Preprints

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”

Publications

· “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.