Waring’s problem for unipotent algebraic groups
Annales de l'Institut Fourier, Volume 69 (2019) no. 4, p. 1857-1877

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.

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.

Received : 2017-07-26
Revised : 2018-04-10
Accepted : 2018-04-26
Published online : 2019-09-16
DOI : https://doi.org/10.5802/aif.3283
Classification:  11P05,  20G15,  14L15
Keywords: Waring’s problem, easier Waring’s problem, unipotent algebraic groups
@article{AIF_2019__69_4_1857_0,
     author = {Larsen, Michael and Nguyen, Dong Quan Ngoc},
     title = {Waring's problem for unipotent algebraic groups},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {69},
     number = {4},
     year = {2019},
     pages = {1857-1877},
     doi = {10.5802/aif.3283},
     language = {en},
     url = {https://aif.centre-mersenne.org/item/AIF_2019__69_4_1857_0}
}
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/item/AIF_2019__69_4_1857_0/

[1] Avni, Nir; Gelander, Tsachik; Kassabov, Martin; Shalev, Aner Word values in p-adic and adelic groups, Bull. Lond. Math. Soc., Tome 45 (2013) no. 6, pp. 1323-1330 | Article | MR 3138499 | Zbl 1288.20062

[2] Birch, Bryan J. Waring’s problem for 𝔭-adic number fields, Acta Arith., Tome 9 (1964), pp. 169-176 | Article | MR 166185 | Zbl 0131.28901

[3] Car, Mireille Le problème de Waring pour les corps de fonctions, Journées Arithmétiques (Luminy, 1989), Société Mathématique de France (Astérisque) Tome 198-200 (1991), pp. 77-82 | Zbl 0758.11040

[4] Chinburg, Ted Infinite easier Waring constants for commutative rings, Topology Appl., Tome 158 (2011) no. 14, pp. 1844-1847 | Article | MR 2823696 | Zbl 1222.11118

[5] Demazure, Michel; Gabriel, Pierre Groupes algébriques. Tome I: Géométrie algébrique, généralités, groupes commutatifs, Masson; North-Holland (1970), xxvi+700 pages (Avec un appendice Corps de classes local par Michiel Hazewinkel) | Zbl 0203.23401

[6] Ellison, William Waring’s problem for fields, Acta Arith., Tome 159 (2013) no. 4, pp. 315-330 | Article | MR 3080795 | Zbl 1310.11101

[7] Gallardo, Luis H.; Vaserstein, Leonid N. The strict Waring problem for polynomial rings, J. Number Theory, Tome 128 (2008) no. 12, pp. 2963-2972 | Article | MR 2464848 | Zbl 1220.11151

[8] Grunewald, Fritz J.; Schwermer, Joachim Free nonabelian quotients of SL 2 over orders of imaginary quadratic numberfields, J. Algebra, Tome 69 (1981) no. 2, pp. 298-304 | MR 617080 | Zbl 0461.20026

[9] Guralnick, Robert M.; Tiep, Pham Huu Effective results on the Waring problem for finite simple groups, Am. J. Math., Tome 137 (2015) no. 5, pp. 1401-1430 | Article | MR 3405871 | Zbl 1338.20009

[10] Im, Bo-Hae; Larsen, Michael Waring’s problems for rational functions in one variable (https://arxiv.org/abs/1801.06770 )

[11] Kamke, Erich Verallgemeinerungen des Waring–Hilbertschen Satzes, Math. Ann., Tome 83 (1921) no. 1-2, pp. 85-112 | Article | MR 1512001 | Zbl 48.0142.06

[12] Larsen, Michael; Shalev, Aner; Tiep, Pham Huu The Waring problem for finite simple groups, Ann. Math., Tome 174 (2011) no. 3, pp. 1885-1950 | Article | MR 2846493 | Zbl 1283.20008

[13] Liu, Yu-Ru; Wooley, Trevor D. Waring’s problem in function fields, J. Reine Angew. Math., Tome 638 (2010), pp. 1-67 | Article | MR 2595334 | Zbl 1221.11203

[14] Rosenlicht, Maxwell Some basic theorems on algebraic groups, Am. J. Math., Tome 78 (1956), pp. 401-443 | Article | MR 82183 | Zbl 0079.25703

[15] Serre, Jean-Pierre Cohomologie galoisienne, Springer, Lecture Notes in Mathematics, Tome 5 (1965), v+212 pages (With a contribution by Jean-Louis Verdier) | MR 201444 | Zbl 0136.02801

[16] Shalev, Aner Word maps, conjugacy classes, and a noncommutative Waring-type theorem, Ann. Math., Tome 170 (2009) no. 3, pp. 1383-1416 | MR 2600876 | Zbl 1203.20013

[17] Siegel, Carl Ludwig Generalization of Waring’s problem to algebraic number fields, Am. J. Math., Tome 66 (1944), pp. 122-136 | Article | MR 9778 | Zbl 0063.07008

[18] Siegel, Carl Ludwig Sums of mth powers of algebraic integers, Ann. Math., Tome 46 (1945), pp. 313-339 | Article | MR 12630 | Zbl 0063.07010

[19] Suzuki, Michio Group theory. II, Springer, Grundlehren der Mathematischen Wissenschaften, Tome 248 (1986), x+621 pages (Translated from the Japanese) | MR 815926 | Zbl 0586.20001

[20] Tavgenʼ, Oleg I. Bounded generation of normal and twisted Chevalley groups over the rings of S-integers, Proceedings of the International Conference on Algebra, Part 1 (Novosibirsk, 1989), American Mathematical Society (Contemporary Mathematics) Tome 131 (1992), pp. 409-421 | MR 1175793 | Zbl 0778.20020

[21] Voloch, José Felipe On the p-adic Waring’s problem, Acta Arith., Tome 90 (1999) no. 1, pp. 91-95 | Article | MR 1708680 | Zbl 0932.11074

[22] Wooley, Trevor D. On simultaneous additive equations. III, Mathematika, Tome 37 (1990) no. 1, pp. 85-96 | MR 1067890 | Zbl 0691.10008

[23] Wooley, Trevor D. On simultaneous additive equations. I, Proc. Lond. Math. Soc., Tome 63 (1991) no. 1, pp. 1-34 | Article | MR 1105717 | Zbl 0691.10007

[24] Wooley, Trevor D. On simultaneous additive equations. II, J. Reine Angew. Math., Tome 419 (1991), pp. 141-198 | MR 1116923 | Zbl 0721.11011

[25] Wright, E. Maitland An easier Waring’s problem, J. Lond. Math. Soc., Tome 9 (1934) no. 4, pp. 267-272 | Article | MR 1574875 | Zbl 0010.10306