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Mathematical Notes, vol. 69, no. 2, 2001, pp. 170-174. Translated from Matematicheskie Zametki, vol. 69, no. 2, 2001, pp. 194-199. Original Russian Text Copyright c 2001 by Grigoryan, Mailybaev

On the Weierstrass Preparation Theorem
S. S. Grigoryan and A. A. Mailybaev
Received May 22, 2000

Abstract--An analytic function of several variables is considered. It is assumed that the function vanishes at some point. According to the Weierstrass preparation theorem, in the neighborhood of this point the function can be represented as a product of a nonvanishing analytic function and a polynomial in one of the variables. The coefficients of the polynomial are analytic functions of the remaining variables. In this paper we construct a method for finding the nonvanishing function and the coefficients of the polynomial in the form of Taylor series whose coefficients are found from an explicit recursive procedure using the derivatives of the initial function. As an application, an explicit formula describing a bifurcation diagram locally up to second-order terms is derived for the case of a double root.
Key words: Weierstrass preparation theorem, analytic function of several variables, bifurcation

diagram.

Consider an analytic function of several variables f (z, p), z C , p = (p1 ,. . .,pn ) Cn , which vanishes at the p oint z0 , p0 = (p0 , . . .,p0 ) . By the Weierstrass preparation theorem [1, 2], in n 1 the neighb orhood of the p oint z0 , p0 the function f (z, p) can b e expressed uniquely as a product of two analytic functions, one of which is a p olynomial in z - z0 with leading coefficient equal to one, and the other function is not equal to zero at the p oint z = z0 , p = p0 . The representation f (z, p) as a product of two factors simplifies the problem of studying the roots of the equation f (z, p) = 0 and their prop erties. For example, if f (z, p) is a p olynomial in z , then it follows from the Weierstrass preparation theorem that we can essentially lower the order of the equation in studying the dep endence of isolated roots of the p olynomial f (z, p) on p . Such an op eration is esp ecially useful in the analysis of multiple roots in problems of stability, in which one tries to define the domain stability (the set of values of the parameters p for which the roots z (p) satisfy a certain condition) [3]. Another area of research where this theorem can b e effectively used is the study of bifurcation diagrams (the sets of values of the parameters p for which there are multiple roots). The applications describ ed ab ove can b e realized under the condition that the factors into which the function f (z, p) is expanded according to the preparation theorem are known. In the present pap er, we construct a method for finding these factors in the form of Taylor series whose coefficients are found from an explicit recursive procedure using the derivatives of the function f (z, p) at the point z = z0 , p = p0 . As an application, an explicit formula describing a bifurcation diagram locally up to second-order terms is derived for the case of a double root. The preparation theorem for the case of infinitely differentiable functions was proved by Malgrange [4]. The results of the present pap er can b e carried over without changes to this case and enable us to find the Taylor series of the desired functions. Weierstrass Preparation Theorem [1, 2]. Suppose that f (z, ing at the point z = z0 , p = p0 , where z = z0 is an m-multiple i.e., m -1 f f = ЗЗЗ = = 0, f (z0 , p0 ) = z z m-1
170 0001-4346/2001/6912-0170 $25.00

p) is an analytic function vanishroot of the equation f (z, p0 ) = 0 , mf = 0, z m

c 2001 Plenum Publishing Corporation


ON THE WEIERSTRASS PREPARATION THEOREM

171

where the derivatives are taken at the point z = z0 , p = p0 . Then there exists a neighborhood U0 Cn+1 of the point (z0 , p0 ) in which the function f (z, p) can be expressed as f (z, p) = (z - z0 )m + am
-1

(p)(z - z0 )m

-1

+ ЗЗЗ + a0 (p) b(z, p) ,

(1)

where a0 (p) ,. . .,am-1 (p) ,b(z, p) are analytic functions uniquely defined by the function f (z, p) and ai (p0 ) = 0 , b(z0 , p0 ) = 0 . To obtain the expansion (1), it is necessary to determine the functions ai (p) and b(z, p). In the neighb orhood of the p oint z = z0 , p = p0 , the functions ai (p) and b(z, p) can b e expressed as the Taylor series ai (p) = 1 ai, h ph , h! b(z, p) =
k,

h

h

1 bk, h z k ph , k ! h!
+

(2)

where the sums are taken over all k Z+ , h = (h1 , . . .,hn ), hi Z integers) and the following notation is used: ai, h = |h| ai ph1 ЗЗЗ p 1
n j =1

(Z

+

is the set of nonnegative

hn n

,

bk, h =

k + | h| b z k ph1 ЗЗЗ p 1

hn n

,

|h| = h1 + ЗЗЗ + hn , h! = h1 ! ЗЗЗ hn ! .

