Abstract
Consider a form g(x 1,...,x s ) of degree d, having coefficients in the completion \(\mathbb{F}_q ((1/t))\) of the field of fractions \(\mathbb{F}_q (t)\) associated to the finite field \(\mathbb{F}_q\). We establish that whenever s > d 2, then the form g takes arbitrarily small values for non-zero arguments \(x \in \mathbb{F}_q [t]^s\). We provide related results for problems involving distribution modulo \(\mathbb{F}_q [t]\), and analogous conclusions for quasi-algebraically closed fields in general.
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CVS was supported in part by NSF Grant DMS-0635607 and NSA Young Investigators Grant H98230-10-1-0155.
TDW was supported first by an NSF grant, and subsequently by a Royal Society Wolfson Research Merit Award.
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Spencer, C.V., Wooley, T.D. Diophantine inequalities and quasi-algebraically closed fields. Isr. J. Math. 191, 721–738 (2012). https://doi.org/10.1007/s11856-012-0004-5
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DOI: https://doi.org/10.1007/s11856-012-0004-5