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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
Published electronically: May 27, 2009
MathSciNet review: 2525777
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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.

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Additional Information

Manjul Bhargava
Affiliation: Department of Mathematics, Princeton University, Princeton, New Jersey 08544
MR Author ID: 623882

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.