The Expected Number of Real Zeros of Random Polynomial

w w w . a j e r . o r g Page 169 American Journal of Engineering Research (AJER)

e-ISSN: 2320-0847 p-ISSN : 2320-0936 Volume-03, Issue-05, pp-169-179 www.ajer.org Research Paper Open Access

The Expected Number of Real Zeros of Random Polynomial A. K. Mansingh¹, P.K.Mishra²

1 Department of Basic Science and Humanities, MITM, BPUT, BBSR, ODISHA, INDIA.

2 Department of Basic Science and Humanities, CET, BPUT, BBSR, ODISHA, INDIA.

Abstract:- In this paper we have estimated the number of real zeros of

Z z





, z X (Z)

Q

n

0 j

j j

n which is a random Gaussian polynomial satisfying the normal distribution with mean zero and variance one i.e. E(X _j)0 and E(X_j)² 1 for j ≥ 0. Much research works has been done on the same polynomial with different co-efficients satisfying the above condition and found that the expected number of zeros is approximated to

 ²_ ^{logn as n}in the interval (-,). Our present work is to estimate the number of zeros in the interval [0, 1] and found that the expected number of real zeros of the above polynomial under same conditions is ^ENn ^0,1~

 

Keywords: - Independent identically distributed random variables, random algebraic polynomial, random algebraic equation, real roots.

INTRODUCTION

Let X0, X1……Xn be a stationary Gaussian process satisfying E(X _j)0 and E(X_j)²1 for j ≥ 0 as sufficient conditions. Then the expected number of real zeros of the above

polynomial 



 ⁿ

j j j

n z X z

Q

0

)

( is

 2

1 ( log n ) as n tends to infinity. The expected number of zeros of a random trigonometric polynomial has been studied by Dunnage[1] and estimated the number of zeros in the interval [0,1] which is approximated to (1/2)logn where





n but in the same problem we consider a polynomial which is piece wise continuous and differentiable in the same interval and used normal distribution with mean zero and variance one and applied Gaussian process. Ibragimov and Maslova [2] had worked with same mean and variance under different conditions and had got the expected number of zeros in the same interval is approximated to the result of Dunnage [1].

Let us consider the Gaussian polynomial



 ⁿ

j

j j

n z X z

Q

0

)

( ₍₁₎

w w w . a j e r . o r g Page 170 Which is a piece-wise continuous and differentiable polynomial within a closed interval [0, 1]

and satisfying same conditions. Suppose that there is an interval [-b,b] , 0<b≤ π, where

 

f is uniformly approximated by the partial sums, Snf  of its Fourier series

development, and 0 ≤ m ≤ f  ≤ M ≤  , θ-π,π where m and M are the lower and upper bounds of f  . Then the expected number of real zeros of the above polynomial is

 

 ^{0,1 ~} ¹_2π ^logn

ENn as n.

Let the number of zeros of the above polynomial is denoted by ENn. .The main aim of our work is to estimateENn  0,1 .

Now we have partitioning the interval [0, 1] into three different sub intervals namely I_n¹, I_n²

& In3

the details of partitions are given bellow,



 













  













 



1 log ), 1 log

(

log ) 1 log

( ), log 1 1

(

) log 1 1

( , 0

3 2 1

n I n

n n n

I

n I

n n n

For n≥3,

Let Nn (a,b) be the number of sign changes of a piece-wise linear approximation to Qn(x).

First we estimateENn(In²). Then at last section we have approximated ENn(In1

) and ENn(In3

) and found that the expected number of zeros of both the intervals are equivalent to σ(logn) as n tends to infinity

i.e. EN_n(I_n¹)EN_n(I₃³) (logn)

COVARIANCE ESTIMATES:

Let k_n(x,y)=E{Q_n(x)Q_n(y)} and r_n(x,y )=Cor {Q_n(x) , Q_n(y)}. We estimate k_n(x,y) and r_n(x,y) for x,y satisfying certain conditions. For ^xIⁿ² we derive upper and lower bounds for k_n(x,x).

