Counting pairs of polynomials whose product is a cube

Abstract

Let Fq be the finite field with q elements. We consider the number of pairs (f1,f2) with f1f2 being a cube, where fi∈Fq[T] and deg⁡fi=n for i=1,2. By contour integration, we obtain an asymptotic formula in terms of either n or q. This is a polynomial analogue of a result for integers due to de la Bretèche, Kurlberg and Shparlinski [2].