Bounds on the number of connected components for tropical prevarieties

TL;DR

This paper establishes upper bounds on the number of connected components of tropical prevarieties in real space, extending classical bounds and demonstrating the dependence on the number of defining polynomials.

Contribution

It provides new combinatorial bounds for the connected components of tropical prevarieties, generalizing previous results to arbitrary numbers of polynomials.

Findings

Derived bounds depend on the number of polynomials and degrees.

Extended Sturmfels' Bezout bound to arbitrary k ≥ n.

Showed bounds are close to sharp, indicating their tightness.

Abstract

For a tropical prevariety in ${R}^n$ given by a system of $k$ tropical polynomials in $n$ variables with degrees at most $d$, we prove that its number of connected components is less than ${k+7n-1 \choose 3n} \cdot \frac{d^{3n}}{k+n+1}$. On a number of $0$-dimensional connected components a better bound ${k+4n \choose 3n} \cdot \frac{d^n}{k+n+1}$ is obtained, which extends the Bezout bound due to B.~Sturmfels from the the case $k=n$ to an arbitrary $k\ge n$. Also we show that the latter bound is close to sharp, in particular, the number of connected components can depend on $k$.