Diophantine inequalities for polynomial rings

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 lambda(1), lam bda(2), lambda(3) are non-zero elements in F-q((1/T)) satisfying lambda(1)/ lambda(2) is not an element of F-q(T) and sgn(lambda(1)) + sgn(lambda(2)) + sgn(lambda(3)) = 0, then the values of the sum lambda(1)P(1) + lambda(2)P(2) + lambda(3)P(3), as P-i (i = 1, 2, 3) run independently through all monic irreducible polyno mials in F-q[T], are everywhere dense on the "non-Archimedean" line F-q((1/ T)), where sgn(f) is an element of F-q denotes the leading coefficient of f is an element of F-q((1/T)). (C) 1999 Academic Press.