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ЧЕБЫШЕВСКИЙ СБОРНИК
Том 16 Выпуск 2 (2015)
УДК 512.77
СТРУКТУРА ДИСКРИМИНАНТНОГО
МНОЖЕСТВА ВЕЩЕСТВЕННОГО МНОГОЧЛЕНА
А. Б. Батхин (г. Москва)
Аннотация
Проблема описания структуры дискриминантного множества вещественного многочлена часто возникает при решении различных прикладных задач, например, при описании множества устойчивости положений равновесия многопараметрических систем, при вычислении нормальной формы системы Гамильтона в окрестности положения равновесия в случае кратных частот. В работе рассматривается структура дискриминантного множества многочлена с вещественными коэффициентами. Предлагается два подхода к его изучению. Первый подход основан на исследовании нулей идеалов, образованных набором субдискриминантов исходного многочлена. Рассмотрены различные способы вычисления субдискриминантов. Во втором подходе предлагается исследовать особые точки дискриминантного множества. Методами компьютерной алгебры показано, что для малых значений степени исходного полинома оба подхода эквивалентны, но первый более предпочтителен из-за меньшего размера идеалов.
Предлагается конструктивный алгоритм построения полиномиальной параметризации дискриминантного множества в пространстве коэффициентов многочлена. С прикладной точки зрения наибольший интерес представляет описание компоненты коразмерности 1 дискриминантного множества. Именно эта компонента делит пространство коэффициентов на области с одинаковой структурой корней многочлена. Набор компонент различных размерностей дискриминантного множества имеет иерархическую структуру. Каждая компонента большей размерности может рассматриваться как некоторая касательная развертывающая поверхность, образованная линейными многообразиями соответствующей размерности. Роль направляющей при этом выполняет компонента дискриминантного множества, имеющая размерность на единицу меньше и на которой исходный многочлен обладает единственным кратным корнем, а остальные его корни простые. Начиная с одномерного алгебраического многообразия размерности 1, на котором исходный многочлен имеет единственный корень максимальной кратности, на следующем шаге алгоритма получаем описание многообразия, на котором многочлен имеет уже 2 корня — один
простой и один кратный. Повторяя последовательно шаги алгоритма, получаем в итоге параметрическое представление компоненты коразмерности 1 дискриминантного множества.
Приведены примеры дискриминантных множеств кубического многочлена и многочлена четвертой степени.
Ключевые слова: дискриминантное множество, особая точка, рациональная параметризация.
Библиография: 15 названий.
STRUCTURE OF DISCRIMINANT SET OF REAL POLYNOMIAL
A. B. Batkhin (Moscow)
Abstract
The problem of description the structure of the discriminant set of a real polynomial often occurs in solving various applied problems, for example, for describing a set of stability of stationary points of multiparameter systems, for computing the normal form of a Hamiltonian system in vicinity of equilibrium in the case of multiple frequencies. This paper considers the structure of the discriminant set of a polynomial with real coefficients. There are two approaches to its study. The first approach is based on the study of zeroes of ideals formed by the set of subdiscriminants of the original polynomial. Different ways of computing subdiscriminants are given. There is proposed to investigate the singular points of the discriminant set in the second approach. By the methods of computer algebra it is shown that for small values of the degree of the original polynomial, both approaches are equivalent, but the first one is preferred because of smaller ideals.
Proposed constructive algorithm for obtaining polynomial parameterization of the discriminant set in the space of coefficients of the polynomial. From the applied point of view the most interesting is the description of the components of codimension 1 of the discriminant set. It is this component divides the space of the coefficients into the domains with the same structure of the roots of the polynomial. The set of components of different dimensions of the discriminant set has a hierarchical structure. Each component of higher dimensions can be considered as some kind of tangent developable surface which is formed by linear varieties of respective dimension. The role of directrix of this component performs a variety of dimension one less than that on which the original polynomial has only multiple zero and the remaining zeroes are simple. Starting with a one-dimensional algebraic variety of dimension 1 on which the original polynomial has the unique zero of maximal multiplicity, in the next step of the algorithm we obtain the description of the variety on which the polynomial has a pair of zeroes: one simple and another multiple. Repeating sequentially the steps of the algorithm, the resulting parametric
representation of components of codimension 1 of the discriminant set can be obtained.
Examples of the discriminant set of a cubic and quartic polynomials are considered.
Keywords: discriminant set, singular point, rational parametrization.
Bibliography: 15 titles.
1. Introduction
Let
f (x) = xn + a,n-ixn-1 + ... + ao, (1)
be a generic monic polynomial of degree n with real coefficients. The n-dimensional real space n of its coefficients (a0, a1,..., an-1) is called the coefficient space.
Definition 1. The discriminant set D(f) of the polynomial f (x) is called the set of all points of coefficient space n, at which discriminant D(f) is vanished.
