arXiv subjects

Rin Gotou

Publications and source records attributed to Rin Gotou.

6 recordsLinked to original sources

arXiv2026

Chow Quotient Moduli Space of Dynamical Systems on the Projective Line and Rescaling Limits

We study Chow-quotient compactifications of the moduli space of degree-$d$ rational self-maps of the projective line. We compute the degree of every three-dimensional orbit cycle in the Clebsch--Gordan compactification in terms of the main component and the hole depths. As consequences, we obtain the generic orbit class and a rational map on Chow quotients induced by iteration. We also introduce a marked Chow quotient designed to record collections of rescaling limits, describe it by decorated trees, and extend the fixed-point multiplier map to the normalization of the unmarked Chow quotient.

math.AG↗

arXiv2026

Semistable reductions and minimalities of invariants for group scheme actions on projective schemes

Let $K$ be an algebraically closed and complete non-archimedean and non-trivially valued field, and let $G$ be a reductive group scheme acting on a flat projective scheme $X$ defined over the base ring of $K$-integers. For every $K$-point $x$ in $X$, we introduce the minimal invariant locus $\operatorname{MinInvLoc}_x$ and the semistable reduction translation locus $\operatorname{SSRL}_x$ in the translation space $\operatorname{BT}_G(K)$ associated with $G_K$, which is a variant of Bruhat-Tits building, and establish not only the coincidence of those loci but, under a mild completeness assumption, also their non-emptiness. In the dynamical setting which has been studied by Szpiro--Tepper--Williams and Rumely, the coincidence result is already new in higher dimensions, and the non-emptiness result includes Rumely's $1$-dimensional result at least in the spherical complete case.

math.AG↗

arXiv2023

Dynamical Systems of Correspondences on the Projective Line II: Degrees of Multiplier Maps

This paper is a sequel of arXiv:2109.06394. In this paper, we consider a kind of inverse problem of multipliers. The problem is to count number of isospectral correspondences, correspondences which has the same combination of multipliers. We give a primitive explicit upper bound. In particular, for a generic rational map of degree $d$, there are at most $O(d^{10d})$ rational maps with the same combination of multipliers for the fixed points and the 3-periodic points. This paper also includes two proofs of a correction in the errata of a Hutz-Tepper's result, which states that the multipliers of the fixed and 2-periodic points determines generic cubic morphism uniquely. One is done by proceeding the computation in Hutz-Tepper's proof. The other is done by more explicit computation with the help of invariant theory.

math.DS↗

arXiv2022

Bracket Polynomial Expression of Discriminant-Resultants as SL2-invariant

We give a bracket polynomial expression for intermediate terms between discriminant and resultant for pair of binary forms. As an application of the bracket polynomial expression, we give an algebraic proof of the algebraic independence of intermediate terms, which was shown in the theory of dynamical systems.

math.AC↗

arXiv2021

Dynamical Systems of Correspondences on the Projective Line I: Moduli Spaces and Multiplier Maps

We consider moduli spaces of dynamical systems of correspondences over the projective line as a generalization of moduli spaces of dynamical systems of endomorphisms on the projective line. We obtain the rationality of the moduli spaces. The rationality of the moduli space of degree $(d,e)$ correspondences is obtained from a representation-theoretic projection to the one for the usual dynamical systems of degree $d+e-1$. We also show that the multiplier maps for the fixed points and the multiplier index theorem (Woods Hole formula) are also reduced through the projection and obtain the reduced form explicitly.

math.DS↗

arXiv2019

Arithmetic Convergence of Double-iterated Polynomials

Let $f$ be a polynomial with integer coefficients such that $f(n)$ positive for any positive integer $n$. We consider diverging sequences $\{ y_n\}$ given by $y_0 = b$ and $y_{n+1} = f^{y_n}(a)$ with positive integers $a$ and $b$. We show such a sequence converges in $\widehat{\mathbb{Z}}$ and the limit is independent of $b$, if and only if $f$ does not become a permutation of length $p$ on $\mathbb{Z}/p\mathbb{Z}$ for any prime number $p$. We also show that $b'$-adic asymptotic approximations of the equation $f^y(a) = y$ holds in $\mathbb{N}$ for some bases $b'$.

math.NT↗