TY - JOUR

T1 - The Waring problem for Lie groups and Chevalley groups

AU - Hui, Chun Yin

AU - Larsen, Michael

AU - Shalev, Aner

N1 - Publisher Copyright:
© 2015, Hebrew University of Jerusalem.

PY - 2015/9/1

Y1 - 2015/9/1

N2 - The classical Waring problem deals with expressing every natural number as a sum of g(k) kth powers. Similar problems were recently studied in group theory, where we aim to present group elements as short products of values of a given word w ≠ 1. In this paper we study this problem for Lie groups and Chevalley groups over infinite fields. We show that for a fixed word w ≠ 1 and for a classical connected real compact Lie group G of sufficiently large rank we have w(G)2 = G, namely every element of G is a product of 2 values of w. We prove a similar result for non-compact Lie groups of arbitrary rank, arising from Chevalley groups over ℝ or over a p-adic field. We also study this problem for Chevalley groups over arbitrary infinite fields, and show in particular that every element in such a group is a product of two squares.

AB - The classical Waring problem deals with expressing every natural number as a sum of g(k) kth powers. Similar problems were recently studied in group theory, where we aim to present group elements as short products of values of a given word w ≠ 1. In this paper we study this problem for Lie groups and Chevalley groups over infinite fields. We show that for a fixed word w ≠ 1 and for a classical connected real compact Lie group G of sufficiently large rank we have w(G)2 = G, namely every element of G is a product of 2 values of w. We prove a similar result for non-compact Lie groups of arbitrary rank, arising from Chevalley groups over ℝ or over a p-adic field. We also study this problem for Chevalley groups over arbitrary infinite fields, and show in particular that every element in such a group is a product of two squares.

UR - http://www.scopus.com/inward/record.url?scp=84945907184&partnerID=8YFLogxK

U2 - 10.1007/s11856-015-1246-9

DO - 10.1007/s11856-015-1246-9

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AN - SCOPUS:84945907184

SN - 0021-2172

VL - 210

SP - 81

EP - 100

JO - Israel Journal of Mathematics

JF - Israel Journal of Mathematics

IS - 1

ER -