Investigation of the structure of the discriminant set D(f) is important for solution of many applied problems, for instance, for study stability of a stationary point of multiparameter mechanical systems (see [1, 2, 3]), for computing normal form of Hamiltonian system in vicinity of stationary point in the case of equal frequences [4].
The set D(f) contains the algebraic hypersurface H of codimension 1 in the space n. This hypersurface divides the n into 1 + [n/2] domains with fixed number of real zeroes of the polynomial f (x) (see [5, Theorem 2]).
Theorem 1. In the space of coefficients n the domains with the same number of real zeroes of the polynomial f (x) are separated from each other by discriminant hypersurface H C D(f).
The goal of the presented paper is to give the description of the discriminant set D(f) of a generic monic polynomial with real coefficients and, in particular, to construct parametrization of the hypersurface H of codimension 1.
Two main approaches can be proposed for study of the discriminant set D(f).
• The first one is based on exploring the subdiscriminants of the polynomial (1).
• The second one is based on investigation of singular points of the discriminant set D(f).
The structure of the paper is as following. In Section 2 we give the definition of subdiscriminant of a polynomial and recall some its useful properties. In Section 3 we provide the description of the discriminant set D(f) based on subdiscriminant approach. In Section 4 we show that the aproach based on investigation of singular points of the set D(f) is equivalent to the approach based on subdiscriminant technique. Finally, some examples are provided in the last Section 5.
2. Subdiscriminant and its properties
Here we recall the definition of the subdiscriminant, which is used in Section 3 for discriminant set description. This section is mainly based on [6, 7]
Definition 2. Let U, i = 1,...,n be the roots of the polynomial f(x). The discriminant D(f) of f (x) is defined by formula
D(f) = 11 (ti - tj)2
1 ^i^j^n
(2)
Usually discriminant D(f) is defined as a resultant computed on the polynomial f (x) and its first derivative f'(x) [7]:
D(f ) = (-1)n(n—1)/2 R(f,f ').
There are several ways for discriminant D(f) computation:
1. as the determinant of Sylvester matrix (see [6, 7, 8] and below);
2. as the determinant of Bezout matrix (see [6, 7, 9]);
3. as the determinant of Hankel matrix constructed of Newtonian sums (see [6,
7]);
4. with the help of pseudo-remainders sequence.
The author's computational experiments demonstrated that the most effective way of the discriminant (and also subdiscriminant) computation is the way with the help of Bezout matrix.
We consider so called Sylvester-Habicht matrix [6, 10]
/1 an-1 an-2
) 1 an-i
SylHab (f, f ') =
ao a1
0 \n
0 •• 1
0 •• 0
n
........... 2a2
an— 1
n
2a2
(n - 1) an— 1
a1 0
0 ■ ■ 0 0
ao ■ ■ 0 0
. a1 ao
. 2a2 a1
0 . 0
0
So, up to the sign D(f ) = det SylHab (f, f). The choice of Sylvester-Habicht matrix makes definition of subdiscriminant easier than using the original Sylvester matrix.
If one wants to formulate a condition that the polynomial f (x) has exactly k distinct zeroes, it is necessary to use subdiscriminant notion.
Definition 3. The k-th subdiscriminant D(k)(/) of the polynomial f (x) is defined by the following formula:
D(k)(f)= £ n (tj - ti)2-
ic{i,...,p} (j,i)ei #(I)=k l>j
Here #(I) is the cardinality of the set I.
The next proposition provides the way of subdiscriminant computation.
Proposition 1 ([11]). The determinant of a matrix, resulting from the Sylvester matrix SylHab (/, /') by deleting the first k and the last k rows, and the first k and the last k columns, is equal to the k-th subdiscriminant of the discriminant D(f ).
When k = 0 one has the discriminant of the polynomial /(x): D(0)(f) = D(f). It is clear that k-th subdiscriminant D(k)(f ) is the k-th inner [12] of the SylvesterHabicht matrix.
Polynomial /(x) has exactly n — 1 non-trivial subdiscriminants each of which is a quasi homogeneous polynomial of its coefficients a0,..., ara-1.
Theorem 2. The polynomial /(x) has exactly d common zeroes with its derivative /'(x) iff the following conditions take place
D(0) (/) = ... = D(d-1) (/) = 0, D(d) (/) = 0.
V-V-'
d
Multiple zeroes of the polynomial /(x) are the zeroes of the polynomial GCD (/, /'), which can be written as follows:
GCD (/, /') = D(iV + det Md1) ßd-1 + ••• + det Mdd),
where matrix Mdj) is the d-th inner of the matrix which is obtained from the Sylvester - Habicht matrix by replacing d + 1-th column counted from, the right by the j-th column counted from the right.
