Abstract
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 n 1+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.
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Croot, E., Hart, D. On sums and products in ℂ[x]. Ramanujan J 22, 33–54 (2010). https://doi.org/10.1007/s11139-010-9219-4
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DOI: https://doi.org/10.1007/s11139-010-9219-4