p h =

(pj - p0 )hj , j

z k = (z - z0 )k ,

All the derivatives are calculated for z = z0 , p = p0 . By the zeroth-order derivative we mean the value of the function at a p oint, in particular, bk, 0 = k b/ z k . The derivatives of the function f (z, p) are denoted in a similar way. The definition of the functions ai (p) ,b(z, p) is equivalent to the determination of the derivatives ai, h ,bk, h defining the coefficients of the expansion (2). In the following theorem, an explicit expression for the desired derivatives is given in terms of the derivatives of the function f (z, p). Theorem. The derivatives ai, h and bk, h of the functions ai (p) ,b(z, p) in relation (1) satisfy the fol lowing recurrence relations :
i j

ai, h =

j =0

ij Fj, h ,

Fj, h = fj, h -
k =0 m -1 m+k,

h +h =h h =0 , h =0

c(j, k ; h , h )ak, h bj

-k,

h,

(3)

bk, h =

k! f (m + k)!

h-

j =0

h +h =h h =0

c(m + k, j ; h , h )aj, h bm

+k -j,

h

,

(4)

where the coefficients ij ( i j ), c(k, j ; h , h ) are determined by the relations jj = m! , j !fm, 0 ij = - m! fm, 0
i -1 k =j

fm+i-k,0 kj (m + i - k)!
n

(i > j ) , (5)

j! c(j, k ; h , h ) = (j - k)!

s=1

(hs + hs )! . hs ! hs !

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S. S. GRIGORYAN, A. A. MAILYBAEV

Proof. Relations (4) for h = 0 are of the form k! fm+k, 0 . (6) bk, 0 = (m + k)! They are obtained by (m + k)-fold differentiation of the identity f (z, p0 ) = (z - z0 )m b(z, p0 ) with resp ect to z . Let us take the derivative i+|h| / z i ph1 ЗЗЗ phn of b oth sides of relation (1): n 1
min(i,m)

fi, h =

j =0

h +h

=

h

c(i, j ; h , h )aj, h bi

-j,

h,

am, h =

1, 0,

h = 0, h = 0.

(7)

Supp ose that i < m . Expressing the summands containing aj, h in terms of the other summands and using (6), we obtain
i j =0

ij aj, h = Fi, h ,

ij =

i! fm+i-j, 0 , (m + i - j )!

(8)

where the Fi, h are defined in (3) form: a0 , h F0 , h . . . . G = . . am-1 , h Fm-1 , h

. Equations (8) for i = 0 ,. . .,m - 1 can b e written in matrix , G= 00 10 . . . (m
-1)0

0 11 . . . (m
-1)1

0 0 .. . ЗЗЗ (m

0 0 0
-1)(m-1)

.

Since, by assumption, ii = i! fm, 0 /m! = 0 , we have det G = 0 and, therefore, the system of equations has a unique solution of the form Fh , 0 a0 , h 00 0 0 . . . .. -1 . . . . (9) G-1 = =G , . . 0 . . (m-1)0 ЗЗЗ (m-1)(m-1) am-1 , h Fh ,m-1 We can prove that the matrix G-1 is of the form (9) with entries ij defined in (5) by directly verifying the identity GG-1 = I ( I is the unit matrix). Relation (9) is the expression for relations (3) of the theorem in matrix form. Consider the case i m . Then relation (4) of the theorem is obtained by rearranging the terms in relation (7) with regard to (6) and by substituting m + k for i . Relations (3)-(5) are recurrence relations allowing us to find the derivatives of ai (p) and b(z, p) from the derivatives of the function f (z, p) in the following order: ai, h , bk, h (|h| = 1) ai, h , bk, h (|h Here, to determine the derivative ai, h , it is necessary to calculate h | - 1)m and h < h at the previous stages ( hs hs for all s some value of s). Thus the value of ai, h can b e determined from k i + |h - h |m and h h . The first steps of the recursive procedure are b
k, 0

| = 2) ЗЗЗ . aj, h ,bj, h for j i + (|h - = 1 ,. . .,n and hs < hs for the derivatives fk, h , where
i

ai (p0 ) = 0 ,

m +k f k b k! = , z k (m + k)! z m+k
m -1 j =0

ai = ps

ij
j =0

j +1 f , z j ps (10) .

m+k+1 f k! k+1 b = - z k ps (m + k)! z m+k ps 2 ai = ps pt
i

(m + k)! aj m+k-j b , (m + k - j )! ps z m+k-j

ij
j =0

j +2 f - z j ps pt

j k =0

ak j -k+1 b ak j -k+1 b j! + (j - k)! ps z j -k pt pt z j -k ps

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Example. The function f (z, p) can b e regarded as a family of functions of a single variable z , where p is the parameter vector. By a bifurcation diagram we mean the set of values of the vector p for which the function f (z, p) has multiple roots z . Consider the double root z = z0 . It follows from the Weierstrass preparation theorem that the surface in the space of p on which this root remains a double root (the value of the root may vary) can b e determined by equating to zero the discriminant D = (a1 (p))2 - 4a0 (p) of the equation (z - z0 )2 + a1 (p)(z - z0 ) + a0 (p) = 0 . Substituting the expansions (2) into the equation D = 0 , we can write the equation of this surface as
n

