arXiv subjects

Giulio Peruginelli

Publications and source records attributed to Giulio Peruginelli.

At least 19 recordsLinked to original sources

arXiv2026

Approximating DVRs by elements of bounded ramification

Let $V$ be a DVR with quotient field $K$ and perfect residue field, $v$ be the valuation on $K$ associated with $V$, $\widehat K$ be the completion of $K$, and $\mathbb{K}$ be the completion of an algebraic closure $\overline{\widehat{K}}$ of $\widehat K$. We show that a DVR of the rational function field $K(X)$ which is a residually algebraic extension of $V$ is necessarily of the form $V_{\alpha}=\{\phi\in K(X)\mid v(\phi(\alpha))\geq0\}$, for an element $\alpha$ of $\mathbb{K}$ transcendental over $K$, and that $\alpha$ is algebraic over $\widehat K$ if and only if the residue field extension is finite. Not every such $V_{\alpha}$ is a DVR, however, and we characterize the $\alpha \in\mathbb{K}$ for which $V_{\alpha}$ is a DVR: they are the elements which can be approximated by algebraic elements in $\overline{\widehat{K}}$ with bounded ramification indexes. Combining the two results, we obtain a complete description of the extensions of $V$ to $K(X)$ which are DVRs and residually algebraic over $V$, together with a criterion for each of the two cases to occur. The proofs rest on a bound for the ramification index in a compositum, valid with no tameness assumption and under a separability hypothesis on one residue field extension only; we show that the inequality cannot be improved to a divisibility and that this hypothesis cannot be dropped. We also show that the hypothesis of discreteness cannot be omitted. Furthermore, we show that the set of $\alpha\in\mathbb K$ for which $V_{\alpha}$ is a DVR is a subfield of $\mathbb K$, which sits properly between $\overline{\widehat{K}}$ and $\mathbb K$, and corresponds to those elements $\alpha$ for which the value group of $\widehat K(\alpha)$ is discrete.

math.AC↗

arXiv2026

Geometric configuration of integrally closed Noetherian domains

In this paper, we completely describe the family of integrally closed Noetherian domains between $\mathbb{Z}[X]$ and $\mathbb{Q}[X]$. We accomplish this result by classifying the Krull domains between these two polynomial rings. To this end, we first describe the DVRs of $\mathbb{Q}(X)$ lying over $\mathbb{Z}_{(p)}$ for some prime $p \in \mathbb{Z}$, by distinguishing them according to whether the extension of the residue fields is algebraic or transcendental. We unify the known descriptions of such valuations by considering ultrametric balls in $\mathbb{C}_p$, the completion of the algebraic closure of the field $\mathbb{Q}_p$ of $p$-adic numbers. We then study when the intersection $R$ of such DVRs with $\mathbb{Q}[X]$ is of finite character, so that $R$ is a Krull domain, and we finally compute the divisor class group of $R$. It turns out that such a ring is formed by those polynomials which simultaneously map a finite union of ultrametric balls of $\mathbb{C}_p$ to its valuation domain $\mathbb{O}_p$, as $p\in\mathbb{Z}$ ranges through the set of primes. By a result of Heinzer, the Krull domains of this class are precisely the integrally closed Noetherian domains between $\mathbb{Z}[X]$ and $\mathbb{Q}[X]$. This novel approach provides a geometric understanding of this class of integrally closed domains. Furthermore, we also describe the UFDs between $\mathbb{Z}[X]$ and $\mathbb{Q}[X]$.

math.AC↗

arXiv2025

A classification of Prufer domains of integer-valued polynomials on algebras

Let $D$ be an integrally closed domain with quotient field $K$ and $A$ a torsion-free $D$-algebra that is finitely generated as a $D$-module and such that $A\cap K=D$. We give a complete classification of those $D$ and $A$ for which the ring $\text{Int}_K(A)=\{f\in K[X] \mid f(A)\subseteq A\}$ is a Pr\"ufer domain. If $D$ is a semiprimitive domain, then we prove that $\text{Int}_K(A)$ is Pr\"ufer if and only if $A$ is commutative and isomorphic to a finite direct product of almost Dedekind domains with finite residue fields, each of them satisfying a double-boundedness condition on its ramification indices and residue field degrees.

