We show that the algebraic invariants multiplicity and depth of a graded
ideal in the polynomial ring are closely connected to the fan structure of its
generic tropical variety in the constant coefficient case. Generically the
multiplicity of the ideal is shown to correspond directly to a natural
definition of multiplicity of cones of tropical varieties. Moreover, we can
recover information on the depth of the ideal from the fan structure of the
generic tropical variety if the depth is known to be greater than 0.