Which curves over $\textbf {Z}$ have points with coordinates in a discrete ordered ring?
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- by Lou van den Dries PDF
- Trans. Amer. Math. Soc. 264 (1981), 181-189 Request permission
Abstract:
A criterion is given for curves defined over ${\mathbf {Z}}$ to have an infinite point in a discrete ordered ring. Using this, one can decide effectively whether a given polynomial in ${\mathbf {Z}}[X,Y]$ has a zero in a model for the axioms of open induction. Riemann-Roch for curves over ${\mathbf {Q}}$ is the main tool used.References
- James Ax, The elementary theory of finite fields, Ann. of Math. (2) 88 (1968), 239–271. MR 229613, DOI 10.2307/1970573
- Claude Chevalley, Introduction to the Theory of Algebraic Functions of One Variable, Mathematical Surveys, No. VI, American Mathematical Society, New York, N. Y., 1951. MR 0042164, DOI 10.1090/surv/006
- L. van den Dries, New decidable fields of algebraic numbers, Proc. Amer. Math. Soc. 77 (1979), no. 2, 251–256. MR 542093, DOI 10.1090/S0002-9939-1979-0542093-6 —, Some model theory and number theory for models of weak systems of arithmetic, Proc. Karpacz 1979, Lecture Notes in Math. (to appear).
- Otto Endler, Valuation theory, Universitext, Springer-Verlag, New York-Heidelberg, 1972. To the memory of Wolfgang Krull (26 August 1899–12 April 1971). MR 0357379, DOI 10.1007/978-3-642-65505-0
- Serge Lang, Introduction to algebraic geometry, Interscience Publishers, Inc., New York-London, 1958. MR 0100591 A. Prestel, Lectures on formally real fields, Monografias Mat., IMPA, Rio de Janeiro, 1975.
- Michael O. Rabin, Computable algebra, general theory and theory of computable fields, Trans. Amer. Math. Soc. 95 (1960), 341–360. MR 113807, DOI 10.1090/S0002-9947-1960-0113807-4
- Paulo Ribenboim, The Riemann-Roch theorem for algebraic curves, Queen’s Papers in Pure and Applied Mathematics, No. 2, Queen’s University, Kingston, Ont., 1965. MR 0197461
- J. C. Shepherdson, A non-standard model for a free variable fragment of number theory, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 12 (1964), 79–86. MR 161798 B. L. van der Waerden, Modern algebra. I, Springer-Verlag, Berlin, 1930.
- A. J. Wilkie, Some results and problems on weak systems of arithmetic, Logic Colloquium ’77 (Proc. Conf., Wrocław, 1977) Studies in Logic and the Foundations of Mathematics, vol. 96, North-Holland, Amsterdam-New York, 1978, pp. 285–296. MR 519823
Additional Information
- © Copyright 1981 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 264 (1981), 181-189
- MSC: Primary 03C65; Secondary 03B25, 10N05
- DOI: https://doi.org/10.1090/S0002-9947-1981-0597875-5
- MathSciNet review: 597875