We establish that all rings of $ S $-integers are universally definable in function fields in one
variable over certain ground fields including global and non-archimedean local fields. That …

For a given number field $ K $, we give a $\forall\exists\forall $-first order description of affine
Darmon points over $\mathbb {P}^ 1_K $, and show that this can be improved to a …

We offer a $\forall\exists $-definition for (affine) Campana points over $\mathbb {P}^ 1_K
$(where $ K $ is a number field), which constitute a set-theoretical filtration between $ K …

In this paper, the author first reviews the history of Hilbert's Tenth Problem, and then study
mixed quantifier prefixes over Diophantine equations with integer variables. For example …

In 1970, building on work of Davis, Putnam, and Robinson, Matiyasevich established a
complete classification of subsets of the integers which can be described by an existential …

In this paper we first review the history of Hilbert's Tenth Problem, and then study mixed
quantifier prefixes over Diophantine equations with integer variables. For example, we prove …

In this paper we first review the history of Hilbert's Tenth Problem, and then study mixed
quantifier prefixes over Diophantine equations with integer variables. For example, we prove …