The simple zeroes of the polynomial GCD (/, /') correspond to the double zeroes of the original polynomial /(x). The zeroes of polynomial GCD (/, /') of multiplicity m > 1 correspond to zeroes of the polynomial / (x) with multiplicity m + 1.
3. Subdiscriminant approach
Here we propose the following constructive algorithm for obtaining polynomial parameterization of the discriminant set in the space n of coefficients of the polynomial / (x). The main idea of the algorithm is to consider each variety Vi of dimension i as a tangent developable surface [13, 14] with the directrix defined by the variety Vi_i.
1. We compute the variety V1 with parametrization
Vi : {a* = (—1)n-iCX-*, i = 0,...,n — 1} ,
which solve the system {D(j)(f) = 0} , j = 0,..., n — 2. The polynomial f (x) has the only zero t1 with multiplicity n on it.
2. Using the variety V1 as a directrix we obtain the variety V2 with parametriza-tion
V2 : {a* = (— 1)n-iCintnl-i + (— 1)n-iCn-itn1-1-it2, i = 0,...,n — 1} ,
which solve the system {D(j)(f) = 0} , j = 0,..., n — 3. The polynomial f (x) has two distinct zeroes on it: zero t1 with multiplicity n — 1 and simple zero t1 + t2. The variety V2 is a tangent developable surface of dimension 2 in the space n.
3. Considering the variety V2 as an envelope of two-dimensional planes one gets variety V3, on which the polynomial f (x) has a zero of mupltiplicity n — 2 and a pair of simple zeroes.
4. Finally, repeating the steps described above one obtains the variety Vn-1, which is the tangent develope hypersurface H in n. On this hypersurface polynomial f (x) has one double zero and n — 2 simple zeroes.
Remark 1. There are other varieties in the discriminant set D(f), on which polynomial f (x) has zeroes with other scheme of multiplicity. It is easy to show that in the generic position case these varieties have codimension more then 1 and therefore they cannot be the parts of the hypersurface H.
Proposition 2. The subset of codimension 1 of the discriminant set D(f) of a generic monic polynomial with real coefficients is a tangent develope hypersurface H in the space n. Therefore it has polynomial paramerization.
4. Singular points approach
It is possible to describe the mentioned above varieties V*, i = 1,..., n — 1 as singular points of the discriminant set D(f) of orders n — 1 — i correspondingly.
Definition 4. Let <^(X) be a polynomial of X = (x1,..., xn). The point X = X0 of the set <^(X) = 0 is said to be a singular point of the k-th order if all the partial derivatives of <^(X) with respect to x1,... ,xn of order k, inclusive, vanish and at least one partial derivative of order k + 1 does not vanish.
Compose the ideal Jk from the discriminant D(/) and all its partial derivatives up to the order k inclusive. The ideal Jk includes all singular points of order k and below. The direct computations show that ideal Jk for certain value k ^ n — 2 does not equal to the ideal Zk = {D(j)(/)|j = 0,..., k}, which include the variety Vn-k-1. With the help of computer algebra algorithms (Grobner basis and procedures for polynomial ideals [15]) it was shown that Rad Jk = RadZk for the following values of n = 3, 4, 5, where RadP is the radical of ideal P. It is quite possible that this statement takes place for any n G N.
Remark 2. If statement, that for any finite value k radicals of the ideals Jk and Ik coincide, is right, then the subdiscriminant approach is equal to the singular point approach. But it is clear that the first approach is easer than the second one as far as the number of polynomials in the ideals Ik depends linear on the degree n and the number of polynomials in the ideals Jk grows as nk.
5. Examples
Example 1. Monic cubic polynomial
Let consider monic cubic polynomial /3(x) = x3 + a2x2+a1x+a0. Its subdiscriminants are:
D(0)(/3) = —4a3a0 + «2^1 + 18a2a1a0 — 4a3 — 27a2, D(1)(/3) = a2 — 3ai. The case of the only zero of multiplicity 3 is described by the ideal
J3(1)d=f{D(0)(/3),D(1)(/3)}
with Hilbert dimension equal to one. Normalization the polynomial GCD (/3, /3) on the ideal J3(1) gives
GCD (/3, /3) = 3x2 + 2a2 + a1 = (3x + a2)2/3,
which is a full square.
One-parametric variety V1 has the following parametrization
V1 : {a0 = — ¿3, a1 = 3t1, a2 = —3^} ,
and polynomial /3(x) = (x — t1)3 on it.
The case of two zeroes with multiplicities 2 and 1 correspondingly is described by the ideal J3(0) =f {D(0)(/3)}, which has Hilbert dimension 2 and its zeroes forms two-parametric variety V2 of codimension 1
V2 : {ao = -t? - t?t2, «1 = 3t? + 2tiÎ2, «2 = -3ti - t2}
phc. 1: Discriminant surface for monic cubic polynomial
and polynomial /3(x) = (x — ti)2(x — (ti + t2)) on it. This variety is a tangent developable surface in n = R3 with envelope V1 and is shown in Fig. 1.
