  In this article we study the tropicalization of the Hilbert scheme and its
suitability as a parameter space for tropical varieties. We prove that the
points of the tropicalization of the Hilbert scheme have a tropical variety
naturally associated to them. To prove this, we find a bound on the degree of
the elements of a tropical basis of an ideal in terms of its Hilbert
polynomial.
  As corollary, we prove that the set of tropical varieties defined over an
algebraically closed valued field only depends on the characteristic pair of
the field and the image group of the valuation.
  In conclusion, we examine some simple examples that suggest that the
definition of tropical variety should include more structure than what is
currently considered.
