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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
DOI: https://doi.org/10.1090/S0894-0347-09-00638-9
Published electronically: May 27, 2009
MathSciNet review: 2525777
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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. Y. Amice, Interpolation $ p$-adique, Bull. Soc. Math. France 92 (1964), 117-180. MR 0188199 (32:5638)
  • 2. D. Barsky, Fonctions $ k$-lipschitziennes sur un anneau local et polynômes à valeurs entières, Bull. Soc. Math. France 101 (1973), 397-411. MR 0371863 (51:8080)
  • 3. S. Bernstein, Leçons sur les propriétés extrémales de la meilleure approximation des fonctions analytiques d'une variable réelle, Paris, 1926.
  • 4. M. Bhargava, $ P$-orderings and polynomial functions on arbitrary subsets of Dedekind rings, J. reine. angew. Math. 490 (1997), 101-127. MR 1468927 (98j:13016)
  • 5. M. Bhargava, The factorial function and generalizations, Amer. Math. Monthly 107 (2000), 783-799. MR 1792411 (2002d:05002)
  • 6. M. Bhargava, P.-J. Cahen, and J. Yeramian, Finite generation properties for rings of integer-valued polynomials, J. Algebra, to appear.
  • 7. M. Bhargava and K. S. Kedlaya, Continuous functions on compact subsets of local fields, Acta Arith. 91 (1999) 191-198. MR 1735672 (2001d:11117)
  • 8. P.-J. Cahen, Polynômes à valeurs entières, Canad. J. Math. 24 (1972), 747-754. MR 0309923 (46:9027)
  • 9. P.-J. Cahen and J.-L. Chabert, Integer-valued polynomials, Mathematical Surveys and Monographs, 48, American Mathematical Society, Providence, RI, 1997. MR 1421321 (98a:13002)
  • 10. P.-J. Cahen, J.-L. Chabert, and K. A. Loper, High dimension Prüfer domains of integer-valued polynomials, Journal Korean Math. Soc. 38 (2001), 915-935. MR 1849332 (2002f:13039)
  • 11. P.-J. Cahen and J.-L. Chabert, On the ultrametric Stone-Weierstrass theorem and Mahler's expansion, Journal de Théorie des Nombres de Bordeaux 14 (2002), 1-15. MR 1925989 (2003g:46086)
  • 12. L. Carlitz, A note on integral-valued polynomials. Indag. Math. 21 (1959), 294-299. MR 0108462 (21:7178)
  • 13. N. G. de Bruijn, Some classes of integer-valued functions, Indag. Math. 17 (1955), 363-367. MR 0071450 (17:128a)
  • 14. J. Dieudonné, Sur les fonctions continues $ p$-adiques, Bull. Sci. Math. $ 2$ème série 68 (1944), 79-95. MR 0013142 (7:111c)
  • 15. J. Fresnel and M. van der Put, Rigid Analytic Geometry and its Applications, Birkhäuser, 2004. MR 2014891 (2004i:14023)
  • 16. G. Gerboud, Polynômes à valeurs entières sur l'anneau des entiers de Gauss, Comptes Rendus Acad. Sci. Paris 307 (1988) 375-378. MR 965801 (89m:11097)
  • 17. H. Gunji and D. L. McQuillan, Polynomials with integral values, Proc. Roy. Irish Acad. A78 (1978) 1-7. MR 0472801 (57:12491)
  • 18. I. Kaplansky, The Weierstrass theorem in fields with valuations, Proc. Amer. Math. Soc. 1 (1950), 356-357. MR 0035760 (12:6e)
  • 19. J. G. Llavona, Approximation of continuously differentiable functions, North-Holland Mathematics Studies, vol. 130, North-Holland, Amsterdam, 1986. MR 870155 (88f:41001)
  • 20. K. Mahler, An interpolation series for a continuous function of a $ p$-adic variable, J. reine angew. Math. 199 (1958), 23-34. MR 0095821 (20:2321)
  • 21. W. Narkiewicz, Polynomial Mappings, Springer-Verlag, Berlin, 1995. MR 1367962 (97e:11037)
  • 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. Schikhof, Ultrametric Calculus: An Introduction to $ p$-adic Analysis, Cambridge Stud. Adv. Math., Vol. 4, Cambridge Univ. Press, Cambridge, UK, 1984. MR 791759 (86j:11104)
  • 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. A. Verdoodt, Orthonormal bases for non-Archimedean Banach spaces of continuous functions, $ p$-adic functional analysis (Poznań, 1998), 323-331, Lecture Notes in Pure and Appl. Math., 207, Dekker, New York, 1999. MR 1703503 (2000h:46095)
  • 28. C. G. Wagner, Interpolation series for continuous functions on $ \pi $-adic completions of $ {\rm GF}(q, x),$ Acta Arith. 17 (1970/1971), 389-406. MR 0282973 (44:207)
  • 29. C. G. Wagner, Polynomials over $ {\rm GF}(q,x)$ with integral-valued differences, Arch. Math. (Basel) 27 (1976), no. 5, 495-501. MR 0417137 (54:5197)
  • 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. Z. Yang, Locally analytic functions over completions of $ F\sb r[U]$, J. Number Theory 73 (1998), no. 2, 451-458. MR 1657996 (99h:11062)
  • 32. J. Yeramian, Anneaux de Bhargava, Ph.D. Thesis, Université Aix-Marseille, 2004.

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Additional Information

Manjul Bhargava
Affiliation: Department of Mathematics, Princeton University, Princeton, New Jersey 08544
Email: bhargava@math.princeton.edu

DOI: https://doi.org/10.1090/S0894-0347-09-00638-9
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.

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