The Decidability of the Existential Theory of 𝔽_{p}[[t]] and Resolution of Singularities

Hans Schoutens

I will present an unpublished proof due to Jan Denef, showing how the decidability of the existential theory of 𝔽_{p}[[t]], follows from Resolution of Singularities in characteristic p. Here t is a single variable and 𝔽_{p} the p-element field. For the positive existential theory (i.e., we only have equations), this is an immediate consequence of a Theorem of Greenberg: if one can find an approximate solution of order N, then a true solution exists, where N depends only on the degree of the equations involved. For the general case one uses the fact that, if a variety V defined over 𝔽_{p}((t)) has a non-singular 𝔽_{p}((t))-rational point, then the 𝔽_{p}((t))-rational points on V are Zariski dense at that point. If V is a projective variety, then any 𝔽_{p}((t))-rational point is also 𝔽_{p}[[t]]-rational, by the Valuative Criterion for Properness. This settles the problem for a non-singular system of equations and inequalities. Finally, one reduces to the non-singular case using Resolution of Singularities.

Address: Department of Mathematics, Hill Center–Bush Campus, Rutgers University, Piscataway, NJ 08855 (USA)

E-mail: hschoute@math.rutgers.edu