We show that the algebraic invariants multiplicity and depth of the quotient ring of a polynomial ring and a graded ideal are closely connected to the fan structure of the generic tropical variety of in the constant coefficient case. Generically the multiplicity of 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 from the fan structure of the generic tropical variety of if the depth is known to be greater than . In particular, in this case we can see if is Cohen–Macaulay or almost-Cohen–Macaulay from the generic tropical variety of .
"Algebraic properties of generic tropical varieties." Algebra Number Theory 4 (4) 465 - 491, 2010. https://doi.org/10.2140/ant.2010.4.465