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Register a new userA valuation theoretic characterization of recursively saturated real closed fields
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We give a valuation theoretic characterization for a real closed field to be recursively saturated. Our result extends the characterization of Harnik and Ressayre cite{hr} for a divisible ordered abelian group to be recursively saturated.
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In an extended abstract Ressayre considered real closed exponential fields and integer parts that respect the exponential function. He outlined a proof that every real closed exponential field has an exponential integer part. In the present paper, we give a detailed account of Ressayres construction, which becomes canonical once we fix the real closed exponential field, a residue field section, and a well ordering of the field. The procedure is constructible over these objects; each step looks effective, but may require many steps. We produce an example of an exponential field $R$ with a residue field $k$ and a well ordering $<$ such that $D^c(R)$ is low and $k$ and $<$ are $Delta^0_3$, and Ressayres construction cannot be completed in $L_{omega_1^{CK}}$.
DAquino, Knight and Starchenko classified the countable real closed fields with integer parts that are nonstandard models of Peano Arithmetic. We rule out some possibilities for extending their results to the uncountable and study real closures of $omega_1$-like models of PA.
We prove that every non-trivial valuation on an infinite superrosy field of positive characteristic has divisible value group and algebraically closed residue field. In fact, we prove the following more general result. Let $K$ be a field such that for every finite extension $L$ of $K$ and for every natural number $n>0$ the index $[L^*:(L^*)^n]$ is finite and, if $char(K)=p>0$ and $f: L to L$ is given by $f(x)=x^p-x$, the index $[L^+:f[L]]$ is also finite. Then either there is a non-trivial definable valuation on $K$, or every non-trivial valuation on $K$ has divisible value group and, if $char(K)>0$, it has algebraically closed residue field. In the zero characteristic case, we get some partial results of this kind. We also notice that minimal fields have the property that every non-trivial valuation has divisible value group and algebraically closed residue field.
Alternating quantifier depth is a natural measure of difficulty required to express first order logical sentences. We define a sequence of first order properties on rooted, locally finite trees in a recursive manner, and provide rigorous arguments for finding the alternating quantifier depth of each property in the sequence, using Ehrenfeucht-Fra{i}ss{e} games.
We consider valued fields with a distinguished contractive map as valued modules over the Ore ring of difference operators. We prove quantifier elimination for separably closed valued fields with the Frobenius map, in the pure module language augmented with functions yielding components for a p-basis and a chain of subgroups indexed by the valuation group.
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