Eine Bemerkung zu den höheren Pythagoraszahlen reeller Körper. (A remark on higher Pythagoras numbers of real fields) (Q1122612)

If K is a field let \(\sum K^ n\) be the set of all sums of n-th powers of elements of K. Let \(\sum^{r}K^ n\) be the subset of everything expressible as a sum of r n-th powers in K. The n-th Pythagoras number is defined to be \(P_ n(K):=\inf \{r\in {\mathbb{N}} |\) \(\sum K^ n=\sum^{r}K^ n\}\). Then \(P_ 2(K)\) is the usual Pythagoras number of K. Becker proved earlier that if \(P_ 2(K)\) is finite then \(P_{2n}(K)\) is finite for every \(n\in {\mathbb{N}}\). As a special case Becker deduced that \(P_ 4({\mathbb{R}}(X))\leq 36\) where \({\mathbb{R}}(X)\) is the field of rational functions in one variable over the reals. The author improves some of Becker's inequalities and deduces that \(P_ 4({\mathbb{R}}(X))\leq 24\). The argument employs Becker's results concerning real valuations and the real holomorphy ring of K. Recently Lam, Choi and Reznick have found an elementary argument proving that \(P_ 4({\mathbb{R}}(X))\leq 6\).