Transcendental extensions of a valuation domain of rank one

Let $V$ be a valuation domain of rank one and quotient field $K$. Let $\overline{\hat{K}}$ be a fixed algebraic closure of the $v$-adic completion $\hat K$ of $K$ and let $\overline{\hat{V}}$ be the integral closure of $\hat V$ in $\overline{\hat{K}}$. We describe a relevant class of valuation domains $W$ of the field of rational functions $K(X)$ which lie over $V$, which are indexed by the elements $\alpha\in\overline{\hat{K}}\cup\{\infty\}$, namely, $W=W_{\alpha}=\{\varphi\in K(X) \mid \varphi(\alpha)\in\overline{\hat{V}}\}$. If $V$ is discrete and $\pi\in V$ is a uniformizer, then a valuation domain $W$ of $K(X)$ is of this form if and only if the residue field degree $[W/M:V/P]$ is finite and $\pi W=M^e$, for some $e\geq 1$, where $M$ is the maximal ideal of $W$. In general, for $\alpha,\beta\in\overline{\hat{K}}$ we have $W_{\alpha}=W_{\beta}$ if and only if $\alpha$ and $\beta$ are conjugated over $\hat K$. Finally, we show that the set $\mathcal{P}^{{\rm irr}}$ of irreducible polynomials over $\hat K$ endowed with an ultrametric distance introduced by Krasner is homeomorphic to the space $\{W_{\alpha} \mid \alpha\in\overline{\hat{K}}\}$ 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 $D$ be an integrally closed domain with quotient field $K$ and $n$ a positive integer. We give a characterization of the polynomials in $K[X]$ which are integer-valued over the set of matrices $M_n(D)$ in terms of their divided differences. A necessary and sufficient condition on $f\in K[X]$ to be integer-valued over $M_n(D)$ is that, for each $k$ less than $n$, the $k$-th divided difference of $f$ is integral-valued on every subset of the roots of any monic polynomial over $D$ of degree $n$. If in addition the intersection of the maximal ideals of finite index is $(0)$ then it is sufficient to check the above conditions on subsets of the roots of monic irreducible polynomials of degree $n$, that is, conjugate integral elements of degree $n$ over $D$.Comment: minor changes, notation made uniform throughout the paper. Fixed a wrong assumption we used in (4), (5) and Thm 4.1: "$D$ has zero Jacobson radical" has to be replaced with "the intersection of the maximal ideals of finite index is $(0)$". 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 $f(X)$ in $\mathbb{Z}[X]$ by looking at the contraction of the powers of the maximal ideals of the overring ${\rm Int}(\mathbb{Z})$ containing $f(X)$. Given a prime $p$ and a positive integer $n$, we also obtain a complete description of the ideal of polynomials in $\mathbb{Z}[X]$ whose fixed divisor is divisible by $p^n$ 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 $V$ be a rank one valuation domain with quotient field $K$. We characterize the subsets $S$ of $V$ for which the ring of integer-valued polynomials ${\rm Int}(S,V)=\{f\in K[X] \mid f(S)\subseteq V\}$ 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 ${\rm Int}(S,V)$ is Pr\"ufer if and only if no element of the algebraic closure $\overline{K}$ of $K$ is a pseudo-limit of a pseudo-monotone sequence contained in $S$, with respect to some extension of $V$ to $\overline{K}$. 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 $O$ be an order in a quadratic number field $K$ with ring of integers $D$, such that the conductor $\mathfrak F = f D$ is a prime ideal of $O$, where $f\in\mathbb Z$ is a prime. We give a complete description of the $\mathfrak F$-primary ideals of $O$. They form a lattice with a particular structure by layers; the first layer, which is the core of the lattice, consists of those $\mathfrak F$-primary ideals not contained in $\mathfrak F^2$. We get three different cases, according to whether the prime number $f$ is split, inert or ramified in $D$.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 $V$ be a valuation domain with quotient field $K$. We show how to describe all extensions of $V$ to $K(X)$ when the $V$-adic completion $\widehat{K}$ 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 $K$ in the topology induced by $V$.Comment: all comments are welcome!

The Zariski-Riemann space of valuation domains associated to pseudo-convergent sequences

Let $V$ be a valuation domain with quotient field $K$. Given a pseudo-convergent sequence $E$ in $K$, we study two constructions associating to $E$ a valuation domain of $K(X)$ lying over $V$, especially when $V$ 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 $K$ 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 $D$ be a commutative domain with field of fractions $K$ and let $A$ be a torsion-free $D$-algebra such that $A \cap K = D$. The ring of integer-valued polynomials on $A$ with coefficients in $K$ 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 $D$. The condition $A \cap K$ 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 $D$, we prove that if $A$ is finitely generated as a $D$-module, then \Int_K(A) is nontrivial if and only if \Int(D) is nontrivial. When $A$ is not necessarily finitely generated but $D$ is Dedekind, we provide necessary and sufficient conditions for \Int_K(A) to be nontrivial. These conditions also allow us to prove that, for $D$ Dedekind, the domain \Int_K(A) has Krull dimension 2