Algebraic properties of a family of Jacobi polynomials
Journal de Théorie des Nombres de Bordeaux, Volume 21 (2009) no. 1, pp. 97-108.

The one-parameter family of polynomials J n (x,y)= j=0 n y+j jx j is a subfamily of the two-parameter family of Jacobi polynomials. We prove that for each n6, the polynomial J n (x,y 0 ) is irreducible over for all but finitely many y 0 . If n is odd, then with the exception of a finite set of y 0 , the Galois group of J n (x,y 0 ) is S n ; if n is even, then the exceptional set is thin.

La famille des polynômes à un seul paramètre J n (x,y)= j=0 n y+j jx j est une sous-famille de la famille (à deux paramètres) des polynômes de Jacobi. On montre que pour chaque n6, quand on spécialise en y 0 , le polynôme J n (x,y 0 ) est irréductible sur , sauf pour un nombre fini des valeurs y 0 . Si n est impair, sauf pour un nombre fini des valeurs y 0 , le groupe de Galois de J n (x,y 0 ) est S n  ; si n est pair, l’ensemble exceptionnel est mince.

Published online:
DOI: 10.5802/jtnb.659
Keywords: Orthogonal polynomials, Jacobi polynomial, Rational point, Riemann-Hurwitz formula, Specialization
John Cullinan 1; Farshid Hajir 2; Elizabeth Sell 3

1 Department of Mathematics Bard College Annandale-On-Hudson, NY 12504
2 Department of Mathematics University of Massachusetts Amherst MA 01003
3 Department of Mathematics Millersville University P.O. Box 1002 Millersville, PA 17551
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John Cullinan; Farshid Hajir; Elizabeth Sell. Algebraic properties of a family of Jacobi polynomials. Journal de Théorie des Nombres de Bordeaux, Volume 21 (2009) no. 1, pp. 97-108. doi : 10.5802/jtnb.659. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.659/

[1] S. Ahlgren, K. Ono, Arithmetic of singular moduli and class polynomials. Compos. Math. 141 (2005), 283–312. | MR: 2134268 | Zbl: 1133.11036

[2] J. Brillhart, P. Morton, Class numbers of quadratic fields, Hasse invariants of elliptic curves, and the supersingular polynomial. J. Number Theory. 106 (2004), 79–111. | MR: 2049594 | Zbl: 1083.11036

[3] R. Coleman, On the Galois groups of the exponential Taylor polynomials. L’Enseignement Math. 33 (1987), 183–189. | MR: 925984 | Zbl: 0672.12004

[4] J.D. Dixon, B. Mortimer, Permutation Groups. Springer-Verlag, 1996. | MR: 1409812 | Zbl: 0951.20001

[5] F. Hajir, Algebraic properties of a family of generalized Laguerre polynomials. To appear in Canad. J. Math. | MR: 2514486 | Zbl: pre05554011

[6] F. Hajir, On the Galois group of generalized Laguerre polynomials. J. Théor. Nombres Bordeaux 17 (2005), 517–525. | Numdam | MR: 2211305 | Zbl: 1094.11042

[7] F. Hajir, S. Wong, Specializations of one-parameter families of polynomials. Annales de L’Institut Fourier. 56 (2006), 1127-1163. | Numdam | MR: 2266886 | Zbl: 1160.12004

[8] M. Hindry, J. Silverman, Diophantine Geometry, An Introduction. Springer-Verlag, 2000. | MR: 1745599 | Zbl: 0948.11023

[9] M. Kaneko, D. Zagier, Supersingular j-invariants, hypergeometric series, and Atkin’s orthogonal polynomials. In Computational perspectives on number theory. AMS/IP Stud. Adv. Math. 7 (1998), 97–126. | MR: 1486833 | Zbl: 0955.11018

[10] M. Liebeck, C. Praeger, J. Saxl, A classification of the maximal subgroups of the finite alternating and symmetric groups. J. Algebra 111 (1987), no. 2, 365–383. | MR: 916173 | Zbl: 0632.20011

[11] K. Mahlburg, K. Ono, Arithmetic of certain hypergeometric modular forms. Acta Arith. 113 (2004), 39–55. | MR: 2046967 | Zbl: 1100.11015

[12] P. Müller, Finiteness results for Hilbert’s irreducibility theorem. Ann. Inst. Fourier 52 (2002), 983–1015. | Numdam | MR: 1926669 | Zbl: 1014.12002

[13] I. Schur, Affektlose Gleichungen in der Theorie der Laguerreschen und Hermitschen Polynome. Gesammelte Abhandlungen. Band III, 227–233, Springer, 1973. | Zbl: 0002.11501

[14] I. Schur. Gessammelte Abhandlungen Vol. 3, Springer, 1973 | Zbl: 0274.01054

[15] E. Sell, On a family of generalized Laguerre polynomials. J. Number Theory 107 (2004), 266–281. | MR: 2072388 | Zbl: 1053.11083

[16] G. Szegö, Orthogonal Polynomials, 4th ed. Providence, RI: Amer. Math. Soc., 1975.

[17] S. Wong, On the genus of generalized Laguerre polynomials. J. Algebra. 288 (2005), no. 2, 392–399. | MR: 2146136 | Zbl: 1078.33009

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