Hilbert's 17th problem and best dual bounds in quadratic minimization
Positive semi-definite polynomials in \(\mathbb{R}[X]= \mathbb{R}[X_1, \dots, X_n]\) can always be written as sums of squares in the function field \(\mathbb{R}(X_1, \dots,X_n)\) (Hilbert's 17-th problem, solved by Artin), but frequently not in the polynomial ring. Many variations of this classsical problem have been studied, resulting in various Positivstellensätze and Nicht-Negativstellensätze. The present paper asks: Given a polynomial \(P\in\mathbb{R} [X]\), when does there exist some \(c\in\mathbb{R}\) such that \(P-c\) is a sum of squares? The question is clearly related to the optimisation problem: Determine \(\inf\{P(x)\mid x\in \mathbb{R}^n\}\). Inflating the number of variables, the author transforms this problem into an optimisation problem about quadratic polynomials. He proves that the above question has an affirmative answer if and only if the dual bound of the quadratic optimisation problem is finite. Variations for polynomial functions on subvarieties of \(\mathbb{R}^n\) or on basic closed semi-algebraic sets are also included. The paper has a large number of typos.
scientific article; zbMATH DE number 1418953