[hal-01078469] IRRATIONAL MIXED DECOMPOSITION AND SHARP FEWNOMIAL BOUNDS FOR TROPICAL POLYNOMIAL SYSTEMS

Given convex polytopes P 1 P r ⊂ R n and finite subsets W I of the Minkowsky sums P I = i∈I P i we consider the quantity N W = I⊂r −1 r−I W I If W I = Z n ∩ P I and P 1 P n are lattice polytopes in R n then N W is the classical mixed volume of P 1 P n giving the number of complex solutions of a general com-plex polynomial system with Newton polytopes P 1 P n 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 W = W I defined by W I = i∈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 ⊂ R n We also prove that the discrete mixed volume associated with W 1 W r is bounded from above by the Kouchnirenko number r i=1 W i − 1 For r = n this number was proposed as a bound for the num-ber of nondegenerate positive solutions of any real polynomial system with supports W 1 W n ⊂ R n This conjecture was disproved but our result shows that the Kouch-nirenko number is a sharp bound for the number of nondegenerate positive solutions of real polynomial systems constructed by means of the combinatorial patchworking



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