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July, 2019 Discriminants of classical quasi-orthogonal polynomials with application to Diophantine equations
Masanori SAWA, Yukihiro UCHIDA
J. Math. Soc. Japan 71(3): 831-860 (July, 2019). DOI: 10.2969/jmsj/79877987


We derive explicit formulas for the discriminants of classical quasi-orthogonal polynomials, as a full generalization of the result of Dilcher and Stolarsky (2005). We consider a certain system of Diophantine equations, originally designed by Hausdorff (1909) as a simplification of Hilbert's solution (1909) of Waring's problem, and then create the relationship to quadrature formulas and quasi-Hermite polynomials. We reduce these equations to the existence problem of rational points on a hyperelliptic curve associated with discriminants of quasi-Hermite polynomials, and show a nonexistence theorem for solutions of Hausdorff-type equations by applying our discriminant formula.

Funding Statement

The first author was supported in part by Grant-in-Aid for Young Scientists (B) 26870259 and Grant-in-Aid for Scientific Research (B) 15H03636 by the Japan Society for the Promotion of Science (JSPS). The second author was also supported by Grant-in-Aid for Young Scientists (B) 25800023 by JSPS.


Download Citation

Masanori SAWA. Yukihiro UCHIDA. "Discriminants of classical quasi-orthogonal polynomials with application to Diophantine equations." J. Math. Soc. Japan 71 (3) 831 - 860, July, 2019.


Received: 14 February 2018; Published: July, 2019
First available in Project Euclid: 17 May 2019

zbMATH: 07121555
MathSciNet: MR3984244
Digital Object Identifier: 10.2969/jmsj/79877987

Primary: 05E99 , 33C45 , 65D32
Secondary: 11E76 , 12E10

Keywords: classical quasi-orthogonal polynomial , compact formula , discriminant , Gaussian design , Hausdorff-type equation , quadrature formula

Rights: Copyright © 2019 Mathematical Society of Japan


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Vol.71 • No. 3 • July, 2019
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