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EDN: MNWSFN УДК 517.55+512.7
Parametrizations of Limit Positions for the Discriminant Locus of a Trinomial System
Irina A. Antipova* Ekaterina A. Kleshkova^
Siberian Federal University Krasnoyarsk, Russian Federation
Received 28.01.2023, received in revised form 20.03.2023, accepted 24.04.2023 Abstract. The paper deals with the discriminant of the reduced system of n trinomial algebraic equations. We study zero loci of truncations of the discriminant on facets of its Newton polytope. The basis of the study is the properties of the parametrization of the discriminant set of the system and the general combinatorial construction of the tropicalization of algebraic varieties.
Keywords: algebraic equation, discriminant, Newton polytope, truncation of the polynomial, discriminant set, parametrization.
Citation: Irina A. Antipova, Ekaterina A. Kleshkova, Parametrizations of Limit Positions for the Discriminant Locus of a Trinomial System, J. Sib. Fed. Univ. Math. Phys., 2023, 16(3), 318-329. EDN: MNWSFN.
1. Introduction and preliminaries
Consider a reduced system of n trinomial algebraic equations
Qi := y^ + xi y- 1 = 0, « = 1,...,n (1)
with unknowns y = (yi,..., yn) G (C \ 0)n and variable complex coefficients x = (xi,..., xn), where := y"1 •... • , yjM := yJl •... • yCJn are monomials in variables y1,... ,yn with integer exponents. The coefficients of the system (1) vary in the vector space Cn. We assume that the matrix w formed by column vectors w(i) is non-degenerate. The universal trinomial system in which all monomials have independent variable coefficients can be reduced to the form (1) by means of monomial transformations of the coefficients in view of the polyhomogeneity property of its solution [1,3].
Denote by V° the set in Cn of all x = (xi) such that the polynomial mapping Q = (Q1;..., Qn) has multiple zeros in the complex algebraic torus (C \ 0)n, i.e.
V° := {x G Cn : Qi(y0) = • • • = Qn(y0) = (y°) = 0, y° G (C \ 0)n J ,
where ^^ is the Jacobian of the mapping Q. dy
* iantipova@sfu-kras.ru https://orcid.org/0000-0003-1382-0799 t ekleshkova@gmail.com https://orcid.org/0000-0002-7443-2979 © Siberian Federal University. All rights reserved
Definition 1. The discriminant locus V of the system (1) is defined to be the closure of V° in the space of coefficients. If V is a hypersurface, then its defining polynomial A(x) is said to be the discriminant of the system (1).
This approach to the definition of the discriminant of the polynomial system was proposed in [1] as an extension of the concept of the A-discriminant developed in the book [5]. Further, we denote the discriminant of the system (1) by An(x) to clarify the number of equations and the dimension of the space of coefficients.
Our main objects of interest are limit positions of the discriminant locus V in the toric compactification of the space (C \ 0)n associated with the Newton polytope of the discriminant An(x). Recall that the Newton polytope NAn of the polynomial An(x) is defined to be the convex hull of its support in Rn. Each monomial xa = x^1 • ... • xan is visualized by the point a = (a1,... ,an) of the lattice Zn. The support of a polynomial is defined to be the set of exponents of all its monomials with non-zero coefficients. The limit positions under study are determined by truncations of the discriminant An(x) to facets of the Newton polytope NAn.
Definition 2. The truncation of the polynomial An(x) to a face h of the Newton polytope NAn is the sum of all monomials of An(x) whose exponents belong to the face h.
In the classical case n = 1, it is known that the Newton polytope of the discriminant of the algebraic equation of degree m is combinatorially equivalent to the (m — 1)-dimensional cube [5, Theorem 2.2, Chapter 12]. The classical discriminant is well-studied and, in particular, a new approach to the proof of factorization identities for its truncations was proposed in recent papers [8,9]. The factorization identities were proven in [5] by means of sophisticated techniques of the theory of A-determinants. Truncations are factorized into the product of discriminants of lower degree equations.
Let us introduce a matrix a whose columns are exponents of the system (1),
matrices ' := w*a and ' := ' — |w|En, where w* is the adjoint matrix to the w, the En is the identity matrix and |w| is the determinant of w. The matrices a and w determine the support of the system (1). As it follows from [2], rows of matrices (—') and ' (we denote them ..., n and ifti,... ,ripn, correspondingly) could define the inner normal directions for facets of the polytope NAn. We assume that matrices ' and ' do not contain zero elements. This is a sufficient condition for the set V to be a hypersurface.
