221 research outputs found
Relative polynomial closure and monadically Krull monoids of integer-valued polynomials
Let D be a Krull domain and Int(D) the ring of integer-valued polynomials on
D. For any f in Int(D), we explicitly construct a divisor homomorphism from
[f], the divisor-closed submonoid of Int(D) generated by f, to a finite sum of
copies of (N_0,+). This implies that [f] is a Krull monoid.
For V a discrete valuation domain, we give explicit divisor theories of
various submonoids of Int(V). In the process, we modify the concept of
polynomial closure in such a way that every subset of D has a finite
polynomially dense subset.
The results generalize to Int(S,V), the ring of integer-valued polynomials on
a subset, provided S doesn't have isolated points in v-adic topology.Comment: 12 pages; v.2 contains corrections, in that some necessary conditions
on those subsets S, for which we consider integer-valued polynomials on
subsets, are impose
Polynomial functions on non-commutative rings - a link between ringsets and null-ideal sets
Regarding polynomial functions on a subset of a non-commutative ring ,
that is, functions induced by polynomials in (whose variable commutes
with the coefficients), we show connections between, on one hand, sets such
that the integer-valued polynomials on form a ring, and, on the other hand,
sets such that the set of polynomials in that are zero on is an
ideal of .Comment: 9 pages, conference paper for "advances in algebra ..." at Ton Duc
Thang University, Vietnam, Dec 18-20, 201
Parametrization of Pythagorean triples by a single triple of polynomials
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
Non-unique factorization of polynomials over residue class rings of the integers
We investigate non-unique factorization of polynomials in Z_{p^n}[x] into
irreducibles. As a Noetherian ring whose zero-divisors are contained in the
Jacobson radical, Z_{p^n}[x] is atomic. We reduce the question of factoring
arbitrary non-zero polynomials into irreducibles to the problem of factoring
monic polynomials into monic irreducibles. The multiplicative monoid of monic
polynomials of Z_{p^n}[x] is a direct sum of monoids corresponding to
irreducible polynomials in Z_p[x], and we show that each of these monoids has
infinite elasticity. Moreover, for every positive integer m, there exists in
each of these monoids a product of 2 irreducibles that can also be represented
as a product of m irreducibles.Comment: 11 page
Sylow -groups of polynomial permutations on the integers mod
We describe the Sylow -groups of the group of polynomial permutations of
the integers mod
A construction of integer-valued polynomials with prescribed sets of lengths of factorizations
For an arbitrary finite set S of natural numbers greater 1, we construct an
integer-valued polynomial f, whose set of lengths in Int(Z) is S. The set of
lengths of f is the set of all natural numbers n, such that f has a
factorization as a product of n irreducibles in Int(Z)={g in Q[x] | g(Z)
contained in Z}.Comment: To appear in Monatshefte f\"ur Mathematik; 11 page
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