Answering a famous question of David Hilbert, it was shown by Davis, Putnam, Robinson
and Matiyasevich in 1970 that there can never be an algorithm which can decide whether or …

arXiv:2102.06941v3 [math.NT] 18 Oct 2021 Page 1 arXiv:2102.06941v3 [math.NT] 18 Oct 2021
EXISTENTIAL RANK AND ESSENTIAL DIMENSION OF DIOPHANTINE SETS NICOLAS …

Let K be a field. The étale open topology on the K-points V (K) of a K-variety V was
introduced in [23]. The étale open topology is non-discrete if and only if K is large. If K is …

Suppose that R is a local domain with fraction field K. If R is Henselian, then the R‐adic
topology over K refines the étale open topology. If R is regular, then the étale open topology …

We give a characterization, in terms of the residue field, of those henselian valuation rings
and those henselian valuation ideals that are diophantine. This characterization gives a …

Nondefinability of Rings of Integers in Most Algebraic Fields Page 1 Notre Dame Journal of
Formal Logic Volume 62, Number 3, 2021 Nondefinability of Rings of Integers in Most Algebraic …

Large fields (also called ample, anti-mordellic) generalize many fields of classical interest,
such as algebraically closed fields, real-closed fields, and $ p $-adic fields. In this note we …

Except for a limited number of cases, a complete classification of the Diophantine sets of
polynomial rings and fields of rational functions seems out of reach at present. We contribute …