Let \(V\subset\mathbb R^n\) be an algebraic set and let \({\mathcal P}(V)\) be the ring of polynomial functions on \(V\). Given \(f_1,\dots, f_m\in{\mathcal P}(V)\), define \[ W:= \{x\in V: f_1(x)\geq0,\dots, f_m(x)\geq0\}, \] and let \(P_{W}\) be the cone of \({\mathcal P}(V)\) generated by \(f_1,\dots, f_m\); that is, the family of those \(h\in{\mathcal P}(V)\) that can be written as \(h:= s+\sum_{i=1}^rs_ip_i\) for some nonnegative integer \(r\), where \(s\) and each \(s_i\) are (finite) sums of squares of elements in \({\mathcal P}(V)\) and each \(p_i\) is a (finite) product of some \(f_j\)’s.
The Nichtnegativstellensatz discovered by G. Stengle [Math. Ann. 207, 87–97 (1973; Zbl 0253.14001)] states that a necessary and sufficient condition for a function \(f\in{\mathcal P}(V)\) to be \(\geq0\) on each point in \(W\) is the existence of a nonnegative integer \(k\) and two functions \(g,h\in P_W\) such that \(fg=f^{2m}+h\).
The main result of the article under review is Theorem 1.9, which can be understood as a noncommutative analogue of Stengle’s Nichtnegativstellensatz. I consider that its statement is too technical to be explained here in detail for nonspecialists.
Of course, this very long article (69 pages) contains many other results, more or less in the same vein as the one just quoted. Fortunately, its excellent writing makes it nearly as detailed and self-contained as a book, and so it is not only an excellent paper for specialists but it can also be used to introduce young researchers to this exciting topic.