A Generalization of Roth’s Theorem in Function Fields
نویسنده
چکیده
Let Fq[t] denote the polynomial ring over the finite field Fq, and let SN denote the subset of Fq[t] containing all polynomials of degree strictly less than N . For non-zero elements r1, ⋅ ⋅ ⋅ , rs of Fq satisfying r1 + ⋅ ⋅ ⋅ + rs = 0, let Dr(SN ) denote the maximal cardinality of a set A ⊆ SN which contains no non-trivial solution of r1x1 + ⋅ ⋅ ⋅+ rsxs = 0 with xi ∈ A (1 ≤ i ≤ s). We prove that Dr(SN )≪ ∣SN ∣/(logq ∣SN ∣)s−2.
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