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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
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.

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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

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

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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