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Byeong Moon Kim

Publications and source records attributed to Byeong Moon Kim.

9 recordsLinked to original sources
arXiv2026

Tight universality of $m$-gonal forms with minimal criterion sets

For integers $m\geq3$ and $n\geq1$, an $m$-gonal form is called tight $\mathcal{T}(n)$-universal if it represents exactly the positive integers $\mathcal{T}(n)=\{ n, n+1, n+2, \ldots \}$. In this paper, we study the minimal criterion set $\mathrm{CS}(m,n)$ for tight $\mathcal{T}(n)$-universality. Our main result determines $\mathrm{CS}(m,n)$ for $n \geq8$, $3 \leq m \leq \left\lfloor \frac{3n+1}{2} \right\rfloor$, except for $(m,n)=(7,9)$ and $(7,10)$. More precisely, $$ \mathrm{CS}(m,n)= \begin{cases} \{ n, n+1, \ldots, 2n-1 \}, & m=5, \newline \{ n, n+1, \ldots, 2n \}, & m\neq5. \end{cases} $$ We also establish the corresponding tight $\mathcal{T}(n)$-universality results and show that the upper bound on $m$ is optimal.

math.NT↗
arXiv2023

The Euler-Glaisher Theorem over Totally Real Number Fields

In this paper, we study the partition theory over totally real number fields. Let $K$ be a totally real number field. A partition of a totally positive algebraic integer $δ$ over $K$ is $λ=(λ_1,λ_2,\ldots,λ_r)$ for some totally positive integers $λ_i$ such that $δ=λ_1+λ_2+\cdots+λ_r$. We find an identity to explain the number of partitions of $δ$ whose parts do not belong to a given ideal $\mathfrak a$. We obtain a generalization of the Euler-Glaisher Theorem over totally real number fields as a corollary. We also prove that the number of solutions to the equation $δ=x_1+2x_2+\cdots+nx_n$ with $x_i$ totally positive or $0$ is equal to that of chain partitions of $δ$. A chain partition of $δ$ is a partition $λ=(λ_1,λ_2,\ldots,λ_r)$ of $δ$ such that $λ_{i+1}-λ_i$ is totally positive or $0$.

math.NT↗
arXiv2020

The rank of universal $m$-gonal forms

In this article, we consider the rank of universal $m$-gonal forms for all sufficiently large $m$. Especially, we determine the minimal rank of universal $m$-gonal form and the maximal rank of kinds of proper universal $m$-gonal form.

math.NT↗
arXiv2020

Regular $m$-gonal forms

In this paper, we show that for a fixed rank $n$, there are only finitely many $m$ for which there is a regular $m$-gonal form of rank $n$ and determine every type of the (generalized) regular $m$-gonal form for every sufficiently large $m$.

math.NT↗
arXiv2009

Even universal binary Hermitian lattices over imaginary quadratic fields

A positive definite even Hermitian lattice is called \emph{even universal} if it represents all even positive integers. We introduce a method to get all even universal binary Hermitian lattices over imaginary quadratic fields $\Q{-m}$ for all positive square-free integers $m$ and we list optimal criterions on even universality of Hermitian lattices over $\Q{-m}$ which admits even universal binary Hermitian lattices.

math.NT↗
arXiv2008

The Fifteen Theorem for Universal Hermitian Lattices over Imaginary Quadratic Fields

We will introduce a method to get all universal Hermitian lattices over imaginary quadratic fields over $\mathbb{Q}(\sqrt{-m})$ for all m. For each imaginary quadratic field $\mathbb{Q}(\sqrt{-m})$, we obtain a criterion on universality of Hermitian lattices: if a Hermitian lattice L represents 1, 2, 3, 5, 6, 7, 10, 13,14 and 15, then L is universal. We call this the fifteen theorem for universal Hermitian lattices. Note that the difference between Conway-Schneeberger's fifteen theorem and ours is the number 13.

math.NT↗
arXiv2008

Binary normal regular Hermitian lattices over imaginary quadratic fields

We call a positive definite Hermitian lattice regular if it represents all integers which can be represented locally by the lattice. We investigate binary regular Hermitian lattices over imaginary quadratic fields $\mathbb{Q}(\sqrt{-m})$ and provide a complete list of the (normal) Hermitian lattices.

math.NT↗