no. 4
Waring’s problem for unipotent algebraic groups
[Le problème de Waring pour les groupes algébriques unipotents]
Larsen, Michael1 ; Nguyen, Dong Quan Ngoc2
1 Department of Mathematics Indiana University Bloomington, Indiana 47405 (USA)
2 Department of Applied and Computational Mathematics and Statistics University of Notre Dame Notre Dame, Indiana 46556 (USA)
Annales de l'Institut Fourier, Tome 69 (2019) no. 4, pp. 1857-1877.

Dans cet article, nous formulons un analogue du problème de Waring pour un groupe algébrique G. Soit K un corps. Nous considérons un morphisme de variétés f:𝔸 1 →G, défini sur K, et nous demandons si chaque élément de G(K) est le produit d’un nombre borné d’éléments de f(𝔸 1 (K))=f(K). Nous donnons une réponse affirmative quand G est unipotent et K est un corps de caractéristique 0 ce qui n’est pas formellement réel.

L’idée est la même au niveau intégral, sauf qu’il faut travailler avec des schémas, et la question est de savoir si chaque élément d’un sous-groupe d’indice fini de G(𝒪) peut être écrit comme un produit d’un nombre borné d’éléments de f(𝒪). Nous prouvons que c’est le cas lorsque G est unipotent et 𝒪 est l’anneau d’entiers d’un corps de nombres totalement imaginaire.

In this paper, we formulate an analogue of Waring’s problem for an algebraic group G. At the field level we consider a morphism of varieties f:𝔸 1 →G and ask whether every element of G(K) is the product of a bounded number of elements of f(𝔸 1 (K))=f(K). We give an affirmative answer when G is unipotent and K is a characteristic zero field which is not formally real.

The idea is the same at the integral level, except one must work with schemes, and the question is whether every element in a finite index subgroup of G(𝒪) can be written as a product of a bounded number of elements of f(𝒪). We prove this is the case when G is unipotent and 𝒪 is the ring of integers of a totally imaginary number field.

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Révisé le :
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DOI : 10.5802/aif.3283
Classification : 11P05, 20G15, 14L15
Keywords: Waring’s problem, easier Waring’s problem, unipotent algebraic groups
Mot clés : problème de Waring, problème facile de Waring, groupes algébriques unipotents

Larsen, Michael 1 ; Nguyen, Dong Quan Ngoc 2

1 Department of Mathematics Indiana University Bloomington, Indiana 47405 (USA)
2 Department of Applied and Computational Mathematics and Statistics University of Notre Dame Notre Dame, Indiana 46556 (USA)
Licence : CC-BY-ND 4.0
Droits d'auteur : Les auteurs conservent leurs droits 
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Larsen, Michael; Nguyen, Dong Quan Ngoc. Waring’s problem for unipotent algebraic groups. Annales de l'Institut Fourier, Tome 69 (2019) no. 4, pp. 1857-1877. doi : 10.5802/aif.3283. https://aif.centre-mersenne.org/articles/10.5802/aif.3283/

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