Abstract
We give a description of the tropical Severi variety of univariate polynomials of degree n having two double roots. We show that, as a set, it is given as the union of three explicit types of cones of maximal dimension n − 1, where only cones of two of these types are cones of the secondary fan of {0,..., n}. Through Kapranov’s theorem, this goal is achieved by a careful study of the possible valuations of the elementary symmetric functions of the roots of a polynomial with two double roots. Despite its apparent simplicity, the computation of the tropical Severi variety has both combinatorial and arithmetic ingredients.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
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.
F. Block, Counting algebraic curves with tropical geometry, in Tropical Geometry and Integrable Systems, Contemp. Math. Vol. 580, 2012, pp. 41–54.
T. Bogart, A. N. Jensen, D. Speyer, B. Sturmfels and R. R. Thomas, Computing tropical varieties, J. Symbolic Comput. 42 (2007), 54–73.
A. Dickenstein, S. di Rocco and R. Piene, Higher order duality and toric embeddings, Ann. Inst. Fourier 64 (2014), 375–400.
A. Dickenstein, M. Feichtner and B. Sturmfels, Tropical discriminants, J. Amer. Math. Soc. 20 (2007), 1111–1133.
A. Dickenstein and L. Tabera, Singular tropical hypersurfaces, Discrete Comput. Geom. 47 (2012) 430–453.
M. Einsiedler, M. Kapranov and D. Lind, Non-Archimedean amoebas and tropical varieties, J. Reine Angew. Math. 601 (2006), 139–157.
A. Esterov, Characteristic classes of affine varieties and Plucker formulas for affine morphisms, J. Eur. Math. Soc 2016, to appear.
I. M. Gelfand, M. M. Kapranov and A. V. Zelevinsky, Discriminants, Resultants, and Multidimensional Determinants, Birkhäuser Boston, Boston, MA, 1994.
E. Katz, Tropical invariants from the secondary fan, Adv. Geom. 9 (2009), 153–180.
D. Maclagan and B. Sturmfels, Introduction to Tropical Geometry, Grad. Studies in Math., Vol. 161, American Mathematical Society, Providence, RI, 2015.
G. Mikhalkin, Enumerative tropical algebraic geometry in R2, J. Amer. Math. Soc. 18 (2005), 313–377.
S. Payne, Fibers of tropicalization, Math. Z. 262 (2009), 301–311.
W. A. Stein et al, Sage Mathematics Software (Version 6.9), The Sage Development Team, 2015, http://www.sagemath.org.
L. F. Tabera, Two Tools in Algebraic Geometry: Construction of Configurations in Tropical Geometry and Hypercircles for the Simplification of Parametric Curves, PhD thesis, Universidad de Cantabria, Université de Rennes I, 2007.
L. F. Tabera, On real tropical bases and real tropical discriminants, Collect. Math. 66 (2015), 77–92.
J. J. Yang, Tropical Severi Varieties, Port. Math. 70 (2013), 59–91.
J. J. Yang, Secondary fans and Tropical Severi varieties, Manuscripta Math. 149 (2016), 93–106.
J. Yu and D. Yuster, Representing tropical linear spaces by circuits, in Formal Power Series and Algebraic Combinatorics (FPSAC’ 07), Proceedings, Tianjin, China, 2007.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Dickenstein, A., Herrero, M.I. & Tabera, L.F. Arithmetics and combinatorics of tropical Severi varieties of univariate polynomials. Isr. J. Math. 221, 741–777 (2017). https://doi.org/10.1007/s11856-017-1573-0
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11856-017-1573-0