Irrational Mixed Decomposition and Sharp Fewnomial Bounds for Tropical Polynomial Systems

Abstract

Given convex polytopes $P_1,\ldots ,P_r \subset \mathbb {R}^n$ and finite subsets $\mathcal {W}_I$ of the Minkowski sums $P_I=\sum _{i \in I} P_i$, we consider the quantity $N(\mathbf {W})=\sum _{I \subset \mathbf{[}r \mathbf{]} } {(-1)}^{r-|I|} \big | \mathcal {W}_I \big |$. If $\mathcal {W}_I=\mathbb {Z}^n \cap P_I$ and $P_1,\ldots ,P_n$ are lattice polytopes in $\mathbb {R}^n$, then $N(\mathbf {W})$ is the classical mixed volume of $P_1,\ldots ,P_n$ giving the number of complex solutions of a general complex polynomial system with Newton polytopes $P_1,\ldots ,P_n$. We develop a technique that we call irrational mixed decomposition which allows us to estimate $N(\mathbf {W})$ under some assumptions on the family $\mathbf {W}=(\mathcal {W}_I)$. In particular, we are able to show the nonnegativity of $N(\mathbf {W})$ in some important cases. A special attention is paid to the family $\mathbf {W}=(\mathcal {W}_I)$ defined by $\mathcal {W}_I=\sum _{i \in I} \mathcal {W}_i$, where $\mathcal {W}_1,\ldots ,\mathcal {W}_r$ are finite subsets of $P_1,\ldots ,P_r$. The associated quantity $N(\mathbf {W})$ is called discrete mixed volume of $\mathcal {W}_1,\ldots ,\mathcal {W}_r$. Using our irrational mixed decomposition technique, we show that for $r=n$ the discrete mixed volume is an upper bound for the number of nondegenerate solutions of a tropical polynomial system with supports $\mathcal {W}_1,\ldots ,\mathcal {W}_n \subset \mathbb {R}^n$. We also prove that the discrete mixed volume associated with $\mathcal {W}_1,\ldots ,\mathcal {W}_r$ is bounded from above by the Kouchnirenko number $\prod _{i=1}^r (|\mathcal {W}_i|-1)$. For $r=n$ this number was proposed as a bound for the number of nondegenerate positive solutions of any real polynomial system with supports $\mathcal {W}_1,\ldots ,\mathcal {W}_n \subset \mathbb {R}^n$. This conjecture was disproved, but our result show 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.