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arXiv:2411.14960 (Published 2024-11-22)
First-order definitions of rings of integral functions over algebraic extensions of function fields and undecidability
Comments: 40 pagesIn this paper, we study questions of definability and decidability for infinite algebraic extensions ${\bf K}$ of $\mathbb{F}_p(t)$ and their subrings of $\mathcal{S}$-integral functions. We focus on fields ${\bf K}$ satisfying a local property which we call $q$-boundedness. This can be considered a function field analogue of prior work of the first author which considered algebraic extensions of $\mathbb{Q}$. One simple consequence of our work states that if ${\bf K}$ is a $q$-bounded Galois extension of $\mathbb{F}_p(t)$, then the integral closure $\mathcal{O}_{\bf K}$ of $\mathbb{F}_p[t]$ inside ${\bf K}$ is first-order definable in ${\bf K}$. Under the additional assumption that the constant subfield of ${\bf K}$ is infinite, it follows that both $\mathcal{O}_{\bf K}$ and ${\bf K}$ have undecidable first-order theories. Our primary tools are norm equations and the Hasse Norm Principle, in the spirit of Rumely. Our paper has an intersection with a recent arXiv preprint by Mart\'inez-Ranero, Salcedo, and Utreras, although our definability results are more extensive and undecidability results are much stronger.
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arXiv:1601.07829 (Published 2016-01-28)
Not having a Root in Number Fields is Diophantine
Comments: 8 pagesGiven a number field $K$ and a positive integer $n$, we present a diophantine criterion for a polynomial in one variable of degree $n$ over $K$ not to have any root in $K$. This strengthens the known result that the set of non-$n$-th-powers in $K$ is diophantine. Our approach is based on a generalisation of the quaternion method used by Poonen and Koenigsmann for first-order definitions of $\mathbb{Z}$ in $\mathbb{Q}$.
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arXiv:math/0602541 (Published 2006-02-24)
First-order definitions in function fields over anti-Mordellic fields
Comments: 12 pagesA field k is called anti-Mordellic if every smooth curve over k with a k-point has infinitely many k-points. We prove that for a function field over an anti-Mordellic field, the subfield of constants is defined by a certain universal first order formula. Under additional hypotheses regarding 2-cohomological dimension we prove that algebraic dependence of an n-tuple of elements in such a function field can be described by a first order formula, for each n. We also give a result that lets one distinguish various classes of fields using first order sentences.