**Vincenzo Mantova**^{[1]}

*
Algebraic equations with lacunary polynomials and the Erdős-Rényi conjecture
*

**Pages:** 239-246

**Received:** 2 January 2016

**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.

**Author address:**

[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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