Now, we formulate the main result of our study without detailing all the structures used.
Theorem 1. Let h be a facet of the Newton polytope NAn with a normal direction n G |— ..., n, ipi,... . The zero locus of the truncation An(x)lh contains the set
{x G (C \ 0)n :An_i(z)U=u(œ) =0} , (2)
where An-1(z) is the discriminant of the reduced system of n — 1 trinomials, and z = u(x) : (C \ 0)n ^ (C \ 0)n 1 is a mapping given by monomial functions of coefficients x = (x1,..., xn) of the system (1).
The constructive proof of Theorem 1 is given in Section 4. It is important to note that the proposed scheme is meaningful in case when matrices 0, 0, k, K (see (9), (10), (16), (17) ) do not contain zero elements.
2. Parametrization of the discriminant locus for the system (1)
The parametrization of the dehomogenized discriminant locus of a system of n Laurent polynomials in n variables was comprehensively studied in the paper [1]. It is applied for computing the tropical discriminant of the system [2] and is a key tool in the proof of Theorem 1 .
Let us introduce two copies of the space Cn. The first one is the space Cn with coordinates x = (x i,..., xn), and the second one is Cn with coordinates s = (s1;...,sn). We interpret
"m ■'x
"m
the Cn as the space of homogeneous coordinates in CPJ-1. Consider the multivalued algebraic
PT1 ^
mapping x = x(s) : CPn 1 ^ Cn with components
x = n (tM) 'i = 1,...,n, (3)
where brackets (,} denote the inner product of vectors. The number of branches in (3) equals to the absolute value of the determinant |w|, however, some branches may coincide. If the discriminant locus of the system (1) is an irreducible hypersurface depending on all variables x1,..., xn, then the mapping (3) parametrizes it with the multiplicity equal to the index |Zn : H| of the sublattice H c Zn generated by the columns of the matrix (w|<r), i.e. by all exponents of the system (1).
3. Tropical discriminant
We start this section with some basic concepts of the tropical geometry following the book [10]. Consider the tropical semiring (R U to, ©, ©), where arithmetic operations of addition and multiplication are defined as follows:
x © y := min(x, y), x © y := x + y.
Consider a field K with a valuation val : K ^ R U {to}. For a polynomial
f = E
u£Zn
in the ring ..., x±], the tropicalization is defined to be the function
trop(/)(w) := minuezn (val(cu) + (u, w)). (4)
The tropical polynomial trop(/)(w) is a piecewise linear function that is obtained by replacing all coefficients cu by their valuations val(cu) and performing all operations in the tropical semiring. The Laurent polynomial / over the field K determines the algebraic hypersurface
^(/) = {y G Tn : /(y) = 0},
where Tn := (K \ 0)n is the algebraic torus.
Definition 3. The tropicalization of the algebraic hypersurface V(/) is defined to be the set trop(V(/)) := {w G Rn : the minimum in (4) is attained at least twice} .
Therefore, the trop(V(f)) is the locus in 1", where the function trop(f)(w) fails to be linear.
The general construction of tropicalization for an algebraic variety that admits a rational parametric representation whose components are Laurent monomials in linear forms is proposed in [7]. Let K be a field with trivial valuation. Consider a m x d matrix U = (uij) over K and a p x m matrix V = (vij) with integer entries. The matrix U determines m linear forms in the ring K[si,...,sd] :
li(s) = unsi + ... + uidsd, i = 1,..., m. (5)
The matrix V encodes p Laurent monomials in
zj • ... • zj,j = l,...,p. (6)
The composition of monomials (6) and forms (5) defines a rational mapping a : Kd ^ K with components
m
aj(s) = U li(s)Vij. i=1
We denote by Y the closure of the image of a. According to [7, Theorem 3.1], the tropicalization of the variety Y is a polyhedral fan trop(Y) that coincides with the image of the Bergman fan BM of the matroid M associated with the matrix U under the linear map defined by V. The Bergman fan being a geometric model of the matroid M, is the tropicalization of the linear variety given by the linear map s ^ Us (see [4,10]).
