Sums of values represented by a quadratic form
Let \(K\) be a field of characteristic not \(2\) and let \(q\) be a nondegenerate quadratic form over \(K\) defined on a finite-dimensional \(K\)-vector space \(V\). The \(q\)-level \(s_q(K)\) (resp. \(q\)-sublevel \(\underline{s}_q(K)\)) is defined to be the smallest \(n\) for which there exist \(v_1,\ldots,v_n\in V\) (resp. \(v_1,\ldots,v_{n+1}\in V\setminus\{ 0\}\)) such that \(-1=q(v_1)+\cdots +q(v_n)\) (resp. \(0=q(v_1)+\cdots +q(v_{n+1})\)). If no such representation exists, the respective value is defined to be \(\infty\). If \(q\) is given by the diagonalized form \(\langle 1\rangle\), then the \(q\)-(sub)level is nothing else but the usual level \(s\) (resp. sublevel \(\underline{s}\)), in which case it is easy to see that \(s=\underline{s}\). A famous result by \textit{A. Pfister} [J. Lond. Math. Soc. 40, 159--165 (1965; Zbl 0131.25002)] states that the level, if finite, is always a \(2\)-power, and that each \(2\)-power can be realized as level of a suitably constructed field. The main purpose of the present paper is to extend these results to \(q\)-(sub)levels and to find upper bounds in terms of other field invariants pertaining to quadratic forms such as the \(u\)-invariant and the Pythagoras number. One first notes that if \(q\) is isotropic, then \(\underline{s}_q(K)=0\) and \(s_q(K)=1\), so that one may assume \(q\) to be anisotropic, in which case \(s_q(K)\leq \underline{s}_q(K)+1\), and \(\underline{s}_q(K)\leq s_q(K)\) if \(q\) represents \(1\). It is shown that to each positive integer \(n\) there exists a field \(K\) and a form \(q\) over \(K\) such that \(s_q(K)=n\). However, if \(q\) is an anisotropic Pfister form, or, more generally, a round form, i.e., a form where the represented values of \(q\) are exactly the similarity factors of \(q\), then \(s_q(K)=\underline{s}_q(K)=2^m\) for some nonnegative integer \(m\). If \(\dim(q)=3\), then, for any nonnegative integer \(k\), both \((2^{2k}+2)/3\) and \((2^{2k+1}+1)/3\) can be realized as \(s_q\) for suitable field extensions of \(K\). Other topics that are studied in the present article are: the Pythagoras \(q\)-number \(p_q(K)\), i.e., the least \(n\) such that each \(a\in K\) that is a sum of values of \(q\) can be written as a some of \(\leq n\) such values, provided such an \(n\) exists, otherwise one puts \(p_q(K)=\infty\); finiteness of the \(q\)-level for formally real fields; the behavior of \(s_q\), \(\underline{s}_q\), \(p_q\) under field extensions; values of \(s_q\) for specific fields such as local and global fields, or rational function fields in one or more variables over such fields or over the reals.
scientific article; zbMATH DE number 6145962
scientific article