Riv. Mat. Univ. Parma, Vol. 7, No. 1, 2016

Vincenzo Mantova[1]

Algebraic equations with lacunary polynomials and the Erdős-Rényi conjecture

Pages: 239-246
Accepted: 10 March 2016
Mathematics Subject Classification (2010): 11C08, 12E05, 12Y05, 14G05, 14J99, 11U10.
Keywords: Lacunary polynomial, sparse polynomial, fewnomial, Vojta's conjecture, Bertini's irreducibility theorem, multiplicative group.
[1] : School of Mathematics, University of Leeds, Leeds, LS2 9JT, United Kingdom

The author acknowledges the support by the ERC-AdG 267273 "Diophantine Problems".

Abstract: In 1947, Rényi, Kalmár and Rédei discovered some special polynomials $$p(x) \in \mathbb{C}[x]$$ for which the square $$p(x)^{2}$$ has fewer non-zero terms than $$p(x)$$. Rényi and Erdős then conjectured that if the number of terms of $$p(x)$$ grows to infinity, then the same happens for $$p(x)^{2}$$. The conjecture was later proved by Schinzel, strengthened by Zannier, and a 'final' generalisation was proved by C. Fuchs, Zannier and the author. This note is a survey of the known results, with a focus on the applications of the latest generalisation.

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