# 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.