Let D be a Dedekind domain with infinitely many maximal ideals, all of finite index, and K its quotient field. LetGiven any finite multiset {k 1 , . . . , kn} of integers greater than 1, we construct a polynomial in Int(D) which has exactly n essentially different factorizations into irreducibles in Int(D), the lengths of these factorizations being k 1 , . . . , kn. We also show that there is no transfer homomorphism from the multiplicative monoid of Int(D) to a block monoid.2010 Mathematics Subject Classification. 13A05; 13B25, 13F20, 11R04, 11C08.
Let D be a domain with quotient field K and A a D-algebra. We call a
polynomial with coefficients in K that maps every element of A to an element of
A "integer-valued on A". For commutative A we also consider integer-valued
polynomials in several variables. For an arbitrary domain D and I an arbitrary
ideal of D we show I-adic continuity of integer-valued polynomials on A. For
Noetherian one-dimensional D, we determine the spectrum and Krull dimension of
the ring Int_D(A) of integer-valued polynomials on A. We do the same for the
ring of polynomials with coefficients in M_n(K), the K-algebra of n x n
matrices, that map every matrix in M_n(D) to a matrix in M_n(D).Comment: 17 pages; a glitch in the published version (J.Algebra 373 (2013)
414-425) has been corrected in this post-preprint, namely, in Prop. 6.2 and
Thm. 6.3, the assumption "zero Jacobson radical" needs to be replaced by the
stronger assumption "intersection of maximal ideals of finite index is zero
It is well known that Pythagorean triples can be parametrized by two triples
of polynomials with integer coefficients. We show that no single triple of
polynomials with integer coefficients in any number of variables is sufficient,
but that there exists a parametrization of Pythagorean triples by a single
triple of integer-valued polynomials.Comment: to appear in J. Pure Appl. Algebr
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