Generating ideals in real holomorphy rings (Q1178871)

Let \(R\) be a real closed field and let \(F\) be a formally real function field over \(R\) of transcendence degree \(n\). The real holomorphy ring \(H(F\mid R)\) of \(F\) over \(R\) is defined to be the intersection of all valuation rings of \(F\) which contain \(R\) and have formally real residue fields. The main theorem proved in this paper is: Let \(R\), \(F\) and \(n\) be as above. Then there exists a finitely generated ideal of \(H(F\mid R)\) which cannot be generated by \(n\) elements. In the special case, when \(R=\mathbb{R}\), these results had been proved by the author in an earlier paper [J. Reine Angew. Math. 395, 171-185 (1989; Zbl 0653.13001)].