Problem: find the number of real solutions of an algebraic system fi( x, y). = 0, j-,(x, y> = 0 satisfying ... the roots: a disturbance f(r) + &x1' with E = 2-23 leads to the appearance of five pairs of complex .... u(A) its signature, and n+(A). [ or n_(A)] its ...
Hermite’s Method of Separation of Solutions of Systems of Algebraic Equations and Its Applications Alexei Yu. Uteshev
and Sergei
G. Shulyak
St. Petersburg State University Department of Applied Mathematics Bibliotechnaya St. Petersburg
PI. 2, Petrodvorets 198904, Russia
Submitted by Stephen Barn&t
ABSTRACT The paper may be considered as a supplement to the book by M. Krein and M. Naimark. Problem: find the number of real solutions of an algebraic system fi( x, y) = 0, j-,(x, y> = 0 satisfying an algebraic condition g(x, y) > 0. We give a review of Hermite’s method, which permits one to solve the problem using a finite number of algebraic operations on the coefficients of fi, fs, g. The method reduces the problem to an investigation of the quadratic form in x0, , zN_ 1:
where (oj, pj> is a solution of the system, and N = degf, deg fs. Methods involved: elimination theory, the theory of symmetric functions, and the theory of Hankel quadratic forms. Applications: analogue of Sturm series in Iw’, calculation of the Kronecker-Poincari index of an algebraic field with respect to an algebraic curve in R2, conditions for the sign-definiteness of a homogeneous higher-order polynomial of three variables. We discuss also the possibility of extension of the method to Iw”.
INTRODUCTION
The problem of localization (separation) of solutions of algebraic equations systems arises in a variety of fields, ranging from numerical methods to control theory.
LlNEARALGEBRAANDZTSAPPLZCATZONS177:49-88
(1992)
49
0 Elsevier Science Publishing Co., Inc., 1992 655 Avenue of the Americas, New York, NY 10010
0024-3795/92/$5.00
50
ALEXEI YU. UTESHEV AND SERGEI G. SHULYAK The importance
of the problem
EXAMPLE [29].
Consider
f(X)
can be seen from the following
a polynomial
= (X + l)( X + 2) ... (X + 20)
and suppose
that we need to calculate
= XZO + 210xig
+ *** +20!
its roots by means
)
of, for example,
Newton’s method. Let us assume in addition that the calculations are being carried out to seven digits after the point, and in particular the coefficients of this polynomial are represented within this accuracy. Will this be sufficient for computing the roots correct to three decimal places? The confusing answer is that this accuracy is insufficient even for preserving the reality of all the roots: a disturbance f(r) + &x1’ with E = 2-23 leads to the appearance of five pairs of complex conjugate roots! Thus, if we want to apply any of the numerical methods for finding the roots of such a polynomial, then we must estimate beforehand the sensitivity of its roots to the variations of its coefficients. In 1936 the excellent book of M. Krein and M. Naimark [18] appeared, which contains a general survey of results due to Jacobi, Sturm, Hermite, Sylvester, Hurwitz, Schur, et al. concerned with the problem of localization of the roots a single polynomial equation. However, in the more general case of a system of equations the problem of separation of solutions is treated in that book somewhat weakly. Our paper following problem is considered:
is aimed
at the filling
this void. The
PROBLEM 1. Let f,,f&x, Y), g(x, Y) (degfj = nj> be polynomials with real coefficients. Find the number of real solutions of the system fi(X> satisfying the condition
g(x,
Y) = 9,
fz.(x,
Y) = 9
(I)
y) > 0.
In 1852-56 Charles Hermite proposed two constructive algorithms for this problem [ll-131. His methods are algebraic, i.e., they permit one to solve the problem by using a finite number of algebraic operations on the coefficients of fi, f2, and g. In this paper we shall give an account of one of these methods [12, 131 and discuss its possibilities, difficulties, and some applications. Section 1 is an auxiliary one and contains some results from the theory of equations. Section 2 is devoted to the application of Hermite’s method to the corresponding problem for one equation in one variable [13, 181: i.e., to find
SEPARATION
OF SOLUTIONS
the number of real roots of polynomial inequality g(x) > reduced to that of finding the form in the variables x,,, . . .)
