Karen Lange : Lengths of generalized power series associated with real closed fields.

Posted by , part of the Logic Seminar.

At

Oct. 27, 2009, 4 p.m.

In

SEO 612

Abstract

An integer part $I$ of a real closed field $R$ is a discrete ordered subring containing $1$ such that for all $r\in R$ there exists a unique $i\in I$ with $i\leq r < i+1$. Mourgues and Ressayre [1] showed that every real closed field $R$ has an integer part. Let $k$ be the residue field of $R$, and let $G$ be the value group of $R$. Let $k\langle\langle G\rangle\rangle$ be the set of generalized power series of the form $\Sigma_{g\in S}a_gg$ where $a_g\in k$ and the support of the power series $S\subseteq G$ is well ordered. Mourgues and Ressayre produce an integer part of $R$ by building an isomorphism between $R$ and a truncation closed subfield of $k\langle\langle G\rangle\rangle$. We refer to the image of $r\in R$ as its development. In order to understand the complexity of integer parts, we analyze an algorithmic version of the Mourgues and Ressayre construction. We consider the case where $R$ is countable, and we consider a list of the elements of a transcendence base for $R$ over $k$. Given such a list $\{r_1, r_2,\ldots \}$, the Mourgues and Ressayre construction becomes canonical. Let $R_n$ be the real closure of $k(r_1,\ldots,r_n)$. By a result of Shepherdson [2], the elements of $R_1$ have developments of length at most $\omega$. We show that elements of $R_n$ have developments of length at most $\omega^{\omega^{(n-1)}}$. Thus, the elements of $R$ have developments of length less than $\omega^{\omega^\omega}$. These bounds are sharp. This is joint work with Julia Knight. References: [1] M. H. Mourgues and J.-P. Ressayre, "Every real closed field has an integer part," J. Symb. Logic, vol. 58 (1993), pp. 641-647. [2] J. Shepherdson, "A non-standard model for the free variable fragment of number theory", Bulletin de l'Academie Polonaise Des Sciences, vol. 12(1964), pp. 79-86.

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