On sums and products in ℂ[x]
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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.
Keywords
Sum-product Mason’s theorem ABC theorem Erdős–Szemerédi conjecture Fermat’s last theoremMathematics Subject Classification (2000)
11B30 11C08Preview
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