LOWER AND UPPER BOUNDS OF NUMBER OF ZEROS OF RANDOM ALGEBRAIC POLYNOMIALS
1, *
Dr.
Department of Mathematics
Department of Basic Science and humanities, MITM,
ARTICLE INFO ABSTRACT
In this paper we have estimate
polynomials
assuming real values only and following the normal dis function
/(log A
satisfying the condition (1.2), have at most
Copyright © 2016, Mishra and Mansingh. This is an open access article distributed under the Creative use, distribution, and reproduction in any medium, provided the original work is properly cited.
INTRODUCTION
Consider the family of equations ( ,1)
n k nx f
where
a
k(
t
)
t
,
0
t
1
,
are dependent random variables assuming real values only and following the normal distribution with mean zero and joint density function.
( 1/2) '
(1.2) exp) 2
( /
2 / 1
M
M a s
When M-1 is the moment matrix with
i
1
,
In this paper we estimate the upper bound of the number of real roots of (1.1). We prove the following theorem.
Theorem:There exists an integer n0 and a set E of measure at most
belonging to E, the equations (1.1) satisfying the condition (1.2), have at most constants.
The transformation
x
x
1 makes the equation f density function.(t))
.a
(a0(t)....
and
0
'
)
(
n0 1
n
r
n
t
x
a
*Corresponding author:Dr. Mishra, P.K.,
Department of Mathematics and Humanities, CET, BBSR, Odisha, India
ISSN: 0975-833X
Article History:
Received 23rd January, 2016 Received in revised form 24th February, 2016 Accepted 18th March, 2016 Published online 26th April,2016
Key words:
Independent identically distributed random variables, Random algebraic polynomial, Random algebraic equation, Real roots.
Citation: Dr. Mishra, P.K. and Mansingh, A.K. 2016.
Journal of Current Research, 8, (04), 29299-29308.
RESEARCH ARTICLE
UPPER BOUNDS OF NUMBER OF ZEROS OF RANDOM ALGEBRAIC POLYNOMIALS
Dr. Mishra, P.K. and
2Mansingh, A.K.
Department of Mathematics and Humanities, CET, BBSR, Odisha,
of Basic Science and humanities, MITM, BBSR, Odisha,
ABSTRACT
In this paper we have estimate bounds of the number of level crossings of the random algebraic
polynomials
n k
k k nx a t x
f
0
0 ) ( ) 1 ,
( where ak(t)t,0t1
assuming real values only and following the normal distribution with mean zero and joint density function M as
M
' ) 2 / 1 ( exp ) 2
( /
2 / 1
. There exists an integer n
) log log log
/(logn0 n0 such that, for each n>n0 and all not belonging to E, the equations (1.1)
satisfying the condition (1.2), have at most (loglogn)2logn roots where α
is an open access article distributed under the Creative Commons Attribution License, which use, distribution, and reproduction in any medium, provided the original work is properly cited.
(1.1) 0 ) ( 0
n
k k t x
a
are dependent random variables assuming real values only and following the normal distribution with
n
j
i
j
i
ij
,
0
,
,
,
0
,
1
....
and d’ is the transpose of the column vector d.In this paper we estimate the upper bound of the number of real roots of (1.1). We prove the following theorem.
and a set E of measure at most A/(logn0logloglogn0)such that, for each n>n belonging to E, the equations (1.1) satisfying the condition (1.2), have at most (loglogn)2log
makes the equation fn(x,t)=0 transformed to
and
(a
n(t),
a
n-1(t),...a
(t))
Humanities, CET, BBSR, Odisha, India.
International Journal of Current Research
Vol. 8, Issue, 04, pp.29299-29308, April, 2016
INTERNATIONAL
2016. “Lower and upper bounds of number of zeros of random algebraic polynomials
UPPER BOUNDS OF NUMBER OF ZEROS OF RANDOM ALGEBRAIC POLYNOMIALS
BBSR, Odisha, India
BBSR, Odisha, India
bounds of the number of level crossings of the random algebraic
,
1 are dependent random variables
tribution with mean zero and joint density There exists an integer n0 and a set E of measure at most
and all not belonging to E, the equations (1.1)
roots where α and A are constants.
ribution License, which permits unrestricted
are dependent random variables assuming real values only and following the normal distribution with
and d’ is the transpose of the column vector d. In this paper we estimate the upper bound of the number of real roots of (1.1). We prove the following theorem.
