Geometry of Factorization Identities for Discriminants
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EMATICS
Geometry of Factorization Identities for Discriminants E. N. Mikhalkina,*, V. A. Stepanenkoa,**, and A. K. Tsikha,*** Presented by Academician of the RAS V.A. Vassiliev May 21, 2020 Received May 22, 2020; revised May 22, 2020; accepted June 4, 2020
Abstract—Let Δn be the discriminant of a general polynomial of degree n and 1 be the Newton polytope of Δn. We give a geometric proof of the fact that the truncations of Δn to faces of 1 are equal to products of discriminants of lesser n degrees. The proof is based on the blow-up property of the logarithmic Gauss map for the zero set of Δn. Keywords: discriminant, Newton polytope, logarithmic Gauss map, Horn–Kapranov parametrization DOI: 10.1134/S1064562420040134
Given a meromorphic mapping f : X → Y of analytic sets (spaces), it is reasonable to treat it as an analytic subset of X × Y that is the closure G f of the graph of f. Fibers in G f over uncertainty points can be regarded as “blow-up” or “shrinkage” [1]. The most transparent scheme for such blow-ups and shrinkages can be seen for mappings that are inverses of the logarithmic Gauss map [2]. In the theory of hypergeometric functions, these inverses are known as Horn– Kapranov parametrizations [3, 4]. Our goal in this paper is to study the Horn– Kapranov parametrization and, using the parametrization of a classical discriminant as an example, to prove identities for truncations of the discriminant. Such identities can also be obtained using a complicated technique of the theory of A-determinants [5, Chapter 10]. The discriminant of a polynomial
f ( y) = a0 + a1 y + … + an y n
(1)
is an irreducible polynomial Δ n = Δ n(a0, a1, … , an ) with integer coefficients that vanishes if and only if f has multiple roots. Discriminants play an important role in mathematics, which is demonstrated in the fundamental monographs [5, 6]. The discriminant identity presented in (2) (see below) is motivated by the study of the structure of a general algebraic function [6–9], in the theory of singularities [10] and in tropical matha Siberian
Federal University, Krasnoyarsk, 660041 Russia *e-mail: [email protected] **e-mail: [email protected] ***e-mail: [email protected]
ematics [11]. The indicated identity in the form of a conjecture was given in [12]. It is well known [5] that the Newton polytope 1(Δ n ) ⊂ Rn+1 of the indicated discriminant is combinatorially equivalent to an (n – 1)-dimensional cube. Since such a cube has 2n−1 vertices, it is natural to encode the vertices of 1(Δ n ) by all possible subsets of the set {1,…, n − 1}. The polytope 1(Δ n ) has n – 1 hyperfaces {hk0} lying in the coordinate hyperplanes {tk = 0}, k = 1, … , n − 1 (it is assumed that the coordinates t = (t0, t1,…, tn−1, tn ) are chosen in the ambient space Rn+1). Each hyperface hk0 has 2n−2 vertices, which are encoded by subsets I ⊂ {1,…, n − 1} that do not contain k. Let hk denote the face opposite to hk0 , whose vertices are encoded by subsets I containing k (an explicit equation for a hyperplane containing hk is given by formula (6) in Sect
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