A Prime Analogue of Roth's Theorem in Function Fields

Abstract

Abstract. Let Fq[t] denote the polynomial ring over the nite eld Fq, and let PR denote the subset of Fq[t] containing all monic irreducible polynomials of degree R. For non-zero elements r = (r1; r2; r3) of Fq satisfying r1 + r2 + r3 = 0, let D(PR) = Dr(PR) denote the maximal cardinality of a set AR PR which contains no non-trivial solution of r1x1 + r2x2 + r3x3 = 0 with xi 2 AR (1 i 3). By applying the polynomial Hardy-Littlewood circle method, we prove that D(PR) q jPRj=(log log log log jPRj).