An explicit polynomial analogue of Romanoff's theorem

Igor E. Shparlinski; Andreas J. Weingartner

doi:10.1016/j.ffa.2016.11.002

An explicit polynomial analogue of Romanoff's theorem

Igor E. Shparlinski^*, Andreas J. Weingartner

^*Corresponding author for this work

Research output: Contribution to journal › Article › peer-review

3 Citations (Scopus)

Abstract

Given a polynomial g of positive degree over a finite field, we show that the proportion of polynomials of degree n, which can be written as h+g^k, where h is an irreducible polynomial of degree n and k is a nonnegative integer, has order of magnitude 1/deg⁡g.

Original language	English
Pages (from-to)	22-33
Number of pages	12
Journal	Finite Fields and their Applications
Volume	44
DOIs	https://doi.org/10.1016/j.ffa.2016.11.002
Publication status	Published - 1 Mar 2017
Externally published	Yes

Keywords

Irreducible polynomial
Multiplicative order
Polynomial ring
Romanoff's theorem

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10.1016/j.ffa.2016.11.002

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title = "An explicit polynomial analogue of Romanoff's theorem",

abstract = "Given a polynomial g of positive degree over a finite field, we show that the proportion of polynomials of degree n, which can be written as h+gk, where h is an irreducible polynomial of degree n and k is a nonnegative integer, has order of magnitude 1/deg⁡g.",

keywords = "Irreducible polynomial, Multiplicative order, Polynomial ring, Romanoff's theorem",

author = "Shparlinski, {Igor E.} and Weingartner, {Andreas J.}",

year = "2017",

month = mar,

day = "1",

doi = "10.1016/j.ffa.2016.11.002",

language = "English",

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journal = "Finite Fields and their Applications",

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AU - Shparlinski, Igor E.

AU - Weingartner, Andreas J.

PY - 2017/3/1

Y1 - 2017/3/1

N2 - Given a polynomial g of positive degree over a finite field, we show that the proportion of polynomials of degree n, which can be written as h+gk, where h is an irreducible polynomial of degree n and k is a nonnegative integer, has order of magnitude 1/deg⁡g.

AB - Given a polynomial g of positive degree over a finite field, we show that the proportion of polynomials of degree n, which can be written as h+gk, where h is an irreducible polynomial of degree n and k is a nonnegative integer, has order of magnitude 1/deg⁡g.

KW - Irreducible polynomial

KW - Multiplicative order

KW - Polynomial ring

KW - Romanoff's theorem

UR - http://www.scopus.com/inward/record.url?scp=84996606913&partnerID=8YFLogxK

U2 - 10.1016/j.ffa.2016.11.002

DO - 10.1016/j.ffa.2016.11.002

M3 - Article

AN - SCOPUS:84996606913

SN - 1071-5797

VL - 44

SP - 22

EP - 33

JO - Finite Fields and their Applications

JF - Finite Fields and their Applications

ER -

An explicit polynomial analogue of Romanoff's theorem

Abstract

Keywords

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