Remote Access Journal of the American Mathematical Society
Green Open Access

Journal of the American Mathematical Society

ISSN 1088-6834(online) ISSN 0894-0347(print)



On $ P$-orderings, rings of integer-valued polynomials, and ultrametric analysis

Author: Manjul Bhargava
Journal: J. Amer. Math. Soc. 22 (2009), 963-993
MSC (2000): Primary 11C08, 11S80; Secondary 13F20, 13B25
Published electronically: May 27, 2009
MathSciNet review: 2525777
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We introduce two new notions of ``$ P$-ordering'' and use them to define a three-parameter generalization of the usual factorial function. We then apply these notions of $ P$-orderings and factorials to some classical problems in two distinct areas, namely: 1) the study of integer-valued polynomials and 2) $ P$-adic analysis.

Specifically, we first use these notions of $ P$-orderings and factorials to construct explicit Pólya-style regular bases for two natural families of rings of integer-valued polynomials defined on an arbitrary subset of a Dedekind domain.

Second, we classify ``smooth'' functions on an arbitrary compact subset $ S$ of a local field, by constructing explicit interpolation series (i.e., orthonormal bases) for the Banach space of functions on $ S$ satisfying any desired conditions of continuous differentiability or local analyticity. Our constructions thus extend Mahler's Theorem (classifying the functions that are continuous on $ \mathbb{Z}_p$) to a very general setting. In particular, our constructions prove that, for any $ \epsilon>0$, the functions in any of the above Banach spaces can be $ \epsilon$-approximated by polynomials (with respect to their respective Banach norms). Thus we obtain the non-Archimedean analogues of the classical polynomial approximation theorems in real and complex analysis proven by Weierstrass, de la Vallée-Poussin, and Bernstein. Our proofs are effective.

References [Enhancements On Off] (What's this?)

