Algebro-geometric aspectsof the Christoffel-Darboux kernelsfor classical orthogonal polynomials

Sawa, Masanori; Uchida, Yukihiro

doi:10.1090/tran/7998

Algebro-geometric aspectsof the Christoffel-Darboux kernelsfor classical orthogonal polynomials
HTML articles powered by AMS MathViewer

by Masanori Sawa and Yukihiro Uchida PDF

Trans. Amer. Math. Soc. 373 (2020), 1243-1264 Request permission

Abstract:

In this paper we study algebro-geometric aspects of the Christoffel-Darboux kernels for classical orthogonal polynomials with rational coefficients. We find a novel connection between a projective curve defined by theChristoffel-Darboux kernel and a system of Diophantine equations, which was originally designed by Hausdorff (1909) for applications to Waring’s problem, and which is closely related to quadrature formulas in numerical analysis and Gaussian designs in algebraic combinatorics. We prove some nonsolvability results of such Hausdorff-type equations.

References

Eiichi Bannai and Etsuko Bannai, Tight Gaussian 4-designs, J. Algebraic Combin. 22 (2005), no. 1, 39–63. MR 2163709, DOI 10.1007/s10801-005-2505-3
Ei. Bannai, Etsu. Bannai, and T. Ito, Introduction to Algebraic Combinatorics, Kyoritsu shuppan, Tokyo, 2016.
E. Bannai and R. M. Damerell, Tight spherical designs. I, J. Math. Soc. Japan 31 (1979), no. 1, 199–207. MR 519045, DOI 10.2969/jmsj/03110199
E. Bannai and R. M. Damerell, Tight spherical designs. II, J. London Math. Soc. (2) 21 (1980), no. 1, 13–30. MR 576179, DOI 10.1112/jlms/s2-21.1.13
E. Bannai and T. Ito, Private communication at the 13th Japan-Korea Workshop on Algebra and Combinatorics held at Kyushu Institute of Tech., Kitakyushu, Japan, Feb. 2015.
S. Bochner, Über Sturm-Liouvillesche Polynomsysteme, Math. Z. 29 (1929), no. 1, 730–736 (German). MR 1545034, DOI 10.1007/BF01180560
Wieb Bosma, John Cannon, and Catherine Playoust, The Magma algebra system. I. The user language, J. Symbolic Comput. 24 (1997), no. 3-4, 235–265. Computational algebra and number theory (London, 1993). MR 1484478, DOI 10.1006/jsco.1996.0125
J. E. Cremona, Elliptic Curve Data, http://johncremona.github.io/ecdata/.
John Cullinan and Farshid Hajir, On the Galois groups of Legendre polynomials, Indag. Math. (N.S.) 25 (2014), no. 3, 534–552. MR 3188847, DOI 10.1016/j.indag.2014.01.004
P. Delsarte, J. M. Goethals, and J. J. Seidel, Spherical codes and designs, Geometriae Dedicata 6 (1977), no. 3, 363–388. MR 485471, DOI 10.1007/bf03187604
G. Dumas, Sur quelques cas d’irréductibilité des polynomes á coefficients rationnels, J. Math. Pures Appl. 2 (1906), 191–258.
Charles F. Dunkl and Yuan Xu, Orthogonal polynomials of several variables, 2nd ed., Encyclopedia of Mathematics and its Applications, vol. 155, Cambridge University Press, Cambridge, 2014. MR 3289583, DOI 10.1017/CBO9781107786134
G. Faltings, Endlichkeitssätze für abelsche Varietäten über Zahlkörpern, Invent. Math. 73 (1983), no. 3, 349–366 (German). MR 718935, DOI 10.1007/BF01388432
William Fulton, Algebraic curves. An introduction to algebraic geometry, Mathematics Lecture Note Series, W. A. Benjamin, Inc., New York-Amsterdam, 1969. Notes written with the collaboration of Richard Weiss. MR 0313252
G. H. Hardy and E. M. Wright, An introduction to the theory of numbers, 6th ed., Oxford University Press, Oxford, 2008. Revised by D. R. Heath-Brown and J. H. Silverman; With a foreword by Andrew Wiles. MR 2445243
F. Hausdorff, Zur Hilbertschen Lösung des Waringschen Problems, Math. Ann. 67 (1909), no. 3, 301–305 (German). MR 1511531, DOI 10.1007/BF01450406
