Lengths of roots of polynomials over k((G))
Mourgues and Ressayre showed that any real closed field $R$ can be mapped isomorphically onto a truncation-closed subfield of the Hahn field $k((G))$, where $G$ is the natural value group of $R$ and $k$ is the residue field. If we fix a section of the residue field and a well ordering < of $R$, then the procedure of Mourgues and Ressayre yields a canonical section of $G$ and a unique embedding $d: R$ → $k((G))$. Julia Knight and I believed we had shown that for a real closed field $R$ with a well ordering < of type ω, the series in $d(R)$ have length less than ωωω, but we found a mistake in our proof. We needed a better understanding of what happens to lengths under root-taking. In this talk, we give a partial answer, which allows us to prove the original statement and generalize it. We make use of unpublished notes of Starchenko on the Newton-Puiseux method for taking roots of polynomials.
Karen Lange studies computable model theory, with a focus on the computational complexity of various algebraic structures, including graphs, free groups, and homogeneous models, and in particular on the complexity of structures associated with real closed fields. She received her doctorate from the University of Chicago in 2008, under the supervision of Robert Soare, and subsequently was awarded a National Science Foundation Postdoctoral Research Fellowship for study at Notre Dame University. Currently she is Assistant Professor of Mathematics at Wellesley College.