D = -4
r =1 n

a0 (pr - p0 ) r pr 2 a0 a1 a1 -2 ps pt ps p
t

+
s, t=1

(ps - p0 )(pt - p0 )+ o p - p s t

0

2

= 0,

(11)

where p is the norm in Cn ; the first and second derivatives of the functions a0 (p) ,a1 (p) are expressed in terms of the derivatives of the function f (z, p) by formulas (10); the coefficients ij are determined, according to (5), by the relations 00 = 11 = 2 , 2 f/ z 2 10 = - 2 3 f/ z 3 . 3 ( 2 f/ z 2 )2 (12)

Equations (10)-(12) describ e the surface D = 0 in explicit form via the derivatives of the function f (z, p) up to second-order terms. A similar method can b e used to obtain relations describing the bifurcation diagram in the neighb orhood of the root of arbitrary multiplicity. The bifurcation diagram of the family of functions f (z, p) has a certain physical meaning. Supp ose that f (z, p) is a p olynomial in z resulting from the substitution z = -2 into the characteristic p olynomial of a linear Hamiltonian or nonconservative (circulation) mechanical system. Then the part of the bifurcation diagram characterized by the p ositive real roots is the b oundary of the domain of stability corresp onding to the oscillatory form of the loss of stability (flutter) [5 6]. Moreover, relation (11) describ es the b oundary of the domain of stability locally up to second-order terms.

Fig. 1. Bifurcation diagram of the family of p olynomials and its approximation For example, consider the two-parameter p olynomial function 1 f (z, p) = z 5 - 13z 4 + 62 + p 4
MATHEMATICAL NOTES Vol. 69
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z 3 - (134 + p2 )z 2 + (129 + p2 )z - 45 + p1 p2 . 1
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S. S. GRIGORYAN, A. A. MAILYBAEV

For p0 = (0 , 0) , the p olynomial f (z, p0 ) has double roots z = 1 , z = 3 and a simple root z = 5 . Therefore, the bifurcation diagram in the neighb orhood of the p oint p0 = (0 , 0) consists of two curves. These curves are given by the equation D = 0 written at the p oints (z0 ,p0 ,p0 ) = (1 , 0 , 0) 1 2 and (3 , 0 , 0) . Using relations (10)-(12), we can find the equations of the curves D = 0 up to second-order terms: z0 = 1 : z0 = 3 : 1 1 15595 2 2525 229 2 p1 + p2 - p1 + p1 p2 + p = 0, 16 4 65536 8192 4096 2 27 3 13815 2 661 49 2 p1 + p2 - p+ p1 p2 + p = 0. 8 2 4096 1 512 256 2

(13)

The exact form of the bifurcation diagram and its approximation (13) are shown in Fig. 1 solid and the dotted line, resp ectively. In Fig. 1 the domain (I) defined by the inequalities corresp onds to the presence of only real roots z (p) . Note that the approximate equations bifurcation diagram (13) were calculated from the values of the derivatives of the function at the p oints (z0 ,p0 ,p0 ) = (1 , 0 , 0) and (3 , 0 , 0) . 1 2 ACKNOWLEDGMENTS

by the D>0 for the f (z, p)

This research was supp orted by the Russian Foundation for Basic Research under grant RFFI- GFEN no. 99-01-39129 and carried out during the visit to Dalyan Technical University (China). REFERENCES
1. K. Weierstrass, "Einige auf die Theorie der analytischen Functionen mehrerer Ver? anderlichen sich beziehende S? ze," in: Mathematische Werke, II, Mayer und Muller, Berlin, 1895, pp. 135-188. at ? 2. B. V. Shabat, Introduction to Complex Analysis [in Russian], Pt. II, Nauka, Moscow, 1976. 3. A. A. Mailybaev, "On the stability of polynomials depending on parameters" Izv. Ross. Akad. Nauk Tekhn. Kibernet. Control Theory and Systems (2000), no. 2, 5-12. 4. B. Malgrange, "The preparation theorem for differentiable functions," in: Singularities of Differentiable Mappings. Col lection of Papers [Russian translation], Mir, Moscow, 1968, pp. 183-189. 5. V. I. Arnol d, V. V. Kozlov, and A. I. Neishtadt, "Mathematical aspects of classical and celestial mechanics," in: Contemporary Problems in Mathematics. Fundamental Directions [in Russian], vol. 3, Itogi Nauki i Tekhniki, VINITI, Moscow 1985. 6. A. P. Seiranyan, "Bifurcation in one-parameter circulation systems," Izv. Ross. Akad. Nauk, Mechanics of Solids (1994), no. 1, 142-148. Institute of Mechanics, M. V. Lomonosov Moscow State University E-mail : (S. S. Grigoryan) grigor@inmech.msu.su, (A. A. Mailybaev) mailybaev@inmech.msu.su

MATHEMATICAL NOTES

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No. 2

2001