such that, for each n>n0 and all not
n
log roots where α and A are
(t)
(t),...a
0 have the same jointINTERNATIONAL JOURNAL OF CURRENT RESEARCH
Therefore number of roots and the measure of the exceptional set in the set [
,
] are twice the corresponding value can be considered and now show that this upper bound is same as in [0, 1]. There are many known asymptotic estimates for the number of real zeros that an algebraic or trigonometric polynomial are expected to have when their coefficients are real random variables. The present paper considers the case where the coefficients are complex. The coefficients are assumed to be independent normally distributed with mean zero. A general formula for the case of any complex non stationary random process is also presented.Introduction
Some years ago Kac (1943) gave an asymptotic estimate for the expected number of real zeros of an algebraic polynomial where the coefficients are real independent normally distributed random variables. Later Ibragimov and Maslova (1971) obtained the same asymptotic estimate for a case which included the results due to Kac(1943, 1949), Littlewood and Offord (1939) and others. They considered the case when the coefficients belong to the domain of attraction of normal law. Recently there has been some interesting development of the subject, a general survey of which, together with references may be found in a book by Bharucha-Reid and Sambandham (1986). These generalizations consider different types of polynomials, see for example Dunnage (1966) or study the number of level crossings rather than axis crossings, see Farahmand (1986, 1990). However, they assume the real valued coefficients only. Dunnage (1968) considered a wide distribution for the complex-valued coefficients, however he only obtained an upper limit for the number of real zeros. Indeed, the limitation of this result, being only in the form of an upper bound, is justified. It is easy to see that for the case of complex coefficients there can be no analogue of the asymptotic formula for the expected number of real zeros. To illustrate this point we use the result due to Dunnage (1968). Suppose
nj0xjij fj(x) has areal root where fj(x) is in the form of xj or cosjθ and
jand
j,
j
0,1...n
are sequences of independent random variables. Thisimplies that the polynomials
n
j
0
jfj
(
x
)
and
nj
0
jfj
(
x
)
have a common root and the elimination of fj(x) leads to theequation
(
0,
1...
n,
0,
1...
.
n)
=0.Thus the number of roots in the range [
,
] and the measure of the exceptional set are each four times the corresponding estimates for the range [0,1]. Evans has considered the case when the random co-efficients are independent and normal. Our technique of proof is analogous to that of Evans.
2
.
We define the circles C0, Cc, Cm and C1 as follows. C0 with centre at z=0 andradius
,
2
1 Ce with centre at
0 0
2
n
log
log
4
3
n
z
And of radius
0 0
2 n log log 4 1
n
Cm with centre at z=Xm=1-2-m and of radius
...M
,m
m
m
for
2
)
1
(
( 1) 0 12
1
mm
Xm
r
Where
n C
and
n M n
and n m
n
n n
3 n 0
0
log log radius and 1 z at centre with 1
2 log
log log log log 1
2 log
log log log log
1 2
log
log log log log log
By Jensen’s theorem the number of zeros of a regular function
(
z
)
in a circle z0 and of radius r does not exceed)
/
log(
))
(
/
(
log
0r
R
z
M
n
Where M is the upper bound of
(
z
)
in a concentric circle of radius R. We use this theorem to find the number of zeros of fn(z,t) in each circle. Summing the number of zeros in each of the circle we get the upper bound of the number of zeros of fn(z, t) in the
circle.
.
3
To estimate the upper bound of the number of zeros of fn(z, t) in the circle C0, we shall use the fact that each ak(t) hasmarginal frequent function.
2
/
22
1
t
e
Now if
max
a
v
(
n
1
)
then
a
v
(
n
1
)
for at least one value ofv
n
,
so that
)
1)(2/
(n
)
1
(
)
1
(max
2 2
) 1 )( 2 / 1 ( 2 / 1 2
1 2 / 1/2
0
n n
t v n
v v
e
dt
e
n
a
P
n
a
P
………. (3.1)Since
fx
(
z
,
t
)
(
n
1
)
z
n
max
a
v
,
in the circle
)
1
(
a
max
)
1
(
1
t)
(x,
f
get
,
1
log log 2 2
v log
log 2 x
log log 2
n n n
n n
n
e
n
n
We
z
………. (3.2)
Outside a set of measure at most (2/)1/2e(1/2)(n1)2 by (3.1).
.
