# On the number of terms of a power of a polynomial

### Umberto Zannier

Scuola Normale Superiore, Pisa, Italy### Andrzej Schinzel

Polish Academy of Sciences, Warszawa, Poland

## Abstract

Let f(x) be a polynomial with complex coefficients. Rényi and independently Erdős in 1949 conjectured that a bound for the number of terms of f(x)2 implies a bound for the number of terms of f(x). In 1987 Schinzel found a proof of this conjecture, actually for all powers f(x)l, and he gave some explicit bounds. The aim of this paper is to improve such inequalities in a substantial way.

## Cite this article

Umberto Zannier, Andrzej Schinzel, On the number of terms of a power of a polynomial. Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. 20 (2009), no. 1, pp. 95–98

DOI 10.4171/RLM/534