arXiv subjects

Malcolm Hoong Wai Chen

Publications and source records attributed to Malcolm Hoong Wai Chen.

6 recordsLinked to original sources

arXiv2026

Abstract regular polytopes of $G_2(q)$ in even characteristic

For every even prime power $q>2$, we explicitly construct a rank five string C-group representation of the Chevalley simple group $G_2(q)$ using its natural seven-dimensional matrix representation. We also give a census for the string C-group representations of $G_2(q)$ of ranks four and above for every prime power $3 \le q \le 11$; providing computational evidence that rank five string C-group representations of $G_2(q)$ are rare and that this construction is exceptional among the currently known examples.

math.GR↗

arXiv2026

Elementary characterization of the Galois groups of $x^{12}+ax^6+b$

Let $f(x)=x^{12}+ax^6+b \in \mathbb{Q}[x]$ be an irreducible polynomial, $g_4(x)=x^4+ax^2+b$, $g_6(x)=x^6+ax^3+b$, and let $G_4$ and $G_6$ be the Galois groups of $g_4(x)$ and $g_6(x)$, respectively. This paper provides a streamlined elementary characterization of all sixteen possible Galois groups of $f(x)$. In particular, we give explicit criteria showing that the Galois group of $f(x)$ can be uniquely determined by the pair $(G_4,G_6)$ along with testing whether at most two expressions involving $a$ and $b$ are rational squares. This contributes to a broader program to systematically study the Galois groups of $x^{2m}+ax^m+b$ for a positive integer $m$.

math.NT↗

arXiv2026

Covering rings by proper ideals

A cover by left ideals of an associative (not necessarily commutative or unital) ring $R$ is a collection of proper left ideals whose set-theoretic union equals $R$. If such a cover exists, then $η_\ell(R)$ is the cardinality of a minimal cover, and $R$ is $η_\ell$-elementary if $η_\ell(R)<η_\ell(R/I)$ for every nonzero two-sided ideal $I$ of $R$. We classify all $η_\ell$-elementary rings, and determine their covering numbers. Covers by right or two-sided ideals are also studied. This completely characterizes rings admitting finite covers by ideals. Our results generalize to finite covers of modules by submodules, and we determine all possible covering numbers.

math.RA↗

arXiv2025

Rings as unions of proper ideals

In this note, we define and investigate ideal covering numbers of associative rings (not assumed to be commutative or unital): three invariants defined as the minimal number of proper left, right, or two-sided ideals whose union equals the ring. For every prime $p$, we construct four infinite families of rings without identity that attain the sharp lower bound $p + 1$ for ideal covering numbers, each exhibiting distinct behavior with respect to left, right, and two-sided ideal coverings. As a consequence of a result by Lucchini and Maróti, we also characterize all rings with ideal covering numbers three. Finally, we make several observations and propose open questions related to these invariants and the structure of rings admitting such ideal coverings.

math.RA↗

arXiv2025

Abstract Regular Polytopes of Finite Irreducible Coxeter Groups

Here, for $W$ the Coxeter group $\mathrm{D}_n$ where $n > 4$, it is proved that the maximal rank of an abstract regular polytope for $W$ is $n - 1$ if $n$ is even and $n$ if $n$ is odd. Further it is shown that $W$ has abstract regular polytopes of rank $r$ for all $r$ such that $3 \leq r \leq n - 1$, if $n$ is even, and $3 \leq r \leq n$, if $n$ is odd. The possible ranks of abstract regular polytopes for the exceptional finite irreducible Coxeter groups are also determined.

math.GR↗

arXiv2022

Galois groups of certain even octic polynomials

Let $f(x)=x^8+ax^4+b \in \mathbb{Q}[x]$ be an irreducible polynomial where $b$ is a square. We give a method that completely describes the factorization patterns of a linear resolvent of $f(x)$ using simple arithmetic conditions on $a$ and $b$. As a result, we determine the exact six possible Galois groups of $f(x)$ and completely classify all of them. As an application, we characterize the Galois groups of irreducible polynomials $x^8+ax^4+1 \in \mathbb{Q}[x]$. We also use similar methods to obtain analogous results for the Galois groups of irreducible polynomials $x^8+ax^6+bx^4+ax^2+1 \in \mathbb{Q}[x]$.

math.NT↗