Sums of even powers in archimedean rings (Q1601783)

A ring \(A\) is totally real if every sum of squares of the form \(1+a^2_1 +\cdots +a^2_k\) is a unit. An element \(a\in A\) is totally positive if \(\varphi (a)\geq 0\) for every homomorphism \(\varphi:A\to R\) into a real closed field; it is strictly totally positive if always \(\varphi(a) >0\). The ring is totally Archimedean if for every \(a\in A\) there is some \(n\in\mathbb{N}\) such \(n\pm a\) are both totally positive. A preordering of level \(r\) is a subset \(T\subseteq A\) that is additively and multiplicatively closed, contains every \(2r\)-th power and does not contain \(-1\). The ring \(A\) is always assumed to be totally real and totally Archimedean. This implies that strictly totally positive elements are units. The author proves the following analogue of Schmüdgen's Positivstellensatz [\textit{K. Schmüdgen}, Math. Ann. 289, 203-206 (1991; Zbl 0744.44008)] for preorderings of level \(r\): Assume that \(T\) is a preordering of level \(r\). Assume that \(a\in A\) and that \(\varphi(a)>0\) whenever \(\varphi: A\to R\) is a homomorphism into a real closed field with \(\varphi(T) \subseteq R^\geq\). Then it follows that \(a\in T\). [Another variant of Schmüdgen's theorem for preorderings of higher level is proved by \textit{D. Zhang}, Arch. Math. 77, 309-312 (2001; see the preceding review Zbl 1015.13008).] It is shown that every strictly totally positive element can be written as a sum of \(2r\)-th powers of strictly positive elements for every \(1\leq r\in\mathbb{N}\). Bounds for the length of such representations are studied.