IRRATIONAL MIXED DECOMPOSITION AND SHARP FEWNOMIAL BOUNDS FOR TROPICAL POLYNOMIAL SYSTEMS

small corrections in version 2.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 $N (W) =\sum_{I \subset [r]} (−1)^{r−|I|} W_I$ . 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. A special attention is paid to the family defined by $W_I =\sum_{i \in I} W_i $, where $W_1 , . . . , W_r$ are finite subsets of $P_1 , . . . , P_r$ . The associated quantity $N (W)$ is called discrete mixed volume of $W_1 , . . . , W_r$ . 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 $W_1 , . . . , W_n$. We also prove that the discrete mixed volume associated with $W_1 , . . . , W_r$ 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