o
“Defining using unit groups”, with Barry Mazur and Karl Rubin

o
“Hilbert’s Tenth Problem and the Inverse
Galois Problem”, with Francesca Baliestrieri and
Jennifer Park

o
“On
definitions of polynomials over function fields of positive characteristic”

·
“Existential definability and Diophantine
stability”, with Barry
Mazur and Karl Rubin,
Journal of Number Theory **254 **(2024), 1-64

·
“On existential definitions of C.E.
subsets of rings of functions of characteristic 0”, with Russell Miller,
Annals of Pure and Applied Logic, **173 **(2022**)**,
Paper No. 103076, 50 pp.

· Review
of "Hilbert's Tenth Problem" by M. Ram Murty
and Brandon Fodden, Notices of AMS, **68** (2021), no 4, 576-579

·
“Divisibility of Order and First Order
Definability and Decidability in Infinite Algebraic Extensions of Rational
Numbers”, Israel Journal of Math, 226 (2018), no. 2,
579–633.

· “A Computable Functor From Graphs to Fields”
with Russell Miller, Bjorn Poonen, Hans Schoutens, Journal
of Symbolic Logic, 83(1), March 2018, 326-348

· “On decidable algebraic fields” with Moshe
Jarden, Journal of Symbolic Logic, 82(2), June 2017, 474-488

· “As Easy
as ℚ:
Hilbert's Tenth Problem for Subrings of the Rationals
and Number Fields” with
Kirsten
Eisentraeger, Russell
Miller, Jennifer Park, Transactions
of the American Mathematical Society, 369(11),
November 2017, 8291–8315

· “Hilbert's Tenth
Problem over Function Fields of Positive Characteristic Not Containing the
Algebraic Closure of a Finite Field” with Kirsten Eisentraeger , Journal of the European Math
Society, 19(7), 2017, 2103-2138

· “Turing degrees of isomorphism types of
geometric objects” with Wesley Calvert and
Valentina Harizanov,
Computability, Volume 3 (2), pages 105-134, September 2014

· “Definability and Decidability in Infinite
Algebraic Extensions” with Carlos Videla,
Annals of Pure and Applied Logic, 165 (2014), no. 7-8, 1243–1262

· “Categoricity Properties for Computable
Algebraic Fields” with Denis
Hirschfeldt, Ken Kramer, and Russell Miller, Transactions of
AMS, published electronically 10/20/14

·
“Computable
Categoricity for Algebraic Fields with Splitting Algorithms” with Russell Miller, Transactions of
AMS, published electronically 10/20/14

·
“Using
Indices of Points on an Elliptic Curve to Construct A Diophantine Model of $\Z$
and Define $\Z$ Using One Universal Quantifier in Very Large Subrings of
Number Fields, Including $\Q$”, International Journal of Number Theory,
Volume No.08, Issue No. 6, (2012) 1335–1365

·
“Hilbert's
Tenth Problem and Mazur's Conjectures in Complementary Subrings of Number
Fields” (with Kirsten Eisentraeger
and Graham Everest), Math. Research Letters, 18 (2011), no. 06,
1141–1162

·
“The
analogue of Büchi's problem for function fields”
(with Xavier Vidaux),
Journal of Algebra,
Volume
330, Issue 1, 15 March 2011, Pages 482-506

·
“Defining
Integers”, Bulletin of Symbolic
Logic, Volume 17, Number 2, June 2011, page 230-251.

·
“Rational
Inseparability of Integral Closure”, Journal of Number Theory, **Volume 129, Issue 10, Pages 2227-2646 (October 2009)**** **

·
“Undecidability in function fields of positive characteristic” (with Kirsten Eisentraeger), IMRN, 2009, no. 21,
4051--4086. Click here for the
link.

·
“Diophantine
Undecidability of Holomorphy Rings of Function Fields
of Characteristic 0” ( with Laurent Moret-Bailly),
les Annales de l'Institut
Fourier, volume 55, number 5, (2009), p.
2103-2118

·
“Rings of Algebraic
Numbers in Infinite Extensions of $\Q$ and Elliptic Curves Retaining Their Rank”,
the Proceedings of the Conference on Model
Theory and Computable Model Theory, University of Florida, 2007, Archive for Mathematical Logic, vol.
48, 2009, 77-114

·
“Defining the
integers in large rings of number fields using one universal quantifier”, Zapiski Nauchnykh Seminarov POMI, volume 358, 2008, p.
199 - 223 (with Gunther Cornelissen)

·
Book:
Hilbert’s
Tenth Problem: Diophantine Classes and Other Extensions to Global Fields,
Cambridge University Press, 2006. To see a preliminary draft, please click on
the following link: linked
pdf file of the book. All reports of errors
are very much appreciated.

