PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 137, Number 11, November 2009, Pages 3647–3656 S 0002-9939(09)09951-1 Article electronically published on June 12, 2009
TROPICALIZATION AND IRREDUCIBILITY OF GENERALIZED VANDERMONDE DETERMINANTS
CARLOS D’ANDREA AND LUIS FELIPE TABERA
(Communicated by Martin Lorenz)
Abstract. We find geometric and arithmetic conditions in order to charac- terize the irreducibility of the determinant of the generic Vandermonde matrix over the algebraic closure of any field k. We also characterize those determi- nants whose tropicalization with respect to the variables of a row is irreducible.
1. Introduction
Let n, N be positive integers, let X1,...,XN be n-tuples of indeterminates, i.e. Xi := Xi1,...,Xin ,i=1,...,N, where each Xij is an indeterminate, and let Γ := γ1,...,γN be an N-tuple of n vectors in N , with γj =(γj1,...,γjn). Set γ11 γ12 γ1n γN1 γN2 γNn X X ···X ...... X X ···X 11 12 1n 11 12 1n γ11 γ12 γ1n γN1 γN2 γNn X X ···X ...... X X ···X 21 22 2n 21 22 2n V (X, Γ) := . ...... γ11 γ12 ··· γ1n γN1 γN2 ··· γNn XN1 XN2 XNn ...... XN1 XN2 XNn We call the polynomial V (X, Γ) ∈ Z (Xi,j)1≤i≤N,1≤j≤n the generalized Vander- monde determinant associated to Γ. Example 1.1. If n =1andΓ=(0, 1,...,N − 1), then V (X, Γ) = ± (Xi1 − Xj1), 1≤i Received by the editors March 14, 2008, and, in revised form, February 27, 2009. 2000 Mathematics Subject Classification. Primary 12E05; Secondary 14M12. Key words and phrases. Vandermonde determinants, multivariate factorization, tropical ge- ometry, valuations. The first author was supported by the Programa Ram´on y Cajal and the Research Project grant MTM2007–67493 from the Ministerio de Educaci´on 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 Educaci´on y Ciencia, Spain. c 2009 American Mathematical Society Reverts to public domain 28 years from publication 3647 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 3648 CARLOS D’ANDREA AND LUIS FELIPE TABERA Example 1.2. It is a classical result (see for instance [Mac]) that if n =1,thenfor any set Γ ⊂ N of N elements, the determinant V (X, Γ) is a multiple of the classical − Vandermonde determinant 1≤i