Abstract
We generalize notions and results obtained by Amice for regular compact subsets S of a local field K and extended by Bhargava to general compact subsets of K. Considering any ultrametric valued field K and subsets S that are regular in a generalized sense (but not necessarily compact), we show that they still have strong properties such as having v-orderings \({\{a_n\}_{n\geq0}}\) which satisfy a generalized Legendre formula, which are very well ordered and well distributed sequences in the sense of Helsmoortel and which remain v-orderings when a finite number of the initial terms of the sequence are deleted.
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References
Amice Y.: Interpolation p-adique. Bull. Soc. Math. Fr. 92, 117–180 (1964)
Bhargava M.: P-orderings and polynomial functions on arbitrary subsets of Dedekind rings. J. Reine Angew. Math. 490, 101–127 (1997)
Bhargava M.: Generalized factorials and fixed divisors over subsets of a Dedekind domain. J. Number Theory 72, 67–75 (1998)
Bhargava M.: On P-orderings, rings of integer valued polynomials and ultrametric analysis. J. Am. Math. Soc. 22, 963–993 (2009)
Boulanger, J., Chabert, J.-L., Evrard, S., Gerboud, G.: The characteristic sequence of integer-valued polynomials on a subset. In: Advances in Commutative Ring Theory. Lecture Notes in Pure and Appl. Math., vol. 205, pp. 161–174. Dekker, New York (1999)
Bourbaki N.: Algèbre Commutative. Hermann, Paris (1964)
Cahen P.-J., Chabert J.-L.: Integer-valued polynomials. Amer. Math. Soc. Surveys and Monographs, vol. 48. Providence (1997)
Cahen P.-J., Chabert J.-L.: On the ultrametric Stone-Weierstrass theorem and Mahler’s expansion. Journal de Théorie des Nombres de Bordeaux 14, 43–57 (2002)
Chabert J.-L.: Generalized factorial ideals. Commut. Algebra Arab. J. Sci. Eng. Sect. C 26, 51–68 (2001)
Chabert, J.-L.: Integer-valued polynomials in valued fields with an application to discrete dynamical systems. In: Commutative Algebra and its Applications, pp. 103–134. de Gruyter, Berlin (2009)
Chabert J.-L.: On the polynomial closure in a valued field. J. Number Theory 130, 458–468 (2010)
Evrard S., Fares Y.: p-Adic subsets whose factorials satisfy a generalized Legendre formula. Lond. Math. Soc. 40, 37–50 (2008)
Helsmoortel E.: Comportement local des fonctions continues sur un compact régulier d’un corps local. C. R. Acad. Sci. Paris Sér. A-B 271, A546–A548 (1970)
Johnson K.: Computing r-removed P-orderings and P-orderings of order h, in the proceedings of the third international meeting on integer-valued polynomials (Marseille 2010). Actes des rencontres du CIRM 2(2), 33–40 (2010)
Mahler K.: An interpolation series for continuous functions of a p-adic variable. J. Reine Angew. Math. 199, 23–34 (1958)
Mahler K.: An interpolation series for continuous functions of a p-adic variable. J. Reine Angew. Math. 208, 70–72 (1961)
Pólya G.: Ueber ganzwertige polynome in algebraischen Zahlkörpern. J. Reine Angew. Math. 149, 97–116 (1919)
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Chabert, JL., Evrard, S. & Fares, Y. Regular subsets of valued fields and Bhargava’s v-orderings. Math. Z. 274, 263–290 (2013). https://doi.org/10.1007/s00209-012-1069-x
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DOI: https://doi.org/10.1007/s00209-012-1069-x
Keywords
- Integer-valued polynomials
- Valued fields
- Regular subsets
- Generalized Legendre formula
- Strong v-orderings
- Integer-valued divided differences