Bounds on the number of connected components for tropical prevarieties

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 ${k+7n-1 \choose 3n} \cdot \frac{d^{3n}}{k+n+1}$\mathrm{.}Onanumberof$0$-dimensional connected components a better bound $\left(\genfrac{}{}{0px}{}{k+4n}{3n}\right)\cdot \frac{d^n}{k+n+1}$ 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 $k$.