Nonlinear Algebraic Systems with Three Unknown Variables

International Journal of Research in Engineering and Science (IJRES)
ISSN (Online): 2320-9364, ISSN (Print): 2320-9356
www.ijres.org Volume 2 Issue 6 ǁ June. 2014 ǁ PP.54-59
www.ijres.org 54 | Page
Nonlinear Algebraic Systems with Three Unknown Variables
R. M. Dzhabarzadeh
Institute Mathematics and Mechanics of NAN Azerbaijan
ABSTRACT: Work is devoted to the study of nonlinear algebraic systems with three unknowns and is a
continuation of the author's research in this area. Previously, for nonlinear algebraic of system of equations
was built analog determinant of Kramer and it is given a necessary and sufficient condition for the existence of
solutions of nonlinear algebraic systems with a complex dependence on the variables. For two-parameter
systems in the abstract case, it is obtained the results to determine the conditions for finding the number of
solutions.
In this paper describes the method of determining the conditions on the coefficients of algebraic systems to
establish the number of solutions of the algebraic system of equations, when an unknowns part in the system as
polynomials in any finite degree. This method can be applied to other algebraic systems with more than
three unknown variables.
Keywords: eigen and associated vectors, finite dimensional space, multiparameter system of operators,
nonlinear algebraic system of equations, Resultant-operator for two pencils
I. Introduction
The technique, developed by the author allowed constructing the spectral theory of non-selfadjoint
multiparameter system of operators, linear and nonlinear depending on parameter in Hilbert spaces. Considered
in this paper the nonlinear algebraic system of equations is a special case multiparameter system when operators
are numbers in Hilbert space .R Naturally, that in case of the algebraic systems the got results can be deeper,
and methodology less difficult. The proof uses the notion of resultant of two polynomial pencils
Let be the nonlinear algebraic system with three unknown variables.
0......),,( 11
1
111
1
1 ,10   zayayaxaxaazyxa nm
n
nmm
m
m
i
0.........),,( 2
22222
2
222
2
2 1110  
r
rnmnm
n
nmm
m
m zbzbybybxbxbbzyxb i
(1)
0.........),,( 3
33333
3
333
3
3 1110  
r
rnmnm
n
nmm
m
m zczcycycxcxcczyxc
In [1,2] for nonlinear algebraic systems with any finite number of unknown variables is given way of
constructing an analogue of the determinant of Cramer, in particular, proved: if the continuant of this Cramer
i
s
determinant is not equal to zero, then the algebraic system has a solution.
In addition [3,4,5], for systems of nonlinear algebraic equations with polynomial dependence on two variables
under certain conditions it is defined the number of solutions.
Give some of well-known definitions and concepts necessary for understanding the subsequent presentation.
For two polynomials is introduced the concept of Resultant as follows. Let
1
1 0
1
1 0
( ), ( ); ( ) ... , 0;
( ) ... , 0;
n n
n n n
m m
m m m
f x g x f x a x a x a a
g x b x b x b b




    
    
be two polynomials. Resultant of these polynomials is the operator acting in the space
n m
R 
or
n m
C 
(probably, in some expansion of a field).

Nonlinear algebraic systems with three unknown variables
Re ( , )s f g 
1 1 0
2 1 0
1 0
1 3 2 1 0
4 3 2 1 0
1 1 0
... 0 ... ... 0 0
0 ... ... ... 0 0
. . ... . . . ... ... . .
. . ... . . . ... ... . .
0 0 ... 0 0 ... ...
... ... 0 0
0 ... ... 0 0
. ... . . . . ... . .
. ... . . . . ... . .
0 0 0 0 ...
n n
n
n
m m
m
m m
a a a a
a a a a
a a a
b b b b b b
b b b b b b
b b b b



 
 
 
 











 