)
1
(
)
/
2
(
)
/
2
(
)
)
1
(
(
and
)
,
0
(
2 2
/ 1 )
1 (
0 / 2 2 / 1 2
0 0
2
n
du
e
n
a
P
a
t
f
n t n n
Hence outside a set of measure at most (2/ )1/2(n1)2
we have
2 0
(
)
(
1
)
)
,
0
(
t
a
t
n
f
n ………. (3.3)2
log
log
log
2
)
1
log(
4
2
log
)
)
1
(
log(
2loglog 40
n
n
n
e
N
n
Outside a set of measure at most
)
)
1
(
)
/
2
(
)
/
2
((
1/2e
(n1)1/2
1/2n
2Thus for all n>n0, we have
2
log
log
log
2
)
1
log(
4
0
n
n
N
Outside a set of measure at most
0 1
1 2
/ 1 ) 1 ( 2 / 1
0
)
)
1
(
[
)
/
2
(
n
C
n
e
n n
n
Where C is an absolute constant
To estimate the upper bound of the number of zeros of fn (x, t) in the circle C0 we proceed as follows. The probability that
)
1
(
)
/
2
(
)
/
2
(
)
1
(
2
log
log
4
3
)
(
1 1 2
/ 1 )
1 (
0 2 / 2 2
/ 1
2
0 0
0
1
n n
n n
v
n
v
n
du
e
is
n
n
n
t
a
………. (4.1)
Where
2
log
log
4
3
1
2
log
log
4
1
)
1
(
2
exp
1
)
1
(
2
log
log
4
3
2
log
log
4
3
)
1
(
2
0 0
0 0
2
0 0
0
0 0
0 1
n
n
n
n
n
n
n
n
n
nv
n n
v
n
n
………. (4.2)
If N0 denotes the number of zeros of fn (z, t) in the circle C0 then Jensen’s theorem (J), (3.2), (4.1) and (4.2) we have
2
log
log
log
2
)
1
log(
4
0n
n
N
2 / 1
2 -0
0 1
2 / 1 0 2
2 ) 1 (
)
(logn
-1
loglogn
)
1
(
1
2
0
n
n n
n
n
C
n
e
To obtain an upper estimate of the number of zeros of fn (x, t) in the circle Cm(m=m0,m1,….M) we need the following lemmas.
LEMMA 1:Let E be an arbitrary set. Then for complex numbers g, we have
m(E)
C
log
log
)
(
log
)
(
)
(
log
0
E
m
E
m
dt
g
t
a
E n
v
v v
Where
2
0 0
)
1
(
2
v
t v v
t
v
g
g
PROOF: Let gv=bv+icvg where bv and cv are real. Also let
2 / 1
0
2 / 1
0 0
2
/
)
(
:
2
/
)
(
:
)
(
:
A
c
t
a
t
H
and
A
b
t
a
t
G
A
g
t
a
t
F
v
v v v
v v v
v v
Now
8 / 1
2 /
2 8 8 / 1 8
/ 1 1
2 / 1 0
2 / 1
)
/
2
(
)
(
G
en
du
en
m
A A
n
n
And
8 / 1 2
2 / 1
)
(
H
en
m
A
Since 1/4
1/2
A
4
m(H)
m(G)
m(F)
and
e
d
H
G
F
. Following Evans [Lemma] we get the proof of the lemma.
LEMMA 2:If gv, v=0,1…..are real and if
g g t a t Gv
v v 0
) ( :
); / ( ) / 2 ( ) 1 ( ) 1 ( 4 / 1 2 0 0 2 2 2 0 0 2 2 n n v v v v n v v v v Q and g g g g
and if E is any set having no point in common with G then
E E m dt m(E) 1 log CQm(E) -log ) ( (t)g a log n 0 v v v
PROOF: Following Evans [Lemma2] we get the proof of the lemma.
Let Nm (r, t) denote the number of zeros of fx(z,t) in the circle with centre xm and radius r. By Jensen’s theorem
3 1 /0
(
,
)
)
,
(
log
2
1
)
,
(
m E m z E mdz
t
x
f
t
z
f
dr
r
t
r
N
m zz n m
n m
Therefore writing 1 1 m4
5
log
2
have
we
t),
,
Nm(1/2
for
)
(
mt
dz
t
x
f
t
z
f
t
E m z E mm zz n m
n m
)
,
(
)
,
(
log
4
5
log
2
)
(
1
and hence we get
2v
0
,
log
,
3
2
5
log
d
4
5
log
2
1
)
(
dt
E
t
m
x
n
f
dt
g
t
i
e
m
m
x
n
f
dt
t
m
By Lemmas 1 and 2, if E has no point in common with a set Gm of measure at most Qmt where
2 / 1 2 0 0 2 2 0 2 0 2 2 / 1 ) 1 ( ) 1 ( 2
v v m v v m v v m v v m m x x x x Q
We get)
(
1
1v
0
log
)
(
)
,
m
logV(x
4
5
log
2
)
(
)
(
E
m
E
m
m
CQ
d
E
m
dt
t
m
2
0 0
2
2
0
2 0
2 2
)
1
(
2
5
2
5
)
1
(
)
,
(
v v m v
v m
v
v i m m v
i m m
m
x
x
e
x
e
x
x
V
Since
x
m
1
in
V(x
m
,0),
the second term in both the numerator and denominator is constant. Therefore2
0
2
0
2
25
)
,
(
v v m
v v
i m m m
x
e
x
x
V
22
2 2
3 8 1
2 5 2
1 1 1
) 2 1 ( 1
F
m m
m
Hence we obtain
) ( 1 log ) ( )
(
E m E m m CQ E
dt t
m
If E has no point in common with a set Gm of measure at most
Qm
/
m
2 , taking
m
2Consider
E t
mt m
dt t M t
M
t
I ,
) ( log ) (
) (
Where
2 1 )
(
) (
m k
.
