arXiv subjects

Takaaki Ito

Publications and source records attributed to Takaaki Ito.

3 recordsLinked to original sources

arXiv2026

Non-finite generatedness of the congruences defined by tropical varieties

In tropical geometry, there are several important classes of ideals and congruences such as tropical ideals, bend congruences, and the congruences of the form $\mathbf E(Z)$. Although they are analogues of the concept of ideals of rings, it is not well known whether they are finitely generated. In this paper, we study whether the congruences of the form $\mathbf E(Z)$ are finitely generated. In particular, we show that when $Z$ is the support of a tropical variety, $\mathbf E(Z)$ is not finitely generated except for a few specific cases. In addition, we give an explicit minimal generating set of $\mathbf E(|L|)$ for the tropical standard line $L$.

math.AC↗

arXiv2024

Homomorphisms between the semirings of Boolean functions on 1-dimensional tropical fans

In a previous work, we stated the conjecture that any semiring homomorphism between the Boolean function semirings on 1-dimensional tropical fans has the property called geometric. In this paper, we show that the conjecture is true. As an application, we establish a way to find all the morphisms between given 1-dimensional tropical fans.

math.AG↗

arXiv2022

Local theory of functions on tropical curves in $\mathbb R^n$

We first develop the local theory of functions on $\mathbb R^n$ defined by tropical Laurent polynomials. We study the structure of the semiring of functions, where two functions are identified when they coincide on a neighborhood of a fixed point. We see that this semiring is closely related to the semiring of functions defined by Boolean Laurent polynomials. Then we develop the local theory of functions on tropical curves. We construct a contravariant functor from the category of 1-dimensional tropical fans to the category of certain homomorphisms of semirings. As an application, we discuss about the smoothness of 1-dimensional tropical fans at the origin.

math.AG↗