math.RA↗

arXiv2024

Nontriviality of rings of integral-valued polynomials

Let $S$ be a subset of $\overline{\mathbb Z}$, the ring of all algebraic integers. A polynomial $f \in \mathbb Q[X]$ is said to be integral-valued on $S$ if $f(s) \in \overline{\mathbb Z}$ for all $s \in S$. The set $\text{Int}_{\mathbb Q}(S,\overline{\mathbb Z})$ of all integral-valued polynomials on $S$ forms a subring of $\mathbb Q[X]$ containing $\mathbb Z[X]$. We say that $\text{Int}_{\mathbb Q}(S,\overline{\mathbb Z})$ is trivial if $\text{Int}_{\mathbb Q}(S,\overline{\mathbb Z}) = \mathbb Z[X]$, and nontrivial otherwise. We give a collection of necessary and sufficient conditions on $S$ in order $\text{Int}_{\mathbb Q}(S,\overline{\mathbb Z})$ to be nontrivial. Our characterizations involve, variously, topological conditions on $S$ with respect to fixed extensions of the $p$-adic valuations to $\overline{\mathbb Q}$; pseudo-monotone sequences contained in $S$; ramification indices and residue field degrees; and the polynomial closure of $S$ in $\overline{\mathbb Z}$.

math.NT↗

arXiv2024

A topological approach to key polynomials

In this paper we present characterizations of the sets of key polynomials and abstract key polynomials for a valuation $\mu$ of $K(x)$, in terms of (ultrametric) balls in the algebraic closure $\overline K$ of $K$ with respect to $v$, a fixed extension of $\mu_{\mid K}$ to $\overline K$. In particular, we show that the ways of augmenting $\mu$, in the sense of Mac Lane, are in one-to-one correspondence with the partition of a fixed closed ball $B(a,\delta)$ associated to $\mu$ into the disjoint union of open balls $B^\circ(a_i,\delta)$, modulo the action of the decomposition group of $v$. We also present a similar characterization for the set of limit key polynomials for an increasing family of valuations of $K(x)$.

math.AC↗

arXiv2023

Stacked Pseudo-Convergent Sequences and Polynomial Dedekind Domains

Let $p\in\mathbb Z$ be a prime, $\overline{\mathbb Q_p}$ a fixed algebraic closure of the field of $p$-adic numbers and $\overline{\mathbb Z_p}$ the absolute integral closure of the ring of $p$-adic integers. Given a residually algebraic torsion extension $W$ of $\mathbb Z_{(p)}$ to $\mathbb Q(X)$, by Kaplansky's characterization of immediate extensions of valued fields, there exists a pseudo-convergent sequence of transcendental type $E=\{s_n\}_{n\in\mathbb N}\subset\overline{\mathbb Q_p}$ such that $W=\mathbb Z_{(p),E}=\{\phi\in\mathbb Q(X)\mid\phi(s_n)\in\overline{\mathbb Z_p},\text{ for all sufficiently large }n\in\mathbb N\}$. We show here that we may assume that $E$ is stacked, in the sense that, for each $n\in\mathbb N$, the residue field (the value group, respectively) of $\overline{\mathbb Z_p}\cap\mathbb Q_p(s_n)$ is contained in the residue field (the value group, respectively) of $\overline{\mathbb Z_p}\cap\mathbb Q_p(s_{n+1})$; this property of $E$ allows us to describe the residue field and value group of $W$. In particular, if $W$ is a DVR, then there exists $\alpha$ in the completion $\mathbb C_p$ of $\overline{\mathbb Q_p}$, $\alpha$ transcendental over $\mathbb Q$, such that $W=\mathbb Z_{(p),\alpha}=\{\phi\in\mathbb Q(X)\mid\phi(\alpha)\in O_p\}$, where $O_p$ is the unique local ring of $\mathbb C_p$; $\alpha$ belongs to $\overline{\mathbb Q_p}$ if and only if the residue field extension $W/M\supseteq\mathbb Z/p\mathbb Z$ is finite. As an application, we provide a full characterization of the Dedekind domains between $\mathbb Z[X]$ and $\mathbb Q[X]$.

math.NT↗

arXiv2022

Polynomial Dedekind domains with finite residue fields of prime characteristic

We show that every Dedekind domain $R$ lying between the polynomial rings $\mathbb Z[X]$ and $\mathbb Q[X]$ 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 $p\in\mathbb Z$ there exists a finite subset $E_p$ of transcendental elements over $\mathbb Q$ in the absolute integral closure $\overline{\mathbb Z_p}$ of the ring of $p$-adic integers such that $R=\{f\in\mathbb Q[X]\mid f(E_p)\subseteq \overline{\mathbb Z_p}, \forall \text{ prime }p\in\mathbb Z\}$. Moreover, we prove that the class group of $R$ 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 $R$ between $\mathbb Z[X]$ and $\mathbb Q[X]$.

math.AC↗

arXiv2021

The polynomial closure is not topological

We characterize the polynomial closure of a pseudo-convergent sequence in a valuation domain $V$ of arbitrary rank, and then we use this result to show that the polynomial closure is never topological when $V$ has rank at least $2$.

math.AC↗

arXiv2020

Metrizability of spaces of valuation domains associated to pseudo-convergent sequences

Let $V$ 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 $V$.

