A generalization of Roth's theorem in function fields

Abstract

Let 𝔽q[t] denote the polynomial ring over the finite field 𝔽q, and let formula 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 formula denote the maximal cardinality of a set formula which contains no non-trivial solution of r1x1 + ⋯ + rsxs = 0 with xi ∈ A (1 ≤ i ≤ s). We prove that formula.