On a tropical dual Nullstellensatz

Since a tropical Nullstellensatz fails even for tropical univariate polynomials we study a conjecture on a tropical {\it dual} Nullstellensatz for tropical polynomial systems in terms of solvability of a tropical linear system with the Cayley matrix associated to the tropical polynomial system. The conjecture on a tropical effective dual Nullstellensatz is proved for tropical univariate polynomials

Probabilistic communication complexity over the reals

Deterministic and probabilistic communication protocols are introduced in which parties can exchange the values of polynomials (rather than bits in the usual setting). It is established a sharp lower bound $2n$ on the communication complexity of recognizing the $2n$-dimensional orthant, on the other hand the probabilistic communication complexity of its recognizing does not exceed 4. A polyhedron and a union of hyperplanes are constructed in \RR^{2n} for which a lower bound $n/2$ on the probabilistic communication complexity of recognizing each is proved. As a consequence this bound holds also for the EMPTINESS and the KNAPSACK problems