**October 25**

**Hans Schoutens**,
CUNY

**Computing away negation using ancients: from existential to Diophantine sentences**

Last semester, I discussed geometric methods for decidability over a complete discrete valuation ring (DVR) in equal characteristic, suggesting that these methods could be applied effectively. In this talk, I aim to clarify the computability issues surrounding this topic while at the same time shifting focus to the case of mixed characteristic. Whereas quantifier elimination (QE) results are established for p-adic numbers, the general landscape remains less explored. I will demonstrate that for any existential sentence over a computable ring, we can effectively construct a positive existential (or Diophantine) sentence which is logically equivalent to the original in every excellent Henselian DVR containing the ring. This construction hinges on Resolution of Singularities, which is feasible in characteristic zero.

Furthermore, I will utilize ultraproducts, specifically the protoproduct variant, to show how Diophantine statements over a DVR can be reduced to those over a residue ring. Since the residue ring is Artinian—and in the case of p-adics, even finite—the associated problems become significantly more manageable. However, it is important to note that this approach does not yet yield a general QE result, as it applies only to sentences, not formulas. The challenge lies in the dependence of certain effective bounds on parameters. I will provide insights into how to derive a bound based on a refined notion of complexity within the equational system—beyond simply considering its degree—using ultraproducts. Additionally, I will address a request from the audience in my last talk by demonstrating that this bound is indeed effective.

And somehow it will also require some delving into the theory of Witt vectors and ancient elements, as I will explain.