Polynomials defining many units

Ángel del Río Mateos - Universidad de Murcia
Date and time
Tuesday, January 19, 2016 at 5:00 PM - 16:45 rinfresco, 17:00 inizio seminario
Ca' Vignal - Piramide, Floor 0, Hall Verde
Programme Director
Francesca Mantese
External reference
Publication date
January 11, 2016
Computer Science  


Let P be a polynomial in one variable and let g be a group element of order n. We say that P defines a unit in order n if P(g) is a unit of the group of integral group ring of the group generated by g with integral coefficients. Marciniak and Sehgal introduced the notion of generic unit and give a characterization of them. A generic unit is a monic polynomial P with integral coefficients such that there is a positive integer D such that P defines a unit in order n for every n coprime with D. Obviously a generic unit defines units in infinitely many orders. We prove a converse of this result. More precisely we prove that if P is a polynomial with integral coefficients, non-necessarily monic, such that P defines units on infinitely many units, then P=Q or P=-Q for Q a generic unit.

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