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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. [ABSTRACT FROM AUTHOR]