Algebra & Number Theory

Algebraic properties of generic tropical varieties

Tim Römer and Kirsten Schmitz

Full-text: Open access


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

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

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

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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)

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


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.

Export citation


  • R. Bieri and J. R. J. Groves, “The geometry of the set of characters induced by valuations”, J. Reine Angew. Math. 347 (1984), 168–195.
  • T. Bogart, A. N. Jensen, D. Speyer, B. Sturmfels, and R. R. Thomas, “Computing tropical varieties”, J. Symbolic Comput. 42:1-2 (2007), 54–73.
  • W. Bruns and A. Conca, “Gröbner bases, initial ideals and initial algebras”, in Homological methods in commutative algebra (Tehran, 2004), edited by L. L. Avramov et al., 2004.
  • W. Bruns and J. Gubeladze, Polytopes, rings, and $K$-theory, Springer, Dordrecht, 2009.
  • W. Bruns and J. Herzog, Cohen–Macaulay rings, Cambridge Studies in Advanced Mathematics 39, Cambridge University Press, 1993.
  • M. Develin and B. Sturmfels, “Tropical convexity”, Doc. Math. 9 (2004), 1–27. Erratum in 9 (2004), 205–206.
  • A. Dickenstein, E. M. Feichtner, and B. Sturmfels, “Tropical discriminants”, J. Amer. Math. Soc. 20:4 (2007), 1111–1133.
  • J. Draisma, “A tropical approach to secant dimensions”, J. Pure Appl. Algebra 212:2 (2008), 349–363.
  • D. Eisenbud, Commutative algebra, Graduate Texts in Mathematics 150, Springer, New York, 1995.
  • D. Eisenbud and S. Goto, “Linear free resolutions and minimal multiplicity”, J. Algebra 88:1 (1984), 89–133.
  • S. Eliahou and M. Kervaire, “Minimal resolutions of some monomial ideals”, J. Algebra 129:1 (1990), 1–25.
  • A. Gathmann, “Tropical algebraic geometry”, Jahresber. Deutsch. Math.-Verein. 108:1 (2006), 3–32.
  • A. Gathmann and H. Markwig, “Kontsevich's formula and the WDVV equations in tropical geometry”, Adv. Math. 217:2 (2008), 537–560. 1131.14057
  • A. Gathmann, M. Kerber, and H. Markwig, “Tropical fans and the moduli spaces of tropical curves”, Compos. Math. 145:1 (2009), 173–195. 1169.51021
  • H.-G. Gräbe, “Two remarks on independent sets”, J. Algebraic Combin. 2:2 (1993), 137–145.
  • J. Herzog and H. Srinivasan, “Bounds for multiplicities”, Trans. Amer. Math. Soc. 350:7 (1998), 2879–2902.
  • I. Itenberg, G. Mikhalkin, and E. Shustin, Tropical algebraic geometry, Oberwolfach Seminars 35, Birkhäuser, Basel, 2007.
  • A. N. Jensen, Algorithmic aspects of Gröbner fans and tropical varieties, Ph.D. thesis, Aarhus University, 2007,
  • A. N. Jensen, “Gfan, a software system for Gröbner fans and tropical varieties”, version 0.4, 2009,
  • A. N. Jensen, H. Markwig, and T. Markwig, “An algorithm for lifting points in a tropical variety”, Collect. Math. 59:2 (2008), 129–165.
  • E. Katz, H. Markwig, and T. Markwig, “The $j$-invariant of a plane tropical cubic”, J. Algebra 320:10 (2008), 3832–3848.
  • D. Maclagan and R. R. Thomas, “Computational algebra and combinatorics of toric ideals”, pp. 1–106 in Commutative algebra and combinatorics (Allahabad, 2003), vol. 1, edited by R. V. Gurjar et al., Ramanujan Math. Soc. Lect. Notes Ser. 4, Ramanujan Math. Soc., Mysore, 2007.
  • G. Mikhalkin, “Tropical geometry and its applications”, pp. 827–852 in International Congress of Mathematicians (Madrid, 2006), vol. 2, edited by M. Sanz-Solé et al., Eur. Math. Soc., Zürich, 2006.
  • T. Mora and L. Robbiano, “The Gröbner fan of an ideal”, J. Symbolic Comput. 6:2-3 (1988), 183–208.
  • T. Römer and K. Schmitz, “Generic tropical varieties”, preprint, 2009.
  • D. E. Speyer, Tropical geometry, ProQuest LLC, Ann Arbor, MI, 2005.
  • D. Speyer and B. Sturmfels, “The tropical Grassmannian”, Adv. Geom. 4:3 (2004), 389–411.
  • B. Sturmfels, Gröbner bases and convex polytopes, University Lecture Series 8, American Mathematical Society, Providence, RI, 1996.
  • W. V. Vasconcelos, Computational methods in commutative algebra and algebraic geometry, Algor. Comput. Math. 2, Springer, Berlin, 1998.