37 research outputs found
Transcendental extensions of a valuation domain of rank one
Let be a valuation domain of rank one and quotient field . Let
be a fixed algebraic closure of the -adic completion
of and let be the integral closure of in . We describe a relevant class of valuation domains
of the field of rational functions which lie over , which are
indexed by the elements , namely,
.
If is discrete and is a uniformizer, then a valuation domain
of is of this form if and only if the residue field degree
is finite and , for some , where is the maximal ideal
of . In general, for we have
if and only if and are conjugated over
. Finally, we show that the set of
irreducible polynomials over endowed with an ultrametric distance
introduced by Krasner is homeomorphic to the space endowed with the Zariski topology.Comment: accepted for publication in the Proceedings of the AMS (2016);
comments are welcome
Integer-valued polynomials over matrices and divided differences
Let be an integrally closed domain with quotient field and a
positive integer. We give a characterization of the polynomials in which
are integer-valued over the set of matrices in terms of their divided
differences. A necessary and sufficient condition on to be
integer-valued over is that, for each less than , the -th
divided difference of is integral-valued on every subset of the roots of
any monic polynomial over of degree . If in addition the intersection of
the maximal ideals of finite index is then it is sufficient to check the
above conditions on subsets of the roots of monic irreducible polynomials of
degree , that is, conjugate integral elements of degree over .Comment: minor changes, notation made uniform throughout the paper. Fixed a
wrong assumption we used in (4), (5) and Thm 4.1: " has zero Jacobson
radical" has to be replaced with "the intersection of the maximal ideals of
finite index is ". Keywords: Integer-valued polynomial, Divided
differences, Matrix, Integral element, Polynomial closure, Pullback. In
Monatshefte f\"ur Mathematik, 201
Primary decomposition of the ideal of polynomials whose fixed divisor is divisible by a prime power
We characterize the fixed divisor of a polynomial in
by looking at the contraction of the powers of the maximal ideals of the
overring containing . Given a prime and a
positive integer , we also obtain a complete description of the ideal of
polynomials in whose fixed divisor is divisible by in
terms of its primary components.Comment: Fixed typos in (9) and (12
Pr\"ufer intersection of valuation domains of a field of rational functions
Let be a rank one valuation domain with quotient field . We
characterize the subsets of for which the ring of integer-valued
polynomials is a Pr\"ufer
domain. The characterization is obtained by means of the notion of
pseudo-monotone sequence and pseudo-limit in the sense of Chabert, which
generalize the classical notions of pseudo-convergent sequence and pseudo-limit
by Ostrowski and Kaplansky, respectively. We show that is
Pr\"ufer if and only if no element of the algebraic closure of
is a pseudo-limit of a pseudo-monotone sequence contained in , with
respect to some extension of to . This result expands a
recent result by Loper and Werner.Comment: to appear in J. Algebra. All comments are welcome. Keywords: Pr\"ufer
domain, pseudo-convergent sequence, pseudo-limit, residually transcendental
extension, integer-valued polynomia
The lattice of primary ideals of orders in quadratic number fields
Let be an order in a quadratic number field with ring of integers
, such that the conductor is a prime ideal of , where
is a prime. We give a complete description of the -primary ideals of . They form a lattice with a particular structure by
layers; the first layer, which is the core of the lattice, consists of those
-primary ideals not contained in . We get three
different cases, according to whether the prime number is split, inert or
ramified in .Comment: Keywords: Orders, Conductor, Primary ideal, Lattice of ideals. to
appear in Int. J. Number Theory (2016
Extending valuations to the field of rational functions using pseudo-monotone sequences
Let be a valuation domain with quotient field . We show how to
describe all extensions of to when the -adic completion
is algebraically closed, generalizing a similar result obtained
by Ostrowski in the case of one-dimensional valuation domains. This is
accomplished by realizing such extensions by means of pseudo-monotone
sequences, a generalization of pseudo-convergent sequences introduced by
Chabert. We also show that the valuation rings associated to pseudo-convergent
and pseudo-divergent sequences (two classes of pseudo-monotone sequences)
roughly correspond, respectively, to the closed and the open balls of in
the topology induced by .Comment: all comments are welcome!
The Zariski-Riemann space of valuation domains associated to pseudo-convergent sequences
Let be a valuation domain with quotient field . Given a
pseudo-convergent sequence in , we study two constructions associating
to a valuation domain of lying over , especially when has
rank one. The first one has been introduced by Ostrowski, the second one more
recently by Loper and Werner. We describe the main properties of these
valuation domains, and we give a notion of equivalence on the set of
pseudo-convergent sequences of characterizing when the associated valuation
domains are equal. Then, we analyze the topological properties of the
Zariski-Riemann spaces formed by these valuation domains.Comment: any comment is welcome! Trans. Amer. Math. Soc. 373 (2020), no. 11,
7959-799
Integral-valued polynomials over sets of algebraic integers of bounded degree
AbstractLet K be a number field of degree n with ring of integers OK. By means of a criterion of Gilmer for polynomially dense subsets of the ring of integers of a number field, we show that, if h∈K[X] maps every element of OK of degree n to an algebraic integer, then h(X) is integral-valued over OK, that is, h(OK)⊂OK. A similar property holds if we consider the set of all algebraic integers of degree n and a polynomial f∈Q[X]: if f(α) is integral over Z for every algebraic integer α of degree n, then f(β) is integral over Z for every algebraic integer β of degree smaller than n. This second result is established by proving that the integral closure of the ring of polynomials in Q[X] which are integer-valued over the set of matrices Mn(Z) is equal to the ring of integral-valued polynomials over the set of algebraic integers of degree equal to n
Non-triviality conditions for integer-valued polynomial rings on algebras
Let be a commutative domain with field of fractions and let be a torsion-free -algebra such that . The ring of integer-valued polynomials on with coefficients in is \Int_K(A) = \{f \in K[X] \mid f(A) \subseteq A\}, which generalizes the classic ring \Int(D) = \{f \in K[X] \mid f(D) \subseteq D\} of integer-valued polynomials on .
The condition implies that D[X] \subseteq \Int_K(A) \subseteq \Int(D), and we say that \Int_K(A) is nontrivial if \Int_K(A) \ne D[X]. For any integral domain , we prove that if is finitely generated as a -module, then \Int_K(A) is nontrivial if and only if \Int(D) is nontrivial. When is not necessarily finitely generated but is Dedekind, we provide necessary and sufficient conditions for \Int_K(A) to be nontrivial. These conditions also allow us to prove that, for Dedekind, the domain \Int_K(A) has Krull dimension 2