OF ALGEBRAIC
SYSTEMS
51
a polynomial f(r) (degf = n> that satisfy a 0. According to Hermite’s idea, this problem is positive and negative indices for the quadratic
x,-1:
here A,, . . . , A, are the roots of f(r) (Theorem 2.2). Coefficients of the form (2) turn out to be symmetric polynomials in X 1, . . . , A,, and so they can be expressed rationally in terms of the coefficients of f and g (Theorem 1.1). Similarly, under some assumptions, Problem 1 is reduced in Section 3 to the investigation of the quadratic form in real x,,, . , xN_ 1 (N = n,n,>:
here (a,, coefficients
PIX.. . ,( (Ye, of
PN) are the solutions of (1) (Theorem 3.1). The turn out to be symmetric polynomials in PN); thus they can be expressed rationally in terms of the
(3)
(o1, PI), . . . ,(ahi, coefficients of fi, fi, and g (Theorem 4.1). Constructive methods for their calculation are due to Poisson and Jacobi, and these are discussed in Section 4. Thus, Problem 1 is reduced to the verification of a finite number of algebraic conditions on the coefficients of fi, fs, and g. Choosing g(x, y> = (x - sX y - t) in (3), we obtain a system of polynomials in s and t, which is an analogue in R2 of a Sturm sequence (Theorem 3.3). In Section method.
5 we treat
some
examples
and discuss
difficulties
of the
Section 6 is devoted to applications of Hermite’s method to the following problems of the qualitative theory of differential equations:
PROBLEM 2. Find the Kronecker-Poincare with respect
PROBLEM definiteness
to a curve ia(x,
index of a vector field (P, Q> y> = 0 if P, Q, @ are polynomials.
3. Give necessary and sufficient of a homogeneous polynomial (form)
By using Theorem
3.1, these problems
conditions for the of three variables.
can be solved algebraically.
sign-
52
ALEXEI YU. UTESHEV
AND SERGEI
G. SHULYAK
In Section 7 we discuss the possibility of an extension of Her-mite’s method to [w3 and point out its relationships with other areas of classical and modem mathematics. In the Appendix the proof of Theorem
4.2 is given.
NOTATION. 1. For the sequence of real nonvanishing numbers A,, . . . , A,,, A,)] denotes the number of permanences [or P(A,,..., A,,) [or V(A,,..., variations] of sign (Section 2). 2. For the quadratic form A(X, X), .&denotes its matrix, r(A) its rank, its positive [or negative] index u(A) its signature, and n+(A) [or n_(A)] (Section 2). 3. n + (s, t) and n _ (s, t) are defined 4. g(f,
in Theorem
3.2.
For polynomials f(x) and g(x), .9(f) is the discriminant g) is the resultant of f and g (Section 1).
of f(x);
f,(x, y), fa(x, y) considered as polynomials in y [or d enotes the resultant of fi and fi (Section 3); the Jacobian of fi and fi is denoted by Afi, fi> in Sections 4 and 6.1 [and by Ax, y) in the Appendix]. 5.
xl, tix)
For polynomials
[or g(y)1
REMARK. Henceforth, any polynomial with real coefficients will be called a real polynomial, and, unless otherwise mentioned, all the polynomials will be of this type.
1.
PRELIMINARY
RESULTS
We recall here some results from the theory of algebraic
equations
[24,
251. Consider a real polynomial f(x) = a,,x” + uix”-i + *** +a, (a, z 0). Denote its roots by A,, . . . , A, and consider them as functions of a,, . . , a,. THEOREM 1.1. L-d I?(%,,.. ., x,) be a symmetric polynomial of its variables, i.e., let its value be unchanged when any two of the variables are interchanged. Zf we substitute in F(h 1, . . . , A,), instead of A,, . . , A,, their representations in terms of a,, . , a,, then the function obtained, a,), is a rational one in its arguments (a polynomial if a0 = 1). 5%,,..., Consider now a real polynomial g(x) = b,,x”’ + blxm-’ + **- +b, (b, # 0). Construct the symmetric polynomial F = g(xl) -*- g(x,). By Theorem 1.1, F( A,, . . . , A,) is a rational function of a,, . . , a,.
SEPARATION
OF SOLUTIONS
OF ALGEBRAIC
SYSTEMS
53
DEFINITION. The expression a,“g(h,) *** g(A,) is a polynomial with respect to a,, . . . , a,, b,, . . . , b,,; it is called the resultant off and g and is denoted by S’(f, g). THEOREM 1.2. COROLLARY.
S(f) f'>= 0
DEFINITION.
The
with respect
to
iff and iff has a
B?(f) g ) = 0 if and only
aO,
expression
. . , a,;
.
if and only
( - 1)“‘” - ‘)/‘S?(f,
g have a comnwn root. multiple root.
f’ )/a,,is a polynomial of f and is denoted
it is called the discriminant
by .Hf 1. From this definition
follows the equality
There exist several methods coefficients
for the representation
of S(f,
g) in terms of the
of f and g. Here is one widely used:
THEOREM 1.3.
