Polynomial separation of points in algebras

Abstract

We show that for a wide variety of domains, including all Dedekind rings with finite residue fields, it is possible to separate any two algebraic elements a, b of an algebra over the quotient field by integer-valued polynomials (i.e. to map a and b to 0 and 1, respectively, with a polynomial in K[x] that maps every element of D to an element of D), provided only that the minimal polynomials of a and b in K[x] are co-prime (which is obviously necessary). In contrast to this, it is impossible to separate a, b ∈ D by a n × n-integermatrix-valued polynomial (a polynomial in K[x] that maps every n×n matrix over D to a matrix with entries inD), except in the trivial case where a−b is a unit ofD. (This is despite the fact that the ring of n× n-integer-matrix-valued polynomials for any fixed n is non-trivial whenever the ring of integer-valued polynomials is non-trivial.) 2000 Math. Subj. Classification: Primary 13F20; Secondary 13B25, 11C08, 15A36, 16B99.