On sums and products in C[x]

TL;DR

This paper explores sum-product phenomena in polynomial rings, establishing conditional and unconditional results related to the Erdős–Szemerédi conjecture, with implications for algebraic structures and number theory.

Contribution

It proves a weak form of the sum-product conjecture in C[x], conditional on a Fermat's Last Theorem variant, and unconditionally for sets of monic polynomials.

Findings

Conditional sum-product bound in C[x] based on Fermat's Last Theorem

Unconditional sum-product result for monic polynomial sets

Extension of Bourgain and Chang's theorem to polynomial rings

Abstract

We show that under the assumption of a 24-term version of Fermat's Last Theorem, there exists an absolute constant c > 0 such that if S is a set of n > n_0 positive integers satisfying |S.S| < n^(1+c), then the sumset S.S satisfies |S+S| >> n^2. In other words, we prove a weak form of the Erdos-Szemeredi sum-product conjecture, conditional on an extension of Fermat's Last Theorem. Unconditionally, we prove this theorem for when S is a set of n monic polynomials. We also prove an analogue of a theorem of Bourgain and Chang for the ring C[x].