Le problème facile de Waring. (The easier Waring problem) (Q1190625)

In this Waring problem, the knowledge of the function \(\nu(k)\) [the smallest integer \(s\) such that every integer is the sum of \(s\) integers of the form \(\pm z^ k\), \(z\) an integer] is far from precise. Except for the trivial majoration \(G(k)+1\) \([G(k)\) being the asymptotic constant associated with the Waring problem], there are only large majorations for small \(k\). Here the author gives classical facts and the particular cases \(k=4\) and \(k=5\). Then certain new identities are presented which arose in a paper of \textit{L. N. Vaserstein} [J. Number Theory 28, 66-68 (1988; Zbl 0634.10042)], who gave a better bound for \(\nu(8)\). A short description of the Tarry-Escott problem is given, which is, for \(k\geq 9\), the only way to get effective majorations of \(\nu(k)\).