Consider the rational map CP"~ ^ CW that is obtained by raising the components of the map (3) to the power |w|. It defines the rational variety V, and by the construction given above the tropicalization trop(V) is encoded by the pair of block matrices
Usnxn = (-ME„|*T| *T)T , and Vnx3n = (ME"!-^!^) . (7)
The study of the tropical discriminant in case of the general polynomial system has been carried out in [2]. In Section 5, using an example, we demonstrate how a tropical variety constructed on the basis of the parametrization of the discriminant set reveals normals to facets of the Newton polytope of the discriminant, while the matroid associated with the matrix U suggests the parametrization of 'hidden' limit positions of the discriminant locus.
4. Zero loci of truncations
Here we present a proof of the main result.
The relation between the discriminant An(x) and its truncation An(x)lh to the facet h is established by means of the function
Hh (X) = T ( Xl ,...,Tn ) ,
where n 1,...,^n are entries of the normal vector n to the facet h, d is the weighted degree of all monomials of the truncation An(x)lh with respect to the weight (j,.
Lemma 1. The function Hh (x) being a homogenization of the discriminant of the system (1) with respect to the weight ( has the following property
HI(x) An(x)lh.
Proof. The weighted degree of all monomials of the An(x) that do not belong to the truncation An(x)|h is strictly less than d. Therefore, as a result of passing to the limit when t ^ 0, all monomials disappear, except for those whose exponents belong on the face h. Lemma 1 is proved. □
Proof of Theorem 1. First, we study the truncation An(x)|h to the facet h having a normal vector n = —ipj. Recall that ipj is the jth row of the matrix ^ and introduce the following constructions. Let 6denote a 2 x 2 minor of the matrix ^ formed by the intersection of rows ^k, 4>j, and columns tp(m), ip(j). Therefore, we have
S
(m)
± (^j — jm),
(8)
where the sign on the right-hand side depends on the choice of k and m. Define the square matrix 0 of the order n — 1 that contains the entries
Q(m)__S(m)
Q к •-S к , Q (m) —Sm+1)
k,m G {1,
, k G{1,...,j — 1} , m G {j,
Q (m) •— , k G{j,...,n — 1} , m G {1,
Q(m)__S(m+1)
Q к S к+1 ,
and matrices
С •— and
Introduce a system of n — 1 trinomials
k,m G {j, »:— Q — Ç.
..,j — 1}, .,n — 1}, ..,j — 1}, .,n — 1},
(9)
(10)
ye + zyd — 1 — 0, i — 1,...,n — 1,
(11)
where y — (y1,..., yn-1), C(i) and Q(l) are columns of matrices С and Q respectively.
Now we are in a position to study a parametrization of the zero locus of the truncation Дп(х)\^. To this end, according to Lemma 1, we consider the sets {x • HJT(x) — 0}, t — 0 that admit the parametrization
Фм Ms.
Xi — — ty j
n f - ) # JM^ A I (фк,sM M i
(физ) УД (фк,s)j ' i
1 , . . . , n.
(12)
In the projective space with coordinates s — (s1 • ... • sn) , we define a plane Yj given by the equation
( Ф j ,s) —0,
and use the parametrization (12) to get restrictions of the monomials Xi • Xj j j to Yj for i — j. Note that
( \^\sj A
\( - ,s)J
-Ф У'/Ф У
Yj
and, for i — j,
( '-i,s)
(—1)
M js
-ф?/ФУ
Yj
E (Ф(т) j — № 4m)) sm — \м\ф jV
m=j
Ms
Moreover, for k = j
k, s)
k, s)
E (^j -
m=j
Yj
£ {^j - W*
m=j
(j)J,(m)
Therefore, in terms of notations (9) and (10) restrictions of monomials xi • Xj plane Yj admit representations
j/j
to the
Xi • Xj
xi • Xj
Yj
Yj
(Öi,s)
£ (j)s■ 1 "-1 (0-1, s)
n7 (¿kA \
Ml (°k >s)j
n ((IkA)
Ml (°k >s)j
Aj)
i = j + 1,.. . ,n,
(13)
where s = (si : ... : sn-1) are homogeneous coordinates of CPn-2. The right-hand sides of formulae (13) determine the parametrization of the discriminant locus for the system (11). Thus, the zero locus of the truncation An(x)\h contains the set
{x e (C \ 0)n : An-i(z)\z=u(x) =0}
(14)
where An-1 (z) is the discriminant of the system (11) and z = u(x) is a monomial mapping with components
-j'/j
Ui (x)
j
(i+1)/j
i = 1,...,j -1, , i = j, ... ,n - 1.
