LAUSR

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 $$\deg (I)$$ deg ( I ) or $$\deg (I)+1$$ deg ( I ) + 1 —we call them paving tropical ideals. We show that paving tropical ideals of degree $$d+1$$ d + 1 are in bijection with $${\mathbb {Z}}^n$$ Z n -invariant d -partitions of $${\mathbb {Z}}^n$$ Z n . This implies that zero-dimensional tropical ideals of degree 3 with Boolean coefficients are in bijection with $${\mathbb {Z}}^n$$ Z n -invariant 2-partitions of quotient groups of the form $${\mathbb {Z}}^n/L$$ 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.