On sums and products in ℂ[x]

We prove that there exists an absolute constant c>0 such that if A is a set of n monic polynomials, and if the product set A.A has at most n1+c elements, then |A+A|≫n2. This can be thought of as step towards proving the Erdős–Szemerédi sum-product conjecture for polynomial rings. We also show that under a suitable generalization of Fermat’s Last Theorem, the same result holds for the integers. The methods we use to prove are a mixture of algebraic (e.g. Mason’s theorem) and combinatorial (e.g. the Ruzsa–Plunnecke inequality) techniques.