arXiv subjects

Dong Quan Ngoc Nguyen

Publications and source records attributed to Dong Quan Ngoc Nguyen.

At least 19 recordsLinked to original sources

arXiv2026

Polynomial parametrization of $\text{SL}_2$ over global function field rings

We combine the universal polynomial matrix identities of Larsen--Nguyen with Trost's arithmetic lemmas to obtain polynomial parametrizations of $\text{SL}_2(\mathcal{O}_{K,S})$ for every global function field $K$ and every nonempty finite set of places $S$. The additional ingredient is a function-field cancellation lemma, proved using ray class fields and Chebotarev in odd characteristic and the perfectness of finite residue fields in characteristic two. The proof also gives a polynomial matrix over $\mathbb{Z}$ that works simultaneously for all these rings.

math.NT↗

arXiv2024

Ultra-Galois theory and an analogue of the Kronecker--Weber theorem for rational function fields over ultra-finite fields

In the first part of this paper, we develop a general framework that permits a comparison between explicit class field theories for a family of rational function fields $\mathbb{F}_s(t)$ over arbitrary constant fields $\mathbb{F}_s$ and explicit class field theory for the rational function field $\mathfrak{K}(t)$ over the nonprincipal ultraproduct $\mathfrak{K}$ of the constant fields $\mathbb{F}_s$. Under an additional assumption that the constant fields $\mathbb{F}_s$ are perfect procyclic fields, we prove a correspondence between ramifications of primes $P$ in $\mathfrak{K}(t)$ and ramifications of primes $P_s$ in $\mathbb{F}_s(t)$, where the $P_s$ are primes in $\mathbb{F}_s(t)$ whose nonprincipal ultraproduct coincides with $P$. In the second part of the paper, we are mainly concerned with rational function fields over a large class of fields, called $n$-th level ultra-finite fields that are a generalization of finite fields. At the $0$-th level, ultra-finite fields are simply finite fields, and for an arbitrary positive integer $n$, an $n$-th level ultra-finite field is inductively defined as a nonprincipal ultraproduct of $(n - 1)$-th level ultra-finite fields. We develop an analogue of cyclotomic function fields for rational function fields over $n$-th level ultra-finite fields that generalize the works of Carlitz and Hayes for rational function fields over finite fields such that these cyclotomic function fields are in complete analogy with the classical cyclotomic fields $\mathbb{Q}(ζ)$ of the rationals $\mathbb{Q}$. The main result in the second part of the paper is an analogue of the Kronecker--Weber theorem for rational function fields over $n$-th level ultra-finite fields that explicitly describes, from a model-theoretic viewpoint, the maximal abelian extension of the rational function field over a given $n$-th level ultra-finite field for all $n \ge 1$.

math.NT↗

arXiv2023

Local symbols and a first-order definition of the polynomial ring over an ultra-finite field in its fraction field

In this paper, we prove the existence of a first-order definition of the polynomial ring over a nonprincipal ultraproduct of finite fields of unbounded cardinalities in its fraction field by a universal-existential formula in the language of rings augmented by an additional constant symbol $t$. As a consequence, we prove that the full first-order theory of the rational function field over a nonprincipal ultraproduct of finite fields of characteristic $0$ is undecidable.

math.NT↗

arXiv2023

Higher reciprocity law and An analogue of the Grunwald--Wang theorem for the ring of polynomials over an ultra-finite field

In this paper, we establish an explicit higher reciprocity law for the polynomial ring over a nonprincipal ultraproduct of finite fields. Such an ultraproduct can be taken over the same finite field, which allows to recover the classical higher reciprocity law for the polynomial ring $\mathbb{F}_q[t]$ over a finite field $\mathbb{F}_q$ that is due to Dedekind, Kühne, Artin, and Schmidt. On the other hand, when the ultraproduct is taken over finite fields of unbounded cardinalities, we obtain an explicit higher reciprocity law for the polynomial ring over an infinite field in both characteristics $0$ and $p >0$ for some prime $p$. We then use the higher reciprocity law to prove an analogue of the Grunwald--Wang theorem for such a polynomial ring in both characteristics $0$ and $p > 0$ for some prime $p$.

