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Counting colorings on varieties.

Fernando Rodríguez-Villegas — 2007

Publicacions Matemàtiques

We present a combinatorial mechanism for counting certain objects associated to a variety over a finite field. The basic example is that of counting conjugacy classes of the general linear group. We discuss how the method applies to counting these and also to counting unipotent matrices and pairs of commuting matrices. [Proceedings of the Primeras Jornadas de Teoría de Números (Vilanova i la Geltrú (Barcelona), 30 June - 2 July 2005)].

Concordant sequences and integral-valued entire functions

Jonathan PilaFernando Rodriguez Villegas — 1999

Acta Arithmetica

A classic theorem of Pólya shows that the function 2 z is the “smallest” integral-valued entire transcendental function. A variant due to Gel’fond applies to entire functions taking integral values on a geometric progression of integers, and Bézivin has given a generalization of both results. We give a sharp formulation of Bézivin’s result together with a further generalization.

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