Polynomial representations via spectral decompositions
The present note is an improvement and a continuation of [\textit{O. Demanze}, J. Math. Anal. Appl. 247, No.~2, 570--587 (2000; Zbl 0961.26007)]. Specifically, the author studies a polynomial \(P(x)\) depending on \(n\) real variables, which is strictly positive (in a strong sense, involving points at infinity in the projective completion) on a basic semialgebraic set \[ \{x\in\mathbb{R}^n;\, P_1(x)\geq 0,\dots, P_k(x)\geq 0\} \] defined by polynomials \(P_j\). The main result asserts that such a polynomial \(P\) can be represented as a weighted sum of squares of rational functions in the variables \(x\) and \(\sqrt{1+ x^2_j}\), \(1\leq j\leq n\), with a common denominator of the form \[ \prod^n_{j=1} \sqrt{1+ x^{2^{m_j}}},\quad m_j\geq 0. \] This is a refinement of known Positivstellensätze in real algebraic geometry. The proof is an adaptation of the functional theoretic technique contained in [\textit{M. Putinar} and \textit{F.-H. Vasilescu}, ``Solving moment problems by dimensional extension'', Ann. Math. (2) 149, No.~3, 1087--1107 (1999; Zbl 0939.44003)].
scientific article; zbMATH DE number 2021989