Paving Tropical Ideals

TL;DR

This paper investigates a special class of zero-dimensional tropical ideals called paving tropical ideals, establishing their combinatorial classification via invariant partitions and providing new examples and applications.

Contribution

It introduces paving tropical ideals, characterizes their structure through invariant partitions, and constructs numerous examples including non-realizable cases.

Findings

Paving tropical ideals correspond to invariant partitions of $\\mathbb Z^n$.

Zero-dimensional degree 3 tropical ideals relate to 2-partitions of quotient groups.

Constructed uncountably many degree-3 tropical ideals with Boolean coefficients.

Abstract

Tropical ideals are a class of ideals in the tropical polynomial semiring that combinatorially abstracts the possible collections of supports of all polynomials in an ideal over a field. We study zero-dimensional tropical ideals I with Boolean coefficients in which all underlying matroids are paving matroids, or equivalently, in which all polynomials of minimal support have support of size deg(I) or deg(I)+1 -- we call them paving tropical ideals. We show that paving tropical ideals of degree d+1 are in bijection with $\mathbb Z^n$-invariant d-partitions of $\mathbb Z^n$. This implies that zero-dimensional tropical ideals of degree 3 with Boolean coefficients are in bijection with $\mathbb Z^n$-invariant 2-partitions of quotient groups of the form $\mathbb Z^n/L$. We provide several applications of these techniques, including a construction of uncountably many zero-dimensional degree-3 tropical ideals in one variable with Boolean coefficients, and new examples of non-realizable zero-dimensional tropical ideals.