Decompositions into sums of two irreducibles in $ {\mathbf{F}}_{q}[t]$

Bender, Andreas O.

doi:10.1016/j.crma.2008.07.025

Number Theory

Decompositions into sums of two irreducibles in

F_{q} [t]

[Décompositions en sommes de deux polynômes irréductibles dans

F_{q} [t]

]

Présenté par : Jean-Pierre Serre

Bender, Andreas O.¹

¹ Korea Institute for Advanced Study, Seoul 130-722, South Korea

Comptes Rendus. Mathématique, Tome 346 (2008) no. 17-18, pp. 931-934.

Résumé
Abstract

Un polynôme unitaire $f \in F_{q} [t]$ de degré n à coefficients dans un corps fini $F_{q}$ de caractéristique différente de 2 s'écrit comme une somme $f = g + h$ , où $g, h \in F_{q} [t]$ sont des polynômes unitaires irréductibles de degrés n et $n - 1$ , dès que q est plus grand qu'une borne explicite dépendant uniquement de n.

A monic polynomial in $F_{q} [t]$ of degree n over a finite field $F_{q}$ of odd characteristic is the sum of two monic irreducibles in $F_{q} [t]$ of degrees n and $n - 1$ , provided q is larger than an explicitly given bound in terms of n.

Reçu le : 2008-05-17
Accepté le : 2008-07-28
Publié le : 2008-08-23

DOI : 10.1016/j.crma.2008.07.025

Affiliations des auteurs :

Bender, Andreas O. ¹

¹ Korea Institute for Advanced Study, Seoul 130-722, South Korea

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Bender, Andreas O. Decompositions into sums of two irreducibles in $ {\mathbf{F}}_{q}[t]$. Comptes Rendus. Mathématique, Tome 346 (2008) no. 17-18, pp. 931-934. doi : 10.1016/j.crma.2008.07.025. http://www.numdam.org/articles/10.1016/j.crma.2008.07.025/

Bibliographie
Cité par

[1] A.O. Bender, Representing an element in $F_{q} [t]$ as the sum of two irreducibles in $F_{q^{s}} [t]$ , submitted for publication

[2] Bender, A.O.; Wittenberg, O. A potential analogue of Schinzel's hypothesis for polynomials with coefficients in $F_{q} [t]$ , Int. Math. Res. Not., Volume 36 (2005), pp. 2237-2248 (also available from) | arXiv

[3] Effinger, G.W.; Hayes, D.R. Additive Number Theory of Polynomials Over a Finite Field, Oxford University Press, New York, NY, 1991

[4] Eisenbud, D. Commutative Algebra With a View Toward Algebraic Geometry, Graduate Texts in Mathematics, vol. 150, Springer-Verlag, New York, NY, 1995

[5] Hurwitz, A. Über Riemann'sche Flächen mit gegebenen Verzweigungspunkten, Math. Ann., Volume 39 (1891), pp. 1-61 (and Math. Werke, Band 1/XXI, Birkhäuser, Basel, 1932)

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