In a matrix ),(Re gfs the number of lines with coefficients ia to equally leading degree of unknown x of
a polynomial )(xg , that is m, the number of lines of a matrices with numbers ib coincides with the leading
degree of unknown x of a polynomial )(xf , that is n. Continuant of this Resultant is a polynomial from
coefficients and equal to zero then and only then when polynomials also have a common roots (probably, in
some expansion of a field).
The continuant of Resultant can be discovered as a continuant of Sylvester Matrix. Sylvester matrix,
allowing calculating continuant of Resultant of two polynomials, is opened by English mathematician James
Sylvester.
For a separable polynomial (in particular, for fields of performance zero) this continuant is equal to product of
values of one of polynomials on radicals of another (as before, product undertakes in view of a multiplicity of
radicals):
The study of such nonlinear algebraic system (1) of equations is spent with help following result from [3]:
let the n bundles depending on the same parameter 
 niBBBB ik
k
iii i
i
,...,2,1,...)( ,,1,0  
)(i
B - operational bundles acting in a finite dimensional Hilbert space i
H correspondingly. Suppose that
n
kkk  ...21
. In the space
21 kk
H 
(the direct sum of 21
kk  tensor product
n
HHH  ...1
of spaces n
HHH ,...,, 21
) are introduced the operators
)1,...,1(  niRi
with the help of operational matrices (3.12)
Let )(i
B be the operational bundles acting in a finite dimensional Hilbert space i
H , correspondingly.
Without loss ofcopies with












































ikii
ikii
ikii
k
kk
k
i
i
i
i
BBB
BBB
BBB
BBB
BBBB
BBB
R
,,1,0
,,1,0
,,1,0
1,1,11,0
1,1.11,11,0
1,1,11,0
1
00
00
0.0
00
00
0
1
11
1
, .,...,2 ni 

The number of lines with operators 11 ,...,1,0, ksBs  in the matrix 1iR is equal to 2k and the number of
lines with operators isi ksB ,...,1,0,  is equal to .1k We designate  )( ip
B the set of eigen values
of an operator )(i
B .From [3] we have the result:
Theorem 1.   

))((
1
ip
n
i
B then and only then when   }).{(, 1
1
1
 


ki
n
i
KerBKerR
In particular, in [5] the two-parameter system operators
0)......(),( 11`
1
11
1
110
 
xAAAAAxA nm
n
mm
m

0)......(),( 22`
2
22
2
110
 
yBBBBByB nm
n
mm
m
 (2)
is studied in tensor product 21
HH  of finite-dimensional spaces 1
H and 2
H .
Dimension of space H is the product of dimensions of spaces 1
H and 2
H . In (2) linear operators
),...,1,0( 11
nmiAi
 act in finite-dimensional space 1
H ; and linear operators
),...,1,0( 22
nmiBi
 act in finite-dimensional space 2
H .
If 2121
HHff  and 2121
HHgg  then inner product of these elements in space
21
HH  is defined by means of the formulae
  2121
),(,],[ 21112121 HHHH
gggfggff  
This definition is spread to other elements of tensor product spaces on linearity.
Let's reduce a series of known positions concerning the spectral theory of multiparameter system.
Definition1. [7]
2
),( C is an eigen value of the system (2) if there are nonzero elements
1 2,x H y H  such that (2) is fulfilled. A tensor yxz  is named an eigenvector of system (2).
Definition2. [7]. An operator 2
EAA kk

(accordingly, kk
BEB 
1
) where 1
E (accordingly,
2
E ) is identical operators in 1
H (accordingly, 2
H ), names operator, induced in 21
HHH  by
operators k
A (accordingly, k
B ).
Definition3.[8] A tensor
21 ,mm
z name  21
,mm - the associated vector to an eigenvector yxz 0,0
if following conditions are satisfied


21
21
),(
!!
1
0
21
rr
rr
kr
A
rrii 
 

 02211 ,
 rkrk
z


21
21
),(
!!
1
,0
21
rr
rr
kr
A
rri 
 

 02211 ,
 rkrk
z
.2,1;2,1;  simk ss
 21
,kk is arrangement from set of the whole nonnegative numbers on 2 with possible recurring and zero.
By means of the approach stated in [5,6], we can establish completeness, multiple completeness of
system of eigen and associated vectors, a possibility of multiple expansions on system of eigen and associated
vectors of multiparameter system (2).

