On a tropical dual Nullstellensatz

Since a tropical Nullstellensatz fails even for univariate polynomials we study
a conjecture on a tropical {\it dual} Nullstellensatz in terms of the tropical
Cayley matrix. The conjecture is proved for tropical univariate polynomials.
Also we produce an algorithm (being a tropical version of the Gram-Schmidt
process) for solving tropical linear systems and a criterion of their solvability
via the tropical and Kapranov ranks