12 citations · 33 across the 23 of their papers we have counts for
27 papers
First-order definitions of rings of integral functions over algebraic extensions of function fields and undecidability
Alexandra Shlapentokh, Caleb Springer
In this paper, we study questions of definability and decidability for infinite algebraic extensions of and their subrings of -integral fun…
Computability in infinite Galois theory and algorithmically random algebraic fields
Wesley Calvert, Valentina Harizanov, Alexandra Shlapentokh
We introduce a notion of algorithmic randomness for algebraic fields. We prove the existence of a continuum of algebraic extensions of that are random according to our…
Defining using unit groups
Barry Mazur, Karl Rubin, Alexandra Shlapentokh
We consider first-order definability and decidability questions over rings of integers of algebraic extensions of $\Q$, paying attention to the uniformity of definitions. The unifo…
Existential definability and diophantine stability
Barry Mazur, Karl Rubin, Alexandra Shlapentokh
Let be a number field, let be an algebraic (possibly infinite degree) extension of , and let be their rings of integers. Suppose is an abelian…
First-order theory of a field and its Inverse Galois Problem
Francesca Balestrieri, Jennifer Park, Alexandra Shlapentokh
Let be a finite group. Then there exists a first-order statement in the language of rings without parameters and depending only on such that, for any field , we h…
On existential definitions of C.E. subsets of rings of functions of characteristic 0
Russell Miller, Alexandra Shlapentokh
We extend results of Denef, Zahidi, Demeyer and the second author to show the following. (1) Rational integers have a single-fold Diophantine definition over the ring of integral f…