Irrational Mixed Decomposition and Sharp Fewnomial Bounds for Tropical Polynomial Systems

Given convex polytopes P1, . . . , Pr ⊂ R n and finite subsets WI of the Minkowsky sums PI = � i∈I Pi, we consider the quantityN(W) = � I⊂[r] (−1) r−|I|� WI � �. If WI = Z n ∩ PI and P1, . . . , Pn are lattice polytopes in R , then N(W) is the classical mixed volume of P1, . . . , Pn giving the number of complex solutions of a general complex polynomial system with Newton polytopes P1, . . . , Pn. We develop a technique that we call irrational mixed decomposition which allows us to estimate N(W) under some assumptions on the family W = (WI). In particular, we are able to show the nonnegativity of N(W) in some important cases. A special attention is paid to the family W = (WI) defined by WI = � i∈I Wi, where W1, . . . ,Wr are finite subsets of P1, . . . , Pr. The associated quantityN(W) is called discrete mixed volume ofW1, . . . ,Wr. 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 W1, . . . ,Wn ⊂ R . We also prove that the discrete mixed volume associated with W1, . . . ,Wr is bounded from above by the Kouchnirenko number � r i=1(|Wi| − 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 W1, . . . ,Wn ⊂ R . 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.