On the Pythagoras numbers of real analytic curves (Q2384736)

For \(X\subset \mathbb R^n\) a real analytic curve, define \({\mathcal O}(X)\) to be the ring of global analytic functions on \(X.\) For \(x\in X\) let \(X_x\) be a reduced real analytic curve germ and \({\mathcal O}(X_x)\) its ring of analytic function germs. \(X\) has the property `psd = sos' if every positive semidefinite analytic function in \({\mathcal O}(X)\) is sum of squares. This property is defined similarly for each curve germ \(X_x\) w.r.t. of \({\mathcal O}(X_x).\) The main results of the paper are: Proposition 1.1. Equivalent are: a. \(X\) satisfies psd = sos. b. Every \(X_x\) satisfies psd = sos. c. Every germ \(X_x\) is the union of nonsingular independent branches. Theorem 1.2. The Pythagoras number of the ring \({\mathcal O}(X)\) is the supremum of the Pythagoras numbers of the rings \({\mathcal O}(X_x),\) \(x\) a singular point of \(X,\) or by at most 1 larger than this. Proposition 2.2. There exist analytic curves in \(\mathbb R^n\), \(n\geq 3\) with infinite Pythagoras numbers. The proofs rely on M. Artin's approximation [see e.g. \textit{J. Bochnak, M. Coste} and \textit{M.-F. Roy}, Ergebn. Math. Grenzgeb. (3) 36 (1998; Zbl 0912.14023)], on results of \textit{J. Ortega} [Math. Ann. 289, No. 1, 111--123 (1991; Zbl 0743.14041)], and \textit{C. Scheiderer} [e.g., J. Reine Angew. Math. 540, 205--227 (2001; Zbl 0991.13014)], and H. Cartan's `theorems A and B' on sheaves, among others.