A real Nullstellensatz and Positivstellensatz for the semipolynomials over an ordered field
Given a totally ordered field \(K\) with real closure \(R\), a variation of Hilbert's original 17-th problem asks whether given a polynomial \(P\in K[ X]= K[X_1, \dots, X_n]\) which takes nonnegative values at each point of \(\mathbb{R}^n\) (i.e., \(P\) is positive semidefinite on \(\mathbb{R}^n\)) is representable in the form \(P= \sum p_i r_i^2\) with \(0\leq p_i\in K\), \(r_i\in K(X)\) rational functions. After this problem had been solved the focus of further investigations shifted to the question of how the \(p_i\) and \(r_i\) depend on the coefficients of \(P\). The most far reaching solution was presented by \textit{C. N. Delzell} [J. Reine Angew. Math. 440, 157-173 (1993; Zbl 0774.12003)] who proved that the \(p_i\) as well as the coefficients of the polynomials in the numerators and the denominators of the \(r_i\) can be chosen to be sup-inf- polynomially definable continuous functions, i.e., certain piecewise polynomial functions. The present paper contains another proof of this result as well as of a related Positivstellensatz. The method of proof is constructive.
sup-inf-polynomially definable continuous functions
scientific article; zbMATH DE number 500592
scientific article