Let T_n() be a trigonometric polynomial of order n with real coefficients. Suppose we define

^{- ,}  ^{and c R}

, e c )

( _v

n

-n v

iv

v  











 ^

Tn ₍₂₎

  



 



 

 



2 / x c 2

, ₀

n

0 v

v

v c

x T

G _n  (3)

Then

     

    _{x,y (0,1]}

1 y - 1 x - y) 1 (x, r

(0,1]

y x, 1

, , ,

,

12 12

2 2

/ 

 

 

 

xy xy

y T G x T y G x

T_n ^{n n}

ka

Taking 0dr^/(x, y)

w w w . a j e r . o r g Page 171 We estimate kn(x,y) satisfying the conditions





d

0 And x, y In

 2

We know from co-variance estimates

have w e

y x, For

n as 0 )

, ( )

, (

2 1

/

0 ) , ( ,

sup

2

I I^{n n}

n

d y x r

y x

y x r y x r Iⁿ















  

THEOREM -1 Suppose that f() can be uniformly approximated by the partial sums of its Fourier series development ^{f() A,}^,& f (0)>0. If there is a constant α and an integer N0 such that G⁽S_nf^,x⁾^{0for n}^N0 ^{, x}  ^0,1 Applying co-variance estimates to find mean and variance of the series _j

m

j jX

a

0

we have

0 m for 2

0 2 2

0



 







 





m

j j

m

j

jX A aj

a

E  _and ^  





 f( )d

) , (

0

 0 

  

 



 







 



  ⁿ

v iv v n

v

iv v

n x y x e y e

k

Q(x)Q(y) ( , )

E y) k(x,

et k x y

L _n

Limn









   





  



 ^





 ye f  d

xe ⁱ 1 ⁱ ( )

1 ^{1 1}  4



 ^m

m v

iv ve

c ^

) ( T

Let _m (5)

We have by Cauchy’s residue theorem that ¹  ¹¹ ^{1 2 (1 )1}





    

^eiv ^{xe i yei d xv xy}











And ¹  ¹¹ ^{1 2 (1 )1}





 

    

^{e iv xe i yei d yv xy}











 ^xe   ^ye  ^{g d C}

y x h

b

b

i

i  









^  1 ^ ()  1

) ,

( ^{1 1} (6)

Where C is a constant depending on B and b.

Taking m=n in equation (5) we have m=n and Tn(θ)=Snf(θ), where Snf(θ) is the nth partial sum of the Fourier series development of f(θ). Now for ^{x, y} ^0,1 we have

) 0

, , ( ) ,

(x y S f x y J

k_n ^k^a _n ^ (7)

Where J₀ J₁J₂J₃J₄ and

w w w . a j e r . o r g Page 172

    ^{ }





  







b

b

n i i

n

a S f x y xe ye S f d

k

J₁ ( , , ) 1 ^{ 1}1 ^{ 1}    ⁸

   





  







b

b

i

i ye f d

xe y

x k

J₂ ( , ) 1 ^{ 1}1 ^{ 1} ()   9

   





   





b

b

n i

i ye S f f d

xe

J₃ 1 ^{ 1}1 ^{ 1}( () ()   10

) , ( ) ,

4 k (x y k x y

J  _n  (11)

From the conditions of Zygmund [7] we found that there is a constant B not depending on n such that

0.

n all for B

d f

S_n   



, )

(







 & g()S_nf() gives J₁C Where C depends on b and B & g(θ)= f(θ) gives

Let

  ( ) ( )

sup )

(

,





 S f f

n

a _n

b b









By Cauchy’s inequality we have

2 / 1 2 2

/ 1

3 (1 2) (1 )

) ( 2

y x

n J a



  

) , , ( )

,

(x y k S f x y

k_n  ^a _n

 

 

   









   



  



