掲載種別：研究論文（学術雑誌）  

<jats:p>In this paper, we deal with two classes of Diophantine equations, $x^2+y^2+z^2+k_3xy+k_1yz+k_2zx=(3+k_1+k_2+k_3)xyz$ and $x^2+y^4+z^4+2xy^2+ky^2z^2+2xz^2=(7+k)xy^2z^2$, where $k_1,k_2,k_3,k$ are nonnegative integers. The former is known as the Markov Diophantine equation if $k_1=k_2=k_3=0$, and the latter is a Diophantine equation recently studied by Lampe if $k=0$. We give algorithms to enumerate all positive integer solutions to these equations, and discuss the structures of the generalized cluster algebras behind them.</jats:p>

DOI： 10.37236/11420

Open Access

記述言語：英語   掲載種別：研究論文（学術雑誌）   出版者・発行元：Informa {UK} Limited  

DOI： 10.1080/00927872.2023.2173763

記述言語：英語   掲載種別：研究論文（学術雑誌）   出版者・発行元：Cellule {MathDoc}/{CEDRAM}  

DOI： 10.5802/aif.3596

Open Access

出版者・発行元：{SPIE}  

DOI： 10.2969/aspm/08810491

記述言語：英語   掲載種別：研究論文（学術雑誌）   出版者・発行元：Oxford University Press ({OUP})  

<jats:title>Abstract</jats:title><jats:p>In the present paper, we first give a characterization for Bongartz completion in $\tau $-tilting theory via $c$-vectors. Motivated by this characterization, we give the definition of Bongartz completion in cluster algebras using $c$-vectors. Then we prove the existence and uniqueness of Bongartz completion in cluster algebras. We also prove that Bongartz completion admits certain commutativity. We give two applications for Bongartz completion in cluster algebras. As the first application, we prove the full subquiver of the exchange quiver (or known as oriented exchange graph) of a cluster algebra $\mathcal A$ whose vertices consist of the seeds of $\mathcal A$ containing particular cluster variables is isomorphic to the exchange quiver of another cluster algebra. As the second application, we prove that in a $Y$-pattern over a universal semifield, each $Y$-seed (up to a $Y$-seed equivalence) is uniquely determined by the negative $y$-variables in this $Y$-seed.</jats:p>

DOI： 10.1093/imrn/rnac205

担当区分：筆頭著者   記述言語：英語   掲載種別：機関テクニカルレポート，技術報告書，プレプリント等  

DOI： 10.48550/arXiv.2512.04547

記述言語：英語   掲載種別：機関テクニカルレポート，技術報告書，プレプリント等  

DOI： 10.48550/arXiv.2507.06900

記述言語：英語  

In this paper, we study positive integer solutions to a generalized form of
the Markov equation, given as $x^2 + y^2 + z^2 + k(yz + zx + xy) = (3 +
3k)xyz$. This equation extends the classical Markov equation $x^2 + y^2 + z^2 =
3xyz$. We generalize the concept of Cohn triples for the classical Markov
equation to the generalized Markov equations. Using this, we provide a
generalization of the uniqueness theorem of Markov numbers that are prime
powers.

arXiv

その他リンク： http://arxiv.org/pdf/2312.07329v1

In this paper, we demonstrate that the lattice structure of a set of clusters
in a cluster algebra of finite type is anti-isomorphic to the torsion lattice
of a certain Geiss-Leclerc-Schr\"oer (GLS) path algebra and to the $c$-Cambrian
lattice. We prove this by explicitly describing the exchange quivers of cluster
algebras of finite type. Specifically, we prove that these quivers are
anti-isomorphic to those formed by support $\tau$-tilting modules in GLS path
algebras and to those formed by $c$-clusters consisting of almost positive
roots.

arXiv

その他リンク： http://arxiv.org/pdf/2211.08935v2

開催年月日： 2026年2月

記述言語：日本語   会議種別：口頭発表（招待・特別）  

開催地：九州大学   国名：日本国  

開催年月日： 2026年2月

記述言語：英語   会議種別：口頭発表（招待・特別）  

開催地：Kavli IPMU  

開催年月日： 2026年1月

記述言語：日本語   会議種別：口頭発表（招待・特別）  

開催地：富山大学   国名：日本国  

開催年月日： 2025年12月

記述言語：日本語   会議種別：公開講演，セミナー，チュートリアル，講習，講義等  

開催地：オンライン   国名：日本国  

開催年月日： 2025年9月

記述言語：日本語   会議種別：口頭発表（招待・特別）  

開催地：東京大学   国名：日本国