arXiv subjects

Victor Lisinski

Publications and source records attributed to Victor Lisinski.

2 recordsLinked to original sources

arXiv2023

Decidability of positive characteristic tame Hahn fields in $\mathcal{L}_t$

We show that any positive characteristic tame Hahn field $\mathbb{F}((t^Γ))$ containing $t$ is decidable in $\mathcal{L}_t$, the language of valued fields with a constant symbol for $t$, if $\mathbb{F}$ and $Γ$ are decidable. In particular, we obtain decidability of $\mathbb{F}_p((t^{1/p^{\infty}}))$ and $\mathbb{F}_p((t^{\mathbb{Q}}))$ in $\mathcal{L}_t$. This uses a new AKE-principle for equal characteristic tame fields in $\mathcal{L}_t$, building on work by Kuhlmann, together with Kedlaya's work on finite automata and algebraic extensions of function fields. In the process, we obtain an AKE-principle for tame fields in mixed characteristic.

math.LO↗

arXiv2023

Approximation and algebraicity in positive characteristic Hahn fields

We study the relative algebraic closure $K$ of $\bar{\mathbb{F}}_p((t))$ inside $\bar{\mathbb{F}}((t^{\mathbb{Q}}))$. We show that the supports of elements in $K$ have order type strictly less than $ω^ω$. We also recover a theorem by Rayner giving a bound to the ramification away from $p$ in the support of elements in $K$, and an analogue of Rayner's result for the residue field. This work has applications to the decidability of the first order theory of $\mathbb{F}_p((t^{\mathbb{Q}}))$, and other tame fields, in the language of valued fields with a constant symbol for $t$.

math.NT↗