Lower Bounds Of The Number Of Real Roots Of Random Algebraicpolynomials

Lower bounds of the number of real Roots of random

algebraic polynomials

1_{Dipty Rani Dhal,} 2_{Dr.P.K.Mishra,}

1_{Associate Professor of Mathematics,} 1, 2

CET, BPUT, BBSR, Odisha, India

Abstract . let Nn( ω ) be the number of real roots of the equation

( )

0

0

=

∑

=

ν ν

a

ν

ξ

ν

ω

x

n

where ξ_ν

( )

ω ’s are identically distributed random variables with mean zero and joint density function particularly defined and aν’s are nonzero real numbers which are finite . It is shown that for any sequence of positive constants ( Єn, n ≥ 0 ) satisfying Єn → 0 and Єn2 log n → ∞ , we have

Pr

( )

(

)

1 0 0

log

log

inf

0

−

∈

<













<∈

>

n

N

n

n

n

n

ω

n

µ

n

For all n0 sufficiently large and positive constant μ .

1991 Mathematics subject classification (amer. Math. Soc.): 60 B 99.

Keywords and phrases: Independent, identically distributed random variables, random

algebraic polynomial, random algebraic equation, real roots, domain of attraction of the

normal law, slowly varying function.

1 . Introduction . Let Nn

( )

ω be the number of real roots of the algebraic equation

( 1.1 ) f ( x , ω ) =

∑

_ν=0

a

ν

ξ

ν

( )

ω

x

ν

=

0

n

where ξν

( )

ω ’s are random variables assuming real values only . Several authors have estimated bounds for Nn ( ω ) when the random variables satisfy different

distribution laws . Littlewood and Offord [ 1 ] made the first attempt in this direction. They considered the cases when the ξν

( )

ω ’s are normally distributed or

uniformly distributed in ( -1 , 1 ) or assume only the values -1 and +1 with equal probability . They obtained in each case that

Pr

(

N

n

( )

ω

>

µ

Λ

n

)

>

1

−

A

/

log

n

for large n where

Λ

_n= ( log n )/ (log log log n ) and μ , A are positive constants.

Samal [ 3 ] has considered the general case when ξν

( )

ω ’s have identical distribution with expectation zero , the variance and the absolute moment finite and nonzero . He has shown that Nn

( )

ω ≥ Єn log n outside an exceptional set whose

measure tends to zero as n tends to infinity where Єn→0 but Єnlog n → ∞ .

Evans [ 4 ] was the first to obtain ‘ strong result ’ for this bound . He showed that the exceptional set could be independent of n , the degree of polynomial . He has proved that in case of normally distributed independent coefficients there exists an integer n 0 such that for n > n 0 ,

( 1.2 )

Pr

(

N

n

( )

ω

>

µ

Λ

'

n

)

>

1

−

A

(

log

log

n

0

)

/

log

n

0

where A is a positive constant and

Λ

'

_n= ( log n )/ (log log n ) . Subsequently ,

Samal and Pratihari ( [ 5 ] & [ 6 ] ) have improved the ‘ strong result ’ for the lower bound . Considering random equations with independent coefficients they have shown that for n > n 0 ,

( 1.3 )

where Єn→0 but Єn2 log n → ∞ . The results of Evans [ 4 ] is a particular case of

( 1.3 ) for it is obtained from it by replacing Єn = ( log log n ) – 1. On the other

hand their exceptional set is much smaller . Recently , Sambadham, M. and Renganathan [ 7 ] have obtained the same lower bound for Nn( ω ) as Evans when

the random coefficients ξν

( )

ω ’s are normally distributed with mean zero and joint density function defined by

( 1.4 )

where M – 1 is the moment matrix with σi=1 ; ρ i j = ρ , 0 < ρ < 1 , i ≠ j ;

i , j = 0 ,1 ,2 , . . . , n and a’ is the transpose of the column vector a. Another estimate of the lower bound for Nn( ω ) is due to Nayak and Patanayak [ 8 ].

Considering the polynomial

( 1 . 5 ) f ( x , ω ) =

( )

ν ν

a

ν

ξ

ν

ω

x

n

∑

=0

( )

(

₀

)

0

log

/

log

inf

Pr

0

n

n

N

n

n

n n

_



<

∈

n









<∈

>

ω

µ

(

)

{

(

)

a

M

a

}

M

2

.

