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 Z n -invariant d-partitions of Z n . This implies that zero-dimensional tropical ideals of degree 3 with Boolean coefficients are in bijection with Z n -invariant 2-partitions of quotient groups of the form 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 zerodimensional tropical ideals. Data Availability Statement. Data sharing not applicable to this article as no datasets were generated or analysed during the current study.

Generalisations of Tropical Geometry over Hyperfields

“…One recent development in tropical geometry has been the study of tropical ideals, chiefly in [MR18] and [MR20], then elsewhere in [FGG ], [AR22], [GG16] and [Zaj18].…”

“…Tropical ideals satisfy extra combinatorial properties, by way of a monomial elimination axiom. Furthermore, other contributions progressing the understanding of tropical ideals can be seen in [GG16], [GG18], [DR21], [AR22], [FGG ],…”

“…A matroid M is a paving matroid if all the circuits are of size rankpMq or rankpMq `1. In [AR22] the authors study zero dimensional tropical ideals, whose underlying matroid is a paving matroid.…”