On Pellet's Theorem Concerning the Roots of a Polynomial

Annals of Mathematics

On Pellet's Theorem Concerning the Roots of a PolynomialAuthor(s): J. L. WalshSource: Annals of Mathematics, Second Series, Vol. 26, No. 1/2 (Sep. - Dec., 1924), pp. 59-64Published by: Annals of MathematicsStable URL: http://www.jstor.org/stable/1967743 .

Accessed: 12/11/2014 20:44

Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at .http://www.jstor.org/page/info/about/policies/terms.jsp

.JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range ofcontent in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new formsof scholarship. For more information about JSTOR, please contact [email protected].

.

Annals of Mathematics is collaborating with JSTOR to digitize, preserve and extend access to Annals ofMathematics.

http://www.jstor.org

This content downloaded from 67.214.247.154 on Wed, 12 Nov 2014 20:44:17 PMAll use subject to JSTOR Terms and Conditions

http://www.jstor.org/action/showPublisher?publisherCode=annals

http://www.jstor.org/stable/1967743?origin=JSTOR-pdf

http://www.jstor.org/page/info/about/policies/terms.jsp


ON PELLET'S THEOREM CONCERNING THE ROOTS OF A POLYNOMIAL.*

BY J. L. WALSH.

1. Introduction. It is the object of this note to present a converse and to indicate some applications of the following theorem due to Pellet:t

If the polynomial

F(x) _ aoi+|aliw+1a21X-2+ ?I* an-iI39b-

|Ian ISX +I an+,I xn+1 + I a"n+2t XJ+2+* + I aklxk

ha2 two positive roots xi and x2c (xi < x2), then the polynomial

(2) f (x) = ao+ al x + a2 x2 +. * * + ak XO

has no roots in the annular region xi < I x I < x2, and has precisely i roots whoms absolute value is not greater than xi.

Pellet's proof depends on the following easily proved theorem due to Rouchd:t

f two functions fi (x) and fi(x) are analytw on and within a closed curve, and if on this curve we have

Ifi (x) I > If2 (X) l then the two equations

i (x) _ 0, fi (X) +f2 (x) = 0,

have the same number of roots interior to the curve. 2. Proof of Pellet's Theorem. Pellet's proof of his theorem is quite

simple, but we give a new? proof which is even more elementary, and which is more in keeping with the methods we shall use later.

The polynomial F(x) has, by Descartes' rule of signs, either no positive root or precisely two positive roots, so if the hypothesis of the theorem is satisfied, there are no positive roots other than xi and x2. We assume,

* Presented to the Amecaathematical Society, Dec. 1928. t A. I. Pellet, Darboux's Bulletin (2), voL 5 (1881), pp. 398-39. + Journal de lItcole Polytechnique, voL 22 (1862), pp. 217-218. ? See, however, Picad, Noun. Ann. (3), vol. 11 (1892), pp. 147-148 and the reference there

given to Mayer. 59



60 J. L. WALSH.

as we may do without loss of generality, that neither as nor ak vanishes. Then F(O) and F(+ c) are both positive, so F(x) is negative if xx < X < X2:

|3 anj l:2 > I ao I + I a, Ix + * * + I a-,Jl x-1

+ I aG,+1Xl + I an+21x++** + ?Iakf x'.

If we do not restrict x to being positive, but require merely that x shall lie in the annular region xi < IxI < X2, we have by (3),

(4) l ane I Ix n > I ft + I a, I:| I * I * - + I is, I I Iftl-l

+ I aw+1 I I X I"+' + I a*+2 1 |XI, |2 + *. *+I| akI | lX l

and thus f(x) cannot vanish. It is clearly true that f(x) cannot vanish in the annular region so long

as (3) is satisfied, even if the coefficients aj are allowed to vary. Let us allow I aol, I a, l, **, I a-i I to vary continuously, and monotonically to approach zero. This variation simply strengthens the inequality (3), and xi (for which (3) becomes an equality) decreases monotonically. We suppose that the coefficients a,,, an+, ** * X ak are kept fixed. The root X2 increases monotonically. However, xi approaches zero, together with n roots of f(x). The variation of the roots of f(x) is continuous, no root x ever lies in the annular region xi < xj< x2, so there are always n roots x of f(x) in the circle Ix I 1 xi. The proof of Pellet's Theorem is now complete.

Cauchy established* a result that is essentially a special case of Pellet's Theorem, namely that all roots of f(x) are in absolute value less than or equal to the positive root xl of the equation

(5) laol}+Iallx+. - .+* aAllrfix.-.lakerk = 0;

we set :xl= c if ak =0. We find by the substitution x = /lx' and application of Cauchy's result that all roots of f(x) are in absolute value greater than or equal to the positive (or zero) root of the equation

(6) jaoI-| a jx-l aIjx- | a | -** ak = 0.

It is a direct corollary to Pellet's Theorem that if (3) is satisfied for a particular real value of x, then that value of I xI separates the n roots of f(x) of smallest moduli from the remaining n-k roots.

'Ezcim de 1lthmtiques, 1829; (Buvrex (2), vol. 9, p. 122.



THE ROOTS OF A POLYNOMIAL. 61

The proof of Pellet's Theorem holds with only minor changes for a function f(x) which is an entire function or a convergent power series.

3. A converse of Pellet's Theorem. We shall add to Pellet's Theorem a converse, to the effect that the theorem (including the results due to Cauchy) gives all possible regions which contain no roots of f(x) but whose determination depends only on the absolute values of the coefficients ao, a,, ***, ak.