L&f, g) = det M, where b0
a0
b, .. ....
a1 a0 . . . . .
. . bo
a0
.
M= %I
bnl
a,-1
b m-1
(1.2)
hrl
an . ......
. .
. . .
. . .
. . . . . .
. .
btn
a, n columns.
rn columns
THEOREM 1.4.
mials p(x)
If SXf, g) z 0, then there exist uniquely defined polynoand 9(x> such that deg p < m - 1, deg 9 < n - 1, and
p(x)f(x)
+ 9(x)&x)
p(x) (or 9(x)) equals the determinant replacing the last row in it with
(x
77-l
,xm-z )...)
1,o )...)
0)
=N_f3
g).
of the matrix obtained from
r
(0,.
. . ,o,
xn-l,
Xn-2,.
. .)
M by
1)
54 2.
ALEXEI YU. UTESHEV AND SERGEI G. SHULYAK HERMITE’S
METHOD
Let us consider
f(x)
=
IN THE
CASE
two real polynomials
Suppose
that f 9(f,
PROBLEM. condition
g(x)
Find > 0.
the
x,
X)
number
5 [x0 + x,/i,
=
j=l
H(
x,
x)
=
t
g(hi)[
j=l
roots, i.e., by Theorem
of
real
roots
s=
Sl ,,_;
of f(x)
let us construct
1.2, their
that
satisfy
two quadratic
+ -** +x n _ 11h”-‘I2 = x*sx,
x,, + xlhi
+ -.*
+x~_~A;-‘]’= X*m
the
forms
(2.1) (2.2)
--
with Hunkel matrices s0
+ **. +b,,.
A,, . . , A, be the roots of f(x).
In order to solve the above problem in real variables x0, . ., x,-1:
S(
blP1
= box”’ +
g(x)
and g have no common g) z 0. Let
EQUATION
(a0 Z 0, b, Z 0)
aoxn+ alxn-’ + *** +a,,
resultant
OF ONE
81
s2
.a*
s2
sg
..a
.s,.
s”‘+.l’
a%?= [ hj+k]fiiO,
. . . ...’
sn-1 S” . .s,,_2’
= [ 1
sj+k]~'~~o,
x* = (xo,...,x”_l).
Here sk = Cy= 1 Aj” is the Newton sum of f(x)
i2-51, and
+ blsk+m-l h, = t g(Aj)A; = bOSk+rn
+ ... +b,,+.
j=l
Since the functions sk and h, (k = 0,. ,2n - 2) are symmetric polynomials in the A,, . . , A,, they permit, by Theorem 1.1,rational representations
SEPARATION
OF SOLUTIONS
in terms of the coefficients
OF ALGEBRAIC
aO, .
, a,. In particular,
(n --
55
SYSTEMS
a1
if
k=O,
if
k=l,
if
2. The number of distinct real roots of fC x) satisfying the condition g( x> > 0 (or g( x> < 0) equals n+(H) - q (or n_(H) - q). Here q = [r(S) - a(S)]/2 is the number of distinct pairs of complex conjugate roots off ( x).
Proof. For simplicity, suppose that f(x) has no multiple roots [r(S) = n]. Then to every real root Aj satisfying g(Aj) > 0 (or < 0) corresponds the positive (negative> square g< A,>XF in the representation of H( X, Xl, where xi = ;Ya + XIAj + *** +x _ An-l is a real linear form in x0,. .., x,-i. Let = Aj = c - id there exist a pair of comp~exic&jugate roots A. = c + id A. (d # 0); let g(A,l = a + ib; then g(Aj+i) La - ib. D’edoih
x0
+
xlAj
+
-**
+~~_~Ajn-l
=
X
+
iY,
56
ALEXEI YU. UTESHEV AND SERGEI G. SHULYAK
where X and Y are real linear forms in x0, . . . , x,_ 1. If a # 0, then for the pair Aj, Aj + 1 we have the sum
and if a = 0, b # 0, then we have the sum
(The
case a = b = 0 cannot
occur,
since
f and
g have no common
roots.)
Denote
x,= I
X-!Y 1 XL
if
a # 0,
if
a=O,
yi=
(
;+y
;;
zr;>
Thus, for the pair Aj, Aj + I we have two squares of real linear forms Xj and 3; one of them possesses a positive coefficient, the other a negative one. Making similar transformations for all the roots of f< x), we obtain a set of linear forms X., Y, which are linearly independent, since they are nonsingu. of complex forms x0 + xlAj + .** +x,_ rA;-’ (j = lar linear corn 6.mations
1>..., n> with distinct
Aj.
