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.

DOI: 10.5802/jtnb.659
Keywords: Orthogonal polynomials, Jacobi polynomial, Rational point, Riemann-Hurwitz formula, Specialization
Cullinan, John 1; Hajir, Farshid 2; Sell, Elizabeth 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
@article{JTNB_2009__21_1_97_0,
     author = {Cullinan, John and Hajir, Farshid and Sell, Elizabeth},
     title = {Algebraic properties of a family of {Jacobi} polynomials},
     journal = {Journal de th\'eorie des nombres de Bordeaux},
     pages = {97--108},
     publisher = {Universit\'e Bordeaux 1},
     volume = {21},
     number = {1},
     year = {2009},
     doi = {10.5802/jtnb.659},
     mrnumber = {2537705},
     language = {en},
     url = {http://archive.numdam.org/articles/10.5802/jtnb.659/}
}
TY  - JOUR
AU  - Cullinan, John
AU  - Hajir, Farshid
AU  - Sell, Elizabeth
TI  - Algebraic properties of a family of Jacobi polynomials
JO  - Journal de théorie des nombres de Bordeaux
PY  - 2009
SP  - 97
EP  - 108
VL  - 21
IS  - 1
PB  - Université Bordeaux 1
UR  - http://archive.numdam.org/articles/10.5802/jtnb.659/
DO  - 10.5802/jtnb.659
LA  - en
ID  - JTNB_2009__21_1_97_0
ER  - 
%0 Journal Article
%A Cullinan, John
%A Hajir, Farshid
%A Sell, Elizabeth
%T Algebraic properties of a family of Jacobi polynomials
%J Journal de théorie des nombres de Bordeaux
%D 2009
%P 97-108
%V 21
%N 1
%I Université Bordeaux 1
%U http://archive.numdam.org/articles/10.5802/jtnb.659/
%R 10.5802/jtnb.659
%G en
%F JTNB_2009__21_1_97_0
Cullinan, John; Hajir, Farshid; Sell, Elizabeth. 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. http://archive.numdam.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 | Zbl

[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 | Zbl

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

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

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

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

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

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

[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 | Zbl

[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 | Zbl

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

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

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

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

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

[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 | Zbl

Cited by Sources: