  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.