Theorem2. Let operators ),...,1,0( 11
nmiAi
 also ),...,1,0( 22
nmiBi
 act in finite-
dimensional spaces 1
H and 2
H , accordingly, and one of three following conditions is fulfilled:
a) 211221
),max( nmnmnm  ,       221
, nmm
KerBKerA ;
221
, nmm
BA 
are self -
conjugate operators everyone in their space
b) 121221
),max( nmnmnm  ,       112
, nmm
KerAKerB
111
, mnm BA  - self-conjugate operators, acting in their spaces
c) 1221
nmnm  , ))1(( 1
2
2
11
211
11
2
1
n
m
n
nm
nnn
nm
n
m BABAKer   ,
221121
,,, nmnmmm
BABA 
are
the self-conjugate operators, acting everyone in finite-dimensional space, correspondingly.
Then the ),max( 1221
nmnm -fold base on system of eigen and associated vectors of (2) takes place.
At the study of the system (1) we will be used essentially the results from [5,6].:
In (1) we fix variables yx, . Let 00 , yyxx  be. Then the system (1) contains three polynomials
depending on one variable .z Using the result of the theorem2, we build operators 1R и 2R .They are the
Resultant- operators of polynomials ),,( 00 zyxa and ),,( 00 zyxb , and also polynomials
),,( 00 zyxa and ),,( 00 zyxc , correspondingly.
Introduce the notations:
1
111
1
1
......),(~
,10
n
nmm
m
m yayaxaxaayxa i

2
222
2
2
......),(
~
110
n
nmm
m
m ybybxbxbbyxb i
  (3)
3
333
3
3
......),(~
110
n
nmm
m
m ycycxcxccyxc  
For the polynomials ),,( 00 zyxa and ),,( 00 zyxb Resultant 1R has a form





















2222222
11
11
11
...),(
~
),(~......
.......
0...),(~0
.....),(~
)},,(),,,({
2100
100
100
100
00001
rnmnmnm
nm
nm
nm
bbbyxb
ayxa
ayxa
ayxa
zyxbzyxaRR
(4)
In the matrices of the operator 1R the number of lines with 100 11
),,(~
nmayxa equal to 2r and the
number of lines with 22222
,...,),,(
~
100 rnmnm bbyxb  is equal to1.
By analogy operator 2R in the space
13 r
R is determined with the help of matrices





















3333333
11
11
11
...),(~
),(~......
.......
0...),(~0
.....),(~
)},,(),,,({
2100
1000
1000
1000
00002
rnmnmnm
nm
nm
nm
cccyxc
ayxa
ayxa
ayxa
zyxczyxaRR (5)

Continuants of Resultants 1R and 2R are equal to zero if and only if matrices of operators 1R and 2R have
nonzero kernels.
So at the expansion of continuants of Resultants 1R and 2R we obtain very bulky forms, in obtained equationswe
will operate with the leading degrees of variables in 1R and 2R . Further we will work only with the members
having the leading degrees. We can write the expansions of continuants of Resultants 1R and 2R in the forms:
2
22
2
11
2
2
2
11
21
222
2
11
21
222
2
010100 ............ r
nm
r
nm
m
m
r
nm
rn
rnm
r
nm
rm
rnm
r
m ybaxbaybaxba   (6)
3
33
3
11
3
3
3
11
31
333
3
11
31
333
3
010100 ............ r
nm
r
nm
m
m
r
nm
rn
rnm
r
nm
rm
rnm
r
m ycaxcaycaxca   (7)
(6) is the expansion of the continuant of the Resultant of the polynomials ),,( 00 zyxa and
),,( 00 zyxb from (1),when 0xx  and 0yy  .
Expression (7) is the expansion of the continuant of the Resultant –operator of polynomials ),,( 00 zyxa
and ).,,( 00 zyxc
If the couple of numbles( 1x , 1y ) turns expression (6) to zero, then from the definition of Resultant follows that
at these meanings of variables 1xx  and 1yy  0),,( 1 zyxa and 0),,( 1 zyxb .Thus the
first and second equations from (1) have the common decision ).,( 111 yxz
If the couple ( 2x , 2y ) turns (7) to zero, then from the definition of Resultant follows that at these meanings of
variables 2xx  and 2yy  we have 0),,( 2 zyxa and 0),,( 2 zyxc .Thus the first and
third equations from (1) have the common decision ).,( 221 yxz
Now we consider the system
: 0............ 2
22
2
11
2
2
2
11
21
222
2
11
21
222
2
11  
n
nm
r
nm
m
m
r
nm
rn
rnm
r
nm
rm
rnm
r
m ybaxbaybaxba
0............ 3
33
3
11
3
3
3
11
31
333
3
11
31
333
3
11  
n
nm
r
nm
m
m
r
nm
rn
rnm
r
nm
rm
rnm
r
m ycaxcaycaxca , (8)
For any solution ),( yx of the system (8) the common solution of 0),,( 1 zyxa и
0),,( 1 zyxb denote ),(1 yxz and the common solution of 0),,( 2 zyxa and
0),,( 2 zyxc denote ),(2 yxz . So variable z enter the equation 0),,( zyxa linearly we
have ),(1 yxz = ),(2 yxz = ),( yxz . Consequently, any solution ),( yx of (8) corresponds to only one
solution ),,( zyx of the system (1), i.e. the number of solutions of (1) equals the number of solutions of a
nonlinear algebraic system (8) with two unknowns.
System (8) is a special case of (2), studied by the author in [5]. In (8) the real numbers
),...,2,1;,...,2,1;1,...,2,1(,, 33322211 rnmkrnmjnmicba kji  are operators in
the space R , variables yx, act as parameters, the eigenvectors of the system (8) is constant
Let 331221331;221 ,,, nrnnrnmrmmrm  be then the system (8) is written in the form
0...... 21
222
2
11
21
222
2
1
 
