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.