Applied Mathematics, 2011, 2, 1356-1358

doi:10.4236/am.2011.211189 Published Online November 2011 (http://www.SciRP.org/journal/am)

Copyright © 2011 SciRes. AM

On the Zeros of a Polynomial

Mohammad Syed Pukhta

Division of Agricultural Engineering, Sher-e-Kashmir University of Agricultural Sc iences

& Technology of Kashmir, Srinagar, India

E-mail: mspukhta_67@yahoo.co.in

Received February 27, 2011; revised May 25, 2011; accepted J une 3, 2011

Abstract

In this paper we consider the problem of finding the estimate of maximum number of zeros in a prescribed

region and the results which we obtain generalizes and improves upon some well known results.

Keywords: Polynomial, Zeros, Complex Number, Prescribed Region

1. Introduction

Let be a polynomial of degree n such

that

0

ni

i

i

pz az

110

0

nn

aa aa

then according to a well known result of Enstrom and

Kakeya, the polynomial

,pz does not vanish in

1.z concerning the number of zeros of the polyno-

mial in the region 1,

2

z the following result is due to

Mohammad [1].

Theorem A. Let be a polynomial of

degree n such that

0

ni

i

i

pz az

110

0,

nn

aa aa

then the number of zeros of

pz in 1,

2

z does not

exceed

1

1log

log 2

n

o

a

a

.

Dewan [2] generalized Theorem A to the polynomials

with complex coefficients and obtained the following

result.

Theorem B. If is a polynomial of de-

0

ni

i

i

pz az

gree n with complex coefficients such that

π

arg,0,1, 2,,

2

i

ai

11

,

nn

aa aa

0

then the number of zeros of in

pz 1

2

z does not

exceed

1

0

0

cossin12sin

1log

log 2

n

ni

i

aa

a

.

Theorem C. Let be a polynomial of

0

ni

i

i

pz az

degree n with complex coefficients. If

Re, Im,

ii ii

aa

for and 0,1, ,in

110

0,

nn

then the number of zeros

of

pz in 1

2

z

does not exceed

0

0

1

1log

log 2

n

ni

i

a

.

In this paper we generalize Theorem B and Theorem C

under less restrictive conditions on the coefficients,

which also improve upon them. More precisely, we

prove the following.

Theorem 1. Let be a polynomial of

0

ni

i

i

pz az

degree n with complex coefficients, such that

π

arg,0,1,2,,.

2

i

ai

n for some real

and

110

0

nn

aa aa

n for some real

and

then the number of zeros of in

pz ,z

does not

exceed