On sums and products in C[x]

Details On sums and products in C[x] On sums and products in C[x]

2008-12-12

arxiv.org/abs/0812.2286v3

Mathematics

Combinatorics

We added a conditional theorem to the paper, which holds under a certain generalization of Fermat's Last Theorem; and, we adde

Scientific paper

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].

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