By the inertia law for quadratic
forms,
n+(H) = m, + q,
n_(H) =
m2
+
4,
where m, [or m,] is the number of real roots of f(x) satisfying the inequality g(x) > 0 [or g(x) < 01,from which follows the assertion of the theorem. w REMARK 1. replaced by
It can be seen from the proof that the form (2.2)
2 g(Aj)[ j=l
ra’P~(Aj)
+ rr*i(Aj)
+ ... +r,-$‘n-i(Aj)12T
may be
(2.4)
57
SEPARATION OF SOLUTIONS OF ALGEBRAIC SYSTEMS where
U,,(x),
.
, qn_ r(x)
are
arbitrary
real
polynomials
satisfying
the
condition
rank[‘JLl(Aj)],“k_l = r(S). REMARK 2. The numbers (T(S), n + ( H >, n _ ( H ) for the quadratic forms s and H may be determined as usual by means of the number P of permanences and the number V of variations of sign in the sequence of leading principal
minors of corresponding =P(l,S1,S,,S,
a(S)
,...,
matrices: S,)
n+(H)
= P(l,Z~,~z,.
. .>%.>,
r~_( H)
= V(LZ~,&>.
. .>T>
[r = r(S)]
if none
of them
is zero.
then the numbers a(S), sequences, forms S and H should be calculated
-v(l,~,>~,>...~~,)~ (2.5)
If there
exists a zero in one of these
n+(H),
n_(H)
for Hankel
quadratic
with the help of the rule of Frobenius
M. EXAMPLE 2.1.
f(r)
= x5 + 4x4 - 2x - 2, g(x)
= x3 - r2 + 1. What
is the number of real roots of f(x) satisfying the condition Solution: The Newton sums off(x) are so = 5, s4
264,
=
s8
Sll
=
=
s5 = -1054, 68,624,
s2
=
16,
s3 = -64,
sg = 4240,
sa = -276,076,
> O?
s7 = -17,056, SlO
=
1,110,676,
-4,468,336.
The leading principal s, = 5,
s1 = -4,
g(x)
minors of the matrix S are
S, = 64,
S, = 512,
s, = -60,160,
S, = -2375,600.
By Theorem 2.1 and the formulae (2.51, f(r) has three distinct real roots and a pair of complex conjugate roots, i.e., 4 = 1. Calculate h,:
ho = -75, h, = 84,626,
h, = 324,
h, = -1302,
h, = -340,460,
h, = 5230,
h, = 1,369,696,
h, = -21,032, h, = -5,510,388.
58
ALEXEI YU. UTESHEV AND SERGEI G. SHULYAK
The leading principal zi
= -75,
&,
minors of the G%?’ are = - 7326,
2s
= 173,460,
S?
= 23,423,800,
Zs = 201,926,OOO. By the formulae (2.51, n + ( H > = 3; th us, by Theorem roots of f(x) satisfy the inequality g(x) > 0. Check: A, = -4.0230
(g
< 0),
h, = 0.9415
&3 = - 0.6680 A consequence case g(x)
f(x)
of Theorem
2.2, exactly two real
(g > 0),
( g > 0).
2.2 is the following
result obtained
for the
= t - x:
THEOREM 2.3 (Ioachimstal [4, 181). lying in the interval (a, b) equals V(1,A,(a),&,(a),...,
&.(a>)
Th e number of distinct real roots of
- V(l,A,(b),A,(b),...,A,(b)),
where
SO
A,(t)
= det [sjikt
- s~+~+~]~~J~ = s::; 1
provided series
that there are no two consecutive
Sl
**a
sP
s2 **Sp-cl . . . . . . . . . . . . . , sp *a- SPp-1 t
*‘*
tp
zeros at t = a or t = b in the
l,A,(t),A,(t),...,A,(t).
(2.6)
Thus, the series (2.6) plays the role of the Sturm series in the problem separation
It is possible to obtain the sequence selection of the functions g(x) and Ye,
REMARK. suitable
of
of the real roots of f(z). of Sturm functions by a (k = 0, . . . , n - 1) in
SEPARATION OF SOLUTIONS OF ALGEBRAIC SYSTEMS (2.4) [18]. See also [8] for a systematic
exploration
59
and historical
review of the
relevant theory. The following
result may be useful for calculations
in Theorems
2.1 and
2.2.
THEOREM 2.4. (a) (b)
S, = det S =.@f)/~~“-~. One has
(2.7) Proof.
We shall prove only (b), since (a) follows from it:
Zn = det
=&w.*
g(A,)det
I
.. . ...
1
1 A . . y :.: . ‘*:,-I
.^t.. A;-’
...
A;-l . . . A",-1 .. ..... . . . A",-1
I *
Ii m
3.
