### Abstract

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 λ_{1}, λ_{2}, λ_{3} are non-zero elements in F_{q}((1/T)) satisfying λ_{1}/λ_{2}∉F_{q}(T) and sgn(λ_{1})+sgn(λ_{2})+sgn(λ _{3})=0,then the values of the sumλ_{1}P_{1}+λ_{2}P _{2}+λ_{3}P_{3}, 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)).

Original language | English |
---|---|

Pages (from-to) | 46-61 |

Number of pages | 16 |

Journal | Journal of Number Theory |

Volume | 78 |

Issue number | 1 |

DOIs | |

Publication status | Published - 1999 Sep 1 |

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### Keywords

- Diophantine inequalities
- Hardy-Littlewood method

### ASJC Scopus subject areas

- Algebra and Number Theory

### Cite this

**Diophantine Inequalities for Polynomial Rings.** / Hsu, Chih-Nung.

Research output: Contribution to journal › Article

*Journal of Number Theory*, vol. 78, no. 1, pp. 46-61. https://doi.org/10.1006/jnth.1999.2390

}

TY - JOUR

T1 - Diophantine Inequalities for Polynomial Rings

AU - Hsu, Chih-Nung

PY - 1999/9/1

Y1 - 1999/9/1

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

UR - http://www.scopus.com/inward/record.url?scp=0007372934&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0007372934&partnerID=8YFLogxK

U2 - 10.1006/jnth.1999.2390

DO - 10.1006/jnth.1999.2390

M3 - Article

AN - SCOPUS:0007372934

VL - 78

SP - 46

EP - 61

JO - Journal of Number Theory

JF - Journal of Number Theory

SN - 0022-314X

IS - 1

ER -