Diophantine Inequalities for Polynomial Rings Original Research Article

We study the Hardy–Littlewood method for the Laurent series field imageq((1/T)) over the finite field imageq with q elements. We show that if λ1, λ2, λ3 are non-zero elements in imageq((1/T)) satisfying λ1/λ2negated set membershipimageq(T) and sgn(λ1)+sgn(λ2)+sgn(λ3)=0,then the values of the sumλ1P1+λ2P2+λ3P3, as Pi (i=1, 2, 3) run independently through all monic irreducible polynomials in imageq[T], are everywhere dense on the “non-Archimedean” line imageq((1/T)), where sgn(f)set membership, variantimageq denotes the leading coefficient of fset membership, variantimageq((1/T)).