HERMITE’S
METHOD
IN THE
Consider now real polynomials x and y, and the system fl(X>Y)
f&x,
=o,
CASE
OF TWO
y), fi in
two
variables
(3.1)
If (LY, j3) is a solution of (3.1) with real LY, p, then it is called red. If a component of solution, say /3, is not real, then ((Y, p> is also a solution of (3.1); these solutions are called complex conjugate.
ALEXEI YU. UTESHEV AND SERGEI G. SHULYAK
60
Let deg fJ = nj (j = 1,2), and let the coefficients of ~“1 and ynj in the expansion of fj in powers of X, y differ from zero. (If this is not the case, it is possible to obtain a new system from (3.1) with the claimed property by means of a suitable linear substitution [2I, p. 131.) Ordering fi,fi in powers of y, we construct the resultant of these polynomials fix) =S,,(fr, fa) (elimination Let ax) solutions. (Bezout
of y ). Similarly, we construct y( y ) = SX(f,, fi), eliminating x. f 0, Y(y) f 0. Th en the system (3.1) has a finite number of This number generally equals N = nIn2 ( = degz = deg y)
theorem)
ASSUMPTION1.
[21, 231. Henceforth
we shall consider
degZ=
y(y);
the case
deg $Y = n1n2.
If ( (Y, p ) is a solution of (3.1) then (Y is a root of SF? x) and p is a root of and vice versa: to every root of Z(r) corresponds at least one of the
roots of y(y)
such that this pair is a solution of (3.1).
If (a + ib, fi) (b z 0) is a solution of the system (3.1) with real p, then /3 is a multiple root of ‘JY( y ). Let at least one of polynomials z 0 or 9_(y) # 0. Let, roots, i.e., g(z)
k?(x) or Y(y) for definiteness,
ASSUMPTION2. multiple
have no this be
true of y( y ). Under‘ this assumption, all the solutions of (3.1) are distinct, and the number 29 of complex conjugate solutions equals the number of complex conjugate roots of y( y) and may be determined by Theorem 2.1. ASSUMPTION3.
Let the polynomial
y) satisfy the condition
(j = l,...,N).
g( aj> Pj) + O Later on we shall give a method
g(x,
of verification
PROBLEM 1. Given the system (3.1) satisijing the condition g(x, y) > 0.
of this condition.
find the number
of its real solutions
SEPARATION
OF SOLUTIONS
OF ALGEBRAIC
In order to solve this problem real variables
let us construct
x,,, . . . , xN_ 1 [analogue
=
61
SYSTEMS the quadratic
form H(l) in
of (2.211:
x*#‘)x,
(3.2)
where
e%e)= [ h’:!,],N,,:,, x* = (XO,...,XN_l)? (k = 0,...,2N
- 2).
(3.3)
j=l
THEOREM 3.1 (Brioschi [41X Th e number of real solutions of (3.1) satisfying g(x, y) > 0 (or g(x, y> < 0) equals n+(H(‘)) - q (or n_(H(‘)) -
4). The proof is similar to that of Theorem Analogue of Theorem 2.3: THEOREM 3.2 (Hermite,
Brioschi
2.2.
[4, 131).
The number of real solutions
of (3.1) lying inside the rectangle
x1
-
12+(x2>
Yl)l.
(3.5)
Here n+(s, t> denotes the positive index of the form (3.2) obtained for the case g(x, y> = (x - sX y - t>. For the choice of g In the notation
of (3.2),
(3.8)
SEPARATION (b)
OF SOLUTIONS
In the notation
OF ALGEBRAIC
of Theorem
63
SYSTEMS
3.3,
(3.9)
Here A,, is the coefficient
of the y
N in the expansion
The proof is similar to that of Theorem
4.
THE VALUE OF A SYMMETRIC OF AN ALGEBRAIC SYSTEM DEFINITION. The function
F(r,,
, j,:
1 < j,
y).
2.4.
POLYNOMIAL
yl;.
. ;x,,
symmetric function ofn pairs of variables (x,, yl), unchanged when any of the pairs are interchanged:
for distinct j,,
of y(
OF
SOLUTIONS
yn) (a32n -+ C’) . , (x,,
is called a
yn) if its value is
< n (k = 1, . . , n).
Let us return number of its
to the system (3.1) supposing, as in Section 3, that the N = n1n2. Consider its solutions solutions equals of fi and fa. The (ai> P1)>. . . >(%J> PN) as functions of the coefficients analogue of Theorem 1.1 is THEOREM4.1 (Schlafli
[22]).
