Irrational mixed decomposition and sharp fewnomial bounds for tropical polynomial systems

Abstract

Given convex polytopes $P_1,...,P_r$ in $R^n$ and finite subsets $W_I$ of the Minkowsky sums $P_I=\sum_{i \in I} P_i$, we consider the quantity $N(W)=\sum_{I \subset {\bf [}r {\bf ]}} {(-1)}^{r-|I|} \big| W_I \big|$. We develop a technique that we call irrational mixed decomposition which allows us to estimate $N(W)$ under some assumptions on the family $W=(W_I)$. In particular, we are able to show the nonnegativity of $N(W)$ in some important cases. The quantity $N(W)$ associated with the family defined by $W_I=\sum_{i \in I} W_i$ is called discrete mixed volume of $W_1,...,W_r$. We show that for $r=n$ the discrete mixed volume provides an upper bound for the number of nondegenerate solutions of a tropical polynomial system with supports $W_1,...,W_n$. We also prove that the discrete mixed volume of $W_1,...,W_r$ is bounded from above by the Kouchnirenko number $\prod_{i=1}^r (|W_i|-1)$. For $r=n$ this number was proposed as a bound for the number of nondegenerate positive solutions of any real polynomial system with supports $W_1,...,W_n$. This conjecture was disproved, but our result shows that the Kouchnirenko number is a sharp bound for the number of nondegenerate positive solutions of real polynomial systems constructed by means of the combinatorial patchworking.