Paving tropical ideals

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.