arXiv subjects

Kohei Takehira

Publications and source records attributed to Kohei Takehira.

4 recordsLinked to original sources

arXiv2025

Irreducibility of polynomials defining parabolic parameters of period 3

Morton and Vivaldi defined the polynomials whose roots are parabolic parameters for a one-parameter family of polynomial maps. We call these polynomials delta factors. They conjectured that delta factors are irreducible for the family $z\mapsto z^2+c$. One can easily show the irreducibility for periods $1$ and $2$ by reducing it to the irreducibility of cyclotomic polynomials. However, for periods $3$ and beyond, this becomes a challenging problem. This paper proves the irreducibility of delta factors for the period $3$ and demonstrates the existence of infinitely many irreducible delta factors for periods greater than $3$.

math.NT↗

arXiv2025

Arithmetic properties of multiplier polynomials for certain polynomial maps

We investigate the arithmetic properties of the multiplier polynomials for certain $1$-parameter families of polynomials. In particular, we prove integrality theorems of multiplier polynomials for $z^d+c$, $(z-c)z^d + c$ and $z^{d+1}+cz$. As a corollary, we obtain the uniform upper bound of the naive height of parabolic parameters of unicritical polynomials. Moreover, we determined the quadratic parabolic parameters for $z^2 + c$. We also conditionally list parabolic parameters for $z^2 + c$ of fixed degrees.

math.DS↗

arXiv2024

On the number of points with bounded dynamical canonical height

This paper discusses the number of points for which the dynamical canonical height is less than or equal to a given value. The height function is a fundamental and important tool in number theory to capture the ``number-theoretic complexity" of a point. Asymptotic formulas for the number of points in projective space below a given height have been studied by Schanuel, for example, and their coefficients can be written by class numbers, regulators, special values of the Dedekind zeta function, and other number theoretically interesting values. We consider an analogous problem for dynamical canonical height, a dynamical analogue of the height function in number theory, introduced by Call-Silverman . The main tool of this study is the dynamical height zeta function studied by Hsia. In this paper, we give explicit formulas for the dynamical height zeta function in special cases, derive general formulas for obtaining asymptotic behavior from certain functions, and combine them to derive asymptotic behavior for the number of points with bounded dynamical canonical height.

math.NT↗

arXiv2021

The rationality of dynamical zeta functions and Woods Hole fixed point formula

For one variable rational function $ϕ\in K(z)$ over a field $K$, we can define a discrete dynamical system by regarding $ϕ$ as a self morphism of $\mathbb{P}_{K}^{1}$. Hatjispyros and Vivaldi defined a dynamical zeta function for this dynamical system using multipliers of periodic points, that is, an invariant which indicates the local behavior of dynamical systems. In this paper, we prove the rationality of dynamical zeta functions of this type for a large class of rational functions $ϕ\in K(z)$. The proof here relies on Woods Hole fixed point formula and some basic facts on the trace of a linear map acting on cohomology of a coherent sheaf on $\mathbb{P}_{K}^{1}$.

math.NT↗