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arXiv2021

Quaternionic analogues of Bers's theorem and Iss'sa's theorem

In their recent work, Gentili and Struppa proposed a different quaternionic analogue of the notion of holomorphic functions in the complex plane, called \textit{slice regular functions}, which has led to several analogues of classical theorems in complex function theory in the quaternionic setting. The quaternionic analogue of meromorphic functions is called \textit{semiregular functions}. Such a function is said to be \textit{slice preserving} if it maps each domain in a complex line to a domain in the same complex line. In this work, we deal with the relation between analytic and algebraic properties of certain fields of semiregular functions in symmetric slice domains of the real quaternions. More precisely, we prove quaternionic analogues of Bers's theorem and Iss'sa's theorem that deal with the algebraic structures of the ring of slice preserving regular functions and of certain fields of slice preserving semiregular functions on symmetric slice domains in the real quaternions.

math.CV↗

arXiv2020

Weight Prediction for Variants of Weighted Directed Networks

A weighted directed network (WDN) is a directed graph in which each edge is associated to a unique value called weight. These networks are very suitable for modeling real-world social networks in which there is an assessment of one vertex toward other vertices. One of the main problems studied in this paper is prediction of edge weights in such networks. We introduce, for the first time, a metric geometry approach to studying edge weight prediction in WDNs. We modify a usual notion of WDNs, and introduce a new type of WDNs which we coin the term \textit{almost-weighted directed networks} (AWDNs). AWDNs can capture the weight information of a network from a given training set. We then construct a class of metrics (or distances) for AWDNs which equips such networks with a metric space structure. Using the metric geometry structure of AWDNs, we propose modified $k$ nearest neighbors (kNN) methods and modified support-vector machine (SVM) methods which will then be used to predict edge weights in AWDNs. In many real-world datasets, in addition to edge weights, one can also associate weights to vertices which capture information of vertices; association of weights to vertices especially plays an important role in graph embedding problems. Adopting a similar approach, we introduce two new types of directed networks in which weights are associated to either a subset of origin vertices or a subset of terminal vertices . We, for the first time, construct novel classes of metrics on such networks, and based on these new metrics propose modified $k$NN and SVM methods for predicting weights of origins and terminals in these networks. We provide experimental results on several real-world datasets, using our geometric methodologies.

cs.SI↗

arXiv2020

Community detection, pattern recognition, and hypergraph-based learning: approaches using metric geometry and persistent homology

Hypergraph data appear and are hidden in many places in the modern age. They are data structure that can be used to model many real data examples since their structures contain information about higher order relations among data points. One of the main contributions of our paper is to introduce a new topological structure to hypergraph data which bears a resemblance to a usual metric space structure. Using this new topological space structure of hypergraph data, we propose several approaches to study community detection problem, detecting persistent features arising from homological structure of hypergraph data. Also based on the topological space structure of hypergraph data introduced in our paper, we introduce a modified nearest neighbors methods which is a generalization of the classical nearest neighbors methods from machine learning. Our modified nearest neighbors methods have an advantage of being very flexible and applicable even for discrete structures as in hypergraphs. We then apply our modified nearest neighbors methods to study sign prediction problem in hypegraph data constructed using our method.

cs.SI↗

arXiv2020

A local geometry of hyperedges in hypergraphs, and its applications to social networks

In many real world datasets arising from social networks, there are hidden higher order relations among data points which cannot be captured using graph modeling. It is natural to use a more general notion of hypergraphs to model such social networks. In this paper, we introduce a new local geometry of hyperdges in hypergraphs which allows to capture higher order relations among data points. Furthermore based on this new geometry, we also introduce new methodology--the nearest neighbors method in hypergraphs--for analyzing datasets arising from sociology.

cs.LG↗

arXiv2020

On the generating polynomials for the distribution of generalized binomial coefficients in discrete valuation domains

