Appears in collection : Combinatorics and Arithmetic for Physics: special days

We define Hilbert function of a semiring ideal of tropical polynomials in n variables. For n=1 we prove that it is the sum of a linear function and a periodic function (for sufficiently large values). The leading coefficient of the linear function equals the tropical entropy of the ideal. For an arbitrary n we discuss a conjecture that the tropical Hilbert function of a radical ideal is a polynomial of degree at most n−1 (for sufficiently large values). For n=1 the conjecture is true, also we have proved it for zero-dimensional ideals and for planar tropical curves.