(15)
Next, we follow the similar way to study the truncation An(x)\h to the facet h with the normal vector j = ijjj. Define the square matrix k of the order n — 1 that contains the entries
(16)
k Am) , | , (m) := sk - MVk \ k,m e {l,.. ■ .,j -1},
k x(m+1) | | , (m+1) k e {1,.. ..,j - 1} , m e{j,.. ., n -1},
k — Ä(m) lMUA(m) k e {j,.. . ,n - 1} , m e {1,.. ■ .,j -1},
k := Ök+1 - \M\*k+1 , k, m e {j, . . ., n -1},
and matrices
n := \^\*jj)En-i, x := k - n.
In this case, the sets {x : Hh(x) =0}, t = 0 admit parametrization
Xi = — TY j
w \^\si -1 / (*k, s) \ (*i,s) l=\ {(*k,s)j
M
i = 1,
(17)
(18)
In the projective space with coordinates s = (s1 : ... : sn) , we define a plane Yj given by the equation
(^j ,s) =0,
s
m
s
m
f\j)
and use the parametrization (18) to get restrictions of monomials xi • x^ j j to Yj for i = j. We obtain the following result
xi ' Xj
Ji)
)
Y U-D (19)
(j)„
/ ( s ) \
i = j + 1,. .. ,n.
j'si-±Y\ (KA.V
!,s) 1_U (Kk ,s)J
(ïti- i , s) ^ V (xk, s
Yj ( i 1 ! k = l V ( k'
The right-hand sides of (19) determine the parametrization of the discriminant set for the trinomial system with the support (n|k). Thus, the zero locus of the truncation An(x)|h contains the set
{x e (C \ 0)n : An _i(^)|z=„(x) = 0} , (20)
where An _ 1 (z) is the discriminant of the system of n — 1 trinomials with the support (n| k) and z = u(x) is a monomial mapping with entries
f _ ifl^ . 1 . 1
xi -x^ , i = 1,...— 1,
u'(x) = i j -^'tw . . , (21)
^xi+1 • xj , i = j,...,n — 1.
The proof of Theorem 1 is completed. □
Remark 1. The described procedure can be applied to the truncation An|ho that lies in a
coordinate plane a.j = 0. The limit set |x : H^o (x) = 0 j as t ^ 0 approaches the discriminant set {An _ 1 (x1,... [j]..., xn) = 0} of the system of n — 1 trinomials of the form
xy (i)[jl— 1 = 0, i = 1,... [j] ...,n, (22)
where ^(i)[j ] := (^ ,... [j] ..., 4°), yfWj] := yf1 • ■■■ [j ]... • yf.
5. Example: 'hidden' facets and truncations
Consider a system of equations
'y1 + ay!2 y2y3 — 1 = 0,
y2 + by1y22y3 — 1 = 0, (23)
y + cy1y2y32 — 1 = 0
with unknowns y1, y2, y3 and variable coefficients a, b, c. The matrix of exponents of the system is
100 (w|CT) = ( 0 10 0 0 1
Its columns generate the lattice Z3. Since matrices
2 1 1 1 2 1 112
2 1 1 1 1 1 § = I 1 2 1 I and § = I 1 1 1 1 1 2 1 1 1
Xi ^ X j
do not contain zero elements, the discriminant set V is a hypersurface. The rational mapping CP2 ^ C30 b c given by formulae
«1
b = -
« 1 + «2 + «3 «1 + «2 + «3 V 2«1 + «2 + «3
«2 «1 + «2 + «3
2
«1 + «2 + «3
«1 + «2 + «3
«1 + «2 + «3 V 2«1 + «2 + «3
«3 «1 + «2 + «3
«1 + 2«2 + «3 J V «1 + «2 + 2«3
2
«1 + «2 + «3 «1 + 2«2 + «3
«1 + «2 + «3
)(
«1 + «2 + «3 «1 + «2 + 2«3
«1 + «2 + «3 V 2«1 + «2 + «3 J \«1 + 2«2 + «3 J \«1 + «2 + 2«3
V «1 + «2 + 2«3 J '
(24)
parametrizes V with the multiplicity one. Here s = (s1, s2, s3) & C3 are homogeneous coordinates
in cp2.