For a discrete valuation domain $V$ with maximal ideal $\mathfrak{m}$ such that the residue field $V/\mathfrak{m}$ is finite, there exists a sequence of polynomials $(F_n(x))_{n \ge 0}$ defined over the quotient field $K$ of $V$ that forms a basis of the $V$-module $\text{Int}(V) = \{f \in K[x] | f(V)\subseteq V\}$. This sequence of polynomials bears many resemblances to the classical binomial polynomials $(\binom{x}{n})_{n \ge 0}$. In this paper, we introduce a generating polynomial to account for the distribution of the $V$-values of the polynomials $F_n(x)$ modulo the maximal ideal $\mathfrak{m}$, and prove a result that provides a method for counting exactly how many $V$-values of the polynomials $(F_n(x))_{n \ge 0}$ fall into each of the residue classes modulo $\mathfrak{m}$. Our main theorem in this paper can be viewed as an analogue of the classical theorem of Garfield and Wilf in the context of discrete valuation domains.

math.NT↗

arXiv2020

Effectively bounded idempotent generation of certain $2 \times 2$ singular matrices by idempotent matrices over real quadratic number rings

Let $k = \mathbb{Q}(\sqrtα)$ be a real quadratic number field, where $α$ is a positive square-free integer. Let $\mathcal{O}_k$ be the ring of integers of $k$. In this paper, we prove that a certain set of $2 \times 2$ singular matrices with entries in $\mathcal{O}_k$ can be written as a product of a bounded number of idempotent matrices. Our main theorem can be viewed as a generalization of the recent result by Cossu and Zanardo, which studies finite generation of certain singular matrices by idempotent matrices over $\mathcal{O}_k$ instead of bounded generation of certain singular matrices by idempotent matrices over $\mathcal{O}_k$ as considered in this paper.

math.NT↗

arXiv2018

Polynomial Parametrization for $\text{SL}_2$ over Quadratic Number Rings

If $R$ is the ring of integers of a number field, then there exists a polynomial parametrization of the set $\text{SL}_2(R)$, i.e., an element $A \in \text{SL}_2(\mathbb{Z}[x_1,\ldots,x_n])$ such that every element of $\text{SL}_2(R)$ is obtained by specializing $A$ via some homomorphism $\mathbb{Z}[x_1,\ldots,x_n]\to R$.

math.NT↗

arXiv2017

The distribution of the Carlitz binomial coefficients modulo a prime

For a nonnegative integer $n$, and a prime $\wp$ in $\mathbb{F}_q[T]$, we prove a result that provides a method for computing the number of integers $m$ with $0 \le m \le n$ for which the Carlitz binomial coefficients $\binom{n}{m}_C$ fall into each of the residue classes modulo $\wp$. Our main result can be viewed as a function field analogue of the Garfield-Wilf theorem.

math.NT↗

arXiv2017

Waring's problem for unipotent algebraic groups

In this paper, we formulate an analogue of Waring's problem for an algebraic group $G$. At the field level we consider a morphism of varieties $f\colon \mathbb{A}^1\to G$ and ask whether every element of $G(K)$ is the product of a bounded number of elements $f(\mathbb{A}^1(K)) = f(K)$. We give an affirmative answer when $G$ is unipotent and $K$ is a characteristic zero field which is not formally real. The idea is the same at the integral level, except one must work with schemes, and the question is whether every element in a finite index subgroup of $G(\mathcal{O})$ can be written as a product of a bounded number of elements of $f(\mathcal{O})$. We prove this is the case when $G$ is unipotent and $\mathcal{O}$ is the ring of integers of a totally imaginary number field.

math.NT↗

arXiv2015

Representation of units in cyclotomic function fields

Hilbert's Satz 90 tells us that for a given cyclic extension $K/k$, a unit of norm $1$ in $K$ can be written as a quotient of conjugate elements in $K$. For the extensions $\mathbb{Q}(ζ_p)/\mathbb{Q}$ with $p$ prime $> 3$, Newman proved a refinement of Hilbert's Satz 90 that gives a sufficient and necessary condition for which a unit of norm $1$ in $\mathbb{Q}(ζ_p)$ can be written as a quotient of conjugate units. In order to obtain this result, Newman proved a stronger result that gives a unique representation of units of norm $1$ as a product of a power of $\dfrac{1 - ζ_p^e}{1 - ζ_p}$ with a quotient of conjugate units, where $e$ is a given primitive root modulo $p$. In this paper, we obtain a function field analogue of Newman's result for the $\wp$-th cyclotomic function field extensions $\mathbb{K}_{\wp}/\mathbb{F}_q(T)$, where $\wp$ is a monic prime in $\mathbb{F}_q[T]$. As a consequence, we proved a refinement of Hilbert's Satz 90 for the extensions $\mathbb{K}_{\wp}/\mathbb{F}_q(T)$ that gives a sufficient and necessary condition for which a unit of norm $1$ in $\mathbb{K}_{\wp}$ can be written as a quotient of conjugate units.

