Diophantine inequalities and quasi-algebraically closed fields

Consider a form $g(x_1,...,x_s)$ of degree $d$, having coefficients in the completion $F_q((1/t))$ of the field of fractions $F_q(t)$ associated to the finite field $F_q$. We establish that whenever $s>d^2$, then the form $g$ takes arbitrarily small values for non-zero arguments $x\in F_q[t]^s$. We provide related results for problems involving distribution modulo $F_q[t]$, and analogous conclusions for quasi-algebraically closed fields in general.

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