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
Fulltext PDF
Abstract 
References 
Similar Articles 
Additional Information
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 .
 [1]
Bjorn
Poonen, Hilbert’s tenth problem and
Mazur’s conjecture for large subrings of ℚ, J. Amer. Math. Soc. 16 (2003), no. 4, 981–990 (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, 344–350. MR 2382821
(2008m:11238)
 [3]
Martin
Davis, On the number of solutions of
Diophantine equations, Proc. Amer. Math.
Soc. 35 (1972),
552–554. MR 0304347
(46 #3482), http://dx.doi.org/10.1090/S00029939197203043471
 [4]
Martin
Davis, Hilary
Putnam, and Julia
Robinson, The decision problem for exponential diophantine
equations, Ann. of Math. (2) 74 (1961),
425–436. MR 0133227
(24 #A3061)
 [5]
Jan
Denef, Leonard
Lipshitz, Thanases
Pheidas, and Jan
Van Geel (eds.), 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 2–5,
1999. MR
1802007 (2001g:00018)
 [6]
Ju.
V. Matijasevič, The Diophantineness of enumerable sets,
Dokl. Akad. Nauk SSSR 191 (1970), 279–282 (Russian).
MR
0258744 (41 #3390)
 [7]
Yuri
V. Matiyasevich, Hilbert’s tenth problem, Foundations of
Computing Series, MIT Press, Cambridge, MA, 1993. Translated from the 1993
Russian original by the author; With a foreword by Martin Davis. MR 1244324
(94m:03002b)
 [8]
Alexandra
Shlapentokh, Hilbert’s tenth problem, New Mathematical
Monographs, vol. 7, Cambridge University Press, Cambridge, 2007.
Diophantine classes and extensions to global fields. 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, 99–106. MR 0485277
(58 #5124)
 [10]
Craig
Smoryński, Logical number theory. I, Universitext,
SpringerVerlag, Berlin, 1991. An introduction. MR 1106853
(92g:03001)
 [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)
 [7]
 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)
 [8]
 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)
Similar Articles
Retrieve articles in Proceedings of the American Mathematical Society
with MSC (2010):
11U05,
03B25,
11D45
Retrieve articles in all journals
with MSC (2010):
11U05,
03B25,
11D45
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
PII:
S 00029939(2013)116989
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
