(Chapel Hill Conf. 2003), Dekker 2005, pp 249-254.

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\in D by a (n × n)-integer-matrix-valued polynomial (a polynomial in K[x] that maps every (n × n) matrix over D to a matrix with entries in D), except in the trivial case where a-b is a unit of D. (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 Mathematics Subject Classification: Primary 13F20; Secondary 13B25, 11C08, 15A36, 16B99.

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