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We show that the algebraic invariants multiplicity and depth of a gradedideal in the polynomial ring are closely connected to the fan structure of itsgeneric tropical variety in the constant coefficient case. Generically themultiplicity of the ideal is shown to correspond directly to a naturaldefinition of multiplicity of cones of tropical varieties. Moreover, we canrecover information on the depth of the ideal from the fan structure of thegeneric tropical variety if the depth is known to be greater than 0. Inparticular, in this case we can see if the ideal is Cohen-Macaulay oralmost-Cohen-Macaulay from its generic tropical variety.

Algebraic Properties of Generic Tropical Varieties