Let f1(x), . . . , fk(x) 2 Fq[t1, . . . ,tr][x1, . . . , xs] be polynomials in x with coecients in Fq[t1, . . . ,tr]. Suppose that fj (0) = 0 (1 ? j ? k), and let {0} ? Al Fq (1 ? l ? s). Provided that the number of variables s is large enough in terms of q, r, the cardinalities of the sets |Al|, and the degrees of the polynomials fj (x), there exists a non-zero common solution to the system of equations fj (x) = 0 (1 ? j ? k), where each xl (1 ? l ? s) is a polynomial in t with coecients in Al. We also establish similar results for systems of congruences over Dedekind domains and systems of inequalities over Fq((1/t)).
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