Unlikely intersections and multiple roots of sparse polynomials

Francesco Amoroso (1) , Martín Sombra (2) , Umberto Zannier (3)

1 LMNO - Laboratoire de Mathématiques Nicolas Oresme
2 UNIROUEN - Université de Rouen Normandie
3 SNS - Scuola Normale Superiore di Pisa

We present a structure theorem for the multiple non-cyclotomic irre-ducible factors appearing in the family of all univariate polynomials with a given set of coefficients and varying exponents. Roughly speaking, this result shows that the multiple non-cyclotomic irreducible factors of a sparse polynomial, are also sparse. To prove this, we give a variant of a theorem of Bombieri and Zannier on the intersection of a fixed subvariety of codimension 2 of the multiplicative group with all the torsion curves, with bounds having an explicit dependence on the height of the subvariety. We also use this latter result to give some evidence on a conjecture of Bolognesi and Pirola.

Domaines: Théorie des nombres [math.NT]

Francesco Amoroso, Martín Sombra, Umberto Zannier. Unlikely intersections and multiple roots of sparse polynomials. Mathematische Zeitschrift, 2017, ⟨10.1007/s00209-017-1860-9⟩. ⟨hal-01081416v4⟩