Smooth Polynomial Solutions to a Ternary Additive Equation

Abstract

AbstractLet Fq[T] be the ring of polynomials over the finite field of q elements and Y a large integer. We say a polynomial in Fq[T] is Y-smooth if all of its irreducible factors are of degree at most Y. We show that a ternary additive equation a + b = c over Y-smooth polynomials has many solutions. As an application, if S is the set of first s primes in Fq[T] and s is large, we prove that the S-unit equation u + v = 1 has at least exp solutions.