Continuous Pythagoras numbers for rational quadratic forms (Q580403)

In a commutative ring R consider the set \(\Sigma\) of elements representable as \(\sum w_ is^ 2_ i\), where \(w_ i\), resp. \(s_ i\), are chosen from subsets W, resp. S, of R. Concrete examples studied in the paper are \(R={\mathbb{R}}[X_ 1,...,X_ n]\), \(W={\mathbb{R}}^+\) or \(W=\{1\}\) and \(S=\{linear\) forms\(\}\). The question is, are there a fixed number m and functions observing special restrictions \(f_ 1,...,f_ m\), resp. \(g_ 1,...,g_ m\), from \(\Sigma\) to W, resp. S, such that for all \(a\in \Sigma\) one has \(a=\sum^{m}_{i=1}f_ i(a)g_ i(a)^ 2 ?\) The minimal possible m is then some sort of Pythagoras number. The restrictions posed on the \(f_ i\), \(g_ i\) are first continuity (of \(f_ i\), \(g_ i\) separately or of \(f_ ig^ 2_ i)\) and secondly some kinds of ''rationality'': \({\mathbb{Q}}\)-rational; semialgebraic, defined by polynomials over \({\mathbb{Q}}\); f(\({\mathbb{Q}})\subset {\mathbb{Q}}\). The author obtains several values or bounds for his Pythagoras numbers.