Number systems over general orders

Kód výsledku v IS VaVaI

<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F61988987%3A17310%2F19%3AA20023O8" target="_blank" >RIV/61988987:17310/19:A20023O8 - isvavai.cz</a>

Výsledek na webu

<a href="https://link.springer.com/article/10.1007%2Fs10474-019-00958-x" target="_blank" >https://link.springer.com/article/10.1007%2Fs10474-019-00958-x</a>

DOI - Digital Object Identifier

<a href="http://dx.doi.org/10.1007/s10474-019-00958-x" target="_blank" >10.1007/s10474-019-00958-x</a>

Jazyk výsledku

angličtina

Název v původním jazyce

Number systems over general orders

Popis výsledku v původním jazyce

Let O be an order, that is a commutative ring with 1 whose additive structure is a free Z-module of finite rank. A generalized number system (GNS for short) over O is a pair (p, D) where p is an element of O[x] is monic with constant term p(0) not a zero divisor of O, and where D is a complete residue system modulo p(0) in O containing 0. We say that (p, D) is a GNS over O with the finiteness property if all elements of O[x]/(p) have a representative in D[x] (the polynomials with coefficients in D). Our purpose is to extend several of the results from a previous paper of Petho and Thuswaldner, where GNS over orders of number fields were considered. We prove that it is algorithmically decidable whether or not for a given order O and GNS (p, D) over O, the pair (p, D) admits the finiteness property. This is closely related to work of Vince on matrix number systems. Let F be a fundamental domain for O circle times(Z) R/O and p is an element of O[X] a monic polynomial. For alpha is an element of O, define p(alpha)(x) := p(x+ alpha) and D-F, p(alpha) := p(alpha) F boolean AND O. Under mild conditions we show that the pairs (p(alpha), D-F, (p(alpha))) are GNS over O with finiteness property provided alpha is an element of O in some sense approximates a sufficiently large positive rational integer. In the opposite direction we prove under different conditions that (p(-m), D-F, p((-m))) does not have the finiteness property for each large enough positive rational integer m. We obtain important relations between power integral bases of ' etale orders and GNS over Z. Their proofs depend on some general effective finiteness results of Evertse and Gyory on monogenic etale orders.

Název v anglickém jazyce

Number systems over general orders

Popis výsledku anglicky

Let O be an order, that is a commutative ring with 1 whose additive structure is a free Z-module of finite rank. A generalized number system (GNS for short) over O is a pair (p, D) where p is an element of O[x] is monic with constant term p(0) not a zero divisor of O, and where D is a complete residue system modulo p(0) in O containing 0. We say that (p, D) is a GNS over O with the finiteness property if all elements of O[x]/(p) have a representative in D[x] (the polynomials with coefficients in D). Our purpose is to extend several of the results from a previous paper of Petho and Thuswaldner, where GNS over orders of number fields were considered. We prove that it is algorithmically decidable whether or not for a given order O and GNS (p, D) over O, the pair (p, D) admits the finiteness property. This is closely related to work of Vince on matrix number systems. Let F be a fundamental domain for O circle times(Z) R/O and p is an element of O[X] a monic polynomial. For alpha is an element of O, define p(alpha)(x) := p(x+ alpha) and D-F, p(alpha) := p(alpha) F boolean AND O. Under mild conditions we show that the pairs (p(alpha), D-F, (p(alpha))) are GNS over O with finiteness property provided alpha is an element of O in some sense approximates a sufficiently large positive rational integer. In the opposite direction we prove under different conditions that (p(-m), D-F, p((-m))) does not have the finiteness property for each large enough positive rational integer m. We obtain important relations between power integral bases of ' etale orders and GNS over Z. Their proofs depend on some general effective finiteness results of Evertse and Gyory on monogenic etale orders.

Druh

J_imp - Článek v periodiku v databázi Web of Science

CEP obor

—

OECD FORD obor

10101 - Pure mathematics

Projekt

<a href="/cs/project/GA17-02804S" target="_blank" >GA17-02804S: Vlastnosti číselných posloupností a jejich aplikace</a><br>

Návaznosti

P - Projekt vyzkumu a vyvoje financovany z verejnych zdroju (s odkazem do CEP)

Rok uplatnění

2019

Kód důvěrnosti údajů

S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů

Název periodika

ACTA MATH HUNG

ISSN

0236-5294

e-ISSN

1588-2632

Svazek periodika

159

Číslo periodika v rámci svazku

1

Stát vydavatele periodika

NL - Nizozemsko

Počet stran výsledku

19

Strana od-do

187-205

Kód UT WoS článku

000486228800012

EID výsledku v databázi Scopus

2-s2.0-85068158786