Khovanskii bases of subalgebras arising from finite distributive lattices

TL;DR

This paper explores when polynomials from distributive lattices form Khovanskii bases in subalgebras, characterizing specific lattice structures like generalized snake and certain free posets.

Contribution

It characterizes distributive lattices and posets for which associated polynomials form Khovanskii bases, extending the theory of Hibi ideals and SAGBI bases.

Findings

Characterization of distributive lattices with Khovanskii bases

Identification of generalized snake and certain free posets

Extension of SAGBI basis theory to subalgebras from lattices

Abstract

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.