arXiv subjects

Yuji Tsuno

Publications and source records attributed to Yuji Tsuno.

4 recordsLinked to original sources

arXiv2025

On the unit group scheme of the group algebra of a certain non-commutative finite flat group scheme over an $\Bbb{F}_{p}$-algebra

Suwa investigated the unit group scheme of the group ring associated with a finite flat group scheme and provided a characterization of torsors possessing the normal base property for such schemes. In this paper, we examine the unit group scheme of the group ring for a specific non-commutative finite flat group scheme and characterize torsors with the normal base property in this context. Moreover, in connection with the Noether problem for Hopf algebras proposed by Kassel and Masuoka, we compute the quotient of the unit group scheme under the action of this non-commutative finite flat group scheme.

math.NT↗

arXiv2024

On the Grothendieck resolution for a certain finite flat commutative group scheme of order $p^{n}$ over an $\Bbb{F}_{p}$-algebra

For any commutative finite flat group scheme, Grothendieck constructed an embedding into some smooth group scheme. This embedding is called the Grothendieck resolution. Let $p$ be a prime number and $n$ a positive integer. In connection with the normal basis problems in the framework of group schemes proposed by Suwa and the author, we consider the Grothendieck resolution for a certain finite flat commutative group scheme of order $p^{n}$ over an $\Bbb{F}_{p}$-algebra.

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arXiv2021

Polynomial values with integer coefficients of the generating functions for Fibonacci polynomials

Fibonacci polynomials are generalizations of Fibonacci numbers, so it is natural to consider polynomial versions of the various results for Fibonacci numbers. According to Hong, Pongsriiam, Bulawa, and Lee, the generating function of the Fibonacci sequence in the domain of rational numbers, $f(t)=t/(1-t-t^2)$, takes an integer value if and only if $t=F_{k}/F_{k+1}$ for some $k \in \N$ or $t=-F_{k+1}/F_{k}$ for some $k \in \N^{+}$, where $F_{k}$ is the $k$th Fibonacci number. This study is built upon their work by considering polynomial sequences that satisfy the recurrence relation $F_{i+2}(x)=axF_{i+1}(x)+bF_{i}(x)$ with initial values $(F_{0}(x), F_{1}(x))=(0, 1)$, where $a$ and $b$ are positive integers such that $b|a$. As an application, for a square-free natural number $d \in \N$, we verify the results are of the same form as the above for the generating function of the sequence satisfying the recurrence relation $F_{i+2}(\sqrt{d})=a\sqrt{d} F_{i+1}(\sqrt{d})+bF_{i}(\sqrt{d})$ with initial values $(F_{0}(\sqrt{d}), F_{1}(\sqrt{d}))=(0, 1)$.

math.NT↗

arXiv2019

On integer values of the generating functions for sequences given by the Pell's equations

D. S. Hong and P. Pongsriiam have provided a necessary and sufficient condition for the generating function for Fibonacci numbers (resp. the Lucas numbers) to be an integer value, for rational numbers. In other words, their results relate to the integer values of the generating functions of the sequences obtained from the integer solutions of Pell's equation $5x^{2}-y^{2}=\pm4$. If we change this Pell's equation to another type of Pell's equation, how will their results change? This is a natural and interesting problem. In this paper, we show that a result similar to theirs is obtained for the generating functions for sequences given by Pell's equation $x^{2}-my^{2}=\pm1 \ (m\text{ is a non-square natural number})$.

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