Roth's theorem on systems of linear forms in function fields

Abstract

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 a matrix Y = ` ai,j ∈ FR×S q satisfying ai,1 + · · ·+ai,S = 0 (1 ≤ i ≤ R), let DY (SN ) denote the maximal cardinality of a set A ⊆ SN for which the equations ai,1x1 + · · ·+ai,SxS = 0 (1 ≤ i ≤ R) are never satisfied simultaneously by distinct elements x1, . . . , xS ∈ A. Under certain assumptions on Y , we prove an upper bound of the form DY (SN ) ≤ q (C/N) for positive constants C and γ.