 

x y

xy xⁿ y^{n n}

n C a

2 2 1

1 1

2 / 2 1

/ 1

1 1 1

For x,y 0,1 and where C is a constant depending upon b, A and B So

   ^{1 (1)}

) , , ( ) ,

( _1/2

2 2 / 21

y x

n y w

x f S k y x

k_n ^a _n



 

 (12)

Where w(n)0 as n For x,yI¹n In 2

Now 





 ⁿ

v v v

nf x r x

S G

1 2

) 1

,

( (13)

We show that there is a constant α>0 and an integer N0 such that

0 2

n ,

x 0

) ,

(^{S f x when} I ^{n N}

G _n     From Abel’s Theorem and Titchmarsh[4]

conditions we see that G(S_nf ,x) is uniformly convergent for x 0,1 and

  

 











n

v v x

f r x

f S G

1 2 1

1

0 )

,

lim ( ^{ (14)}

PRROF OF THEOREM -2 Using the two conditions f()A,, and f(0)>0 of Theorem-1 and applying the uniform convergence of the series within a certain interval

  



) ,

(  

f

S_n and S_nf(0)G(S_nf,0) by adopting similar procedure as in the proof of Theorem -1 we see that J₁=J₂=0 holds for ^xI^1nIⁿ². For some µ we construct an

w w w . a j e r . o r g Page 173 interval^^-,^{and f() 0}. Such an interval exists as f()is continuous at θ=0 and

) 0 (

f >0. We consider the case when ^x





The following inequality holds for x,y0,1





1 ^{1 2 -1/2}

0

sup ^{     }







b xe ⁱ

b









2 for

1

1 ¹

0

sup ^{  }











 ^{ }







b xe ⁱ

b

We have

0,1

x for

0 0

sup ^



 



  ^



 n iv

v v

b

e x

Substitution in kn(x,x) with 1xe^{iv 2} replacedby

2

0

 n iv

v

ve x



 We got

0,1

x C ) ( -

x) (x,

k ²

b

b -

n

0 v

n  _ ^{xveiv f  d}   ⁽¹⁵⁾

Substitute f () 1/2 in (15) we have C n

0 n

0 ^-b

2

-

2





 



  

 x^ve ^{iv d b} x^ve ^{iv d}



 



 



(16)

Where C depends only on b and A.

By simple calculation we have

 0,1 x , 2

n

0 v

2 2

- n

0 v



 

  

iv v

ve d x

x  





 (17)



 

  

 ~2 , n

n

0 v

2 2

b -

n

0 v

v b

iv

ve d x

x ^   (18)

and ^x





From our construction of [-b,b] we have







 ^

 f d x e d

e x

b

iv v b

iv v

2

b -

n

0 v 2

b -

n

0 v

)

(  

 _ ^{ } _ ^{ (19)}

So the desired result follows for^x^{1(logn)1/2,1}^.

When x=1 we have

 ^{1,1 1 2}  ¹ ²  ¹  ⁰

1

f n r j n

n n

n

j n j

k  ^ 















 (20)

w w w . a j e r . o r g Page 174 Where σn f(θ) is the nth Cesaro sum associated with f(θ). As f(θ) is continuous at θ=0 we have σn f(0) f(0) as n. Hence we

have ^  





 a e f d

e a x

a E

v iv v v

iv v j

j ( )

-

m

0 m

0 m 2

0

j   

 _ ^{ }_^{ }_^_^{ }_^

















given that f()A. Then by using Cauchy’s inequality we have

 



 d

e a A x

a

E  _{j j}   _v ^iv







 



 





- m 2

0 v m 2

0 j

(21)

By simple calculation we have

 _ ^{ } ^m_

j iv j

ve d a

a

0 2

- m 2

0 v

2

 



 (22)

GENERAL APPROXIMATION FORMULA : - For x a,b we approximate Q_n(x) by a process which linearly interpolates between Q_n(a) and Q_n(b). It is convenient to count the sign changes of this process in [a,b] by

