Algebraic properties of a family of Jacobi polynomials

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

Journal de Théorie des Nombres de Bordeaux (2009)

  • Volume: 21, Issue: 1, page 97-108
  • ISSN: 1246-7405

Abstract

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The one-parameter family of polynomials J n ( x , y ) = ∑ j = 0 n y + j j x j is a subfamily of the two-parameter family of Jacobi polynomials. We prove that for each n ≥ 6 , the polynomial J n ( x , y 0 ) is irreducible over ℚ for all but finitely many y 0 ∈ ℚ . 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.

How to cite

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Cullinan, John, Hajir, Farshid, and Sell, Elizabeth. "Algebraic properties of a family of Jacobi polynomials." Journal de Théorie des Nombres de Bordeaux 21.1 (2009): 97-108. <http://eudml.org/doc/10878>.

@article{Cullinan2009,
abstract = {The one-parameter family of polynomials $J_\{n\}(x,y) = \sum _\{j=0\}^\{n\} \binom\{y+j\}\{j\}x^\{j\}$ is a subfamily of the two-parameter family of Jacobi polynomials. We prove that for each $n \ge 6$, the polynomial $J_\{n\}(x,y_\{0\})$ is irreducible over $\mathbb\{Q\}$ for all but finitely many $y_\{0\} \in \mathbb\{Q\}$. 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.},
affiliation = {Department of Mathematics Bard College Annandale-On-Hudson, NY 12504; Department of Mathematics University of Massachusetts Amherst MA 01003; Department of Mathematics Millersville University P.O. Box 1002 Millersville, PA 17551},
author = {Cullinan, John, Hajir, Farshid, Sell, Elizabeth},
journal = {Journal de Théorie des Nombres de Bordeaux},
keywords = {Orthogonal polynomials; Jacobi polynomial; Rational point; Riemann-Hurwitz formula; Specialization; Jacobi polynomials; irreducibility; Galois groups},
language = {eng},
number = {1},
pages = {97-108},
publisher = {Université Bordeaux 1},
title = {Algebraic properties of a family of Jacobi polynomials},
url = {http://eudml.org/doc/10878},
volume = {21},
year = {2009},
}

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
PB - Université Bordeaux 1
VL - 21
IS - 1
SP - 97
EP - 108
AB - The one-parameter family of polynomials $J_{n}(x,y) = \sum _{j=0}^{n} \binom{y+j}{j}x^{j}$ is a subfamily of the two-parameter family of Jacobi polynomials. We prove that for each $n \ge 6$, the polynomial $J_{n}(x,y_{0})$ is irreducible over $\mathbb{Q}$ for all but finitely many $y_{0} \in \mathbb{Q}$. 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 - eng
KW - Orthogonal polynomials; Jacobi polynomial; Rational point; Riemann-Hurwitz formula; Specialization; Jacobi polynomials; irreducibility; Galois groups
UR - http://eudml.org/doc/10878
ER -

References

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