Graduate Texts in Mathematics bridge the gap between passive study and creative
understanding, offering graduate-level introductions to advanced topics in mathematics. The …

We consider the problem of finding the unconstrained global minimum of a real-valued
polynomial p (x): R^ n → R ,aswellastheglobalminimumofp(x),inacompactsetKdefinedbypolynomialinequ …

Serving both as an introduction to the subject and as a reference, this book presents the
theory in elegant form and with modern concepts and notation. It covers the general theory …

The study of positive polynomials brings together algebra, geometry and analysis. The
subject is of fundamental importance in real algebraic geometry when studying the …

Let β≡ β (2n) be an N-dimensional real multi-sequence of degree 2n, with associated
moment matrix 𝓜 (n)≡ 𝓜 (n)(β), and let r:= rank 𝓜 (n). We prove that if 𝓜 (n) is positive …

Let $\gamma\equiv\gamma^{\left (2n\right)} $ denote a sequence of complex numbers
$\gamma _ {00},\gamma _ {01},\gamma _ {10},\dots,\gamma _ {0, 2n},\dots,\gamma _ {2n, 0} …

A non-commutative polynomial which is positive on a bounded semi-algebraic set of
operators has a weighted sum of squares representation. This Positivstellensatz parallels …

This paper gives a survey, with detailed references to the literature, on recent developments
in real algebra and geometry concerning the polarity between positivity and sums of …

This paper considers polynomial optimization with unbounded sets. We give a
homogenization formulation and propose a hierarchy of Moment-SOS relaxations to solve it …

Let A be a commutative ring with 1, let P⊂A be a preordering of higher level (ie 0,1∈P,-
1\not∈P,P+P⊂P,P⋅P⊂P and A^2n⊂P for some n∈\N) and let M⊂A be an archimedean P …