Wednesday, 05 February, 2014
SPEAKER: Prof. Giulio Peruginelli, UT
TITLE: About some new rings of integer-valued polynomials.
ABSTRACT: In this talk we will introduce some new rings of integer-valued polynomials introduced and studied recently by Loper, Frisch and Werner. Given a domain D with quotient field K and a
torsion-free D-algebra A (possibly with zero divisors and non-commutative) we consider polynomials in K[X] which map every element of A into A. The set of such polynomials forms a ring denoted by Int_K(A). If D is a residually finite ring and A is finitely generated as a D-module, we show that the integral closure of Int_K(A) is equal to another ring of integer-valued polynomials, namely those polynomials in K[X] which map the roots of the minimal polynomials of all the elements of A into elements which are integral over D (joint result with N. Werner). This result generalizes a result of Gilmer, Heinzer and Lantz for the integral closure of the classical ring Int(D), in the case that D is a residually finite domain. Finally, we will show an application of this result to polynomially dense subsets of the set of integral elements over a residually finite domain D whose degree over K is bounded.