So, variety V2 is the hypersurface H dividing coefficient space n into two domains with different numbers of real zeroes.
Example 2. Monic quartic polynomial.
Let /4(x) =f x4 + a3x3 + a2x2 + a1x + a0 be a monic quartic polynomial. Its subdiscriminants are the following:
D(0) (/4) = —27a4a2 + 18a3a2a1a0 — 4a3a1 — 4a2a2a0 + a2a2a2 +
30 22
+ 144a3a2a0 — 6a3«2a0 — 80a3a2 a1a0 + 18a3a2a3 + 16a2a0 —
— 4a3a1 — 192a3a1a0 — 128a2a0 + 144a2a2a0 — 27a1 + 256a0, D(1) (/4) = —6a3a1 + 2aija2 — 12a2 a0 + 28a3a2a1 — 8a2 + 32a2a0 — 36a1 D(2) (/4) = 3a3 — 8a2.
One-parameter variety V1 has parametrization
V1 : {a0 = ¿1, a1 = — 4tf, a2 = 6t1, a3 = —4t1} On this variety /4(x) = (x — t1)4.
Discriminant set D(/4) contains two varieties, on which the polynomial /4(x) has only two distinct zeroes with multiplicities (3, 1) and (2, 2) correspondingly. The variety V21) is a tangent developable surface in coefficient space n = R4 as was shown in Section 3. Its parametrization is the following
V21) : t ao = t4 + t3t2, a1 = —4t1 — 3t1 ¿2, a2 = 6t2 + 3t1t2,a3 = — 4^ — ¿2,}
on which /4(x) = (x — t1)3(x — t1 — t2). Considering variety V'21) as an envelope of hyperplanes one can obtain the parametrization of those part of variety V31) of codimension 1, on which the polynomial /4(x) has only 3 distinct real zeroes. The parametriation of this veriety can be given in the following form:
V31) : {ao = ¿2 (¿2 — ¿3) , a1 = 2*1 (¿33 — ¿2 — M2) ,
a2 = ¿1 + 4^2 + ¿2 + ¿3, a3 = —2(i1 + ¿2)}, (3)
on which /4(x) = (x — ¿1)2(x — (¿2 + ¿3))(x — (¿2 — ¿3)). Setting in (3) ¿3 = 0 one
(2)
can get the parametric representation of two-dimensional variety V2 , on which the
polynomial /4(x) has a pair of real distinct zeroes with multiplicity two. Moreover,
(2)
setting ¿3 = tt3 one gets parametrization of 3-dimensional variety V3 on which the polynomial /4(x) has one real zero ¿1 with multiplicity 2 and a pair of simple complex mutually conjugated zeroes ¿2 ± ii3. So, both varieties V'31) and V32) form the hypersurface H C D(/4) dividing the coefficient space n into 3 domains with different numbers of real zeroes.
Let us make a linear Tschirnhaus transformation x = y — a3/4 and we obtain polynomial /4(y) = y4 + 62y2 + b1y + 60 with the coefficient space n = R3. Then the variety V1 shrinks to the origin, the variety V721) : {60 = — 3^, b1 = 8i3, 62 = — 6i2}, on which /4(y) = (y — ¿1)3(y+3i1), becomes one-dimensional, and, finally, the variety
V3 : tbo = ¿2 (¿1 — ¿3) , 61 = 2^3, 62 = — 2i1 — ¿3, } which is a tangent developable surface, on which
/4 (y) = (y — ¿1)2(y — (¿1 + ¿3))(y — (¿2 — ¿3)),
and the variety V2(2) : {60 = ¿2, b1 = 0, 62 = — 2i2}.
Variety V21) is one-parameter set of singular points of the first order, because they form the set of casps. The part of variety V22) is one-parameter set of singular points of the first order, because one branch of the parabola is isolated (for ¿2 > 0) and the other branch is the curve of selfintersection of the variety V3. The last variety is hypersurface H of codimention 1 in the coefficient space II and it divides this space into three domains with different numbers of real zeroes of the polynomial /4(y). All the varieties described above are shown in Figure 2.
Рис. 2: Discriminant surface for monic quartic polynomial /4(y).
6. Resume
The described in Section 3 algorithm of polynomial parametrization of the discriminant set D(/) of the real polynomial /(x) allows to represent this parametrization in a such form that it is possible to obtain the structure of the discriminant set D(/) as a finite set of algebraic varieties V of different dimensions from 1 to n — 1.
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Институт прикладной математики им. М. В. Келдыша.
Поступило 30.04.2015
