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Third Italian Number Theory Meeting

Algebraic equations with lacunary polynomials and the Erd\H{o}s-R\'enyi conjecture

speaker: Vincenzo Mantova (Università di Pisa)

abstract: In 1949, R\'enyi and Erd\H{o}s 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 following more 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.

Tue 22 Sep, 15:30 - 15:55, Aula Contini
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