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A classical tool in the study of real closed fields are the fields $K((G))$ of generalized power series (i.e., formal sums with well-ordered support) with coefficients in a field $K$ of characteristic 0 and exponents in an ordered abelian group $G$. In this paper we enlarge the family of ordinals $\alpha$ of non-additively principal Cantor degree for which $K((\mathbb{R}^{\le 0}))$ admits irreducibles of order type $\alpha$ far beyond $\alpha=\omega^2 $ and $\alpha = \omega^3$ known prior to this work.

This paper generalizes Shelah's generic pair conjecture (now theorem) for the measurable cardinal case from first order theories to finite diagrams. We use homogeneous models in the place of saturated models., Comment: Accepted for publication in the Israel Journal of Mathematics

The purpose of this paper is to give a characterization for polynomials and rational functions which admit only non-negative values on definable sets in real closed valued fields. That is, generalizing the relative positivstellens\"atze for sets defined also by valuation terms. For this, we use model theoretic tools, together with existence of canonical valuations., Comment: Accepted for publication in Proceedings of the American Mathematical Society

Let $\langle K,\nu \rangle$ be a real closed valued field, and let $S\subseteq K^n$ be an open semi-algebraic set. Using tools from model theory, we find an algebraic characterization of rational functions which admit, on $S$, only values in the valuation ring. We use this result to deduce a criterion for a rational function to be bounded on an open semi algebraic subset of some irreducible variety over a real closed field or over an ordered field which is dense in its real closure., Comment: Accepted to publication in Communications in Algebra