rn
rnm
r
nm
rm
rnm
r
m ybaxba
0...... 31
333
3
11
31
333
3
1
 
rn
rnm
r
nm
rm
rnm
r
m ycaxca (9)
Theorem3. Suppose the following conditions satisfy:

1) 331221331;221 ,,, nrnnrnmrmmrm  (10)
2) 0)()()1()()( 31
333
3
1
31
222
2
11
21
2
121
33311
331
222
3
1
 
rn
rnm
r
m
rn
rnm
r
nm
rrnrn
rnmnm
rrn
rnm
r
m cabacaba
Then the system (1) has 2111 rrnm solutions.
Using this approach, we can consider different cases ( for example: ,, 331;221 mrmmrm 
331221 , nrnnrn  and so on)
II. Conclusion
This paper presents a method of determining the number of solutions of nonlinear algebraic equations.
Partiсularly, it is considered the nonlinear algebraic system of equations with three unknowns.
Since the proof of Theorem 3 we are faced with an unwieldy calculations we restrict ourselves to a
single nonlinear algebraic system.
. If necessary, this method will allow to reader to determine the number of solutions of the interested algebraic
system.
References
Theses:
[1] Dzhabarzadeh R.M. Materials of International conferences:”The theory of functions and problems of a Fourier analysis”,
devoted to 100 years anniversary of acad. I.I.Ibragimov, pp.92-94, Baku-2012. Not linear algebraic systems and analogs of
continuants of Cramer.
Chapters in Books:
[2] Dzhabarzadeh R.M. Multiparameter spectral theory. Lambert Academic Publishing, 2012, p. 184 (on Russian) ,pp.78-81
Theses:
[3] Dzhabarzadeh R.M. Baku International Topology conference, 3-9 оct., 1987, Теz. 2, Baku, 1987, p.93 On existence of
common eigen value of some operator-bundles, that depends polynomial on parameter.
Proceedings Papers:
[4] Dzhabarzadeh R.M. , Salmanova G.H. American Journal of Mathematics and Mathematical Sciences.-2012, vol.1, №1, pp.32-
37, 2013 Multiparameter system of operators, not linearly depending on parameters.
Proceedings Papers:
[5] Dzhabarzadeh R.M. Pure and Applied Mathematics Journal, vol. 2, No. 1, pp. 32-37, 2013 About solutions of nonlinear algebraic
system with two variables.
Chapters in Books:
[6] Dzhabarzadeh R.M. Nonlinear algebraic system of equations. Lambert Academic Publishing, 2013, p. 101 (on Russian), pp.80-
92.
Proceedings Papers:
[7]. Atkinson F.V. Bull.A mer.Math.Soc.1968, 74, 1-271. Multiparameter spectral theory.
Proceedings Papers:
[8] Dzhabarzadeh R.M. Transactions of NAS Azerbaijan, v. 3-4 1998, p.12-18 Spectral theory of two parameter system in finite-
dimensional spa’ce.

Nonlinear Algebraic Systems with Three Unknown Variables

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