Vol. 4, No. 4, 2010

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Algebraic properties of generic tropical varieties

Tim Römer and Kirsten Schmitz

Vol. 4 (2010), No. 4, 465–491
Abstract

We show that the algebraic invariants multiplicity and depth of the quotient ring SI 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 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.

Keywords
tropical variety, constant coefficient case, Gröbner fan, generic initial ideals, Cohen–Macaulay, multiplicity, depth
Mathematical Subject Classification 2000
Primary: 13F20
Secondary: 14Q99, 13P10
Milestones
Received: 11 September 2009
Revised: 5 February 2010
Accepted: 6 April 2010
Published: 13 June 2010
Authors
Tim Römer
Institut für Mathematik
Universität Osnabrück
49069 Osnabrück
Germany
Kirsten Schmitz
Institut für Mathematik
Universität Osnabrück
49069 Osnabrück
Germany