1,720,986 research outputs found
Polynomial Dedekind domains with finite residue fields of prime characteristic
Full text linkWe show that every Dedekind domain lying between the polynomial rings
and with the property that its residue fields of
prime characteristic are finite fields is equal to a generalized ring of
integer-valued polynomials, that is, for each prime there
exists a finite subset of transcendental elements over in the
absolute integral closure of the ring of -adic
integers such that . Moreover, we prove that the class
group of is isomorphic to a direct sum of a countable family of finitely
generated abelian groups. Conversely, any group of this kind is the class group
of a Dedekind domain between and .Comment: to appear in the Pacific Journal of Math. (2023
Integral-valued polynomials over sets of algebraic integers of bounded degree
Full text linkLet be a number field of degree with ring of integers . By means of criterion of Gilmer for polynomially dense subsets of the ring of integers of a number field, we show that, if maps every element of of degree to an algebraic integer, then is integral-valued over , that is . A similar property holds if we consider the set of all algebraic integers of degree and a polynomial : if is integral over for every algebraic integer of degree , then is integral over for every algebraic integer of degree smaller than . This second result is established by proving that the integral closure of the ring of polynomials in which are integer-valued over the set of matrices is equal to the ring of integral-valued polynomials over the set of algebraic integers of degree equal to
Primary decomposition of the ideal of polynomials whose fixed divisor is divisible by a prime power
Full text linkWe characterize the fixed divisor of a polynomial in by looking at the contraction of the powers of the maximal ideals of the overring \IZ 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
Transcendental extensions of a valuation domain of rank one
Full text linkLet 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, the valuation domains . 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
Factorization of Integer-Valued Polynomials with Square-Free Denominator
Full text linkWe describe an algorithm to compute the different factorizations of a given image
primitive integer-valued polynomial f(X) = g(X)/d ∈ Q[X], where g ∈ Z[X] and
d ∈ N is square-free, assuming that the factorizations of g(X) in Z[X] and d in Z are
known. We translate this problem into a combinatorial one
The ring of polynomials integral-valued over a finite set of integral elements
Full text linkLet be an integral domain with quotient field and a finite subset of . McQuillan proved that the ring \Int(\Omega,D) of polynomials in which are integer-valued over , that is, such that , is a Pr\"ufer domain if and only if is Pr\"ufer. Under the further assumption that is integrally closed, we generalize his result by considering a finite set of a -algebra which is finitely generated and torsion-free as a -module, and the ring \Int_K(S,A) of integer-valued polynomials over , that is, polynomials over whose image over is contained in . We show that the integral closure of \Int_K(S,A) is equal to the contraction to of \Int(\Omega_S,D_F), for some finite subset of integral elements over contained in an algebraic closure \olK of , where is the integral closure of in . Moreover, the integral closure of \Int_K(S,A) is Pr\"ufer if and only if is Pr\"ufer. The result is obtained by means of the study of pullbacks of the form , where is a monic non-constant polynomial over : we prove that the integral closure of such a pullback is equal to the ring of polynomials over which are integral-valued over the set of roots of in
Parametrization of integral values of polynomials
Full text linkWe will recall a recent result about the classification of those polynomial in one variable with rational coefficients whose image over the integer is equal to the image of an integer coefficients polynomial in possibly many variables. These set is polynomially generated over the integers by a family of polynomials whose denominator is and they have a symmetry with respect to a particular axis. We will also give a description of the linear factors of the bivariate separated polynomial over a number field , which we need to formulate a conjecture for a generalization of the previous result over a generic number fiel
Metrizability of spaces of valuation domains associated to pseudo-convergent sequences
Full text linkLet V be a valuation domain of rank one with quotient field K. We study the set of extensions of V to the field of rational functions K(X) induced by pseudo-convergent sequences of K from a topological point of view, endowing this set either with the Zariski or with the constructible topology. In particular, we consider the two subspaces induced by sequences with a prescribed breadth or with a prescribed pseudo-limit. We give some necessary conditions for the Zariski space to be metrizable (under the constructible topology) in terms of the value group and the residue field of V
The lattice of primary ideals of orders in quadratic number fields
Full text linkLet be an order in a quadratic number field with ring of integers , such that the conductor \f = f D is a prime ideal of , where is a prime. We give a complete description of the \f-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 \f-primary ideals not contained in \f^2. We get three different cases, according to whether the prime number is split, inert or ramified in
Metrizability of spaces of valuation domains associated to pseudo-convergent sequences
Full text linkLet be a valuation domain of rank one with quotient field . We study
the set of extensions of to the field of rational functions induced
by pseudo-convergent sequences of from a topological point of view,
endowing this set either with the Zariski or with the constructible topology.
In particular, we consider the two subspaces induced by sequences with a
prescribed breadth or with a prescribed pseudo-limit. We give some necessary
conditions for the Zariski space to be metrizable (under the constructible
topology) in terms of the value group and the residue field of .Comment: pp. 1-22, final version, to appear in J. Algebra Appl. (2021). arXiv
admin note: substantial text overlap with arXiv:1809.0953
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