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

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 $(1-t^2)^{-1/2}dt/\pi$ on $(-1,1)$. 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 $\mathbb{Q}$, such ideal solutions are included in the famous parametric solutions found by Borwein (2002).