2

n2

.

exp

1

2

'

1

−

Π

−

where a_ν’s are nonzero real numbers such that ( kn / t n ) = O ( log n ) ;

k n = max ν | aν | ,tn = min ν | aν | and ξν

( )

ω ’s are identically distributed dependent

random variables with mean zero and the joint density function given by ( 1.4 ) , they have shown that there exists a positive integer n0 such that for n > n 0 ,

the exceptional set being at most μ’{ log ( ( k n0 / tn0 ) log n 0 ) / log n0 } 1 / 2.

In this paper our object is to improve the ‘ strong result ’ for the lower bound of Nn ( ω ) in case of the dependent coefficients . In fact , considering the

random polynomial ( 1. 5 ) with dependent coefficients under the condition ( 1.4 ) we establish that for any sequence of positive constants (Єn , n ≥ 0 ) satisfying

Єn→0 and Єn2 log n → ∞ , we have

Pr

( )

(

)

1 0 0

log

log

inf

0

−

∈

<













<∈

>

n

N

n

n

n

n

ω

n

µ

n

for all n0 sufficiently large and a positive constant μ .

Throughout the paper n is considered to be sufficiently large in order that the inequalities are satisfied and μ ’ s denote positive absolute constants assuming not necessarily the same value at every place of occurrence , [ x ] denote the greatest integer not exceeding x . We wish to prove the following theorem .

Theorem . Let Nn ( ω ) be the number of real roots of the random algebraic

equation f ( x , ω ) =

∑

₌₀

( )

=

0

ν ν

a

ν

ξ

ν

ω

x

n

in which ξν

( )

ω ’s are identically distributed random variables with mean zero and joint density function given by ( 1.4 ) and a_ν’s are nonzero real numbers such that ( kn / t n ) = O ( log n ) ;where k n = max ν | aν | ,tn = minν | aν |. Then , for any sequence of

positive constants (Єn , n ≥ 0 ) satisfying Єn→0 and Єn2 log n → ∞ , we have

Pr

( )

(

)

1 0 0

log

log

inf

0

−

∈

<













<∈

>

n

N

n

n

n

n

ω

n

µ

n

for all n 0 sufficiently large and positive constant μ.

2 . Proof of the theorem . Let A and B be constants such that

( )

n

{

(

k

t

)

n

}

N

_n

ω

>

µ

log

log

_{n n}

log

( 2.1 ) 0 < B < 1 and A > 1 Let

( 2.2 ) n

(

t

n

k

n

)

exp

{

c

'

(

n

log

n

)

}

2

∈

=

β

where c’ is a constant to be chosen later , and

( 2.3 )

[

(

) (

)

]

1

2

2

₊

=

k

t

Ae

B

M

_n

β

_{n n n , where e = exp ( 1 )}

so

( 2.4 )

We define

( 2.5 ) φ ( x ) = x x . Let k be an integer determined by

( 2.6 )

(

)

(

8 7

)

₍

₎

(

8 11

)

11

8

7

8

k

+

M

_n k+

≤

n

≤

φ

k

+

M

_n k+

φ

The first inequality gives

( 8k + 7 ) . { log ( 8k +7 ) + log M n } ≤ log n

and ultimately k ≤ μ ’’ ( log n ) / ( log β n ) . The second inequality gives

log n < ( 8k + 11 ) . { log ( 8k +11 ) + log M n }

< ( 8k + 11 )2 + log ( 8k +11 ) + log M n

< μk2 log M n

so that

k > μ’ {( log n ) / (log βn )}1 / 2 .

Thus

(

)

2 2

1

log

.

'

log

'

n

k

c

n

c

∈

n

≤

≤

∈

n

µ

µ

, so that k approaches infinity less rapidly than n . We consider f ( xm , ω ) = U m ( ω ) + R m ( ω )

at the points

(

)

2 2

(

)

2 2

'

_{n n n} n

n n

n

t

M

k

t

k

β

µ

β

µ

≤

≤

( 2.7 ) xm =

(

)

2 1

4

1

4

1

1





























+

−

n

m

M

m

φ

for m = [ k / 2 ] + 1 , [ k / 2 ] + 2 , … , k where Um ( ω ) =

( )

ν ν ν

ξ

ν

ω

m

v

v

a

x

∑

= +

2 1 1

and

( )

( )

ν ν

ν

ξ

ω

ω

n _m

v v v

v

m

a

x

R













₊

=

∑

∑

+ = = 2 1

1

0

where ν1 =

(

)

n

m

M

m

1

4

1

4

−

−

φ

_{, ν2 =}

(

)

n

m

M

m

3

4

3

4

+

+

φ

2.1 . We shall use the fact that each ξ _ν ( ω ) has marginal frequency function ( 1 / 2Π ) exp ( - ω2 / 2 ) .