,
log
)
(
E
)
(
:
2
log
n
log
log
log
log
)
(
)
(
0 0
n
m
k K
t
mt m
k n k
dt
k
k
t
I
and
E
Then
k
t
M
E
t
E
Put
n
n
t
M
where
1 contains the terms for which
(
)
(
)
/
.
2k
E
m
E
m
k
First consider
1 . The function x log x -1
is increasing with x for
0
x
e
1 and thereforem(E)
1
log
)
(
1
log
)
(
2
)
m(E
1
log
)
(
22 k
E
m
k
k
k
E
m
E
m
k
If
m
e
(
).
e
1
k
)
m(E
2e 2
k
or
m
E
Now
m(E) 1 log ) ( log )
(
) m(E
1 )log m(E Q max log
, ) ( log
1 log
) (
2 2
K k
n
0
E m k K
P E m K P C
k C
dt t k
k t d k k
t
k k
k m m K
m E
t mt
m
K
If Ek has no point in common with a set
m k
m m
k G
H
0
of measure at most
k
m m
k
m
P
0
2
1
where
max
.
0 m k m
m
Q
Pk
Now consider
2 , where(
)
(
)
/
.
2k
E
m
E
m
k
Then
m(E)
1
log
)
(
log
)
(
)
m(E
1
)log
m(E
log
C
,
)
(
log
1
log
)
(
2
K k
0
E
m
k
k
P
E
m
P
C
k
P
dt
t
k
k
t
d
k
k
t
k k
k k
k
m
m E
m E
t
mt m
K K
If Ek has no point in common with a set
H
k . Hence
2 ) ( 0
2 ) ( 0
) (
) ( ) (
) (
2 2
m(E)
1
log
)
(
log
)
(
m(E)
1
log
)
(
log
)
(
k E m k
k t m k
E m
n k m
k k
k k E
m n k m
k k
E
m
k
P
E
m
P
E
m
k
K
P
E
m
K
P
C
I
If E has no point in common with a set H of measure at most
.
max
1
0 0
m k m
m
Q
m
Now
2 2
2
/
2
n m
Q
1
22
0 2
0 0
2
0 2
0
2
0 2
1
1
4
4
)
1
(
)
1
(
2
n m
v v m v
v m
v v
v m v
m
v v
v m v
m
x
x
x
x
x
x
x
Since the second term in both the numerator and denominator is dominant. Therefore
2 1/ 0 2
/ 1
1 1 0 2
/ 1
1 1 ) ( 2
/ 1
e
n
if
2
4
)
(log
1
4
)
2
1
(
1
4
e
n
Pk
n n
Therefore we have
)
(
1
log
)
(
E
m
E
C
I
mif E has no point in common with a set H of measure at most
.
2
0 2 / 1
0 2
m
C
m
e
Thus we obtain that for n>n0 and arbitrary E
E
m t
m
m m
m
E
C
dt
t
M
t
m(E)
1
log
)
(
M(t)
log
)
(
)
(
) (
0
If
M
(
t
)
(
n
)
and E has no point in common with a set H of measure at most.
0m
C
Now let
. ) ( log ) (
) ( )
(
) (
0 )
(
sup
0
M t M t
t t
F
t m
m m
m n
M m
REFERENCES
Bharucha-Reid, A. and M. Sambandham, 1986. Random Polynomials (Academic Press, New York). Cramer H. and M.R. Leadbetter. Stationary and Related Stochastic. Processes. (Wiley, New York)
Dunnage, J.E.A. 1966. The number of real zeros of a random algebraic polynomial, Proc. London. Math. Soc. 16, 53-84.
Dunnage, J.E.A. 1968. The number of real zeros of a class of random algebraic equation , Proc. London. Math. Soc. 18, 439-460. Farahmand, K. 1986. On the average number of real roots of a random algebraic equation. Ann. Probab. 14: 702-709
