arXiv subjects

Eunju Shin

Publications and source records attributed to Eunju Shin.

3 recordsLinked to original sources

arXiv2026

Density of subsets of squarefree elements in certain Dedekind domains

We consider polynomial rings over finite fields and rings of integers of imaginary quadratic fields $\mathbb{Q}(\sqrt{-D})$. In this paper, we formulate the density of squarefree elements divisible by all elements of $T$ but by none of $P$, where $T$ and $P$ are subsets of squarefree elements and $T$ is finite. We also define Mersenne irreducibles in order to estimate the density of squarefree elements divisible by none of $P$.

math.NT↗

arXiv2023

Fundamental domain for the Markoff-Hurwitz equation

For integers $a\neq0$, $k$, and $n\geq3$, we consider the Markoff-Hurwitz equation given by $x_1^1+\cdots+x_n^2-ax_1\cdots x_n=k$. By defining graphs associated with a height function and by using their properties, we find an exact fundamental domain for a symmetric group generated by involution maps sending $(x_1,\dots,x_n)$ to $(x_1,\dots,ax_1\cdots x_{i-1}x_{i+1}\cdots x_n-x_i,\dots,x_n)$, permutations, and double sign changes on the set of integral solutions for the Markoff-Hurwitz equation.

math.CO↗

arXiv2023

Character varieties on a four-holed sphere

For each $\mathbf{k}\in\mathbb{C}^4$, let $V_\mathbf{k}$ be the character variety on a four-holed sphere and $\Gamma_\mathbf{k}$ the group generated by the Vieta involution maps. First, under a certain condition, we find a fundamental domain for $\Gamma_\mathbf{k}$-action on $V_\mathbf{k}$, expressed via inequalities. Second, we show that it is decidable whether or not two integral solutions for $V_\mathbf{k}$ are in the same $\Gamma_\mathbf{k}$-orbit and in the same mapping class group orbit. To achieve our goals, we introduce graphs corresponding to the $\Gamma_\mathbf{k}$-orbits and the mapping class group orbits, and classify their restricted global shapes by analyzing the limited local edge configurations at each vertex, using a descent argument.

math.NT↗