  Grigoriev and Podolskii (2018) have established a tropical analogue of the
effective Nullstellensatz, showing that a system of tropical polynomial
equations is solvable if and only if a linearized system obtained from a
truncated Macaulay matrix is solvable. They provided an upper bound of the
minimal admissible truncation degree, as a function of the degrees of the
tropical polynomials. We establish a tropical Nullstellensatz adapted to {\em
sparse} tropical polynomial systems. Our approach is inspired by a construction
of Canny-Emiris (1993), refined by Sturmfels (1994). This leads to an improved
bound of the truncation degree, which coincides with the classical Macaulay
degree in the case of $n+1$ equations in $n$ unknowns. We also establish a
tropical Positivstellensatz, allowing one to decide the inclusion of tropical
basic semialgebraic sets. This allows one to reduce decision problems for
tropical semi-algebraic sets to the solution of systems of tropical linear
equalities and inequalities.
