no. 2
A note on the spectrum of a rational function
[Une note sur le spectre d’une fonction rationnelle]
Benelmekki, Mohamed1 ; Najib, Salah2
1 Université Sultan Moulay Slimane Laboratoire de Mathématiques et Applications, FST Campus Mghilla, BP 523, 23000 Béni Mellal MAROC
2 Université Sultan Moulay Slimane, Faculté Polydisciplinaire de Khouribga Laboratoire multidisciplinaire de recherche et d’innovation BP 145, Hay Ezzaytoune, 25000 Khouribga MAROC
Annales mathématiques Blaise Pascal, Tome 30 (2023) no. 2, pp. 107-114

In 2008, Bodin provided an alternative approach for bounding the total reducibility order of a non-composite rational function. His proof used some properties of jacobian derivation. In this note, we revisit this proof and eliminate the jacobian derivation aspect. The new ingredient in our presentation is a version of Lüroth’s theorem.

En 2008, Bodin a fourni une approche alternative pour borner l’ordre total de réductibilité d’une fonction rationnelle indécomposable. Sa preuve a utilisé certaines propriétés de dérivation jacobienne. Dans cette note, nous revisitons cette preuve et éliminer l’aspect de dérivation jacobienne. Le nouvel ingrédient de notre présentation est une version du théorème de Lüroth.

Publié le :
DOI : 10.5802/ambp.418
Classification : 12E05, 12F20, 11C08
Keywords: Irreducible polynomials, Indecomposable rational function, Spectrum of a rational function, Lüroth’s theorem

Benelmekki, Mohamed 1 ; Najib, Salah 2

1 Université Sultan Moulay Slimane Laboratoire de Mathématiques et Applications, FST Campus Mghilla, BP 523, 23000 Béni Mellal MAROC
2 Université Sultan Moulay Slimane, Faculté Polydisciplinaire de Khouribga Laboratoire multidisciplinaire de recherche et d’innovation BP 145, Hay Ezzaytoune, 25000 Khouribga MAROC
Licence : CC-BY 4.0
Droits d'auteur : Les auteurs conservent leurs droits 
@article{AMBP_2023__30_2_107_0,
     author = {Benelmekki, Mohamed and Najib, Salah},
     title = {A note on the spectrum of a rational function},
     journal = {Annales math\'ematiques Blaise Pascal},
     pages = {107--114},
     year = {2023},
     publisher = {Universit\'e Clermont Auvergne, Laboratoire de math\'ematiques Blaise Pascal},
     volume = {30},
     number = {2},
     doi = {10.5802/ambp.418},
     language = {en},
     url = {https://www.numdam.org/articles/10.5802/ambp.418/}
}
TY  - JOUR
AU  - Benelmekki, Mohamed
AU  - Najib, Salah
TI  - A note on the spectrum of a rational function
JO  - Annales mathématiques Blaise Pascal
PY  - 2023
SP  - 107
EP  - 114
VL  - 30
IS  - 2
PB  - Université Clermont Auvergne, Laboratoire de mathématiques Blaise Pascal
UR  - https://www.numdam.org/articles/10.5802/ambp.418/
DO  - 10.5802/ambp.418
LA  - en
ID  - AMBP_2023__30_2_107_0
ER  - 
%0 Journal Article
%A Benelmekki, Mohamed
%A Najib, Salah
%T A note on the spectrum of a rational function
%J Annales mathématiques Blaise Pascal
%D 2023
%P 107-114
%V 30
%N 2
%I Université Clermont Auvergne, Laboratoire de mathématiques Blaise Pascal
%U https://www.numdam.org/articles/10.5802/ambp.418/
%R 10.5802/ambp.418
%G en
%F AMBP_2023__30_2_107_0
Benelmekki, Mohamed; Najib, Salah. A note on the spectrum of a rational function. Annales mathématiques Blaise Pascal, Tome 30 (2023) no. 2, pp. 107-114. doi: 10.5802/ambp.418

[1] Bodin, Arnaud Reducibility of rational functions in several variables, Isr. J. Math., Volume 164 (2008), pp. 333-348 | DOI | MR

[2] Bodin, Arnaud; Dèbes, Pierre; Najib, Salah Irreducibility of hypersurfaces, Commun. Algebra, Volume 37 (2009) no. 6, pp. 1884-1900 | DOI | MR | Zbl

[3] Bodin, Arnaud; Dèbes, Pierre; Najib, Salah Families of polynomials and their specializations, J. Number Theory, Volume 170 (2017), pp. 390-408 | DOI | MR

[4] Bodin, Arnaud; Dèbes, Pierre; Najib, Salah The Schinzel Hypothesis for Polynomials, Trans. Am. Math. Soc., Volume 373 (2020) no. 12, pp. 8339-8364 | DOI | MR

[5] Busé, Laurent; Chèze, Guillaume; Najib, Salah Noether’s forms for the study of non-composite rational functions and their spectrum, Acta Arith., Volume 147 (2011) no. 3, pp. 217-231 | DOI | MR

[6] Cygan, Ewa Factorization of polynomials, Bull. Pol. Acad. Sci., Math., Volume 40 (1992) no. 1, pp. 45-52 | MR

[7] Fulton, William Algebraic curves. An introduction to algebraic geometry, Addison-Wesley Publishing Group, 1989

[8] Kaliman, Shulim Two remarks on polynomials in two variables, Pac. J. Math., Volume 154 (1992) no. 2, pp. 285-295 | MR | Zbl | DOI

[9] Lefshetz, Solomon Algebraic Geometry, Princeton Mathematical Series, 18, Princeton University Press, 1953 | DOI

[10] Lorenzini, Dino Reducibility of polynomials in two variables, J. Algebra, Volume 156 (1993) no. 1, pp. 65-75 | DOI | MR

[11] Najib, Salah Sur le spectre d’un polynôme à plusieurs variables, Acta Arith., Volume 114 (2004), pp. 169-181 | DOI | MR | Zbl

[12] Najib, Salah Factorisation des polynômes P(X 1 ,...,X n )-λ et théorème de Stein, Ph. D. Thesis, University of Lille (2005)

[13] Najib, Salah Une généralisation de l’inégalité de Stein-Lorenzini, J. Algebra, Volume 292 (2005) no. 2, pp. 566-573 | DOI | MR

[14] Najib, Salah The spectrum of a rational function, Algebra Colloq., Volume 27 (2020) no. 3, pp. 477-482 | DOI | MR

[15] Petravchuk, Anatoliy P.; Iena, Oleksandr G. On closed rational functions in several variables, Algebra Discrete Math., Volume 6 (2007) no. 2, pp. 115-124 | Zbl | MR

[16] Ruppert, Wolfgang Reduzibilität ebener Kurven, J. Reine Angew. Math., Volume 369 (1986), pp. 167-191 | MR | Zbl

[17] Schinzel, Andrzej Polynomials with special regard to reducibility, Cambridge University Press, 2000 | DOI

[18] Shafarevich, Igor R. Basic algebraic geometry 1, Springer, 1994 | DOI

[19] Stein, Yosef The total reducibility order of a polynomial in two variables, Isr. J. Math., Volume 68 (1989) no. 1, pp. 109-122 | DOI | MR | Zbl

[20] Vistoli, Angelo The number of reducible hypersurfaces in a pencil, Invent. Math., Volume 112 (1993), pp. 247-262 | DOI | MR

Cité par Sources :