In this paper, we formulate an analogue of Waring’s problem for an algebraic group . At the field level we consider a morphism of varieties and ask whether every element of is the product of a bounded number of elements of . We give an affirmative answer when is unipotent and 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 can be written as a product of a bounded number of elements of . We prove this is the case when is unipotent and is the ring of integers of a totally imaginary number field.
Dans cet article, nous formulons un analogue du problème de Waring pour un groupe algébrique . Soit un corps. Nous considérons un morphisme de variétés , défini sur , et nous demandons si chaque élément de est le produit d’un nombre borné d’éléments de . Nous donnons une réponse affirmative quand est unipotent et est un corps de caractéristique 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 peut être écrit comme un produit d’un nombre borné d’éléments de . Nous prouvons que c’est le cas lorsque est unipotent et est l’anneau d’entiers d’un corps de nombres totalement imaginaire.
Revised:
Accepted:
Published online:
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
@article{AIF_2019__69_4_1857_0, author = {Larsen, Michael and Nguyen, Dong Quan Ngoc}, title = {Waring{\textquoteright}s problem for unipotent algebraic groups}, journal = {Annales de l'Institut Fourier}, pages = {1857--1877}, publisher = {Association des Annales de l{\textquoteright}institut Fourier}, volume = {69}, number = {4}, year = {2019}, doi = {10.5802/aif.3283}, language = {en}, url = {https://aif.centre-mersenne.org/articles/10.5802/aif.3283/} }
TY - JOUR AU - Larsen, Michael AU - Nguyen, Dong Quan Ngoc TI - Waring’s problem for unipotent algebraic groups JO - Annales de l'Institut Fourier PY - 2019 SP - 1857 EP - 1877 VL - 69 IS - 4 PB - Association des Annales de l’institut Fourier UR - https://aif.centre-mersenne.org/articles/10.5802/aif.3283/ DO - 10.5802/aif.3283 LA - en ID - AIF_2019__69_4_1857_0 ER -
%0 Journal Article %A Larsen, Michael %A Nguyen, Dong Quan Ngoc %T Waring’s problem for unipotent algebraic groups %J Annales de l'Institut Fourier %D 2019 %P 1857-1877 %V 69 %N 4 %I Association des Annales de l’institut Fourier %U https://aif.centre-mersenne.org/articles/10.5802/aif.3283/ %R 10.5802/aif.3283 %G en %F AIF_2019__69_4_1857_0
Larsen, Michael; Nguyen, Dong Quan Ngoc. Waring’s problem for unipotent algebraic groups. Annales de l'Institut Fourier, Volume 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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