On a global analytic Positivstellensatz
The authors study Positivstellensätze for global analytic functions. They consider an open subset \(\Omega\subseteq\mathbb{R}^n\) and an analytic set \(X\subseteq\Omega\), as well as real analytic functions \(g_1, \dots,g_r\), \(m:X\to\mathbb{R}\). Let \(Y\) be the zero set of \(M\) and let \(S= \{x\in X\mid\forall i= 1,\dots, r: g_i(x)\geq 0\}\). The authors ask whether the following two conditions are equivalent: \(\bullet\) \(S\subseteq Y\). \(\bullet\) There are sums of squares \(s_\nu\) of analytic functions with zero set contained in \(Y\) such that \(\sum_{\nu\in\{0, 1\}^r} s_\nu\cdot g^\nu= 0\). (The sums of squares may be finite or infinite.) The answer is affirmative if \(X\) is a curve (in this case the \(s_\nu\) are sums of 2 squares), or if \(X\) is a normal surface (in this case the \(s_\nu\) are sums of 5 squares), or if \(X\) is a real coherent set whose connected components are all compact.
scientific article; zbMATH DE number 5603927
scientific article