A tropical version of Hilbert function

For a tropical prevariety V ⁿ (being a finite union of rational polyhedra) two tropical polynomials belong to the same congruence class iff they coincide on V. We define a tropical Hilbert function THV (k) as the dimension of the family of congruence classes of tropical polynomials with tropical degree at most k. It is shown that THV (k) coincides with a polynomial of degree V for sufficiently big k when V is a polyhedron. For an arbitrary tropical prevariety we formulate it as a conjecture. The conjecture is justified in case V =0.