Th e value
of a symmetric
N pairs of variables of the solutions ((Y,, p,>, . . , ( aN, &) (3.1) is a rational function of the coefficients of fi and f2. We shall not discuss here arbitrary symmetric polynomial
polynomial
F of
of the system
the problem of calculating the value of an on solutions of (3.1) (see Theorem 4.3). For
the needs of the previous section it is sufficient sums
s,, = ; a$;
to determine
(k,l=O,l,...
).
the values of the
(4.1)
j=l
In this case we shall consider
two approaches
due to Poisson and Jacobi.
ALEXEI
64
YU. UTESHEV
4.1. Poisson’s Method 1231 Let us introduce two new variables Replacing
x in (3.1) by t -
Considering
AND SERGEI G. SHULYAK
0 and t by the formula
0 y, we obtain the algebraic
t = 0 y + x.
system
f1(t - By, y) = 0,
f2(t - ey, y) = 0.
here t and y are variables
and 8 as a parameter,
(4.2) and construct-
ing the resultant
qCfi(t - oy, y),f& we eliminate
vt> O>>
y from (4.2):
q(t,t?) Here Wt,O) degrj ,fi(t, y)> =tit>, rj(0) is a polynomial in 8, and In p a rt’ICU1ar, t-,, = const and, by Assumption 1,
=S%?,,(f,(t, =O,...,N).
?-a # 0. The component tj of a solution (t,(e), ~~(0)) of the system (4.2) is a root of 1Ir(t, 6). Let us construct Newton sums for Wt, 0) using the formulae (2.3):
Sk(B) = Et;
=q-()(e),rl(e),...
)
'N@>>
j=l
where 9 is a rational function of rO, . . . , rN and thus a polynomial in 8 (since t-,, = const). Let us expand .F in powers of 8 and remember that tj = aj +
Opj, with (aj,
pj> being a solution of (3.1):
5 (cYj + epj)”=_fi& +f&e
+f&12 + ***.
j=l
Since this is an identity in 8, comparing
coefficients
(4.4)
gives
(4.5)
SEPARATION
OF SOLUTIONS
The latter formula permits
one to calculate
Calculate
EXAMPLE 4.1.
OF ALGEBRAIC
65
SYSTEMS
every sum (4.1).
sums (4.1) for the system 7xy + y2 + 13x - 2y - 3 =
0,
fi
= 4x2 -
fi
= 9x2 - 14xy + y2 + 28x - 4y - 5 = 0.
Solution:
482 + 7% f 1
0
-(80+7)t-130-2 *(t,
L
*
- (lse + 14)t - 288 - 4
\
e) = 4t2 + 13t - 3
* L
*
9t” + 2& - 5
\
* L
0
= -24(t4
0
9e2 + 148 + i
L
*
+ r,t3 + r,t2 + r,t + r,,),
where rl=
-(50+1),
?-,=582-e-4, r, = -6e4
rg = 503 + 502 + 88 + 4, + 5e3 + 1202 + 48;
hence s,(0) = -rl and S,,, = B, = 5, S,,, = 1; s,(0) = rf - 2r, S 2.0 = 9, S,,, =r,=6, So,, = B, = 15; s,(0) = -rf + 3r,r, - 3r, S =r =20 s = B = 35 etc. 3zoC=hel&!b!&~~ ?ihe s&em: il, “if, (2,3”,, (0, L l), ( - 2,l). REMARK.
Using
the preceding
the calculation of B, and rj( 0)/r, in powers of 0:
'j(O)
r0
rk (from
=Aj,o + OAj-l,l
considerations,
Theorem
3.3).
and and
E. Hess [14] simplified Order the polynomial
+ 02Aj_2,2 + *** +B’Ao,j.
66
ALEXEI YU. UTESHEV AND SERGEI G. SHULYAK
Then
N -(
B, =
A,,,Bk_l
i -(A,,P-1 [Newton
+ A,,,BI,_,
+ *+*+A,,,_,B,
+Ao,zBk-z
+ *.* +A,,.Bk-iv)
sums of the resultant
+ kA,,k)
-(
A,,,rk-l
-(A,,,&-,
if
k>N
if
k=O,
+ A0,21Yk-2 + v-0 +A,,kl?O + A,,,)
if
0 < k < N,
+ Ao,zL
if
k>N.
