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Alexandra Shlapentokh

Publications and source records attributed to Alexandra Shlapentokh.

28 records · Page 2Linked to original sources
arXiv2011

Definability and Decidability in Infinite Algebraic Extensions

We use a generalization of a construction by Ziegler to show that for any field $F$ and any countable collection of countable subsets $A_i \subseteq F, i \in \calI \subset \Z_{>0}$ there exist infinitely many fields $K$ of arbitrary positive transcendence degree over $F$ and of infinite algebraic degree such that each $A_i$ is first-order definable over $K$. We also use the construction to show that many infinitely axiomatizable theories of fields which are not compatible with the theory of algebraically closed fields are finitely hereditarily undecidable.

math.LO↗
arXiv2010

Defining Integers

We survey the recent developments in the area that grew out of attempts to extend an analog of Hilbert's Tenth Problem to the field of rational numbers and rings of integers of number fields. The paper is based on a plenary talk the author gave at the North American Annual ASL meeting at the University of Notre Dame in May of 2009.

math.LO↗
arXiv2010

The analogue of Büchi's problem for function fields

Büchi's $n$ Squares Problem asks for an integer $M$ such that any sequence $(x_0,...,x_{M-1})$, whose second difference of squares is the constant sequence $(2)$ (i.e. $x^2_n-2x^2_{n-1}+x_{n-2}^2=2$ for all $n$), satisfies $x_n^2=(x+n)^2$ for some integer $x$. Hensley's problem for $r$-th powers (where $r$ is an integer $\geq2$) is a generalization of Büchi's problem asking for an integer $M$ such that, given integers $ν$ and $a$, the quantity $(ν+n)^r-a$ cannot be an $r$-th power for $M$ or more values of the integer $n$, unless $a=0$. The analogues of these problems for rings of functions consider only sequences with at least one non-constant term. Let $K$ be a function field of a curve of genus $g$. We prove that Hensley's problem for $r$-th powers has a positive answer for any $r$ if $K$ has characteristic zero, improving results by Pasten and Vojta. In positive characteristic $p$ we obtain a weaker result, but which is enough to prove that Büchi's problem has a positive answer if $p\geq 312g+169$ (improving results by Pheidas and the second author).

math.NT↗
arXiv2009

Diophantine Undecidability of Holomorphy Rings of Function Fields of Characteristic 0

Let $K$ be a one-variable function field over a field of constants of characteristic 0. Let $R$ be a holomorphy subring of $K$, not equal to $K$. We prove the following undecidability results for $R$: If $K$ is recursive, then Hilbert's Tenth Problem is undecidable in $R$. In general, there exist $x_1,...,x_n \in R$ such that there is no algorithm to tell whether a polynomial equation with coefficients in $\Q(x_1,...,x_n)$ has solutions in $R$.

math.LO↗
arXiv2008

Undecidability in function fields of positive characteristic

We prove that the first-order theory of any function field K of characteristic p>2 is undecidable in the language of rings without parameters. When K is a function field in one variable whose constant field is algebraic over a finite field, we can also prove undecidability in characteristic 2. The proof uses a result by Moret-Bailly about ranks of elliptic curves over function fields.

math.NT↗
arXiv2008

Defining the integers in large rings of number fields using one universal quantifier

Julia Robinson has given a first-order definition of the rational integers $\mathbb Z$ in the rational numbers $\mathbb Q$ by a formula $(\forall \exists \forall \exists)(F=0)$ where the $\forall$-quantifiers run over a total of 8 variables, and where F is a polynomial. We show that for a large class of number fields, not including $\mathbb Q$, for every $ε>0$, there exists a set of primes $\cal S$ of natural density exceeding $1-ε$, such that $\mathbb Z$ can be defined as a subset of the ``large'' subring $$\{x \in K : \ord_{\mathfrak p}x >0, \forall \mathfrak p \not \in \cal S \}$$ of K by a formula of the form $(\exists \forall \exists)(F=0)$ where there is only one $\forall$-quantifier, and where F is a polynomial.

math.LO↗
arXiv2006

Diophantine Definability and Decidability in the Extensions of Degree 2 of Totally Real Fields

We investigate Diophantine definability and decidability over some subrings of algebraic numbers contained in quadratic extensions of totally real algebraic extensions of $\mathbb Q$. Among other results we prove the following. The big subring definability and undecidability results previously shown by the author to hold over totally complex extensions of degree 2 of totally real number fields, are shown to hold for {\it all} extensions of degree 2 of totally real number fields. The definability and undecidability results for integral closures of ``small'' and ``big'' subrings of number fields in the infinite algebraic extensions of $\mathbb Q$, previously shown by the author to hold for totally real fields, are extended to a large class of extensions of degree 2 of totally real fields. This class includes infinite cyclotomics and abelian extensions with finitely many ramified rational primes.

math.NT↗
arXiv2005

Turing Degrees of Isomorphism Types of Algebraic Objects

The Turing degree spectrum of a countable structure $\mathcal{A}$ is the set of all Turing degrees of isomorphic copies of $\mathcal{A}$. The Turing degree of the isomorphism type of $\mathcal{A}$, if it exists, is the least Turing degree in its degree spectrum. We show there are countable fields, rings, and torsion-free abelian groups of arbitrary rank, whose isomorphism types have arbitrary Turing degrees. We also show that there are structures in each of these classes whose isomorphism types do not have Turing degrees.

math.LO↗
arXiv2004

Diophantine definability of infinite discrete non-archimedean sets and Diophantine models over large subrings of number fields

We prove that infinite p-adically discrete sets have Diophantine definitions in large subrings of some number fields. First, if K is a totally real number field or a totally complex degree-2 extension of a totally real number field, then there exists a prime p of K and a set of K-primes S of density arbitrarily close to 1 such that there is an infinite p-adically discrete set that is Diophantine over the ring O_{K,S} of S-integers in K. Second, if K is a number field over which there exists an elliptic curve of rank 1, then there exists a set of K-primes S of density 1 and an infinite Diophantine subset of O_{K,S} that is v-adically discrete for every place v of K. Third, if K is a number field over which there exists an elliptic curve of rank 1, then there exists a set of K-primes S of density 1 such that there exists a Diophantine model of Z over O_{K,S}. This line of research is motivated by a question of Mazur concerning the distribution of rational points on varieties in a non-archimedean topology and questions concerning extensions of Hilbert's Tenth Problem to subrings of number fields.

math.NT↗