Ordered fields dense in their real closure and definable convex valuations

In this paper, we undertake a systematic model- and valuation-theoretic study of the class of ordered fields which are dense in their real closure. We apply this study to determine definable henselian valuations on ordered fields, in the language of ordered rings. In light of our results, we re-examine the Shelah–Hasson Conjecture (specialized to ordered fields) and provide an example limiting its valuation-theoretic conclusions.

Теорема Артина для f-колец

The main result states that each positive polynomial p in N variables with coefficients in a unital Archimedean f-ring K is representable as a sum of squares of rational functions over the complete ring of quotients of K provided that p is positive on the real closure of K. This is proved by means of Boolean valued interpretation of Artin's famous theorem which answers Hilbert's 17th problem affirmatively.

Use of Optical Methods for Dimensional Analysis of Multi-Part Moulds

The paper deals with the possibilities of using non-contact 3D measurement systems for the digital capture and subsequent dimensional and shape analysis of multi-part forms. The paper illustrates a practical method of the 3D digitization of two-part moulds using a combination of TRITOP and ATOS optical systems. It introduces dimensional control of the cavity of a mould during its real closure and compares this method to traditional methods of measurement. The analyses carried out show the undeniable benefits of these innovative approaches in practice, because the appropriate use of optical methods often enables the detection of the errors of products and tools which would be impossible to detect using traditional approaches.

Real closed fields and models of Peano arithmetic

Shepherdson [14] showed that for a discrete ordered ring I, I is a model of I Open iff I is an integer part of a real closed ordered field. In this paper, we consider integer parts satisfying PA. We show that if a real closed ordered field R has an integer part I that is a nonstandard model of PA (or even IΣ4), then R must be recursively saturated. In particular, the real closure of I, RC (I), is recursively saturated. We also show that if R is a countable recursively saturated real closed ordered field, then there is an integer part I such that R = RC(I) and I is a nonstandard model of PA.

Plongement dense d'un corps ordonné dans sa clôture réelle

We study the structures (K ⊂ Kr), where K is an ordered field and Kr its real closure, in the language of ordered fields with an additional unary predicate for the subfield K. Two such structures (K ⊂ Kr) and (L ⊂ Lr) are not necessarily elementary equivalent when K and L are. But with some saturation assumption on K and L, then the two structures become equivalent, and we give a description of the complete theory.