Diophantine Inequalities for Polynomial Rings

Chih Nung Hsu

doi:10.1006/jnth.1999.2390

Diophantine Inequalities for Polynomial Rings

Chih Nung Hsu^*

^*此作品的通信作者

數學系

研究成果: 雜誌貢獻 › 期刊論文 › 同行評審

2 引文斯高帕斯（Scopus）

摘要

We study the Hardy-Littlewood method for the Laurent series field F_q((1/T)) over the finite field F_q with q elements. We show that if λ₁, λ₂, λ₃ are non-zero elements in F_q((1/T)) satisfying λ₁/λ₂∉F_q(T) and sgn(λ₁)+sgn(λ₂)+sgn(λ ₃)=0,then the values of the sumλ₁P₁+λ₂P ₂+λ₃P₃, as P_i (i=1, 2, 3) run independently through all monic irreducible polynomials in F_q[T], are everywhere dense on the "non-Archimedean" line F_q((1/T)), where sgn(f)∈F_q denotes the leading coefficient of f∈F_q((1/T)).

原文	英語
頁（從 - 到）	46-61
頁數	16
期刊	Journal of Number Theory
卷	78
發行號	1
DOIs	https://doi.org/10.1006/jnth.1999.2390
出版狀態	已發佈 - 1999 9月

ASJC Scopus subject areas

代數與數理論

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10.1006/jnth.1999.2390

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title = "Diophantine Inequalities for Polynomial Rings",

abstract = "We study the Hardy-Littlewood method for the Laurent series field Fq((1/T)) over the finite field Fq with q elements. We show that if λ1, λ2, λ3 are non-zero elements in Fq((1/T)) satisfying λ1/λ2∉Fq(T) and sgn(λ1)+sgn(λ2)+sgn(λ 3)=0,then the values of the sumλ1P1+λ2P 2+λ3P3, as Pi (i=1, 2, 3) run independently through all monic irreducible polynomials in Fq[T], are everywhere dense on the {"}non-Archimedean{"} line Fq((1/T)), where sgn(f)∈Fq denotes the leading coefficient of f∈Fq((1/T)).",

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AU - Hsu, Chih Nung

PY - 1999/9

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N2 - We study the Hardy-Littlewood method for the Laurent series field Fq((1/T)) over the finite field Fq with q elements. We show that if λ1, λ2, λ3 are non-zero elements in Fq((1/T)) satisfying λ1/λ2∉Fq(T) and sgn(λ1)+sgn(λ2)+sgn(λ 3)=0,then the values of the sumλ1P1+λ2P 2+λ3P3, as Pi (i=1, 2, 3) run independently through all monic irreducible polynomials in Fq[T], are everywhere dense on the "non-Archimedean" line Fq((1/T)), where sgn(f)∈Fq denotes the leading coefficient of f∈Fq((1/T)).

AB - We study the Hardy-Littlewood method for the Laurent series field Fq((1/T)) over the finite field Fq with q elements. We show that if λ1, λ2, λ3 are non-zero elements in Fq((1/T)) satisfying λ1/λ2∉Fq(T) and sgn(λ1)+sgn(λ2)+sgn(λ 3)=0,then the values of the sumλ1P1+λ2P 2+λ3P3, as Pi (i=1, 2, 3) run independently through all monic irreducible polynomials in Fq[T], are everywhere dense on the "non-Archimedean" line Fq((1/T)), where sgn(f)∈Fq denotes the leading coefficient of f∈Fq((1/T)).

KW - Diophantine inequalities

KW - Hardy-Littlewood method

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Diophantine Inequalities for Polynomial Rings

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