+ ***+A~,J’-N)
4.2. Jacobi’s Method [151 K. G. I. Jacobi proposed another the value of the sum (4.1). According
k=O,
0 < k < N,
y(y)],
-A10 r, =
if
if
to Theorem
1.4,
there
(and very elegant) exist
uniquely
defined
y), -&X, y), .53X, y), and &‘(x, y) such that deg, nl, deg, 9 < n2, deg, & < nl, and
4(X,
method
L
for finding polynomials
< n2, degYH
=
+ 12) + (38x2 - 148x + 144), - 12) + (-2x2+4x),
1 -3
0 1
ya-y
-3
x =x(6y @(x, y) = x(6y y(x,y)
-1 7-6~
0 -1
y'+lly-12
7-6~
0
1
0
- 4) + (38~" - 56y + 16), - 4) + (2y2
-
By),
= 80[xy(3x2 - 3y2 +xy)+
(6y3 -2x3
+2(13xy +4x2 A_flTf2) 1 -=%(xX) 1 -=Y(Y)
- 16x2y +xy2)
- 3y2)-
4(3y +2x)],
= 2[2(3ya - 3 x2 + 2xy) + (19x - 13y) - 131, 1
)
,+L+$+E+...
40x4 (
X
i
1
1+2+L+10+... 4oy4 i Y Y2
Y3
, I
and Theorem 4.2 gives S 1,o=ro=7>
S,,1=B,=2,
s,,=17,
s,,=r,=2,
Check: Solutions of the system:(0, l), (3,0), (2,2),(2,-1).
etc.
SEPARATION 4.3.
OF SOLUTIONS
Verijcation
THEOREM
OF ALGEBRAIC
of Assumption
3
4.3 [22].
Every symmetric polynomial F(x,, as a polynomial using the expressions
may be represented
ik, =
69
SYSTEMS
2 xi”y;
(k,l=0,1,2
)...
yl;. . . ;x,,
y,,)
).
j=l
REMARK.
In connection
with the concrete
tion we refer to [28, p. 4711 and the references This theorem Section
method for such a representacited therein.
allows one to verify the fulfillment
3. Indeed,
it is equivalent
of Assumption
3 from
to
(4.8) and the left-hand side in (4.8) is a symmetric polynomial with respect to the solutions of (3.1). By using Theorems 4.2 and 4.3, it may be expressed rationally in terms of the coefficients of fi and fa. On the other hand, such a preliminary verification is not necessary, since, under Assumption 2, the condition
(4.8) is equivalent
to det&i)
# 0
(4.9)
(see Theorem 3.4). Conclusion: Assumption 3 follows from the Assumption 2 and the condition (4.9): we were in need of it only for convenience in presentation.
5.
EXAMPLES
USING
THEOREMS
3.1 AND 3.3
EXAMPLE 5.1. How many real solutions of the system of the Example 4.2 satisfy the inequality g(r,y)=x"+y'-6x>O? Solution: First, let us determine the number 29 of complex conjugate solutions of the system. According to Theorem 2.1, all the roots of the resultant y( y> are real and distinct. Thus, all the solutions of the system are real and distinct, i.e., 4 = 0.
70
ALEXEI YU. UTESHEV AND SERGEI G. SHULYAK
The elements
of the matrix A?” of the quadratic form (3.2) are calculated
by the formula
(k = 0,...,6).
hjl) = Sz,k + S, k+2 - 6S, k The
Sjk were calculated
in Example
4.2. We obtain
the leading principal P(l,
- -70 120 I . - 262
-70 -24 - 120
7;; -70 Calculating
-24
-22
0
minors and applying Theorem
-19,418,
-7668,
-36288)
3.1,
= 1.
Thus, exactly one real solution of the system satisfies the given inequality.
EXAMPLE 5.2.
Let
us separate
the
real
solutions
Example 4.1 by means of Theorem 3.3. Solution: ‘jY(y) = -24(y” - 5y3 + 5y” + 5y - 6). all the solutions of the system are real and distinct. Calculate the determinants (3.7): &1”( s, t)
= 4st - 5s - t + 6,
@‘(s,
= 35s2t2
t)
- 35st’
-
65s’t
-
50s2 -
of the
system
By Theorem
of 2.1,
16t2
+ 85st + 90s + 60t - 40, ti$‘(s,
t) = 440s3t3 -
- 400s2t3
104s3t
+ 1680s2t2
- 96t2 - 2496s’ By (3.9),
- 1440s3t2
- 824st3
+ 16t3 + 1680~~ - 512s’t
+ 1208st + 176t - 1392s
-
+ 2160st” 96,
71
SEPARATION OFSOLUTIONSOFALGEBRAICSYSTEMS where ax)
= -24(x4
[or of yk in Y(y)1
4x2 + 4x).
- x3 -
coincides
from Example 4.1.) By Theorem 3.3, in the rectangle ;{n+(3,2.5)
+ rt+( -1,
coefficient
-28(c'j' ‘j)
(by
Lemma
1)
SEPARATION
OF SOLUTIONS
OF ALGEBRAIC
83
SYSTEMS
Proof of Theorem A.2 in the case g(Z) = 0, g(_$V) f 0. Suppose now that Hx) has a multiple root cxr = (Y% of multiplicity 2 with the other roots being distinct. ff3,...,aN
LEMMA 2. In the equalities
(A.2)
that deg,
1, deg,
chosen
such
76
N -
polynomials 7~
N -
&,N,
9,
d
may
be
1, deg Z/G 2N - n1 -
n2.
