## Algebra & Number Theory

### Algebraic properties of generic tropical varieties

#### Abstract

We show that the algebraic invariants multiplicity and depth of the quotient ring $S∕I$ of a polynomial ring $S$ and a graded ideal $I⊂S$ are closely connected to the fan structure of the generic tropical variety of $I$ in the constant coefficient case. Generically the multiplicity of $S∕I$ 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 $S∕I$ from the fan structure of the generic tropical variety of $I$ if the depth is known to be greater than $0$. In particular, in this case we can see if $S∕I$ is Cohen–Macaulay or almost-Cohen–Macaulay from the generic tropical variety of $I$.

#### Article information

Source
Algebra Number Theory, Volume 4, Number 4 (2010), 465-491.

Dates
Revised: 5 February 2010
Accepted: 6 April 2010
First available in Project Euclid: 20 December 2017

https://projecteuclid.org/euclid.ant/1513729539

Digital Object Identifier
doi:10.2140/ant.2010.4.465

Mathematical Reviews number (MathSciNet)
MR2661539

Zentralblatt MATH identifier
1193.13020

#### Citation

Römer, Tim; Schmitz, Kirsten. Algebraic properties of generic tropical varieties. Algebra Number Theory 4 (2010), no. 4, 465--491. doi:10.2140/ant.2010.4.465. https://projecteuclid.org/euclid.ant/1513729539

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