## Abstract

The classical Waring problem deals with expressing every natural number as a sum of g(k) k^{th} 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.

Original language | American English |
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Pages (from-to) | 81-100 |

Number of pages | 20 |

Journal | Israel Journal of Mathematics |

Volume | 210 |

Issue number | 1 |

DOIs | |

State | Published - 1 Sep 2015 |

### Bibliographical note

Publisher Copyright:© 2015, Hebrew University of Jerusalem.