The number of solutions of a diophantine equation over a recursive ring
Author:
Christoph Baxa
Journal:
Proc. Amer. Math. Soc. 141 (2013), 41754178
MSC (2010):
Primary 11U05; Secondary 03B25, 11D45
Published electronically:
August 20, 2013
MathSciNet review:
3105860
Fulltext PDF
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Abstract: Let be a recursive ring whose quotient field is not algebraically closed with the property that Hilbert's Tenth Problem over is undecidable, and let be a nonempty proper subset of . We prove that it is not decidable whether the number of solutions of a diophantine equation with coefficients in is in .
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 [1]
 Bjorn Poonen, Hilbert's tenth problem and Mazur's conjecture for large subrings of , J. Amer. Math. Soc. 16 (2003), no. 4, 981990 (electronic). MR 1992832 (2004f:11145), http://dx.doi.org/10.1090/S0894034703004338
 [2]
 Bjorn Poonen, Undecidability in number theory, Notices Amer. Math. Soc. 55 (2008), no. 3, 344350. MR 2382821 (2008m:11238)
 [3]
 Martin Davis, On the number of solutions of Diophantine equations, Proc. Amer. Math. Soc. 35 (1972), 552554. MR 0304347 (46 #3482)
 [4]
 Martin Davis, Hilary Putnam, and Julia Robinson, The decision problem for exponential diophantine equations, Ann. of Math. (2) 74 (1961), 425436. MR 0133227 (24 #A3061)
 [5]
 Hilbert's tenth problem: relations with arithmetic and algebraic geometry, Contemporary Mathematics, vol. 270, American Mathematical Society, Providence, RI, 2000. Papers from the workshop held at Ghent University, Ghent, November 25, 1999; Edited by Jan Denef, Leonard Lipshitz, Thanases Pheidas and Jan Van Geel. MR 1802007 (2001g:00018)
 [6]
 Ju. V. Matijasevič, The Diophantineness of enumerable sets, Dokl. Akad. Nauk SSSR 191 (1970), 279282 (Russian). MR 0258744 (41 #3390)
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 Yuri V. Matiyasevich, Hilbert's tenth problem, translated from the 1993 Russian original by the author; with a foreword by Martin Davis. Foundations of Computing Series, MIT Press, Cambridge, MA, 1993. MR 1244324 (94m:03002b)
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 Alexandra Shlapentokh, Hilbert's tenth problem. Diophantine classes and extensions to global fields, New Mathematical Monographs, vol. 7, Cambridge University Press, Cambridge, 2007. MR 2297245 (2009e:11235)
 [9]
 C. Smoryński, A note on the number of zeros of polynomials and exponential polynomials, J. Symbolic Logic 42 (1977), no. 1, 99106. MR 0485277 (58 #5124)
 [10]
 C. Smoryński, Logical Number Theory I, SpringerVerlag, Berlin, 1991. MR 1106853 (92g:03001)
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Additional Information
Christoph Baxa
Affiliation:
Department of Mathematics, University of Vienna, Nordbergstraße 15, A–1090 Wien, Austria
Email:
christoph.baxa@univie.ac.at
DOI:
http://dx.doi.org/10.1090/S000299392013116989
Keywords:
Hilbert's Tenth Problem,
recursive ring
Received by editor(s):
December 1, 2011
Received by editor(s) in revised form:
February 12, 2012
Published electronically:
August 20, 2013
Communicated by:
Julia Knight
Article copyright:
© Copyright 2013
American Mathematical Society
