Atypical values at infinity of real polynomial maps with $2$-dimensional fibers

Masaharu Ishikawa; Tat-Thang Nguyen

doi:10.5802/crmath.761

Article de recherche - Théorie des singularités

Atypical values at infinity of real polynomial maps with $2$-dimensional fibers
[Valeurs atypiques à l’infini des applications polynomiales réelles à fibres bidimensionnelles]

Masaharu Ishikawa ¹ ; Tat-Thang Nguyen ²

¹ Faculty of Economics, Keio University, 4-1-1, Hiyoshi, Kouhoku, Yokohama, Kanagawa 223-8521, Japan
² Institute of Mathematics, Vietnam Academy of Science and Technology, 18 Hoang Quoc Viet road, Cau Giay district, 10072 Hanoi, Vietnam

Comptes Rendus. Mathématique, Volume 363 (2025), pp. 917-932

Abstract (VO)
Résumé

We characterize atypical values at infinity of a real polynomial function of three variables by a certain sum of indices of the gradient vector field of the function restricted to a sphere with a sufficiently large radius. This is a three-variable analogue of a result of Coste and de la Puente for real polynomial functions with two variables. We also give a characterization of atypical values at infinity of a real polynomial map whose regular fibers are $2$-dimensional surfaces.

Nous caractérisons les valeurs atypiques à l’infini d’une fonction polynomiale réelle de trois variables par une certaine somme d’indices du champ de vecteurs gradient de la fonction restreint à une sphère de rayon suffisamment grand. Il s’agit d’un analogue à trois variables d’un résultat de Coste et de la Puente pour les fonctions polynomiales réelles à deux variables. Nous donnons également une caractérisation des valeurs atypiques à l’infini d’une application polynomiale réelle dont les fibres régulières sont des surfaces bidimensionnelles.

Reçu le : 2024-12-17
Révisé le : 2025-05-21
Accepté le : 2025-05-21
Publié le : 2025-07-22

DOI : 10.5802/crmath.761

Classification : 58K15, 58K05, 58K30, 58K45
Keywords: Polynomial map, singularity at infinity, atypical fiber, bifurcation set
Mots-clés : Application polynomiale, singularité à l’infini, fibre atypique, ensemble de bifurcation

Affiliations des auteurs :

Masaharu Ishikawa ¹ ; Tat-Thang Nguyen ²

¹ Faculty of Economics, Keio University, 4-1-1, Hiyoshi, Kouhoku, Yokohama, Kanagawa 223-8521, Japan
² Institute of Mathematics, Vietnam Academy of Science and Technology, 18 Hoang Quoc Viet road, Cau Giay district, 10072 Hanoi, Vietnam

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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     title = {Atypical values at infinity of real polynomial maps with $2$-dimensional fibers},
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Masaharu Ishikawa; Tat-Thang Nguyen. Atypical values at infinity of real polynomial maps with $2$-dimensional fibers. Comptes Rendus. Mathématique, Volume 363 (2025), pp. 917-932. doi: 10.5802/crmath.761

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