·
“Diophantine
Definability and Decidability in Extensions of Degree 2 of Totally Real
Fields”, Journal of Algebra,
volume 313(2), 2007, pages 846-896. Click here for a preliminary version.

·
“Turing
Degrees of Isomorphism Types of Algebraic Objects” (with
Wesley Calvert and Valentina Harizanov),
Journal of the London Mathematical Society, 75
(2), 2007, pages 273-286

·
“Elliptic Curves Retaining Their Rank in Finite Extensions and
Hilbert’s Tenth Problem” , Transactions of AMS, volume 360 (7), 2008

·
“First Order
Definitions of Rational Functions and $S$-integers over Holomorphy
Rings of Algebraic Functions of Characteristic 0”, **Annals of Pure and Applied Logic, pages** 267-283,
Volume 136, Issue 3, November 2005 (Click
here for a preliminary version)

·
“Diophantine
Definability of Infinite Discrete non-Archimedean Sets and Diophantine Models
for Large Subrings of Number Fields” (with Bjorn Poonen),
Journal für die Reine und Angewandte Mathematik,
Volume 588, pages 27-47

·
“Diophantine
undecidability for some function fields of infinite transcendence degree and
positive characteristic”, (in
Russian) in Zapiski Nauchnykh Seminarov POMI, volume 304, (2003),
pages 141-167. (Click here for the Russian
version) English translation of the paper can be
found in Journal of Mathematical
Sciences, Volume 130, Number 2, pages 4631 – 4642. (Click here for the link to the
English version)

·
“On Diophantine
definability and decidability in some infinite totally real extensions of Q”, Transactions of AMS, volume 356, number 8,
3189-3207. Click here for the link to the
article

·
“Existential Definability
with Bounds on Archimedean Valuations”, Journal
of Symbolic Logic, volume 68, Issue 3, September 2003, 860-878. Click
here for the article

·
“A Ring Version of
Mazur’s Conjecture on Topology of Rational Points”, IMRN, 2003:7, 2003, pages 411-423. Click
here for the link to the article.

·
“On Diophantine
definability and decidability in large subrings of totally real number fields
and their totally complex extensions of degree 2”, Journal of Number Theory (95), 2002, pages 227-252. Click
here for the link to the article.

·
“Diophantine
undecidability of function fields of characteristic greater than 2 finitely
generated over a field algebraic over a finite field”, Compositio Mathematica, volume 132(1), May
2002, 99-120. Click
here for a link to the article.

·
“Generalized Weak Presentations”,
Journal of Symbolic Logic, volume
67, number 2, 2002, pages 787-819

·
“On Diophantine
decidability and definability in some rings of algebraic functions of
characteristic 0”, Journal of Symbolic
Logic, volume 67, number 2, 2002, pages 759-786

·
“Defining Integrality
at Prime Sets of High Density over Function Fields”, Monatshefte fuer Mathematik, vol 135
(2002), 59-67

·
“Hilbert’s Tenth
Problem over Number Fields, a Survey”, **Proceedings of Workshop on
Hilbert’s** **Tenth Problem: Relation with Arithmetic and
Algebraic Geometry**, **Ghent, Belgium, November 1999.**** ****Contemporary Mathematics, volume
270, pages 107-137.**

·
“Diophantine
Definability over Non-finitely Generated Non-degenerate Modules of Algebraic
Extensions of **Q**”,
**Archive for Mathematical Logic, volume 40, number 4,
(2001), pages 297-328. ****Click
here for the link to the article.**

·
“Towards Hilbert’s
Tenth Problem for Algebraic Number Fields”, **Proceedings of
joint AMS-IMS-SIAM Summer Conference on Computability Theory and Applications,
June 1999, Boulder, Co, Contemporary Mathematics**, Volume 257,
2000, AMS, Providence, RI

·
“Defining Integrality
at Prime Sets of High Density in Number Fields”, **Duke Mathematical Journal, **Volume 101, Number 1, 2000,
117-134** **

·
“Hilbert’s Tenth
Problem for Algebraic Function Fields over Infinite Fields of Constants of
Positive Characteristic”, **Pacific Journal of Mathematics**,
Volume 193, Number 2, 2000, pages 463-500

·
“Weak Presentations of
Fields Not Extendible to Recursive Presentations”, **Recursion Theory and Complexity, Proceedings of the Kazan 97 Workshop**,
Kazan, July 1997, Walter de Gruyter, New York, 1999, pages 131-156.

