JournalsrlmVol. 20 , No. 1DOI 10.4171/rlm/534

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

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.