# Simultaneous interpolation and P-adic approximation by integer-valued polynomials

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## Abstract

Let $D$ be a Dedekind domain with finite residue fields
and $\F$ a finite set of maximal ideals of $D$.
Let $r_0$, $\ldots$, $r_n$ be distinct elements of $D$,
pairwise incongruent modulo $P^\kP$ for each $P\in\F$,
and $s_0$, $\ldots$, $s_n$ arbitrary elements of $D$.

We show that there is an interpolating $P^\kP$-congruence
preserving integer-valued polynomial, that is,
$f\in \Int(D)=\{g\in K[x]\mid g(D)\subseteq D\}$
with $f(r_i)=s_i$ for $0\le i \le n$, such that, moreover, the
function $f\colon D\rightarrow D$ is constant modulo $P^\kP$
on each residue class of $P^\kP$ for all $P\in\F$.
Translated title of the contribution Simultane Interpolation und P-adische Approximation durch ganzwertige Polynome English Advances in Rings, Modules and Factorizations Graz, Austria, February 19-23, 2018 Alberto Facchini, Marco Fontana, Alfred Geroldinger, Bruce Olberding Cham Springer 183-192 978-3-030-43416-8 978-3-030-43415-1 https://doi.org/10.1007/978-3-030-43416-8_10 Published - 2020 Conference on Rings and Factorizations - Karl-Franzens Universitaet Graz, Graz, AustriaDuration: 19 Feb 2018 → 23 Feb 2018https://imsc.uni-graz.at/rings2018/

### Publication series

Name Springer Proceedings in Mathematics & Statistics 321

### Conference

Conference Conference on Rings and Factorizations Austria Graz 19/02/18 → 23/02/18 https://imsc.uni-graz.at/rings2018/

## Keywords

• Interpolation
• polynomials
• polynomial functions
• polynomial mappings
• congruence preserving
• integer-valued polynomials
• Dedekind domains
• commutative rings
• integral domains

## ASJC Scopus subject areas

• Algebra and Number Theory

## Fields of Expertise

• Information, Communication & Computing

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