<p>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 <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$(1-t^2)^{-1/2}dt/\pi $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>-</mo> <msup> <mi>t</mi> <mn>2</mn> </msup> <mo stretchy="false">)</mo> </mrow> <mrow> <mo>-</mo> <mn>1</mn> <mo stretchy="false">/</mo> <mn>2</mn> </mrow> </msup> <mi>d</mi> <mi>t</mi> <mo stretchy="false">/</mo> <mi>π</mi> </mrow> </math></EquationSource> </InlineEquation> on <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$(-1,1)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mo>-</mo> <mn>1</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>. Our proof is based on the collaboration of a certain polynomial identity and some advanced techniques on the computation of the genus of a certain irreducible curve. Next, we prove a necessary and sufficient condition for the existence of rational 5-designs for the Chebyshev measure. Moreover, as one of our main theorems, we construct an infinite family of ideal solutions for the Prouhet-Tarry-Escott (PTE) problem by utilizing rational 5-designs for the Chebyshev measure, and then establish that, up to affine equivalence over <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\mathbb {Q}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="double-struck">Q</mi> </math></EquationSource> </InlineEquation>, such ideal solutions are included in the famous parametric solutions found by Borwein (2002).</p>

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

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2026年3月31日

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

摘要

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