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

Sophie Frisch

doi:10.1007/978-3-030-43416-8_10

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

Sophie Frisch

Institute of Analysis and Number Theory (5010)

Research output: Chapter in Book/Report/Conference proceeding › Conference paper › peer-review

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
Original language	English
Title of host publication	Advances in Rings, Modules and Factorizations
Subtitle of host publication	Graz, Austria, February 19-23, 2018
Editors	Alberto Facchini, Marco Fontana, Alfred Geroldinger, Bruce Olberding
Place of Publication	Cham
Publisher	Springer
Pages	183-192
ISBN (Electronic)	978-3-030-43416-8
ISBN (Print)	978-3-030-43415-1
DOIs	https://doi.org/10.1007/978-3-030-43416-8_10
Publication status	Published - 2020
Event	Conference on Rings and Factorizations - Karl-Franzens Universitaet Graz, Graz, Austria Duration: 19 Feb 2018 → 23 Feb 2018 https://imsc.uni-graz.at/rings2018/

Publication series

Name	Springer Proceedings in Mathematics & Statistics
Volume	321

Conference

Conference	Conference on Rings and Factorizations
Country/Territory	Austria
City	Graz
Period	19/02/18 → 23/02/18
Internet address	https://imsc.uni-graz.at/rings2018/

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

Access to Document

10.1007/978-3-030-43416-8_10

Cite this

Frisch, S. (2020). Simultaneous interpolation and P-adic approximation by integer-valued polynomials. In A. Facchini, M. Fontana, A. Geroldinger, & B. Olberding (Eds.), Advances in Rings, Modules and Factorizations: Graz, Austria, February 19-23, 2018 (pp. 183-192). (Springer Proceedings in Mathematics & Statistics; Vol. 321). Springer. https://doi.org/10.1007/978-3-030-43416-8_10

Simultaneous interpolation and P-adic approximation by integer-valued polynomials. / Frisch, Sophie.
Advances in Rings, Modules and Factorizations: Graz, Austria, February 19-23, 2018. ed. / Alberto Facchini; Marco Fontana; Alfred Geroldinger; Bruce Olberding. Cham: Springer, 2020. p. 183-192 (Springer Proceedings in Mathematics & Statistics; Vol. 321).

Research output: Chapter in Book/Report/Conference proceeding › Conference paper › peer-review

Frisch, S 2020, Simultaneous interpolation and P-adic approximation by integer-valued polynomials. in A Facchini, M Fontana, A Geroldinger & B Olberding (eds), Advances in Rings, Modules and Factorizations: Graz, Austria, February 19-23, 2018. Springer Proceedings in Mathematics & Statistics, vol. 321, Springer, Cham, pp. 183-192, Conference on Rings and Factorizations, Graz, Austria, 19/02/18. https://doi.org/10.1007/978-3-030-43416-8_10

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title = "Simultaneous interpolation and P-adic approximation by integer-valued polynomials",

abstract = "Let $D$ be a Dedekind domain with finite residue fieldsand $\F$ 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 $P^\kP$-congruencepreserving 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 $P\in\F$.",

keywords = "Interpolation, Polynomials, P-adic approximation, P-adic Lipschitz, integer-valued polynomials, Dedekind domains, polynomial mappings, polynomial functions, commutative rings, Interpolation, polynomials, P-adic approximation, P-adic Lipschitz functions, polynomial functions, polynomial mappings, congruence preserving, integer-valued polynomials, Dedekind domains, commutative rings, integral domains",

author = "Sophie Frisch",

year = "2020",

doi = "10.1007/978-3-030-43416-8_10",

language = "English",

isbn = "978-3-030-43415-1",

series = "Springer Proceedings in Mathematics & Statistics",

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pages = "183--192",

editor = "Alberto Facchini and Marco Fontana and Alfred Geroldinger and Bruce Olberding",

booktitle = "Advances in Rings, Modules and Factorizations",

note = "Conference on Rings and Factorizations ; Conference date: 19-02-2018 Through 23-02-2018",

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TY - GEN

T1 - Simultaneous interpolation and P-adic approximation by integer-valued polynomials

AU - Frisch, Sophie

PY - 2020

Y1 - 2020

N2 - Let $D$ be a Dedekind domain with finite residue fieldsand $\F$ 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 $P^\kP$-congruencepreserving 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 $P\in\F$.

AB - Let $D$ be a Dedekind domain with finite residue fieldsand $\F$ 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 $P^\kP$-congruencepreserving 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 $P\in\F$.

KW - Interpolation

KW - Polynomials

KW - P-adic approximation

KW - P-adic Lipschitz

KW - integer-valued polynomials

KW - Dedekind domains

KW - polynomial mappings

KW - polynomial functions

KW - commutative rings

KW - Interpolation

KW - polynomials

KW - P-adic approximation

KW - P-adic Lipschitz functions

KW - polynomial functions

KW - polynomial mappings

KW - congruence preserving

KW - integer-valued polynomials

KW - Dedekind domains

KW - commutative rings

KW - integral domains

UR - http://blah.math.tu-graz.ac.at/~frisch/wwwpdf/sf-intapprox.pdf

U2 - 10.1007/978-3-030-43416-8_10

DO - 10.1007/978-3-030-43416-8_10

M3 - Conference paper

SN - 978-3-030-43415-1

T3 - Springer Proceedings in Mathematics & Statistics

SP - 183

EP - 192

BT - Advances in Rings, Modules and Factorizations

A2 - Facchini, Alberto

A2 - Fontana, Marco

A2 - Geroldinger, Alfred

A2 - Olberding, Bruce

PB - Springer

CY - Cham

T2 - Conference on Rings and Factorizations

Y2 - 19 February 2018 through 23 February 2018

ER -

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

Abstract

Publication series

Conference

Keywords

ASJC Scopus subject areas

Fields of Expertise

Access to Document

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Cite this