Geodesic
The Waring problem for finite simple groups
Michael Larsen1 ; Aner Shalev2 ; Pham Huu Tiep3
1 Department of Mathematics<br/> Indiana University<br/> Bloomington, IN 47405
2 Einstein Institute of Mathematics<br/>The Hebrew University of Jerusalem<br/>Edmond J. Safra Campus, Givat Ram<br/> Jerusalem 91904<br/>Israel
3 Department of Mathematics<br/>University of Arizona<br/>617 N. Santa Rita Ave.<br/>Tucson, AZ 85721-0089
Annals of mathematics, Tome 174 (2011) no. 3, pp. 1885-1950.

Voir la notice de l'article provenant de la source Annals of Mathematics website

The classical Waring problem deals with expressing every natural number as a sum of $g(k)$ $k$-th powers. Recently there has been considerable interest in similar questions for non-abelian groups, and simple groups in particular. Here the $k$-th power word can be replaced by an arbitrary group word $w \ne 1$, and the goal is to express group elements as short products of values of $w$.
We give a best possible and somewhat surprising solution for this Waring type problem for (non-abelian) finite simple groups of sufficiently high order, showing that a product of length two suffices to express all elements.
Along the way we also obtain new results, possibly of independent interest, on character values in classical groups over finite fields, on regular semisimple elements lying in the image of word maps, and on products of conjugacy classes.
Our methods involve algebraic geometry and representation theory, especially Lusztig’s theory of representations of groups of Lie type.
DOI : 10.4007/annals.2011.174.3.10

Michael Larsen 1 ; Aner Shalev 2 ; Pham Huu Tiep 3

1 Department of Mathematics<br/> Indiana University<br/> Bloomington, IN 47405
2 Einstein Institute of Mathematics<br/>The Hebrew University of Jerusalem<br/>Edmond J. Safra Campus, Givat Ram<br/> Jerusalem 91904<br/>Israel
3 Department of Mathematics<br/>University of Arizona<br/>617 N. Santa Rita Ave.<br/>Tucson, AZ 85721-0089 
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Michael Larsen; Aner Shalev; Pham Huu Tiep. The Waring problem for finite simple groups. Annals of mathematics, Tome 174 (2011) no. 3, pp. 1885-1950. doi : 10.4007/annals.2011.174.3.10. http://geodesic.mathdoc.fr/articles/10.4007/annals.2011.174.3.10/

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