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Javier Utreras

Publications and source records attributed to Javier Utreras.

4 recordsLinked to original sources
arXiv2024

Undecidability of infinite towers of Kummer extensions of $\mathbb{F}_p(t)$

We prove, assuming resolution of singularities in positive characteristic, an analogue of Siegel's theorem on sum of squares in positive characteristic. The method of proof combines techniques from central simple algebras with model theory and builds on work of Anscombe, Dittmann and Fehm. As an application, we show that, for each finite field $\mathbb{F}$ of odd characteristic and any positive integer $n$ coprime with the characteristic of $\mathbb{F}$, the first-order theory of the field given by the compositum of the fields generated by adjoining the $n$--th roots of all monic irreducible polynomials in $\mathbb{F}[t]$, of degree divisible by $n$ is undecidable in the language of rings with the variable $t$ as a constant.

math.LO↗
arXiv2024

Undecidability of infinite algebraic extensions of $\mathbb{F}_p(t)$

Building on work of J. Robinson and A. Shlapentokh, we develop a general framework to obtain definability and decidability results of large classes of infinite algebraic extensions of $\mathbb{F}_p(t)$. As an application, we show that for every odd rational prime $p$ there exist infinitely many primes $r$ such that the fields $\mathbb{F}_{p^a}\left(t^{r^{-\infty}}\right)$ have undecidable first-order theory in the language of rings without parameters. Our method uses character theory to construct families of non-isotrivial elliptic curves whose Mordell-Weil group is finitely generated and of positive rank in $\mathbb{Z}_r$-towers.

math.LO↗