Places, cuts and orderings of function fields

Abstract

Abstract. In this paper we investigate the space of R-places of an algebraic function field of one variable. We deal with the problem of determining when two orderings of such a field correspond to a single R-place. To this end we introduce and study the space of cuts on a real curve and prove that the space is homeomorphic to the space of orderings. Finally, we prove that two cuts (consequently, two orderings) correspond to a single R-place if they are induced by a single ultrametric ball.

Fatigue design of selected details in steel bridges

“…The following result is also proved in [35]: In the paper [26] P. Koprowski and K. Kuhlmann consider the more general case of an algebraic function field F of transcendence degree 1 over a real closed field R.…”

“…In this case B determines a cut (always more then one) on the curve. In [26] it is shown that such a cut is a ball cut, and the following theorem is proved:…”

Selected methods for the classification of cuts, and their applications

“…In the paper [28] P. Koprowski and K. Kuhlmann consider the more general case of an algebraic function field F of transcendence degree 1 over a real closed field R. Choose any smooth projective model of F , i.e., a smooth, projective algebraic curve over R with function field F . In [26,27] M. Knebusch shows that the curve consists of finitely many semialgebraic connected components, each of which can be endowed with a cyclic order.…”

“…In [26,27] M. Knebusch shows that the curve consists of finitely many semialgebraic connected components, each of which can be endowed with a cyclic order. In [28] this is used to define cuts in these components; the collection of all of them is taken to be the set of cuts on the curve. The following result is proved: By the Baer-Krull Theorem, in the first case there is no other ordering on F that induces the same place as the given one.…”

“…Recently, ball cuts have been used to study these spaces (cf. [37,35,36,28]). We describe some results in Section 3.2.…”