April 12
Hans Schoutens,
CUNY
Geometric tools for the decidability of the existential theory of $F_p[[t]]$
I will give a brief survey how tools from algebraic geometry can be used in finding solutions to Diophantine equations over $F_p[[t]]$ and similar rings. These tools include Artin approximation, arc spaces, motives and resolution of singularities. This approach yields the definability of the existential theory of $F_p[[t]]$ (in the ring language with a constant for $t$) contingent upon the validity of resolution of singularities (Denef-Schoutens). Anscombe-Fehm proved a weaker result using model-theoretic tools and together with Dittmann, they gave a proof assuming only the weaker 'local uniformization conjecture.'