2024/11/22 by Shlapentokh, Alexandra, Springer, Caleb
#11U05 (Primary) 12L05 #11U09 (Secondary) #FOS: Mathematics #Logic (math.LO) #Number Theory (math.NT)
paper · doi:10.48550/arxiv.2411.14960
In this paper, we study questions of definability and decidability for infinite algebraic extensions \bf K of \mathbbFp(t) and their subrings of 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 ℚ. One simple consequence of our work states that if \bf K is a q-bounded Galois extension of \mathbbFp(t), then for infinitely many non-constant u the integral closure O\bf K of \mathbbFp[u] 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 O\bf K and \bf K have undecidable first-order theories, and that \mathbbFp[w] is definable in \bf K for every non-constant w in \bf K. 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 Martínez-Ranero, Salcedo, and Utreras, although our definability results are more extensive and undecidability results are much stronger.