Hostname: page-component-cd9895bd7-jkksz Total loading time: 0 Render date: 2024-12-25T18:37:20.288Z Has data issue: false hasContentIssue false

An isomorphism between monoids of external embeddings: About definability in arithmetic

Published online by Cambridge University Press:  12 March 2014

Mihai Prunescu*
Affiliation:
Institut Für Mathematik und Informatik, Universität Greifswald, Germany Institute of Mathematics of the Romanian Academy, Bucharest, Romania, E-mail: prunescu@mail.uni-greifswald.de

Abstract

We use a new version of the Definability Theorem of Beth in order to unify classical theorems of Yuri Matiyasevich and Jan Denef in one structural statement. We give similar forms for other important definability results from Arithmetic and Number Theory.

Type
Research Article
Copyright
Copyright © Association for Symbolic Logic 2002

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[1]Beth, Evert W., On Padoa's method in the theory of definition, Proceedings of the Royal Academy of Sciences (Amsterdam), A 56, 1953.Google Scholar
[2]Beth, Evert W., The foundations of mathematics, North Holland, 1959.Google Scholar
[3]Davis, Martin and Putnam, Hilary, Diophantine sets over polynomial rings, Illinois Journal of Mathematics, vol. 7 (1963), pp. 251256.CrossRefGoogle Scholar
[4]Davis, Martin, Putnam, Hilary, and Robinson, Julia, The decison problem for exponential diophantine equations, Annals of Mathematics, Second series, vol. 74 (1961), no. 3, pp. 425436.CrossRefGoogle Scholar
[5]Denef, Jan, Hilbert's tenth problem for quadratic rings, Proceedings of the American Mathematical Society, vol. 48 (1975), no. 1, pp. 214220.Google Scholar
[6]Denef, Jan, The diophantine problem for polynomial rings and fields of rational functions, Transactions of the American Mathematical Society, (1978), no. 242, pp. 391399.Google Scholar
[7]Denef, Jan, Diophantine sets over ℤ[T], Proceedings of the American Mathematical Society, vol. 69 (1978), no. 1, pp. 148150.Google Scholar
[8]Denef, Jan and Lipshitz, Leonard, Diophantine sets over some rings of algebraic integers, The Journal of the London Mathematical Society, vol. 18 (1978), no. 2, pp. 385391.CrossRefGoogle Scholar
[9]Gödel, Kurt, Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme, Monatshefte für Mathematik und Physik, vol. 38 (1931), no. 1, pp. 173198.CrossRefGoogle Scholar
[10]Kaye, Richard, Models of Peano arithmetic, Oxford Logic Guides, no. 15, 1991.CrossRefGoogle Scholar
[11]Lipshitz, Leonard, Diophantine correct models of arithmetic, Procedings of the American Mathematical Society, vol. 73 (1979), no. 1, pp. 107108.CrossRefGoogle Scholar
[12]Marcus, Daniel A., Number fields, Springer Verlag, 1977.CrossRefGoogle Scholar
[13]Matiyasevich, Yuri V., Hilbert's tenth problem, MIT Press, 1993.Google Scholar
[14]Mazur, Barry, On the diophantine sets over the rationals, Experimental Mathematics, (1990), no. 1, pp. 121.Google Scholar
[15]Pheidas, Thanases, Hilbert's tenth problem for a class of rings of algebraic integers, Proceedings of the American Mathematical Society, (1988), no. 104, pp. 611620.Google Scholar
[16]Pourchet, Yves, Sur la representation en somme de carres des polynomes sur un corps de nombres algebriques, Acta Arithmeticae, (1971), no. 19, pp. 89104.Google Scholar
[17]Prestel, Alexander, Einführung in die mathematische Logik und Modelltheorie, Vieweg Verlag, 1992.Google Scholar
[18]Prunescu, Mihai, A structural approach to diophantine definability. Dissertation. Universität Konstanz, Hartungs-Gorre Verlag, Konstanz, 1999.Google Scholar
[19]Prunescu, Mihai, Defining constant polynomials, Hilbert's tenth problem: Relations with arithmetic and algebraic geometry, Contemporary Mathematics, (2000), no. 270, pp. 139145.Google Scholar
[20]Rabin, Michael O., Computable algebra, general theory and theory of computable fields, Transactions of the American Mathematical Society, (1960), no. 95, pp. 341360.Google Scholar
[21]Robinson, Abraham, Non-standard analysis, Studies in Logic and the Foundations of Mathematics, North-Holland, 1974.Google Scholar
[22]Robinson, Julia, The undecidability for algebraic rings and fields, Proceedings of the American Mathematical Society, (1959), no. 10, pp. 950957.Google Scholar
[23]Rumely, Robert S., Undecidability and definability for the theory of global fields, Transactions of the American Mathematical Society, vol. 262 (1980), no. 1, pp. 195217.CrossRefGoogle Scholar
[24]Sauerland, Ulrich, Entscheidbarkeitsprobleme in Ringen algebraischer Zahlkörper, Diplomarbeit, Universität Konstanz, 1993.Google Scholar
[25]Shlapentokh, Alexandra, Extension of Hilbert's tenth problem to some algebraic number fields, Communications of Pure and Applied Mathematics, vol. XLII (1989), pp. 11131122.Google Scholar
[26]Shlapentokh, Alexandra, Diophantine definitions for some polynomial rings, Communications of Pure and Applied Mathematics, vol. XLIII (1990), pp. 10551066.CrossRefGoogle Scholar
[27]Shoenfield, Joseph R., Mathematical logic, Addison-Wesley, Reading, Massachusetts, 1976.Google Scholar
[28]Smorynski, Craig, Logical number theory, Springer Verlag, 1991.CrossRefGoogle Scholar
[29]Zahidi, Karim, On diophantine sets over polynomial rings, Proceedings of the American Mathematical Society, (2000), no. 128, pp. 877884.Google Scholar