By Theorem deg,JY
Proof.
anddeg & are divisible by x - crl.
n
84
ALEXJZI YU. UTESHEV AND SERGEI G. SHULYAK
Denote by Y’i(x, y), Ji(r, y),Jy,(x, y>,tz;(xl the quotients obtained dividing respectively Y-(x, y), &x, y>,&x, yl,Z%xl by x - (pi.
on
LEMMA 4.
q(aj,fjk)
=
G("j)y'(4)
In place of (A.2) consider
Proof.
The
proof of the theorem
j#k,
q
j =k,
the equations
and apply to them the same arguments To prove now Theorem
if
as in the proof of Lemma
A.2 let us transform
is completed
previous case.
3.
W
the fraction (A.11 to
in exactly the same way as in the W
The above reasoning may be extended without particular difficulty to the case where ax) possesses several multiple roots of distinct multiplicities.
To prove Theorem COROLLARY. = xkyl in Theorem A.2. Let us note here an interesting Corollary
2 to Theorem
THEOREM
A.3.
4.2, it is now suflcient modification
of Theorem
to take Z(x,
y>
A.2 (analogue
of
A.l):
The Jacobi equality
holds for any polynomial
%Cx, y > of degree not exceeding n, + n, - 3.
SEPARATION
deg
OF SOLUTIONS
OF ALGEBRAIC
85
SYSTEMS
The proof is the same as that of Theorem A.2 and uses the estimate Lemma 2 (instead of the fraction (A.l), Z(x, ‘Y from
V(x,
y)/[~?Tx)yC
y)] should be considered).
A. 3. Jacobi’s Theorem in R3 The notation in this section coincides Consider
of
y)
the system (7.1),
with that introduced
and denote bytix),
y(y),
Z(z)
in Section
7.
the resultants
of
jr, fi, f3 obtained by elimination of y and z, of x and z, and of x and y respectively (these resultants are sometimes called the eliminants 1211).
ASSUMPTION A.l. this number
Let
ASSUMPTION A.2.
deg y = deg Z = n1n2n3
Let S@iY) # 0, 9(%/j
THEOREMA.4 [21]. 1,2,3) such that
(we denote
# 0, .&S(Z) # 0.
There exist real polynomials
M,,f,
+ M,,f,
+
M,3f3
=
p(
x>
M,J-1
+
+
M33f3
=
2(
z>.
Let Y’X-(x, y, z) = det AX>
degZ?=
by N).
M32f2
[ Mjk
1;. k = 1
Mj,(x,
y, z> (j, k =
1
and denote the Jacobian
off,
, fi, f3 by
y, z). THEOREM A.5. For any polyndmial in the expans, ion of the fraction
Z!( x, y, z>
the
coeficient
of
X-5) -lz-l
z’( x> y> z)T_( x, y, Z)Z(X>
y, 2)
~C”)HYP-‘(Z) in powers of x-l,
y-l,
and 2-l
equals Cl~zl %(cu,, /Sj, yj>.
Several papers have been devoted to the problem of estimating the degrees of the polynomials Mjk in the equalities (A.4) (B&out equalities); we refer to [3], where also other references can be found. B&out proposed the
ALEXEI
86
in the general
following estimate estimate,
YU. UTESHEV
case:
AND SERGEI
deg Mj,,. < N -
G. SHULYAK
nk [21].
Using this
one has:
THEOREM A.6 [16, 211.
holo!s for any polynomial 4.
The jacobi
equality
‘?Y(x, y, z) of degree
The proofs of Theorems and A.3.
not exceeding
n, + n2 + n3 -
A.5 and A.6 are similar to those of Theorems
A.2
REMARK. Kronecker proposed in [19] a proof of Theorem A.6 that was based on another idea, and showed the validity of Assumption A.l.
We would like to thank Elisabeth Kalinina for helpful suggestions, comments, and assistance in typing. We are also grateful to the referee for useful comments which helped to improve the presentation. Most of the examples were computed with the help of a program written by our student Ivan Tkachenko. His sudden death was a hard blow to us. This paper
is rledicated
to his memory.
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A criterion
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Vahlen,
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YU. UTESHEV
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zeros
of ill-conditioned
(1959).
Received 11 ]uly 1991; final manuscript accepted 1 October 1991
polynomials,
