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 ∈ 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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