  • 1. Yvette Amice, Interpolation 𝑝-adique, Bull. Soc. Math. France 92 (1964), 117–180 (French). MR 0188199
  • 2. Daniel Barsky, Fonctions 𝑘-lipschitziennes sur un anneau local et polynômes à valeurs entières, Bull. Soc. Math. France 101 (1973), 397–411 (French). MR 0371863
  • 3. S. Bernstein, Leçons sur les propriétés extrémales de la meilleure approximation des fonctions analytiques d'une variable réelle, Paris, 1926.
  • 4. Manjul Bhargava, 𝑃-orderings and polynomial functions on arbitrary subsets of Dedekind rings, J. Reine Angew. Math. 490 (1997), 101–127. MR 1468927, 10.1515/crll.1997.490.101
  • 5. Manjul Bhargava, The factorial function and generalizations, Amer. Math. Monthly 107 (2000), no. 9, 783–799. MR 1792411, 10.2307/2695734
  • 6. M. Bhargava, P.-J. Cahen, and J. Yeramian, Finite generation properties for rings of integer-valued polynomials, J. Algebra, to appear.
  • 7. Manjul Bhargava and Kiran S. Kedlaya, Continuous functions on compact subsets of local fields, Acta Arith. 91 (1999), no. 3, 191–198. MR 1735672
  • 8. Paul-Jean Cahen, Polynomes à valeurs entières, Canad. J. Math. 24 (1972), 747–754 (French). MR 0309923
  • 9. Paul-Jean Cahen and Jean-Luc Chabert, Integer-valued polynomials, Mathematical Surveys and Monographs, vol. 48, American Mathematical Society, Providence, RI, 1997. MR 1421321
  • 10. Paul-Jean Cahen, Jean-Luc Chabert, and K. Alan Loper, High dimension Prüfer domains of integer-valued polynomials, J. Korean Math. Soc. 38 (2001), no. 5, 915–935. Mathematics in the new millennium (Seoul, 2000). MR 1849332
  • 11. Paul-Jean Cahen and Jean-Luc Chabert, On the ultrametric Stone-Weierstrass theorem and Mahler’s expansion, J. Théor. Nombres Bordeaux 14 (2002), no. 1, 43–57 (English, with English and French summaries). MR 1925989
  • 12. L. Carlitz, A note on integral-valued polynomials, Nederl. Akad. Wetensch. Proc. Ser. A 62 = Indag. Math. 21 (1959), 294–299. MR 0108462
  • 13. N. G. de Bruijn, Some classes of integer-valued functions, Nederl. Akad. Wetensch. Proc. Ser. A. 58=Indag. Math. 17 (1955), 363–367. MR 0071450
  • 14. Jean Dieudonné, Sur les fonctions continues 𝑝-adiques, Bull. Sci. Math. (2) 68 (1944), 79–95 (French). MR 0013142
  • 15. Jean Fresnel and Marius van der Put, Rigid analytic geometry and its applications, Progress in Mathematics, vol. 218, Birkhäuser Boston, Inc., Boston, MA, 2004. MR 2014891
  • 16. Gilbert Gerboud, Polynômes à valeurs entières sur l’anneau des entiers de Gauss, C. R. Acad. Sci. Paris Sér. I Math. 307 (1988), no. 8, 375–378 (French, with English summary). MR 965801
  • 17. H. Gunji and D. L. McQuillan, Polynomials with integral values, Proc. Roy. Irish Acad. Sect. A 78 (1978), no. 1, 1–7. MR 0472801
  • 18. Irving Kaplansky, The Weierstrass theorem in fields with valuations, Proc. Amer. Math. Soc. 1 (1950), 356–357. MR 0035760, 10.1090/S0002-9939-1950-0035760-3
  • 19. José G. Llavona, Approximation of continuously differentiable functions, North-Holland Mathematics Studies, vol. 130, North-Holland Publishing Co., Amsterdam, 1986. Notas de Matemática [Mathematical Notes], 112. MR 870155
  • 20. K. Mahler, An interpolation series for continuous functions of a 𝑝-adic variable, J. Reine Angew. Math. 199 (1958), 23–34. MR 0095821
  • 21. Władysław Narkiewicz, Polynomial mappings, Lecture Notes in Mathematics, vol. 1600, Springer-Verlag, Berlin, 1995. MR 1367962
  • 22. A. Ostrowski, Über ganzwertige Polynome in algebraischen Zahlkörpern, J. reine angew. Math. 149 (1919) 117-124.
  • 23. G. Pólya, Über ganzwertige ganze Funktionen, Rend. Circ. Mat. Palermo 40 (1915) 1-16.
  • 24. G. Pólya, Über ganzwertige Polynome in algebraischen Zahlkörpern, J. reine angew. Math. 149 (1919) 97-116.
  • 25. W. H. Schikhof, Ultrametric calculus, Cambridge Studies in Advanced Mathematics, vol. 4, Cambridge University Press, Cambridge, 1984. An introduction to 𝑝-adic analysis. MR 791759
  • 26. C. de la Vallée-Poussin, Sur l'approximation des fonctions d'une variable réelle et de leurs dérivées par des polynômes et des suites finies de Fourier, Bull. Acad. Sci. Belgique (1908), 193-254.
  • 27. Ann Verdoodt, Orthonormal bases for non-Archimedean Banach spaces of continuous functions, 𝑝-adic functional analysis (Poznań, 1998) Lecture Notes in Pure and Appl. Math., vol. 207, Dekker, New York, 1999, pp. 323–331. MR 1703503
  • 28. Carl G. Wagner, Interpolation series for continuous functions on 𝜋-adic completions of 𝐺𝐹(𝑞,𝑥)., Acta Arith. 17 (1970/1971), 389–406. MR 0282973
  • 29. Carl G. Wagner, Polynomials over 𝐺𝐹(𝑞,𝑥) with integral-valued differences, Arch. Math. (Basel) 27 (1976), no. 5, 495–501. MR 0417137
  • 30. K. Weierstrass, Über die analytische Darstellbarkeit sogenannter willkürlicher Functionen einer reellen Veränderlichen, Sitzungsberichte der Königlich Preu$ \beta$ischen Akademie der Wissenschaften zu Berlin, 1885 (II).
  • 31. Zifeng Yang, Locally analytic functions over completions of 𝐹ᵣ[𝑈], J. Number Theory 73 (1998), no. 2, 451–458. MR 1657996, 10.1006/jnth.1998.2308
  • 32. J. Yeramian, Anneaux de Bhargava, Ph.D. Thesis, Université Aix-Marseille, 2004.

Similar Articles

Retrieve articles in Journal of the American Mathematical Society with MSC (2000): 11C08, 11S80, 13F20, 13B25

Retrieve articles in all journals with MSC (2000): 11C08, 11S80, 13F20, 13B25

Additional Information

Manjul Bhargava
Affiliation: Department of Mathematics, Princeton University, Princeton, New Jersey 08544

Keywords: $p$-ordering, factorial function, integer-valued polynomials, $p$-adic analysis, ultrametric analysis
Received by editor(s): February 25, 2008
Published electronically: May 27, 2009
Article copyright: © Copyright 2009 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.