J. B. Holt, The Irreducibility of Legendre’s Polynomials, Proc. London Math. Soc. (2) 11 (1913), 351–356. MR 1577230, DOI 10.1112/plms/s2-11.1.351
Peter Lesky, Die charakterisierung der klassischen orthogonalen Polynome durch Sturm-Liouvillesche Differentialgleichungen, Arch. Rational Mech. Anal. 10 (1962), 341–351 (German). MR 141818, DOI 10.1007/BF00281200
Yu. V. Nesterenko, On Waring’s problem (elementary methods), Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 322 (2005), no. Trudy po Teorii Chisel, 149–175, 254 (Russian, with English and Russian summaries); English transl., J. Math. Sci. (N.Y.) 137 (2006), no. 2, 4699–4715. MR 2138457, DOI 10.1007/s10958-006-0266-8
Jürgen Neukirch, Algebraic number theory, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 322, Springer-Verlag, Berlin, 1999. Translated from the 1992 German original and with a note by Norbert Schappacher; With a foreword by G. Harder. MR 1697859, DOI 10.1007/978-3-662-03983-0
Victor V. Prasolov, Polynomials, Algorithms and Computation in Mathematics, vol. 11, Springer-Verlag, Berlin, 2004. Translated from the 2001 Russian second edition by Dimitry Leites. MR 2082772, DOI 10.1007/978-3-642-03980-5
Bruce Reznick, Sums of even powers of real linear forms, Mem. Amer. Math. Soc. 96 (1992), no. 463, viii+155. MR 1096187, DOI 10.1090/memo/0463
M. Riesz, Sur le problème des moments, III, Ark. Math. Fys. 17 (1923), 1–52.
Masanori Sawa and Yukihiro Uchida, Discriminants of classical quasi-orthogonal polynomials with application to Diophantine equations, J. Math. Soc. Japan 71 (2019), no. 3, 831–860. MR 3984244, DOI 10.2969/jmsj/79877987
Masanori Sawa and Yuan Xu, On positive cubature rules on the simplex and isometric embeddings, Math. Comp. 83 (2014), no. 287, 1251–1277. MR 3167458, DOI 10.1090/S0025-5718-2013-02762-7
I. Schur, Einige Sätze über Primzahlen mit Anwendungen auf Irreduzibilitätsfragen, II, Sitzungsber. Preuss. Akad. Wiss. Berlin Phys.-Math. Kl., 14 (1929), 370–391.
J. Shohat, On mechanical quadratures, in particular, with positive coefficients, Trans. Amer. Math. Soc. 42 (1937), no. 3, 461–496. MR 1501930, DOI 10.1090/S0002-9947-1937-1501930-6
J. A. Shohat and J. D. Tamarkin, The Problem of Moments, American Mathematical Society Mathematical Surveys, Vol. I, American Mathematical Society, New York, 1943. MR 0008438, DOI 10.1090/surv/001
Barry Simon, The Christoffel-Darboux kernel, Perspectives in partial differential equations, harmonic analysis and applications, Proc. Sympos. Pure Math., vol. 79, Amer. Math. Soc., Providence, RI, 2008, pp. 295–335. MR 2500498, DOI 10.1090/pspum/079/2500498
A. H. Stroud, Approximate calculation of multiple integrals, Prentice-Hall Series in Automatic Computation, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1971. MR 0327006
P. R. Subramanian, Nonzero zeros of the Hermite polynomials are irrational, Fibonacci Quart. 33 (1995), no. 2, 131–134. MR 1329018
Gabor Szegö, Orthogonal polynomials, American Mathematical Society Colloquium Publications, Vol. 23, American Mathematical Society, Providence, R.I., 1959. Revised ed. MR 0106295
Yuan Xu, A characterization of positive quadrature formulae, Math. Comp. 62 (1994), no. 206, 703–718. MR 1223234, DOI 10.1090/S0025-5718-1994-1223234-0
J. H. Wahab, New cases of irreducibility for Legendre polynomials, Duke Math. J. 19 (1952), 165–176. MR 45864, DOI 10.1215/S0012-7094-52-01917-0
Lawrence C. Washington, Elliptic curves, 2nd ed., Discrete Mathematics and its Applications (Boca Raton), Chapman & Hall/CRC, Boca Raton, FL, 2008. Number theory and cryptography. MR 2404461, DOI 10.1201/9781420071474