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<j≤N the classical Vandermonde determinant. 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<j≤N (Xi1 Xj1). Example 1.3. Suppose n =2,N = 3 and Γ := ((2, 0), (0, 2), (2, 2)). By computing the 3 × 3 determinant we have that V (X, Γ) is equal to 2 2 2 2 − 2 2 2 2 − 2 2 2 2 X11X22X31X32 X11X32X21X22 X12X21X31X32 2 2 2 2 2 2 2 2 − 2 2 2 2 + X12X31X21X22 + X11X12X21X32 X11X12X31X22. { } Set Γ := (1, 0), (0, 1), (1, 1) ;itiseasytoseethat • If char(k) =2,V(X, Γ) is absolutely irreducible over k (Xi,j )1≤i≤N,1≤j≤n (i.e. irreducible over k (Xi,j )1≤i≤N,1≤j≤n , k being the algebraic closure of k). 2 • If char(k)=2,thenV (X, Γ) = V (X, Γ ) in k (Xi,j )1≤i≤N,1≤j≤n . Example 1.4. As an easy exercise, it can be proved that if Γ ⊂ Nn is contained in an affine line, then V (X, Γ) factorizes in a similar way as in Example 1.2. In the univariate case (n = 1), the Vandermonde determinant is associated with the interpolation problem, and it has been extensively studied (see [GV, Gow, ElM] and the references therein). The multivariate interpolation problem is naturally as- sociated with generalized Vandermonde determinants, and there is also an extensive and current literature on the topic. See for instance [CL, GS, LS, Olv, Zhu]. The purpose of this article is to study the irreducibility of V (X, Γ). As Ex- ample 1.3 suggests, the answer will depend on the characteristic of k. Also, our intuition from the univariate case may lead us to believe that generalized Vander- monde determinants have lots of irreducible factors. Our main result essentially tells us that in general these polynomials are absolutely irreducible. There are some trivial factors that can already be read from the set of exponents. { } Let γ := (g1 ...,gn)whereeachgi is defined as min γ1i,...,γN i , i =1,...,n. It is easy to check that the following factorization holds: n N gi V X, (γ1,...,γN ) = Xij V X, (γ1 − γ,...,γN − γ) , i=1 j=1 where V X, (γ1 − γ,...,γN − γ) has no monomial factor. Let dΓ be the largest 1 n n integer such that {γ1 − γ,...,γN − γ}⊂N , and let LΓ ⊂ R be the affine dΓ subspace spanned by Γ. Theorem 1.5. Let N ≥ 3. The Vandermonde polynomial V (X, Γ) is irreducible in k (Xij)1≤i,j,≤N if and only if the following three conditions apply: • dim(LΓ)≥ 2, • γi gcd X 1≤i≤N =1or equivalently γ =(0,...,0), • char(k) does not divide dΓ. Note that dim(LΓ) ≥ 2 implies N ≥ 3andn>1. When n =2andN = 3 or 4, Theorem 1.5 can also be obtained from an application of Ostrowski’s work [Ost] on the irreducibility of fewnomials (see also [BP]). We will prove the general case by making use of Bertini’s theorem on the variety defined by V (X, Γ), and applying some results concerning algebraic independence of maximal Vandermonde minors obtained in [Tab]. License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use GENERALIZED VANDERMONDE DETERMINANTS 3649 When dealing with the problem of factorizing multivariate polynomials, several approaches like those given in [Ost, Gao] focus on the irreducibility of the Newton polytope of the polynomial with respect to the operation of computing Minkowski sums, which gives sufficient conditions to show irreducibility. A refinement of this method can be obtained with the aid of tropical geometry: instead of working with Newton polytopes, we can study regular subdivisions of them. So, we can cover more general families of polynomials, but at the cost of losing track of the characteristic of the ground field k. In our case, the problem can be dealt with as follows: by expanding the determi- nant of the generalized Vandermonde determinant with respect to the first row, we N γ − σi i get the following expansion: V (X, Γ) = i=1( 1) ∆iX1 . For the irreducibility problem, it is enough to consider a dehomogenized version of V (X, Γ) as follows: N−1 γN γi V (X, Γ)aff := X1 + AiX1 , i=1 σ ∆i where Ai := (−1) i ,i=1,...N − 1. ∆N WecanthenregardV (X, Γ)aff as a polynomial in K[X1], K beingnowafield containing all the Ai’s, i =1,...,N − 1. Given any rank one valuation v : K → R, we can extend it to Kn componentwise as follows: v : Kn → Rn (z1,...,zn) → (v(z1),...,v(zn)). The tropicalization of V (X, Γ) is then defined as n Trop(V (X, Γ)) = v({V (X, Γ)aff =0}) ⊆ R , where the closure on the right hand side is taken with respect to the standard topology in Rn. It turns out (see for instance [BG] or [EKL]) that Trop(V (X, Γ)) is a connected polyhedral complex of codimension 1. If V (X, Γ)aff is reducible over K[X1], then Trop(V (X, Γ)) is a reducible tropical hypersurface; i.e. it can be expressed as the union of two proper tropical hypersurfaces. So, if we prove that for a special valuation v,Trop(V (X, Γ)) is irreducible, then V (X, Γ) will be irreducible over any field k. Theorem 1.6. Let N ≥ 3. Given any field K ⊇ k(A1,...,AN−1),thereexists a valuation v defined over K such that Trop(V (X, Γ)) is an irreducible tropical variety if and only if: • dim(LΓ)≥ 2, • γi gcd X 1≤i≤N =1or equivalently γ =(0,...,0), • dΓ =1. This result is optimal in the following sense: it is known that Trop({f =0}) does not depend on the field but only on the values v(Ai). N i Take for instance f = i=0 AiY over a field of characteristic zero with dΓ > 1, N i and let g := i=0 BiY be a polynomial with the same support and with coefficients in a field of characteristic p|dΓ. Give to these polynomials valuations v and v such that v (Bi)=v(Ai). In these conditions, we will have Trop({g =0})=Trop({f =0}), License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 3650 CARLOS D’ANDREA AND LUIS FELIPE TABERA N 1/p i/p p but g =( i=0 Bi Y ) factorizes in the algebraic closure of its field of definition; hence Trop({f =0}) will always be reducible. So, the tropical criteria will not help to deduce the irreducibility of f. The paper is organized as follows: in Section 2, we give explicit conditions for the irreducibility of the Vandermonde variety. In Section 3 we prove Theorem 1.5. We conclude by introducing some tools from tropical geometry and by proving Theorem 1.6 in Section 4. 2. Bertini’s theorem and the irreducibility of the Vandermonde variety We begin by studying the geometric irreducibility of the variety defined by Nn V (X, Γ) in k . In order to do this, we will apply one of the several versions of Bertini’s theorem given in [Jou].