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

Teruyuki Mishima, Xiao-Nan Lu, Masanori Sawa, Yukihiro Uchida

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 $(1-t^2)^{-1/2}dt/π$ 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).

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 -design with rational points for a symmetric classical measure is parametrized by rational functions, then the corresponding measure should be the Chebyshev measure on . 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 -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 -designs for the Chebyshev measure, and then establish that, up to affine equivalence over , such ideal solutions are included in the famous parametric solutions found by Borwein (2002).

Paper Structure

This paper contains 14 sections, 34 theorems, 136 equations, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Quadrature and Hilbert-Kamke equations
  4. Hilbert-Kamke equations and Hausdorff's construction
  5. Rational points and the genus of a curve
  6. A geometric characterization of Hilbert-Kamke equations
  7. Proof of Theorem \ref{['thm:deg3']}
  8. Proof of Theorem \ref{['thm:parametrization0']}
  9. Proof of Theorem \ref{['thm:fivesix']}
  10. Proof of Theorem \ref{['thm:spectrum0']}
  11. Proof of Proposition \ref{['thm:spectrum_if']}
  12. Proof of Proposition \ref{['thm:spectrum_onlyif']}
  13. Proof of Theorem \ref{['thm:PTEparametric']}
  14. Concluding remarks and future works

Key Result

Theorem 1.1

There exists an antipodal $3$-design with $n$ rational points for the Hermite measure $e^{-t^2}dt/\sqrt{\pi}$ on $(-\infty,\infty)$ if and only if $n \notin \{1,2,3,7\}$. $\blacktriangleleft$$\blacktriangleleft$

Theorems & Definitions (72)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.6
  • Definition 2.1
  • Definition 2.2
  • Example 2.3: Newton-Cotes formula
  • Example 2.4: Chebyshev-Gauss quadrature
  • Example 2.5
  • ...and 62 more