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On Facets of the Newton Polytope for the Discriminant of the Polynomial System

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On Facets of the Newton Polytope for the Discriminant of the Polynomial System On facets of the Newton polytope for the discriminant of the polynomial system Irina Antipova,∗ Ekaterina Kleshkova† Abstract We study normal directions to facets of the Newton polytope of the discriminant of the Laurent polynomial system via the tropical approach. We use the combinato- rial construction proposed by Dickenstein, Feichtner and Sturmfels for the tropical- ization of algebraic varieties admitting a parametrization by a linear map followed by a monomial map. Keywords: discriminant, Newton polytope, tropical variety, Bergman fan, matroid. 2020 Mathematical Subject Classification: 14M25, 14T15, 32S25. 1 Introduction Consider a system of n polynomial equations of the form X (i) λ Pi := aλ y = 0; i = 1; : : : ; n (1) λ2A(i) n (i) with unknowns y = (y1; : : : ; yn) 2 (C n 0) , variable complex coefficients a = (aλ ), where (i) n the sets A ⊂ Z are fixed, finite and contain the zero element 0, λ = (λ1; : : : ; λn), λ λ1 λn y = y1 · ::: · yn . Solution y(a) = (y1(a); : : : ; yn(a)) of (1) has a polyhomogeneity property, and thus the system usually can be written in a homogenized form by means of monomial transformations x = x(a) of coefficients (see [1]). To do this, it is necessary to distinguish a collection of n exponents !(i) 2 A(i) such that the matrix ! = !(1);:::;!(n) is non-degenerated. As a result, we obtain a reduced system of the form !(i) X (i) λ Qi := y + xλ y − 1 = 0; i = 1; : : : ; n; (2) arXiv:2107.04978v1 [math.AG] 11 Jul 2021 λ2Λ(i) (i) (i) (i) (i) where Λ := A n f! ; 0g and x = (xλ ) are variable complex coefficients. Denote by Λ the disjoint union of the sets Λ(i) and by N the cardinality of this union, that is the number of variable coefficients in the system (2). The coefficients of the system vary in N the vector space Cx in which points x = (xλ) are indexed by elements λ 2 Λ. ∗The first author was supported by the Theoretical Physics and Mathematics Advancement Foun- dation “BASIS” (№ 18-1-7-60-1), and the Krasnoyarsk Mathematical Center, funded by the Ministry of Science and Higher Education of the Russian Federation within the framework of the establishment and development of regional Centers for Mathematics Research and Education (Agreement No. 075-02-2021- 1388). †The second author was supported by the Theoretical Physics and Mathematics Advancement Foun- dation “BASIS” (№ 18-1-7-60-2). 1 ◦ (i) Denote by r the set of all coefficients x = xλ such that the polynomial mapping n Q = (Q1;:::;Qn) has multiple zeroes in the complex algebraic torus (C n 0) , that is @Q zeroes for which the determinant of the Jacobi matrix @y of the mapping Q equals zero. The discriminant set r of the system (2) is defined to be the closure of the set r◦ in the space of coefficients. If r is a hypersurface, then the polynomial ∆(x) that defines it is said to be the discriminant of the system (2). The set r is also called the reduced discriminant set of the system (1). According to the theory of A–discriminants developed in the book [8], the discriminant of a system of equations is appropriate to call the (A(1);:::;A(n))–discriminant. The goal our research is to construct the tropicalization of the discriminant set r of the system (2) and to find normal vectors to facets of the Newton polytope of the discriminant ∆(x). Recall that the Newton polytope N∆ of the polynomial ∆(x) is defined to be the convex hull of its support in Rn. The idea of computing the tropical discriminant is adopted from the paper [5], where the general combinatorial construction of the tropicalization is given for algebraic vari- eties that admit a parametrization in the form of the product of linear forms. We use the parametrization of the discriminant set for the system of n Laurent polynomials (2) proposed and comprehensively studied in [1]. We write the set Λ in the form of the block matrix Λ = (Λ(1)j ::: jΛ(n)) whose columns are the vectors of exponents of monomials of the system. Introduce (n × N)–matrices Ψ := !∗Λ; Ψ~ := Ψ − j!jχ, where !