Let us study the tropicalization t(V) of the rational variety V C C3. As it was pointed out in Section 3, the mapping (24) is encoded by two matrices
U =
( -1 0 0 \ 0 -1 0 001
1 1 2 1
1 1
100
0 0 -2 -1 -1111 and V = ( 0 1 0 -1 -2 -1 1 1 1 0 0 1 1 1 2 1 1 1
111 111 11
a = —
c = —
Consider the matroid M on the set E = {1,2,3,4, 5, 6,7,8, 9} associated with the set of rows of the matrix U. The tropical linear space related to the matroid M is the Bergman fan B(M). It is a two-dimensional fan in R9/R1 or a graph depicted in Fig. 1a. The graph has ten vertices, corresponding to seven flats of the rank one (1, 2, 3,4,5,6, 789) and three circuits (14789, 25789, 36789) of the matroid M. The edges 12, 13, 15, 16, 23, 24, 26, 43, 35, 45, 46, 56 of the graph correspond to flats of the rank two.
The image of B(M) under V is a two-dimensional fan т(V) С R3 (see Fig. 1b). It is the tropical variety т( V) that consists of all codimension one cones of the normal fan of the Newton polytope Ад3 for the discriminant of the system (23). Seven rays ¡j(1), ... of the fan т(V) generated by columns V are predicted explicitly by the parametrization (24) and determine normal directions
= (1,0,0), M(5) = (-1, -2, -1),
M(2) = (0,1,0), M(6) = (-1,-1, -2),
M(3) = (0,0,1), M(7) = (1,1,1).
M(4) = (-2,-1, -1),
There are three more 'hidden' rays that are revealed as a result of intersection of two-dimensional cones of the fan t(V). More precisely, images of cones 14789 and 23 intersect along the ray R^0(0,1,1)T (j(10) in Fig. 1b); images of 25789 and 13 intersect along R^0(l, 0,1)T (j(9) in Fig. 1b); images of 36789 and 12 intersect along R^0(1, l, 0)T (j(8) in Fig. 1b). Therefore, all inner normals to facets of the Newton polytope of the discriminant for the system (23) are
found. The discriminant of (23) is as follows
A3(a, b, c) = 4a5b2 — 8a5bc + 4a5c2 — 27a4b4 + 36a4b3c — 6a4b3 — 2a4b2c2 + 6a4b2c + a4b2 + 36a4bc3+ + 6a4bc2 — 2a4 bc — 27a4 c4 — 6a4c3 + a4c2 + 36a3b4c — 6a3b4 — 256a3b3c3 — 52a3b3c2 + 16a3b3c—
— 2a3b3 — 52a3b2c3 — 16a3b2c2 + 2a3b2c + 36a3 bc4 + 16a3bc3 + 2a3bc2 — 6a3c4 — 2a3c3 + 4a2 b5 —
2 4 2 2 4 2 4 2 3 3 2 3 2 2 3 2 2 4 2 2 3 2 2 2 2 4
— 2a b c + 6a b c + ab — 52a b c — 16a bc +2abc — 2a b c — 16a b c — 6a b c + 6a bc +
+ 2a2bc3 + 4a2 c5 + a2c4 — 8ab5c + 36ab4c3 + 6ab4c2 — 2ab4c + 36ab3 c4 + 16ab3c3 + 2ab3c2 +
+ 6ab2c4 + 2ab2c3 — 8abc5 — 2abc4 + 4b5 c2 — 27b4 c4 — 6b4 c3 + b4c2 — 6b3 c4 — 2b3 c3 + 4b2 c5 + b2c4.
25789
a)
b)
Fig. 1. a) The Bergman fan B(M). b) The tropical variety t(V)
Fig. 2. The Newton polytope Na3
The Newton polytope Na3 has 10 facets (Fig. 2) enumerated by h(j), j = 1,..., 10, in accordance with normal vectors j(j).
The truncations of the discriminant A3(a, b, c) to coordinate faces h(1), h(2), h(3) are as follows
A3\h(i) = b2c2(-27b2c2 + 4b3 - 6b2 c - 6bc2 + 4c3 + b2 - 2bc + c2),
A3\h(2) = a2c2(-27a2c2 + 4a3 — 6a2c — 6ac2 + 4c3 + a2 — 2ac + c2), (25)
A3\h(s) = a2b2(-27a2b2 + 4a3 - 6a2b - 6ab2 + 4b3 + a2 - 2ab + b2).
