arXiv subjects

Koichiro Tani

Publications and source records attributed to Koichiro Tani.

5 recordsLinked to original sources

arXiv2026

Algebras with straightening laws on join- or meet-semidistributive lattices

We study algebras with straightening laws on join- or meet-semidistributive lattices. We show that for a join-semidistributive (resp. meet-semidistributive) lattice, meet-distributivity (resp. join-distributivity) and Cohen--Macaulayness are equivalent, and that integrality implies these conditions. Thus, Hibi's conjecture that every integral lattice is Cohen--Macaulay holds for join- or meet-semidistributive lattices. For semidistributive lattices, distributivity, integrality and Cohen--Macaulayness are equivalent.

math.AC↗

arXiv2025

Standard multigraded Hibi rings and Cartwright-Sturmfels ideals

In this paper, we introduce standard multigradings on Hibi rings, which are algebras arising from posets. We show that any standard multigrading on a Hibi ring that makes its defining ideal (called the Hibi ideal) homogeneous is induced by a chain of the underlying poset. After that, we calculate the multigraded Hilbert series of Hibi rings by generalizing the theory of $P$-partition and we compute the multidegree polynomials of Hibi rings. Furthermore, we characterize Hibi ideals that are Cartwright-Sturmfels ideals.

math.AC↗

arXiv2025

Initial ideals of generic ideals and variations of Moreno-Soc\'{i}as conjecture

It is known that the initial ideals of generic ideals are the same. Moreno-Soc\'{i}as conjectured that the initial ideal of generic ideals with respect to the degree reverse lexicographic order is weakly reverse lexicographic. In the first half of this paper, we study the initial ideal of generic ideals for arbitrary monomial order and prove that the initial ideal of generic ideals is Borel-fixed. It can be considered as a weakened version of Moreno-Soc\'{i}as conjecture. In the second half, we propose a new method of the computation of the initial ideal of generic ideals using stability condition of Gr\"{o}bner bases. We apply the method in the case of lexicographic order and study the relationship between the lexsegment ideal and the initial ideal of generic ideals. This study aims to bound the maximal degree of Gr\"{o}bner basis. At the last, we propose questions that can be considered as a lexicographic analogue of Moreno-Soc\'{i}as conjecture.

math.AC↗

arXiv2025

Khovanskii bases of subalgebras arising from finite distributive lattices

The notion of Khovanskii bases was introduced by Kaveh and Manon. It is a generalization of the notion of SAGBI bases for a subalgebra of polynomials. The notion of SAGBI bases was introduced by Robbiano and Sweedler as an analogue of Gr\"{o}bner bases in the context of subalgebras. A Hibi ideal is an ideal of a polynomial ring that arises from a distributive lattice. For the development of an analogy of the theory of Hibi ideals and Gr\"{o}bner bases within the framework of subalgebras, in this paper, we investigate when the set of the polynomials associated with a distributive lattice forms a Khovanskii basis of the subalgebras it generates. We characterize such distributive lattices and their underlying posets. In particular, generalized snake posets and $\{(2+2),(1+1+1)\}$-free posets appear as the characterization.

math.AC↗

arXiv2024

Non-finitely generated monoids corresponding to finitely generated homogeneous subalgebras

The goal of this paper is to study the possible monoids appearing as the associated monoids of the initial algebra of a finitely generated homogeneous $\Bbbk$-subalgebra of a polynomial ring $\Bbbk[x_1,\ldots,x_n]$. Clearly, any affine monoid can be realized since the initial algebra of the affine monoid $\Bbbk$-algebra is itself. On the other hand, the initial algebra of a finitely generated homogeneous $\Bbbk$-algebra is not necessarily finitely generated. In this paper, we provide a new family of non-finitely generated monoids which can be realized as the initial algebras of finitely generated homogeneous $\Bbbk$-algebras. Moreover, we also provide an example of a non-finitely generated monoid which cannot be realized as the initial algebra of any finitely generated homogeneous $\Bbbk$-algebra.

math.AC↗