### 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 $P\in\F$.

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 |
---|---|

Original language | English |

Title of host publication | Rings and Factorizations |

Editors | Alberto Facchini, Marco Fontana, Alfred Geroldinger, Bruce Olberding |

Publisher | Springer, Cham |

Publication status | Accepted/In press - 2020 |

### Keywords

- Interpolation
- polynomials
- P-adic approximation
- P-adic Lipschitz functions
- 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

## Fingerprint Dive into the research topics of 'Simultaneous interpolation and P-adic approximation by integer-valued polynomials'. Together they form a unique fingerprint.

## Cite this

Frisch, S. (Accepted/In press). Simultaneous interpolation and P-adic approximation by integer-valued polynomials. In A. Facchini, M. Fontana, A. Geroldinger, & B. Olberding (Eds.),

*Rings and Factorizations*Springer, Cham.