Algebra & Number Theory

Algebraic properties of generic tropical varieties

Tim Römer and Kirsten Schmitz

Full-text: Open access

Abstract

We show that the algebraic invariants multiplicity and depth of the quotient ring SI of a polynomial ring S and a graded ideal IS are closely connected to the fan structure of the generic tropical variety of I in the constant coefficient case. Generically the multiplicity of SI 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 SI 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 SI 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
Received: 11 September 2009
Revised: 5 February 2010
Accepted: 6 April 2010
First available in Project Euclid: 20 December 2017

Permanent link to this document
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

Subjects
Primary: 13F20: Polynomial rings and ideals; rings of integer-valued polynomials [See also 11C08, 13B25]
Secondary: 14Q99: None of the above, but in this section 13P10: Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases)

Keywords
tropical variety constant coefficient case Gröbner fan generic initial ideals Cohen–Macaulay multiplicity depth

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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