Diophantine sets of polynomials over number fields

Demeyer, Jeroen

doi:10.1090/S0002-9939-10-10329-3

Diophantine sets of polynomials over number fields
HTML articles powered by AMS MathViewer

by Jeroen Demeyer PDF

Proc. Amer. Math. Soc. 138 (2010), 2715-2728 Request permission

Abstract:

Let $\mathcal {R}$ be a number field or a recursive subring of a number field and consider the polynomial ring $\mathcal {R}[T]$. We show that the set of polynomials with integer coefficients is diophantine over $\mathcal {R}[T]$. Applying a result by Denef, this implies that every recursively enumerable subset of $\mathcal {R}[T]^k$ is diophantine over $\mathcal {R}[T]$.

References

Martin Davis, Hilbert’s tenth problem is unsolvable, Amer. Math. Monthly 80 (1973), 233–269. MR 317916, DOI 10.2307/2318447
Jeroen Demeyer, Recursively enumerable sets of polynomials over a finite field, J. Algebra 310 (2007), no. 2, 801–828. MR 2308181, DOI 10.1016/j.jalgebra.2006.09.030
Jeroen Demeyer, Recursively enumerable sets of polynomials over a finite field are Diophantine, Invent. Math. 170 (2007), no. 3, 655–670. MR 2357505, DOI 10.1007/s00222-007-0078-6
J. Denef, Diophantine sets over $\textbf {Z}[T]$, Proc. Amer. Math. Soc. 69 (1978), no. 1, 148–150. MR 462934, DOI 10.1090/S0002-9939-1978-0462934-X
J. Denef, The Diophantine problem for polynomial rings of positive characteristic, Logic Colloquium ’78 (Mons, 1978) Studies in Logic and the Foundations of Mathematics, vol. 97, North-Holland, Amsterdam-New York, 1979, pp. 131–145. MR 567668
Kirsten Eisenträger, Hilbert’s tenth problem for function fields of varieties over number fields and $p$-adic fields, J. Algebra 310 (2007), no. 2, 775–792. MR 2308179, DOI 10.1016/j.jalgebra.2006.09.032
A. Fröhlich and J. C. Shepherdson, Effective procedures in field theory, Philos. Trans. Roy. Soc. London Ser. A 248 (1956), 407–432. MR 74349, DOI 10.1098/rsta.1956.0003
Kenneth Ireland and Michael Rosen, A classical introduction to modern number theory, 2nd ed., Graduate Texts in Mathematics, vol. 84, Springer-Verlag, New York, 1990. MR 1070716, DOI 10.1007/978-1-4757-2103-4
K. H. Kim and F. W. Roush, Diophantine unsolvability over $p$-adic function fields, J. Algebra 176 (1995), no. 1, 83–110. MR 1345295, DOI 10.1006/jabr.1995.1234
Yuri Matiyasevich, Enumerable sets are Diophantine, Soviet Math. Dokl. 11 (1970), 354–358.
Laurent Moret-Bailly, Elliptic curves and Hilbert’s tenth problem for algebraic function fields over real and $p$-adic fields, J. Reine Angew. Math. 587 (2005), 77–143. MR 2186976, DOI 10.1515/crll.2005.2005.587.77
Laurent Moret-Bailly, Sur la définissabilité existentielle de la non-nullité dans les anneaux, Algebra Number Theory 1 (2007), no. 3, 331–346 (French, with English and French summaries). MR 2361937, DOI 10.2140/ant.2007.1.331
Jürgen Neukirch, Algebraic number theory, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 322, Springer-Verlag, Berlin, 1999. Translated from the 1992 German original and with a note by Norbert Schappacher; With a foreword by G. Harder. MR 1697859, DOI 10.1007/978-3-662-03983-0
Thanases Pheidas and Karim Zahidi, Undecidability of existential theories of rings and fields: a survey, Hilbert’s tenth problem: relations with arithmetic and algebraic geometry (Ghent, 1999) Contemp. Math., vol. 270, Amer. Math. Soc., Providence, RI, 2000, pp. 49–105. MR 1802009, DOI 10.1090/conm/270/04369
Bjorn Poonen, Undecidability in number theory, Notices Amer. Math. Soc. 55 (2008), no. 3, 344–350. MR 2382821
Alexandra Shlapentokh, Diophantine definitions for some polynomial rings, Comm. Pure Appl. Math. 43 (1990), no. 8, 1055–1066. MR 1075078, DOI 10.1002/cpa.3160430807
Karim Zahidi, Existential undecidability for rings of algebraic functions, Ph.D. thesis, Ghent University, 1999.
Karim Zahidi, On Diophantine sets over polynomial rings, Proc. Amer. Math. Soc. 128 (2000), no. 3, 877–884. MR 1641141, DOI 10.1090/S0002-9939-99-05179-5