We shall need the following lemmas .

Lemma 1 . For α1 > 0 , σ m > α1 t n

(

)

n

m

M

m

1

4

4

+

φ

_where

(

1

)

;

2 2 1 1 2 1 1 2 2

2

∑

∑

+ = = +













+

−

=

v v v m m

m

a

x

a

x

ν ν ν ν ν ν ν

ρ

ρ

σ

_{0 < ρ < 1.}

Proof.

(

)

(

)

m

x

t

x

a

n m M m n m M m n m

ν

φ φ ν ν ν ν ν ν

∑

∑

+ + + − − + =

>

) 3 4 ( 3 4 1 ) 1 4 ( 1 4 = 2 1 1

(

)

(

)

.

1

4

1

4

4

1

4

2 1













−

−

−

+

>

n

m

M

m

n

m

M

m

t

_n

φ

φ

(

)

>tn

(

)

n m M m 1 4 4 +

φ ( B / ( A e ) ) ; e = exp ( 1 ) , where m is large . Again

(

)



_









+

>

∑

+

=

Ae

B

n

m

M

m

t

x

a

v

v v

m

.

4

1

4

2

1 1

2 2

2 ν

φ

ν

so that

(

1

) (

4

1

)

4

.

(

4

1

)

8

₂

;

2 2

2 2

2













+

+













+

−

>

e

A

B

n

m

M

m

t

Ae

B

n

m

M

m

t

_{n n}

m

ρ

φ

ρ

φ

σ

(

)

_n

m

M

m

t

_n2 2

4

1

8

2

1

+

>

α

φ

_{; α}

1 being a positive constant .

Hence the proof of lemma 1 is complete .

Lemma 2 .

( )

{

:|

|

~

}

2

0

Pr

2 1

0

2

=

<

>

−

=

∑

n m

n m

n

e

x

a

λ

π

σ

λ

ω

ξ

ω

ν ν λ

ν ν ν

where λ n = m 2 β n and

(

)

(

)

2 1

0

1 0 2

2 2

1

~

∑

∑

=

+

=

−

=

ρ

ν_{ν ν} ν

ρ

ν_{ν ν} ν

σ

_m

a

x

_m

a

x

_m ;

0 < ρ < 1 .

Proof . Let F ( x ) be the distribution function of

( )

ν ν

ν

a

ν

ξ

ν

ω

x

m

∑

=

1 0

Then

{

ω

a

ξ

( )

ω

x

m

λ

n

σ

m

}

ν ν

ν ν ν

|

~

:|

Pr

1

0

>

∑

=

{

F

(

λ

n

σ

m

)

F

(

λ

n

σ

m

)

}

~

~

1

−

−

−

=

e

dt

n

t

∫

∞ −

=

λ

π

2

2

2

.

2

2 2

n

n

e

λ

π

λ

−

≤

hence the proof of Lemma 2 is complete . Lemma-3.

( )

{

}

n m n m

n

e

n

x

a

λ

π

σ

λ

ω

ξ

ω

ν λ ν ν ν ν 2 1 2 2

2

~

~

|

:|

Pr

− +

=

>

<

∑

Where

(

)

(

)

2 1 2 1 2 2 2 2

1

~

~

∑

∑

+ = + =

+

−

=

n m n m

m

a

x

_{ν ν}

a

x

ν ν ν ν ν ν

ρ

ρ

σ

_;

0 < ρ <1 .

The proof is similar to that of Lemma 2 .

Lemma 4 . For a fixed m ,

{

( )

}

.

2

2

1

|

:|

Pr

2 2 n m m n

e

R

λ

π

σ

ω

ω

<

>

−

−λ

Proof . For a given m we have

R

_m

( )

ω

λ

_n

(

σ

_m

σ

_m

)

~

~

~

₊

<

.