∗ is the adjoint matrix to the matrix !, χ is the matrix whose i-th row represents the characteristic function of the subset Λ(i) ⊂ Λ and j!j is the determinant of the ~ matrix !. In what follows, we denote the rows of the matrices Ψ and Ψ by 1; : : : ; n ~ ~ and 1;:::; n correspondingly, and we denote the rows of the identity matrix EN by e1; : : : ; eN . One of the main results of this paper is Theorem 1. The vectors ~ ~ N e1; : : : ; eN ; − 1;:::; − n; 1;:::; n 2 Z (3) define the normal directions of facets of the Newton polytope of the discriminant for the system (2). The discriminant of the system of polynomials studied in the current paper is the generalization of the notion of A–discriminant [8]. In [9], Kapranov proved that A– discriminants characterize hypersurfaces with the birational logarithmic Gauss mapping, and that reduced discriminant hypersurfaces admit the parametrization by monomials de- pending on linear forms, which is called the Horn-Kapranov uniformization. The tropical analogue of the Horn-Kapranov uniformization was obtained in [5]. In the present research, we use the parametrization of the discriminant set that occurs to be the inverse of the logarithmic Gauss mapping in the case when the set r is an irreducible hypersurface that depends on all groups of the variable coefficients of the system (see [1]). However, the logarithmic Gauss mapping for the studied hypersurface r is not always birational, thus, in general, (A(1);:::;A(n))–discriminants are not reduced to A–discriminants, and consequently require special research. It is important to note that Theorem 1 does not give all normal directions to facets of the Newton polytope but only those that are represented in the parametrization of the discriminant set r explicitly as vectors of coefficients in linear forms. Properties of the parametrization are described in Section 2 of the paper. Further, the construction of the 2 tropical variety and the proof of Theorem 1 are given in Section 3. In final Section 4, we explain how the constructed tropical variety brings out the so-called ¾hidden¿ facets of the Newton polytope for the discriminant of the system. 2 Parametrization of the discriminant set r N N We introduce two copies of the space C : one copy Cx with coordinates x = (xλ), N and another copy Cs with coordinates s = (sλ). In both cases, coordinates of points are N indexed by elements λ 2 Λ. The space Cs is considered as the space of homogeneous co- N−1 ordinates for CPs . It is proved in [1] that the parametrization of the discriminant set r of the system (2) is determined by the multivalued algebraic mapping from the projective N−1 N space CPs into the space Cx of the coefficients of the system with components (i) n 'kλ (i) sλ Y h'~k; si (i) xλ = − ; λ 2 Λ ; i = 1; : : : ; n; (4) h'~i; si h'k; si k=1 −1 ~ where 'k and '~k are rows of matrices Φ := ! Λ and Φ := Φ − χ correspondingly, 'kλ is (i) a coordinate of the row 'k indexed by λ 2 Λ ⊂ Λ, the angle brackets denote the inner product. The number of branches in (4) equals to the absolute value of the determinant j!j but some branches may coincide. If the discriminant set r of the system (2) is an irreducible hypersurface that depends on all groups of variables, then the mapping (4) parametrizes it with the multiplicity that is equal to the index jZn : Hj of the sublattice H ⊂ Zn generated by columns of the matrix (!jΛ), i.e., by all exponents of the monomials from the system (2). The image of the mapping (4) is a hypersurface if all coordinates of vectors 'k, '~k are non-zero, k = 1; : : : ; n: N−1 N We consider the rational mapping CPs ! Cw that is obtained from the mapping (4) as a result of raising of all its coordinates to the power j!j. It has components (i) !j!j n ~ ! kλ (i) j!jsλ Y h k; si wλ = − ; (5) ~ h k; si h i; si k=1 (i) where kλ is a coordinate of the row k indexed by λ 2 Λ ⊂ Λ. The mapping (5) ~ N defines the hypersurface r ⊂ Cw whose amoeba has the same asymptotic directions as the amoeba of the discriminant hypersurface r. Recall that the amoeba Ar of the algebraic hypersurface N r = fx 2 (C n 0) : ∆(x) = 0g N is the image of r by the mapping Log :(C n 0) ! RN defined by means of the formula Log :(x1; : : : ; xN ) ! (log jx1j;:::; log jxN j): Each component of the mapping (5) is a monomial with integer exponents composed with linear functions. It is convenient to associate the mapping (5) with the pair of two block matrices T T ~ T T ~ T U = −|!jEN Ψ Ψ ;V = j!jEN −Ψ Ψ ; (6) where EN is the identity matrix. The rows of the matrix U determine linear functions, and the rows of V determine exponents of monomials in the parametrization (5). 3 3 Tropicalization of the rational variety Further, we pay attention to the algebraic variety r~ ⊂ CN . Let us study its tropical- ization τ(r~ ). According to need, we recall notions and facts from the tropical geometry. Elements of this theory and numerous references to fundamental research can be found in the book [10].
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