| Résume | Call non-standard fewnomial (or sparse/lacunary polynomial) a non-standard polynomial whose number of non-zero terms is finite. The non-standard translation of a conjecture of Rényi and Erdöt;s, proved by Schinzel and then improved by Zannier, says that if the square of a non-standard polynomial is a fewnomial, then the polynomial itself is a fewnomial. With C. Fuchs and Zannier, we proved the more general statement that the ring of fewnomials is integrally closed in the ring of non-standard polynomials. This can be used to show certain properties of covers of multiplicative groups, such as a kind of Bertini irreducibility theorem. I will discuss both standard and non-standard formulations of the theorem, some of the applications, and give a sketch of a new non-standard proof. |
