Logic Seminar
Karen Lange
University of Notre Dame
Lengths of generalized power series associated with real closed fields.
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.
seminar begins with tea
Tuesday October 27, 2009 at 4:00 PM in SEO 612