Bounded meromorphic functions on compact real analytic sets (Q1805911)

The real holomorphy ring of a field \(K\) is the intersection of all real valuation rings of \(K\). A main merit of the paper under review is to interpret natural rings of meromorphic functions (or germs at compact sets), without nonsingularity assumptions, as holomorphy rings. In particular, if \(K=M(X)\) is the field of global meromorphic functions on the compact and irreducible real analytic set \(X\), then \(H(K)=M_b(X)\) is the ring of meromorphic functions on \(X\) bounded on the subset \(Y\) of regular points of \(X\) of maximum dimension. This provides, after a paper by \textit{R. C. Heitman} [Pac. J. Math. 62, 117-126 (1976; Zbl 0308.13014)], the upper bound \(1+\dim (X)\) for the number of generators of the finitely generated ideals of \(M_b(X)\). Moreover, the authors prove another crucial result: If \({\mathfrak m}_x\) is the ideal of analytic functions on \(X\) vanishing at the point \(x\) of \(Y\), then the ideal \(J_x={\mathfrak m}_xM_b(X)\) cannot be generated by fewer than \(1+\dim(X)\) elements. The proofs, which are quite subtle and intrincate, are transparently written, in a very direct way, and involve the theory of real spectra and their tilde operators developed by \textit{J. Bochnak}, \textit{M. Coste} and \textit{M.-F. Roy} [``Géometrie algébrique réelle'' (Ergeb. Math. Vol. 12, Berlin 1987; Zbl 0633.14016; see also the English edition 1998; Zbl 0912.14023)] and \textit{C. Andradas}, \textit{L. Bröcker} and \textit{J. M. Ruiz} [``Constructible sets in real geometry'' (Ergeb. Math. Vol. 33, Berlin 1996; Zbl 0873.14044)]. Also, some approximation results appearing in a paper by \textit{M. Coste}, \textit{J. M. Ruiz} and \textit{M. Shiota} [Am. J. Math. 117, No. 4, 905-927 (1995; Zbl 0873.32007)] are necessary, not only to extend the results to the Nash category, but to reduce the analytic situation to the Nash or algebraic one, which was better known after the work of Buchner, Kucharz and Rusek, among others. Using the connection between holomorphy rings and sums of even powers, first observed by \textit{E. Becker} [in: Géometrie algébrique réelle et formes quadratiques, Journ. S. M. F., Univ. Rennes 1981, Lect. Notes Math. 959, 139-181 (1982; Zbl 0508.12020)], the authors obtain the following: If \(d=\dim(X)\), there exist meromorphic functions \(f_1,\dots,f_{d+1}\) on \(X\) such that for every positive integer \(n\), the sum \(f=f_1^{2n}+ \cdots+ f_{d+1}^{2n}\) is not a sum of less than \(d+1\) \(2n\)-th powers. the case of germs at compact sets is more involved since \(J_x\) can be generated by \(d\) elements but not by \(d-1\), and so the minimal number \(\mu\) of generators of the finitely generated ideals of the corresponding holomorphy ring is still unknown, although \(d\leq \mu\leq d+1\).