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Tropicalization and irreducibility of generalized Vandermonde determinants


Authors: Carlos D'Andrea and Luis Felipe Tabera
Journal: Proc. Amer. Math. Soc. 137 (2009), 3647-3656
MSC (2000): Primary 12E05; Secondary 14M12
DOI: https://doi.org/10.1090/S0002-9939-09-09951-1
Published electronically: June 12, 2009
MathSciNet review: 2529871
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Abstract: We find geometric and arithmetic conditions in order to characterize the irreducibility of the determinant of the generic Vandermonde matrix over the algebraic closure of any field $ k$. We also characterize those determinants whose tropicalization with respect to the variables of a row is irreducible.


References [Enhancements On Off] (What's this?)

  • [BP] Beelen, Peter; Pellikaan, Ruud.
    The Newton polygon of plane curves with many rational points.
    Special issue dedicated to Dr. Jaap Seidel on the occasion of his 80th birthday (Oisterwijk, 1999). Des. Codes Cryptogr. 21 (2000), no. 1-3, 41-67. MR 1801161 (2003c:14024)
  • [BG] Bieri, Robert; Groves, J. R. J.
    The geometry of the set of characters induced by valuations.
    J. Reine Angew. Math. 347 (1984), 168-195. MR 733052 (86c:14001)
  • [CL] Chui, Charles K.; Lai, Hang-Chin.
    Vandermonde determinant and Lagrange interpolation in $ R\sp s$.
    Nonlinear and convex analysis (Santa Barbara, Calif., 1985), 23-35, Lecture Notes in Pure and Appl. Math., 107. Dekker, New York, 1987. MR 892779 (88j:41011)
  • [EKL] Einsiedler, Manfred; Kapranov, Mikhail; Lind, Douglas.
    Non-Archimedean amoebas and tropical varieties.
    J. Reine Angew. Math. 601 (2006), 139-157. MR 2289207 (2007k:14038)
  • [ElM] El-Mikkawy, Moawwad.
    Vandermonde interpolation using special associated matrices.
    Appl. Math. Comput. 141 (2003), no. 2-3, 589-595. MR 1972934 (2004a:65011)
  • [Gao] Gao, Shuhong.
    Absolute irreducibility of polynomials via Newton polytopes.
    J. Algebra 237 (2001), no. 2, 501-520. MR 1816701 (2002f:52013)
  • [GS] Gasca, Mariano; Sauer, Thomas.
    Polynomial interpolation in several variables.
    Adv. Comput. Math. 12 (2000), no. 4, 377-410. MR 1768957 (2001d:41010)
  • [GKZ] Gel$ ^\prime$fand, I. M.; Kapranov, M. M.; Zelevinsky, A. V.
    Discriminants, resultants, and multidimensional determinants.
    Mathematics: Theory & Applications. Birkhäuser Boston, Inc., Boston, MA, 1994. MR 1264417 (95e:14045)
  • [GV] Giosan, Traian; Voicu, M.
    Formules d'approximation et d'interpolation par polynômes obtenues à l'aide du déterminant Vandermonde.
    Bul. Inst. Politehn. Iaşi (N.S.) 16 (20) (1970), facs. 3-4, sect. III, 13-19. MR 0282497 (43:8208)
  • [GR] Goodman, Jacob E.; O'Rourke, Joseph (editors).
    Handbook of discrete and computational geometry.
    Discrete Mathematics and its Applications. Chapman & Hall/CRC, Boca Raton, FL, second edition, 2004. MR 2082993 (2005j:52001)
  • [Gow] Gow, R.
    Cauchy's matrix, the Vandermonde matrix and polynomial interpolation.
    Irish Math. Soc. Bull. 28 (1992), 45-52. MR 1205443 (94i:15002)
  • [Jou] Jouanolou, Jean-Pierre.
    Théorèmes de Bertini et applications.
    Progress in Mathematics, 42. Birkhäuser Boston, Inc., Boston, MA, 1983. MR 725671 (86b:13007)
  • [LS] Little, John; Schwarz, Ryan.
    On toric codes and multivariate Vandermonde matrices.
    Appl. Algebra Engrg. Comm. Comput. 18 (2007), no. 4, 349-367. MR 2322944 (2008h:94117)
  • [Mac] Macdonald, I. G.
    Symmetric functions and orthogonal polynomials.
    Dean Jacqueline B. Lewis Memorial Lectures presented at Rutgers University, New Brunswick, NJ. University Lecture Series, 12. American Mathematical Society, Providence, RI, 1998. MR 1488699 (99f:05116)
  • [Mik] Mikhalkin, Grigory.
    Enumerative tropical algebraic geometry in $ \mathbb{R}\sp 2$.
    J. Amer. Math. Soc. 18 (2005), no. 2, 313-377. MR 2137980 (2006b:14097)
  • [Olv] Olver, Peter J.
    On multivariate interpolation.
    Stud. Appl. Math. 116 (2006), no. 2, 201-240. MR 2200737 (2007b:41055)
  • [Ost] Ostrowski, A. M.
    On multiplication and factorization of polynomials. II. Irreducibility discussion.
    Aequationes Math. 14 (1976), no. 1/2, 1-31. MR 0422229 (54:10220)
  • [Tab] Tabera, L. F.
    Two tools in algebraic geometry: Construction of configurations in tropical geometry and hypercircles for the simplification of parametric curves.
    Ph.D. Thesis, Universidad de Cantabria/Université de Rennes I, 2007.
  • [TS] Tevelev, Jenia; Sturmfels, Bernd.
    Elimination theory for tropical varieties.
    Math. Res. Lett. 15 (2008), no. 3, 543-562. MR 2407231
  • [Zhu] Zhu, Ping.
    Multivariate Vandermonde determinants and general Birkhoff interpolation.
    Northeast. Math. J. 21 (2005), no. 3, 336-344. MR 2173614 (2006d:41003)

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Additional Information

Carlos D'Andrea
Affiliation: Departament d’Àlgebra i Geometria, Universitat de Barcelona, Gran Via 585, E-08007 Barcelona, Spain
Email: cdandrea@ub.edu

Luis Felipe Tabera
Affiliation: Departamento de Matemáticas, Universidad de Cantabria, Estadística y Computación, Av. los Castros s/n, E-39005, Santander, Spain
Address at time of publication: Department of Mathematics, University of California, Berkeley, 893 Evans Hall #3840, Berkeley, California 94720-3840
Email: tabera@berkeley.edu

DOI: https://doi.org/10.1090/S0002-9939-09-09951-1
Keywords: Vandermonde determinants, multivariate factorization, tropical geometry, valuations
Received by editor(s): March 14, 2008
Received by editor(s) in revised form: February 27, 2009
Published electronically: June 12, 2009
Additional Notes: The first author was supported by the Programa Ramón y Cajal and the Research Project grant MTM2007–67493 from the Ministerio de Educación y Ciencia, Spain.
The second author was supported by the Research Project grant MTM2005-08690-CO2-02 and a postdoctoral fellowship from the Ministerio de Educación y Ciencia, Spain
Communicated by: Martin Lorenz
Article copyright: © Copyright 2009 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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