arXiv subjects

Hyuga Yoshizaki

Publications and source records attributed to Hyuga Yoshizaki.

8 recordsLinked to original sources

arXiv2025

On some periodic continued fractions along the $\mathbb{Z}_2$ extension over $\mathbb{Q}$

In 2021, Brock, Elkies, and Jordan generalized the theory of periodic continued fractions (PCFs) over $\mathbb{Z}$ to the ring of integers in a number field. In particular, they considered the case where the number field is an intermediate field of the $\mathbb{Z}_2$-extension over $\mathbb{Q}$ and asked whether a $(N, \ell)$-type PCF for $X_n = 2\cos(2π/2^{n+2})$ exists. In this paper, we construct $(1,2)$ and $(0,3)$-type PCFs for $X_n$ for all $n\geq1$. To the best of our knowledge, this is the first explicit construction of type (0,3) continued fractions for all $n\geq1$. To obtain such results, for each type, we construct a bijection between a certain subset of the group of relative units in each layer of the $\mathbb{Z}_2$-extension and the set of PCFs for $X_n$. While our result confirms the existence of such PCFs for all $n\geq1$ in types $(1,2)$ and $(0,3)$, determining all PCFs remains an open problem. The bijections constructed in our result translate this problem into the study of the subsets of the relative units. As a second main result, we give explicit bounds for the logarithms of the relative units corresponding to $(1,2)$ or $(0,3)$-type PCFs for $X_n$. These bounds allow us to explain interesting phenomena observed in the distribution of such points.

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arXiv2025

The $p$-adic limits of iterated $p$-power cyclic resultants of multivariable polynomials

Let $p$ be a prime number. The $p$-power cyclic resultant of a polynomial is the determinant of the Sylvester matrix of $t^{p^n}-1$ and the polynomial. It is known that the sequence of $p$-power cyclic resultants and its non-$p$-parts converge in $\mathbb{Z}_p$. This article shows the $p$-adic convergence of the iterated $p$-power cyclic resultants of multivariable polynomials. As an application, we show the $p$-adic convergence of the torsion numbers of $\mathbb{Z}_p^d$-coverings of links. We also explicitly calculate the $p$-adic limits for the twisted Whitehead links as concrete examples. Moreover, in a specific case, we show that our $p$-adic limit of torsion numbers coincides with the $p$-adic torsion, which is a homotopy invariant of a CW-complex introduced by S. Kionke.

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arXiv2025

The $p$-adic limits of class numbers in $\mathbb{Z}_p$-towers

This article discusses variants of Weber's class number problem in the spirit of arithmetic topology to connect the results of Sinnott--Kisilevsky and Kionke. Let $p$ be a prime number. We first prove the $p$-adic convergence of class numbers in a $\mathbb{Z}_p$-extension of a global field and a similar result in a $\mathbb{Z}_p$-cover of a compact 3-manifold. Secondly, we establish an explicit formula for the $p$-adic limit of the $p$-power-th cyclic resultants of a polynomial using roots of unity of orders prime to $p$, the $p$-adic logarithm, and the Iwasawa invariants. Finally, we give thorough investigations of torus knots, twist knots, and elliptic curves; we complete the list of the cases with $p$-adic limits being in $\mathbb{Z}$ and find the cases such that the base $p$-class numbers are small and $ν$'s are arbitrarily large.

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arXiv2023

Some periodic integer continued fraction expansions of $\sqrt{m}$ and application to the Pell equations

Periodic integer continued fractions (PICFs) are generalization of the regular periodic continued fractions (RPCFs). It is classical that a RPCF expansion of an irrational number is unique. However, it is no longer unique for a PICF expansion. Hence it is a natural problem to determine all PICF expansions of irrational numbers. In this paper, we determine certain type PICF expansions of square roots of positive square-free integers. To obtain this result, it plays an important role to determine integer points on certain PCF varieties appeared in Brock-Elkies-Jordan. As an application of these results, we obtain fundamental solutions of the Pell equations from PICF expansions of square roots of positive square-free integers as well as the RPCF expansions.

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arXiv2022

Weber's class number problem and its variants

We survey Weber's class number problem and its variants in the spirit of arithmetic topology; we recollect some history, present a relation to certain units and generalized Pell's equation, and overview a study of the $p$-adic limits of class numbers in $\mathbb{Z}_p$-towers together with numerical investigation for knots and elliptic curves.

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arXiv2022

Generalized Pell's equations and Weber's class number problem

We study a generalization of Pell's equation, whose coefficients are certain algebraic integers. Let $X_0=0$ and $X_n=\sqrt{2+X_{n-1}}$ for each $n\in \mathbb{Z}_{\ge 1}$. We study the $\mathbb{Z}[X_{n-1}]$-solutions of the equation $x^2-X_n^2y^2=1$. By imitating the solution to the classical Pell's equation, we introduce new continued fraction expansions for $X_n$ over $\mathbb{Z}[X_{n-1}]$ and obtain an explicit solution of the generalized Pell's equation. In addition, we show that our explicit solution generates all the solutions if and only if the answer to Weber's class number problem is affirmative. We also obtain a congruence relation for the ratios of the class numbers of the $\mathbb{Z}_2$-extension over the rationals and show the convergence of the class numbers in $\mathbb{Z}_2$.

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arXiv2022

Minimal relative units of the cyclotomic $\mathbb Z_2$-extension

Let $\mathbb B_n:=\mathbb Q(\cos(π/2^{n+1}))$. For the relative norm map $\mathrm{N}_{n/n-1} \colon \mathcal O_{\mathbb B_n}^\times \rightarrow \mathcal O_{\mathbb B_{n-1}}^\times$ on the units group, we define $RE_n:=\mathrm{N}_{n/n-1}^{-1}(\{\pm 1\})$, $RE_n^+:=\mathrm{N}_{n/n-1}^{-1}(\{1\})$. Komatsu conjectured that $\mathrm{Tr} ε^2 \geq 2^n(2^{n+1}-1)$ for $ε\in RE_n -\{\pm 1\}$. Morisawa and Okazaki showed that it holds for $ε\in RE_n -RE_n^+$. In this paper we study the case $ε\in RE_n^+$. We conjecture that $\min \{\mathrm{Tr} ε^2 \mid ε\in RE_n^+-\{\pm 1\}\}= 2^n(1+8c_n)$, where $c_1:=2$ and $c_n:=2\cdot \mathrm{round}(2^n/5)$ ($n\geq 2$). We show that this holds for $n\leq 6$ and that a "half" of this: $\min \{\mathrm{Tr} ε^2 \mid ε\in RE_n^+-\{\pm 1\}\} \leq 2^n(1+8c_n)$ holds for even $n$. We also observe a relation to the class number problem.

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arXiv2021

Bijective enumerations for symmetrized poly-Bernoulli polynomials

Recently, Bényi and the second author introduced two combinatorial interpretations for symmetrized poly-Bernoulli polynomials. In the present study, we construct bijections between these combinatorial objects. We also define various combinatorial polynomials and prove that all of these polynomials coincide with symmetrized poly-Bernoulli polynomials.

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