arXiv subjects

Yasuhito Nakajima

Publications and source records attributed to Yasuhito Nakajima.

5 recordsLinked to original sources

arXiv2026

The Vámos Matroid Has No Second Symmetric Power

Symmetric powers of matroids were introduced by Lovász and Mason. The subject has recently attracted renewed interest through its connection with the realizability of tropical linear spaces by tropical ideals. Anderson proved that the tropical linear space associated with a matroid $M$ is the variety of a tropical ideal if and only if $M$ has a $d$th symmetric power for every positive integer $d$. He also asked whether the Vámos matroid $V_8$ has a second symmetric power. We answer this question in the negative by proving that $V_8$ has no second symmetric power. By Anderson's characterization, the associated tropical linear space cannot be realized as the variety of a tropical ideal.

math.CO↗

arXiv2026

Decompositions of good involutions on quandles

We study the behavior of good involutions under two fundamental constructions of quandles: interaction-free unions and direct products. In particular, we show that the set of good involutions on a quandle decomposes naturally into the set of involutions on its maximal trivial component and the set of good involutions on the complement. Moreover, we construct an example of a connected noninvolutory symmetric quandle with multiple good involutions, which serves as a counterexample to a conjecture by Ta.

math.GT↗

arXiv2026

On the nonexistence of good involutions of symplectic quandles

We investigate the necessary and sufficient condition for the existence of good involutions of symplectic quandles, which are defined on free $R$-modules with an antisymmetric bilinear form. In particular, we discuss the nonexistence of good involutions of symplectic quandles.

math.GT↗

arXiv2024

A geometric interpretation of Krull dimensions of $\boldsymbol{T}$-algebras

We investigate Krull dimensions of semirings and semifields dealt in tropical geometry. For a congruence $C$ on a tropical Laurent polynomial semiring $\boldsymbol{T}[X_1^{\pm}, \ldots, X_n^{\pm}]$, a finite subset $T$ of $C$ is called a finite congruence tropical basis of $C$ if the congruence variety $\boldsymbol{V}(T)$ associated with $T$ coincides with $\boldsymbol{V}(C)$. For $C$ proper, we prove that the Krull dimension of the quotient semiring $\boldsymbol{T}[X_1^{\pm}, \ldots, X_n^{\pm}] / C$ coincides with the maximum of the dimension of $\boldsymbol{V}(C)$ as a polyhedral complex plus one and that of $\boldsymbol{V}(C_{\boldsymbol{B}})$ when both $C$ and $C_{\boldsymbol{B}}$ have finite congruence tropical bases, respectively. Here $C_{\boldsymbol{B}}$ is the congruence on $\boldsymbol{T}[X_1^{\pm}, \ldots, X_n^{\pm}]$ generated by $\{ (f_{\boldsymbol{B}}, g_{\boldsymbol{B}}) \,|\, (f, g) \in C \}$ and $f_{\boldsymbol{B}}$ is defined as the tropical Laurent polynomial obtained from $f$ by replacing the coefficients of all non $-\infty$ terms of $f$ with the real number zero. With this fact, we also show that rational function semifields of tropical curves that do not consist of only one point have Krull dimension two.

math.AG↗

arXiv2019

Minimal tropical basis for Bergman fan of matroid

The Bergman fan of a matroid is the intersection of tropical hyperplanes defined by the circuits. A tropical basis is a subset of the circuits set that defines the Bergman fan. Yu and Yuster posed a question whether every simple regular matroid has a unique minimal tropical basis of its Bergman fan, and verified it for graphic, cographic matroids and $R_{10}$. We show every simple binary matroid has a unique minimal tropical basis. Since the regular matroid is binary, we positively answered the question.

math.CO↗