Algebraic equations with spare polynomials and the Erdős-Rényi conjecture

Vincenzo Mantova (Scuola Normale Superiore)

Frank Adams 1,

In 1949, Rényi and Erdős independently conjectured that given a polynomial g(X) over the complex numbers, if we have a bound on the number of non-zero terms of the square of g(X), then there is a bound on the number of terms of g(X) itself. This was proved by Schinzel in 1987, actually with any power in place of the square, and he asked whether the same is true for the composition f(g(X)), where f(Y) is another given polynomial. This was finally proved by Zannier in 2008.    

In a joint work with C. Fuchs and U. Zannier, we extend the result to the most general case: if g(X) is the root of a polynomial F(Y) whose coefficients are themselves polynomials in X with a bounded number of terms, then g(X) is at least the ratio of two polynomials with a bounded number of terms. This can be shown to imply the previous statements. The result has also several implications regarding irreducibility problems and integral points, and a nice non-standard interpretation

Import this event to your Outlook calendar
▲ Up to the top