Polynomials in brackets in (25) are irreducible and coincide with the discriminant A2(z1,z2) of the system
jyi + ziyi2y2 - 1 = 0,
[V2 + Z2yiy22 - 1 = 0,
under condition of a suitable determination of variable coefficients.
The factorized truncations of the polynomial A3(a, b, c) to facets h(4), h(5), h(6) are as follows
A3\h(4) = -a3(256b3c3 + 27ab4 - 36ab3c + 2ab2c2 - 36abc3 + 27ac4 - 4a2b2 + 8a2bc - 4a2c2), A3\h(5) = -b3(256a3c3 + 27a4b - 36a3bc + 2a2bc2 - 36abc3 + 27bc4 - 4a2b2 + 8ab2c - 4b2c2), A3\h(6) = -c3(256a3b3 + 27a4c - 36a3bc + 2a2b2c - 36ab3c + 27b4c - 4a2c2 + 8abc2 - 4b2c2).
Implementing the construction proposed in Theorem 1 for these truncations, we obtain the following representations
A3(a,b,c)|h(4) = a6 ■ A2(z)|z=„(D(a,6,c) A3(a, b, c)|h(5) = 66 ■ A2 (z) ^z=u(2) (a,b,c) A3(a,6,c)|h(6) = c6 ■ A2(z)|z=u(3)(a,b,c)
where
A2(z) = 256z?z| + 27z4 - 36z?z2 + 2z2z| - 36ziz| + 27z| - 4z2 + 8z1z2 - 4z| is the discriminant of the system
(y 12 + z 1 y3y2 - 1 = 0,
\ 2 \ (26)
[y2 + Z2yiy2 - 1 = 0,
and
u(1)(a,b, c) = (ba-1/2,ca-1/2),
(2)(a, b, c) = (ab-1/2,cb-1/2),
(3)(a, b, c) = (ac-1/2,ac-1/2).
As applied to the truncation A3 (a, b, c)\h(7), the construction degenerates. If we consider parametrizations of sets {(a, b, c) : #h(7) (a, b, c) = 0}, r = 0, and, according to (19), express the restrictions of monomials b ■ a-1, c ■ a-1 to the plane s) = 0, then we get the equations
bb = c = 1,
aa
that define the zero locus of the truncation A3\h(7). The truncation itself is as follows
As|h(r) = (b - c)2(a - c)2(a - b)2. (27)
The limit positions of the discriminant locus associated with facets h(8), h(9), h(1°) can be investigated by means of the the tropical fan r(V). For example, consider the ray R^°(1,1,0)T
that defines the normal to the facet M8). It is the intersection of cones associated with flats 36789 and 12. This means that the zero locus of the truncation A3 |h(8) can be obtained by the restriction of the parametrization for the family {(a,b,c) : H^T(8) (a,b,c) =0}, t = 0 to the planes
Yi : si = S2 =0 and 72 : S3 = 0, si + S2 + S3 = 0.
As a result, we get that the zero locus of the truncation A3|h(8) consists of three components c = 0, a = b and c = — 4 which agrees with the expression
A3 |fc(s) = c4(a — b)2(4c +1).
Similarly, we study limit positions of the discriminant locus associated with facets They are given by polynomials
A3U) = b4(a — c)2(4b +1), A3LU0) = a4(b — c)2(4a + 1).
All the discriminants from the example are computed using the computer algebra system for polynomial computations Singular [6].
The research is supported by the Krasnoyarsk Mathematical Center funded by the Ministry of Science and Higher Education of the Russian Federation (Agreement no. 075-02-2023-936).
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Параметризации предельных положений дискриминантного множества системы триномов
Ирина А. Антипова Екатерина А. Клешкова
Сибирский федеральный университет Красноярск, Российская Федерация
Аннотация. Рассматривается дискриминант приведенной системы п триномиальных алгебраических уравнений. Исследуются срезки дискриминанта на гиперграни его многогранника Ньютона. Основой исследования являются свойства параметризации дискриминантного множества системы и общая комбинаторная конструкция тропикализации алгебраических многообразий.
Ключевые слова: алгебраическое уравнение, дискриминант, многогранник Ньютона, срезка, дис-криминантное множество, параметризация.
