Upper bounds on Betti numbers of tropical prevarieties

TL;DR

This paper establishes upper bounds on the Betti numbers of tropical prevarieties, relating topological complexity to geometric and combinatorial properties of defining polynomials in both dense and sparse cases.

Contribution

It provides the first explicit bounds on Betti numbers of tropical prevarieties based on Newton polytope volume and monomial count, advancing understanding of their topological complexity.

Findings

Bound in dense case depends on Minkowski sum volume of Newton polytopes.

Bound in sparse case depends on the number of monomials.

Results connect tropical geometry with combinatorial and algebraic properties.

Abstract

We prove upper bounds on the sum of Betti numbers of tropical prevarieties in dense and sparse settings. In the dense setting the bound is in terms of the volume of Minkowski sum of Newton polytopes of defining tropical polynomials, or, alternatively, via the maximal degree of these polynomials. In sparse setting, the bound involves the number of the monomials.