Again we have

(

)

(

)

2

2 1 0 2 2

1

4

1

4

1

4

2

+

−

+

<

∑

=

m

n

m

M

m

k

x

a

n m

φ

ν ν ν ν and

(

)

(

)

2

1 0 2

1

4

1

4

1

4

2

+

−

+

<

∑

=

m

n

m

M

m

k

x

a

n m

φ

ν ν ν ν

Hence , for positive constants α2and α 3 ,

(

)

(

) (

)

2 2 2 2 2

2

4

1

1

4

1

4

~













−

+

+

≤

n

m

M

m

m

k

_n m

φ

α

σ

and similarly

( )

(

)

(

) (

)

(

)

2 3

2

1

4

1

4

1

4

+

−

+

+

<

m

n

m

M

m

k

R

n n

m

φ

α

α

λ

ω

n m n

n n

M

t

k

m

σ

α

α

α

λ

.

.

16

₁

3 2

2













+

<

_{( by Lemma 1 )}

using the statement ( 2.3 ) . Thus | R m ( ω ) | < σ m except for a set of measure at

most

2

2

π

exp

(

−

λ

_n2

2

)

/

λ

_n for m = [ k / 2 ] + 1 , [ k / 2 ] + 2 , . . . , k . Hence , Lemma 4 is proved .

Lemma 5 . Let η 1 ,η 2 , η 3 , . . . be a sequence of independent random variables

with variance V ( η i ) < 1 for all i . Then , for each Є > 0 ,

{

( )

}

0 2 0

0

1

sup

Pr

K

D

E

K

k

k

K

i

i

i

≤

_∈













≥∈

−

≥

∑

₌

η

η

where D is a positive constant.

This form of ‘ the strong law of large numbers ’ is a consequence of

Hajek-Renye inequality [ 2 ] .

2 . 2 . We define Sm+ , S m- as sets of ω in which , respectively

Um( ω ) > σ m , Um( ω ) < - σ m .

Hence

Em Ǜ F m = ( S 2m+∩ S2m+1- ) Ǜ ( S 2m-∩ S 2m+1+ ) , say ,

As in Samal and Pratihari ( [ 5 ] & [ 6 ] ) . Obviously , the two sets within the braces on the right hand side are disjoint . Therefore

( 2 .8 ) Pr ( Em Ǜ F m ) = ( S 2m+∩ S2m+1- ) Ǜ ( S 2m-∩ S 2m+1+ ) .

We shall use the fact that ξ ν( ω ) has marginal frequency functions

(

1

2

π

)

exp

(

−

t

2

2

)

.

The distribution function is

(

)

∫

(

)

∞ −

−

x

dt

t

2

.

exp

2

1

π

2

The distribution function of

( )

ω

_ν ν

ξ

ν

( )

ω

mν v

v

m

a

x

U

=

∑

2_{= 1+1} is

∫

{

(

)

}

∞ −

−













x

m m

dt

t

2

.

exp

2

1

₂ 2

σ

σ

π

where

(

1

)

;

2 2 1 1 2 1 1 2 2

2

∑

∑

+ = = +













+

−

=

v v v m m

m

a

x

a

x

ν ν ν ν ν ν ν

ρ

ρ

σ

Therefore , the distribution function of Um / σm is

( )

(

)

∫

(

)

∞ −

−

=

x

m

x

t

dt

F

1

2

π

exp

2

2

.

Thus

1

(

1

2

)

exp

(

2

)

.

2

( )

1

1

2

2 2

2



=

−

=

−











−

<

∫

− ∞ − m m m

m

t

dt

F

U

F

π

σ

and

1

1

1

1

2

( )

1

2 2 2 2 2 2 m m m m m m m

F

U

F

U

F

_

=

−











≤

−

=













>

σ

σ

Now

(

)













−

≤

>

=

∩

+ + − + +

1

1

Pr

Pr

1 2 1 2 2 2 1 2 2 m m m m m m

U

and

U

S

S

σ

σ

=

∫ ∫

{

(

)

}

∞ = − −∞ =

=

+

−

1 1 2 2

'

2

/

exp

2

1

x y

dxdy

y

x

δ

π

, ( say ) ,

where δ’ is independent of m. Similarly Pr ( S 2m-∩ S2m+1+ ) = δ’

Hence from ( 2.8 ) we get that

Pr ( EmǛ F m ) = 2 δ’ = δ , ( say )

Obviously δ > 0.

