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

Bhargava, Manjul ¹

¹ Department of Mathematics, Princeton University, Princeton, New Jersey 08544

Journal of the American Mathematical Society, Tome 22 (2009) no. 4, pp. 963-993

Résumé

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.

Détail

DOI : 10.1090/S0894-0347-09-00638-9

Bibliographie

[1] Amice, Yvette Interpolation 𝑝-adique Bull. Soc. Math. France 1964 117 180

[2] Barsky, Daniel Fonctions 𝑘-lipschitziennes sur un anneau local et polynômes à valeurs entières Bull. Soc. Math. France 1973 397 411

[3] Bhargava, Manjul 𝑃-orderings and polynomial functions on arbitrary subsets of Dedekind rings J. Reine Angew. Math. 1997 101 127

[4] Bhargava, Manjul The factorial function and generalizations Amer. Math. Monthly 2000 783 799

[5] Bhargava, Manjul, Kedlaya, Kiran S. Continuous functions on compact subsets of local fields Acta Arith. 1999 191 198

[6] Cahen, Paul-Jean Polynomes à valeurs entières Canadian J. Math. 1972 747 754

[7] Cahen, Paul-Jean, Chabert, Jean-Luc Integer-valued polynomials 1997

[8] Cahen, Paul-Jean, Chabert, Jean-Luc, Loper, K. Alan High dimension Prüfer domains of integer-valued polynomials J. Korean Math. Soc. 2001 915 935

[9] Cahen, Paul-Jean, Chabert, Jean-Luc On the ultrametric Stone-Weierstrass theorem and Mahler’s expansion J. Théor. Nombres Bordeaux 2002 43 57

[10] Carlitz, L. A note on integral-valued polynomials Indag. Math. 1959 294 299

[11] De Bruijn, N. G. Some classes of integer-valued functions Indag. Math. 1955 363 367

[12] Dieudonné, Jean Sur les fonctions continues 𝑝-adiques Bull. Sci. Math. (2) 1944 79 95

[13] Fresnel, Jean, Van Der Put, Marius Rigid analytic geometry and its applications 2004

[14] Gerboud, Gilbert Polynômes à valeurs entières sur l’anneau des entiers de Gauss C. R. Acad. Sci. Paris Sér. I Math. 1988 375 378

[15] Gunji, H., Mcquillan, D. L. Polynomials with integral values Proc. Roy. Irish Acad. Sect. A 1978 1 7

[16] Kaplansky, Irving The Weierstrass theorem in fields with valuations Proc. Amer. Math. Soc. 1950 356 357

[17] Llavona, José G. Approximation of continuously differentiable functions 1986

[18] Mahler, K. An interpolation series for continuous functions of a 𝑝-adic variable J. Reine Angew. Math. 1958 23 34

[19] Narkiewicz, Władysław Polynomial mappings 1995

[20] Schikhof, W. H. Ultrametric calculus 1984

[21] Verdoodt, Ann Orthonormal bases for non-Archimedean Banach spaces of continuous functions 1999 323 331

[22] Wagner, Carl G. Interpolation series for continuous functions on 𝜋-adic completions of 𝐺𝐹(𝑞,𝑥). Acta Arith. 1970/71 389 406

[23] Wagner, Carl G. Polynomials over 𝐺𝐹(𝑞,𝑥) with integral-valued differences Arch. Math. (Basel) 1976 495 501

[24] Yang, Zifeng Locally analytic functions over completions of 𝐹ᵣ[𝑈] J. Number Theory 1998 451 458