^{Q (a)Q (b)}

2 sgn 1 2 ) 1 ,

(  _{n n}



 







 



 b a

N_n (23)

Where sgn{x} =









 0 x 1 -

0 x 0

0

>

x 1

The results of this section depend on an upper bound for the number of zeros of Qn(x) in an interval [a,b]. Define the event

b]

[a, in zeros more k has ) (x Q

U_k _n (24)

For any interval [a,b] _[0,1] let

 

^{1-(n 1) ,1}^,

b ), )(

1 (

and , 1) (n - 0,1 b , ) 1 )(

(

1 -

-1 1



















 ^

a b n

b a b





LEMMA -1 let  2^30 and

(i) f() is continuous at =0 (ii) f(0)0

(iii) f()A,-,

Then there is an integer N1 and an absolute constant C such that 0

k N n )

(U_k C ^3k/5  ₁ 

P 

COROLLARY-1 Under the conditions of Lemma-1 we have

N n , ])

, ([

) ,

(a b EN a b C^6/5  ₁

EN_{n n}

Where C is an absolute constant.

Now we have to estimate the number of zeros in the three sub intervals.

(A). ESTIMATION OF NUMBER OF ZEROS IN THE INTERVAL:- I¹ⁿ

w w w . a j e r . o r g Page 175 LEMMA -2 Let X₀, X₁……X_n be a set of n points satisfying E(X _j)0 and E(X_j)²1 for j ≥ 0 .

(i)Let us consider a function f(θ)which is uniformly approximated by the partial sums, S_nf(θ)in the interval [-π,π]

(ii) In the interval 0 ≤ m ≤ f(θ) ≤ M ≤  and θ-π,π where m and M are the lower and upper bounds off(θ). Then expected number of real zeros in the interval











 

 )

log 1 1 ( ,

1 0

In n is ^ENn(_I¹_n^)^C1/2/2^{loglogn n2 (25)}

Where C is a finite constant

PROOF: - From the Kac-Rice [3] formula and using the postulates of Shankar [4] which states that

 ^{a b}   ^dx

EN C B Rⁿ

b

a n n n

2 2 2 / 1 2 /

1 1

1) ( ) ,

(  _  ^  ⁽²⁶⁾

 ^{( )}

E C , )) (

(Q x ² _n Q ^/ x ²

E

Bn n  n (27)

^{Q (x),Q /(x)}

Cor

Rn n n (28)

Where ( )

dx

d Q_n x =Qn^/(x)

Now we have to find an upper bound for ^{Cn/Bn with x}I¹ⁿ

We represent Bn=kn(x,x) by Noting that f^()m^0,^, and applying Lemma-1 gives 2

0

 2



 ⁿ

v v

n m x

B  for ^x ^1,0 From Lemma-2 we have

  



 



 

 n  1

1 2

2 2 v

iv v v

n A v x ae

C  ^ summing 

 n

1 2

v

x v and using that ^{2( 1)} 2(1 ²) ³

1

2  





^{n v x v}  ^x

v

^0,1

x (29)

 ^1-x

) 1

)(

/ 2 (

/ _n  ^{2(n1) 1 2 -2}

n B A m x

C (30)

2 n for x

,

1 I¹ⁿ

) 1 (

2 ^n   

x







 

 x^{2(n 1)} I¹ⁿas n

x 0,x

sup I

and 1

n

We deduce that





bounded above by a constant. So

 ^{1-x -2 n 2and x} I¹ⁿ

/ _n  

n B

C (31)

Substituting the value of Cn/Bn and using the relation (1 R²n) 1 gives the expected number of zeros in the interval I^{1n is ENn(}_I¹_n^)^C1/2/2^{loglogn for n2} (32) Hence the theorem is proved.

LEMMA-3 Let X0, X1……Xn be a set of n stationary points satisfying E(X_j)0 and 1

) (X_j ² 

E . for j ≥ 0 Suppose that (i) there is an interval , and there is a constant