Let η m , ρ m , θ m be defined as in Samal and Pratihari [ 6 ] , that is ,

η m =













−

δ

δ

1

0

1

y

probabilit

with

y

probabilit

with

Then quite obviously η m’ s are independent random variables with E (η m ) = δ and

Var (η m ) = δ – δ 2 < 1.

Let













<

<

=

+ +

otherwise

R

and

R

if

_{m m m m}

m

1

0

₂

σ

_{2 2 1}

σ

_{2 1}

ρ

And

θ

m

=

η

m

−

η

m

ρ

m.

Then the number of roots in the interval ( x 2m0 , x 2k + 1 ) where m0 =[ k /2 ] +1, must

exceed

∑

₌

k m

m ₀

θ

m .

2 . 3 . We define ω-sets A ( ω ) , B ( ω ) and C ( ω ) as in [ 6 ]. We have

E

( )

ρ

m

≤

Pr

(

R

2m

≥

σ

2m

)

+

Pr

(

R

2m+1

≥

σ

2m+1

)

.

By lemmas

{

}

.

2

2

|

|

Pr

2

2 2

2

n m

m

n

e

R

λ

π

σ

<

−λ

≥

and

{

}

.

2

2

|

|

Pr

2

1 2 1

2

2

n m

m

n

e

R

λ

π

σ

₊ −λ

+

≥

<

Thus

( )

(

)

(

)

2 2

2 2 2

2

exp

'

2

exp

'

n n n

n m

m

m

E

β

β

µ

λ

λ

µ

ρ

<

−

=

−

< μ / m 2

, ( by definition of β n ).

Hence

( )

₁

(

)

.

1

1

1

2

0 2

0

0 0 0

m

m

m

k

E

m

k

k

m m k

m m

m

µ

µ

ρ

<

+

−

≤

+

−

∑

=

∑

=

Therefore

{

( )

}

(

1

)

.

'

2

Pr

0

0 1

2 0

∑

− + ≥

∈

<

k m

k

m

C

ω

µ

Applying Lemma 5 , we have

Pr { B ( ω ) } ≤ 4 D / ( ε2k0 ) = μ / k 0 .

Since A ( ω ) ⊆ B ( ω ) Ǜ C ( ω ) ,

( 2.9 )

{

( )

}

(

1

)

.

'

2

Pr

0

0 1

2 0

0

+

_∈

∑

_{− + ≥}

≤

k m

k

m

k

A

ω

µ

µ

2 . 4. If ω ∉ A ( ω ) ,

∑

>

∑

( ) (

−

−

+

)

∈

= =

1

0

0 0

m

k

E

k m m

m k

m m

m

η

θ

for all k such that k – m0 +1 ≥ k 0 . Therefore

( )

(

)

(

)

n

c

k

N

_{n n}

log

'

2

2

1

₋

_∈

_≥

−

∈

_∈

>

δ

µ

δ

ω

_.

For n > n 0 ,

{

( )

}

(

1

)

.

'

2

Pr

0

0 1

2 0

0

+

_∈

∑

_{− + ≥}

<

k m

k

m

k

A

ω

µ

µ

_'

(

log

)

.

'

0

0 ₀

n

c

k

_



∈

_n













<

<

µ

µ

µ

Taking c’ = ( δ – ε )2 _μ

12 / 4 , the desired result follows .

REFERENCES

1. Littlewood, J.E. and Offord, A.C. On the number of real zeros of random algebraic equation (II), Math. Science 12 (1943), 277-286.

2. Hajek-Renye .”Inequality”, Cambridge University press.

3. Samal G. On the number of real roots of random algebraic equation,

Proc. Cambridge Phil. Soc. 58 (1962), 433-422.

4. Evans, E.A. On the number of real roots of random algebraic equation,

proc. London Math. Soc. (3) 15 (1965), 731-749.

5. Samal and Pratihari . Real zeros of a random algebraic polynomial,

Quar. Jour. Math. Oxford. 2 (1993), 169-175.

6. Samal and Pratihari . The number of real zeros of a class of random algebraic polynomials (I), Proc. London Math. Soc. (3) 18 (1989), 439-460.

7. Sambadham, M. and Renganathan. On the number of real zeros of a random trigonometric polynomial, coefficient with non-zero mean,

Jour. Indian. Math. Soc. 45 (1989), 193-203.

8. Nayak N.N. and Patanayak, S. Strong results for real zeros of random polynomial,

Pac. Jour Math. 103 (1982), 509-522.