TY - JOUR
T1 - Diophantine Inequalities for Polynomial Rings
AU - Hsu, Chih Nung
PY - 1999/9
Y1 - 1999/9
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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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 -