math.AC↗

arXiv2020

On the Lucas and Lehmer sequences in Dedekind domains

In this paper, we first obtain the strong divisibility property for the Lucas and Lehmer sequences in Dedekind domains, and then establish analogues of Zsigmondy's theorem and the primitive divisor results for such sequences in function fields.

math.NT↗

arXiv2019

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 $V$.

math.RA↗

arXiv2018

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.

math.AC↗

arXiv2017

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.

math.AC↗

arXiv2017

Adelic versions of the Weierstrass approximation theorem

Let $\underline{E}=\prod_{p\in\mathbb{P}}E_p$ be a compact subset of $\widehat{\mathbb{Z}}=\prod_{p\in\mathbb{P}}\mathbb{Z}_p$ and denote by $\mathcal C(\underline{E},\widehat{\mathbb{Z}})$ the ring of continuous functions from $\underline{E}$ into $\widehat{\mathbb{Z}}$. We obtain two kinds of adelic versions of the Weierstrass approximation theorem. Firstly, we prove that the ring ${\rm Int}_{\mathbb{Q}}(\underline{E},\widehat{\mathbb{Z}}):=\{f(x)\in\mathbb{Q}[x]\mid \forall p\in\mathbb{P},\;\;f(E_p)\subseteq \mathbb{Z}_p\}$ is dense in the direct product $\prod_{p\in\mathbb{P}}\mathcal C(E_p,\mathbb{Z}_p)\,$ for the uniform convergence topology. Secondly, under the hypothesis that, for each $n\geq 0$, $\#(E_p\pmod{p})>n$ for all but finitely many $p$, we prove the existence of regular bases of the $\mathbb{Z}$-module ${\rm Int}_{\mathbb{Q}}(\underline{E},\widehat{\mathbb{Z}})$, and show that, for such a basis $\{f_n\}_{n\geq 0}$, every function $\underlineφ$ in $\prod_{p\in\mathbb{P}}\mathcal{C}(E_p,\mathbb{Z}_p)$ may be uniquely written as a series $\sum_{n\geq 0}\underline{c}_n f_n$ where $\underline{c}_n\in\widehat{\mathbb{Z}}$ and $\lim_{n\to \infty}\underline{c}_n\to 0$.

math.NT↗

arXiv2016

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.

math.AC↗

arXiv2016

Galois structure on integral valued polynomials

We characterize finite Galois extensions $K$ of the field of rational numbers in terms of the rings ${\rm Int}_{\mathbb{Q}}(\mathcal O_K)$, recently introduced by Loper and Werner, consisting of those polynomials which have coefficients in $\mathbb{Q}$ and such that $f(\mathcal O_K)$ is contained in $\mathcal O_K$. We also address the problem of constructing a basis for ${\rm Int}_{\mathbb{Q}}(\mathcal O_K)$ as a $\mathbb{Z}$-module.

math.NT↗

arXiv2016

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 $D$.

math.AC↗

arXiv2016

Decomposition of integer-valued polynomial algebras

Let $D$ be a commutative domain with field of fractions $K$, let $A$ be a torsion-free $D$-algebra, and let $B$ be the extension of $A$ to a $K$-algebra. The set of integer-valued polynomials on $A$ is ${\rm Int}(A) = \{f \in B[X] \mid f(A) \subseteq A\}$, and the intersection of ${\rm Int}(A)$ with $K[X]$ is ${\rm Int}_K(A)$, which is a commutative subring of $K[X]$. The set ${\rm Int}(A)$ may or may not be a ring, but it always has the structure of a left ${\rm Int}_K(A)$-module. A $D$-algebra $A$ which is free as a $D$-module and of finite rank is called ${\rm Int}_K$-decomposable if a $D$-module basis for $A$ is also an ${\rm Int}_K(A)$-module basis for ${\rm Int}(A)$; in other words, if ${\rm Int}(A)$ can be generated by ${\rm Int}_K(A)$ and $A$. A classification of such algebras has been given when $D$ is a Dedekind domain with finite residue rings. In the present article, we modify the definition of ${\rm Int}_K$-decomposable so that it can be applied to $D$-algebras that are not necessarily free by defining $A$ to be ${\rm Int}_K$-decomposable when ${\rm Int}(A) \cong {\rm Int}_K(A) \otimes_D A$. We then provide multiple characterizations of such algebras in the case where $D$ is a discrete valuation ring or a Dedekind domain with finite residue rings. In particular, if $D$ is the ring of integers of a number field $K$, we show that ${\rm Int}_K$-decomposable algebras $A$ correspond to maximal $D$-orders in a separable $K$-algebra $B$, whose simple components have as center the same finite unramified Galois extension $F$ of $K$ and are unramified at each finite place of $F$. Finally, when both $D$ and $A$ are rings of integers in number fields, we show that ${\rm Int}_K$-decomposable algebras correspond to unramified Galois extensions of $K$.

math.RA↗