·
“Weak Presentations of
Non-finitely Generated Fields”, **Annals of Pure and Applied Logic,
**Volume 94, (1998), 223-252**.**

·
“Diophantine
Definability Over Holomorphy Rings of Algebraic
Function Fields with Infinite Number of Primes allowed as Poles”, **International Journal of Mathematics, **Volume 9, No. 8,
(1998), 1041-1066**.**

·
“Holomorphy Modules
of Number Fields and Function Fields”,with
Heather Ries, **Algebra Colloquium**,
vol. 5(2), 1998, 219-240.

·
“Diophantine
Definability over Some Rings of Algebraic Numbers with Infinite Number of
Primes Allowed in the Denominator”, **Inventiones**** Mathematicae,
**vol 129(3), 1997, 489-507*.*

·
“The Logic of Pseudo
S-integers”, **Israel Journal of Mathematics, **vol
101, (1997), 229-254**.**

·
“Polynomials with a Given
Discriminant over Fields of Functions of Positive Characteristic”, **Pacific Journal of Mathematics**, volume 173, Number 2,
April 1996.

·
“Algebraic and Turing
Separability of Rings”, **Journal of Algebra**,
volume 185, 229-257.

·
“Diophantine
Undecidability over Algebraic Function Fields over Finite Fields of Constants”,
**Journal of Number Theory**, volume 58,
Number 2, June 1996, 317-342.

·
“Rational Separability
over a Global Field”, **Annals of Pure and Applied Logic ,** vol 79,
1996, p. 93-108.

·
“Non-Standard Extensions
of Weak Presentations”, **Journal of Algebra**,
vol 176, 1995, p. 735-749

·
“Weak Presentations of
Computable Fields”, **Journal of Symbolic Logic**,
Volume 60, Number 1, March 1995, pp. 199-208 (with Carl Jockusch).

·
“Diophantine Classes
of Holomorphy Rings of Global Fields”, **Journal of Algebra**, 169(1), October 1994, pp. 139-175.

·
“Diophantine
Equivalence and Countable Rings”, **Journal of Symbolic Logic**,
59(3), September 1994, pp. 1068-1095.

·
“Diophantine
Undecidability for Some Holomorphy Rings of Algebraic
Functions of Characteristic O”, **Communications in Algebra**,
22(11), August 1994, pp. 4379-4404.

·
“Diophantine
Undecidability in Some Rings of Algebraic Numbers of Totally Real Infinite
Extensions of Q”, **Annals of Pure and Applied Logic**,
68 (1994), pp. 299-325.

·
“Diophantine Relations
Between the Rings of S-integers of Fields of Algebraic Functions in One
Variable Over Constant Fields of Positive Characteristic”, **Journal of Symbolic Logic**, 58(1), March 1993, pp.
158-192.

·
“A Diophantine
Definition of Rational Integers in Some Rings of Algebraic Numbers”, **Notre Dame Journal of Formal Logic**, 33(3), Summer 1992,
pp. 299-321.

·
“Hilbert’s Tenth
Problem for Rings of Algebraic Functions of Positive Characteristic”, **Transactions of American Mathematical Society**, 333(1)
September 1992, pp. 275-298.

·
“Hilbert’s Tenth
Problem for Rings of Algebraic Functions of Characteristic Zero”, **Journal of Number Theory**, 40(2), (February 1992), pp.
218-236.

·
“Diophantine
Definitions for Rings of Rational Numbers”, **Communications
on Pure and Applied Mathematics**, Vol. XLIV, (1991), pp.
853-867.

·
“Diophantine
Definitions for Some Polynomial Rings”, **Communications on Pure and Applied
Mathematics**, Vol. XLIII, (1990), pp. 1055-1066.

·
“Diophantine Relations
Between Algebraic Number Fields”, (with Harold Shapiro), **Communications on Pure and Applied Mathematics**, Vol. XLII
(1989), pp. 1113- 1122.

·
“Extension of
Hilbert’s Tenth Problem to Some Algebraic Number Fields”, **Communications on Pure and Applied Mathematics**, Vol. XLII
(1989), pp. 939-962.

This page was last modified on 9/4/23.