</mml:mn> <mml:mo>/</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:msup> <mml:mi>d</mml:mi> <mml:mi>t</mml:mi> <mml:mo>/</mml:mo> <mml:mi>π</mml:mi> </mml:mrow> </mml:math> on $$(-1,1)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>(</mml:mo> <mml:mo>-</mml:mo> <mml:mn>1</mml:mn> <mml:mo>,</mml:mo> <mml:mn>1</mml:mn> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> . Our proof is based on the collaboration of a certain polynomial identity and some advanced techniques on the computation of the genus of a certain irreducible curve. Next, we prove a necessary and sufficient condition for the existence of rational 5-designs for the Chebyshev measure. Moreover, as one of our main theorems, we construct an infinite family of ideal solutions for the Prouhet-Tarry-Escott (PTE) problem by utilizing rational 5-designs for the Chebyshev measure, and then establish that, up to affine equivalence over $$\mathbb {Q}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>Q</mml:mi> </mml:math> , such ideal solutions are included in the famous parametric solutions found by Borwein (2002).