math.NT↗

arXiv2015

The classical umbral calculus, and the flow of a Drinfeld module

David Goss developed a very general Fourier transform in additive harmonic analysis in the function field setting. In order to introduce the Fourier transform for continuous characteristic $p$ valued functions on $\mathbb{Z}_p$, Goss introduced and studied an analogue of flows in finite characteristic. In this paper, we use another approach to study flows in finite characteristic. We recast the notion of a flow in the language of the classical umbral calculus, which allows to generalize the formula for flows first proved by Goss to a more general setting. We study duality between flows using the classical umbral calculus, and show that the duality notion introduced by Goss seems a natural one. We also formulate a question of Goss about the exact relationship between two flows of a Drinfeld module in the language of the classical umbral calculus, and give a partial answer to it.

math.NT↗

arXiv2015

Function field analogues of Bang-Zsigmondy's theorem and Feit's theorem

In the number field context, Bang-Zsigmondy's theorem states that for any integers $u, m > 1$, there exists a prime divisor $p$ of $u^m - 1$ such that $p$ does not divide $u^n - 1$ for every integer $0 < n < m$ except in some exceptional cases that can be explicitly determined. A prime $p$ satisfying the conditions in Bang-Zsigmondy's theorem is called a Zsigmondy prime for $(u, m)$. In 1988, Feit introduced the notion of large Zsigmondy primes as follows: A Zsigmondy prime $p$ for $(u, m)$ is called a large Zsigmondy prime if either $p > m + 1$ or $p^2$ divides $u^m - 1$. In the same year, Feit proved a refinement of Bang-Zsigmondy's theorem which states that for any integers $u, m > 1$, there exists a large Zsigmondy prime for $(u, m)$ except in some exceptional cases that can be explicitly determined. In this paper, we introduce notions of Zsigmondy primes and large Zsigmondy primes in the Carlitz module context, and prove function field analogues of Bang-Zsigmondy's theorem and Feit's theorem.

math.NT↗

arXiv2015

Certain sets over function fields are polynomial families

In 1938, Skolem conjectured that $\mathbf{SL}_n(\mathbb{Z})$ is not a polynomial family for any $n \ge 2$. Carter and Keller disproved Skolem's conjecture for all $n \ge 3$ by proving that $\mathbf{SL}_n(\mathbb{Z})$ is boundedly generated by the elementary matrices, and hence a polynomial family for any $n \ge 3$. Only recently, Vaserstein refuted Skolem's conjecture completely by showing that $\mathbf{SL}_2(\mathbb{Z})$ is a polynomial family. An immediate consequence of Vaserstein's theorem also implies that $\mathbf{SL}_n(\mathbb{Z})$ is a polynomial family for any $n \ge 3$. In this paper, we prove a function field analogue of Vaserstein's theorem: that is, if $\mathbf{A}$ is the ring of polynomials over a finite field of odd characteristic, then $\mathbf{SL}_2(\mathbf{A})$ is a polynomial family in 52 variables. A consequence of our main result also implies that $\mathbf{SL}_n(\mathbf{A})$ is a polynomial family for any $n \ge 3$.

math.NT↗

arXiv2014

A Carlitz module analogue of the Grunwald--Wang theorem

The classical Grunwald--Wang theorem is an example of a local--global (or Hasse) principle stating that except in some special cases which are precisely determined, an element $m$ in a number field $\mathbb{K}$ is an $a$-th power in $\mathbb{K}$ if and only if it is an $a$-th power in the completion $\mathbb{K}_{v}$ for all but finitely many primes $v$ of $\mathbb{K}$. In this paper, we prove a Carlitz module analogue of the Grunwald--Wang theorem.

math.NT↗