We assume ak + O, which involves no loss of generality- We shall suppose, then, that the absolute values of the coefficients of f(x) are given, and we are to determine the locus of the roots of f(x) as the coefficients vary independently in all possible ways while having the prescribed moduli.

The points Q of the locus of the roots of f(x) do not comprise the entire plane, by Cauchy's result. The point set composed of all possible points Q is closed, for the roots of f(x) are continuous functions of the coefficients. This point set has circular symmetry about the origin, for if a: is a root of f(x), the substitution "' = eox, where I aI = 1, does not alter the absolute values of the coefficients of (2) yet can be chosen so as to trans- form x into any preassigned point x' equidistant with x from the origin.

The limit given by Cauchy is the best possible upper limit for the absolute values of the roots which depends only on the absolute values of the coefficients. For that limit is an actual root of a polynomial of type (2) whose coefficients have the prescribed absolute values. Similarly the lower limit defined by (6) is the best possible lower limit for the absolute values of the roots which depends merely on the absolute values of the coefficients.

Let us now suppose that there is no polynomial f(x') whose coefficients have the prescribed absolute values and which has a root of modulus a; we shall prove that (3) is satisfied for a = a for some choice of n. The number of roots of f(X) whose modulus is less than a does not depend on the particular equation f(X) chosen, the coefficients restricted as prescribed, for any two such equations can be changed one into the other by a continuous change of the coefficients (always satisfying the prescribed condition), the roots then vary continuously, and no root can cross the circle ax' = a.

If there are n roots of f(x) whose moduli are less than a, let us consider the expression,

-F(a) = laIO-ao-I al-i|a-. .*- awl ail

-I a*+, I a*+l - I a%+2 1 a*+2 - . . . I1 ak I ak;

we shall prove it to be positive. Allow the n roots of f(a') which are in absolute value less than a to vary continuously and monotonically (in



62 J. L. WALSH.

absolute value), and to approach zero; this variation alters the coefficients of f(x) but does not alter the remaining k- n roots of f(x) whose absolute values are greater than a. During the variation as described, the expression (7) never vanishes, and at the end of the process (7) is positive, by the test connected with (6). Thus (7) is positive throughout the process, (3) is satisfied for x = a, F(z) has two distinct positive roots x1 and x,, we have x1 <a <x2, and Pellet's Theorem gives an annulus containing the circle Ixj = a such that f(x) has no root in the annulus.

The converse of Pellet's Theorem is now completely established. 4. A limiting case of Pellet's Theorem. It is interesting to consider

a limiting case of Pellet's Theorem, namely the case that F(x) has one double positive root xl; we have

la,| + jajjxj + ***+ Ju(n-ll,,&-' (8)

a?, I xII+ Ia,,+, I * + I akIX- = 0.

Of course F(x) has no other positive root. We shall find it convenient to suppose as >0, which involves no loss in generality. Let us determine the circumstances under which f(x) has roots with the modulus x1. We must have

ant = (ao+ Ea, x + + X}, 1- X-

(9) + at,+, x+1 + *. a4k

When we notice (8), the equation jxz = xi, and the inequality a0 >0, the following equations are seen to be necessary and sufficient in order that (9) should obtain:

al ;r = lal lxl , c e,+1

xn+I = IUO......+1 l ;U+ll

ax = la= 11, x' = K +a.+1 , a29= a2Ix41 (x~2~+2 =

a,,1 M-l = a,-,. I elx, ak -1e a= , I 4.

an X - laR= 4



THE ROOTS OF A POLYNOMIAL. 63

The substitution x =x1 (where I o =-- 1) reduces these equations to

al Wal, I , n anI(l~lt

= Ia2j&)2, Cat1= -janj1+1, (10)

An-1 | n-1 I Cn,- (Lk = ak I Wk.

These equations are a necessary and sufficient condition that f(x) should have one or more roots whose modulus is xi. We shall now determine the precise number of such roots.

If equations (10) are satisfied by a number w, a necessary and sufficient condition that (10) should be satisfied by (*I is the system

(l11) #""act- f~th( M=1 ( ea - ( =,t% (d My = co-1430

where a,,, a,,,, ..., a.,,, are those numbers as which are different from zero. If 6 is the greatest common divisor of ml, tn,, *.., m., a necessary and sufficient condition for equations (11) is the single equation

That is, there are precisely 6 values of c which lead to a root of f(x),* and each of these is a double root of f(x), since that polynomial can be transformed into F(x) by a rotation of the plane.

If equations (10) have at least one solution, let us vary ao, al,***, a( by decreasing monotonically their moduli and by keeping fixed their arguments. Inequality (4) for x = xi thereby replaces (8), so F(x) has two positive roots, caused by the separation of the two roots which coin- cided at x1; these two roots become respectively greater and less than xl. Equations (10) are not affected by the process, so the roots of f(x) whose moduli are xi divide, and each double root separates into two distinct roots whose moduli are respectively greater and less than xi. Thus we have, with the aid of Pellet's Theorem,

f F(x) has two coincident positive roots at x1, then either f(x) has preciedy n roots whose moduli are less than xl and precisely k -n roots

* The reasoning thus far applies whether P(x) has one positive root or two positive roots.



64 J. L. WALSH.

whose moduli are greater titan xl, or f(x) has precisely 6 double roots whose modudi are equal to xi, precisely -n -d roots those moduli are less than xi, and precisely k - n -6 roots whose moduli are greater than xi.

In the latter case, of course, only powers of xa appear in f(x). 5. Applications of Pellet's Theorem. A number of applications

of Pellet's Theorem have been made in the literature. The following application is not difficult to make:*

An equation of k + 2 terms those terms of least degree are 1 and xP has always p roots whose moduli are not greater than a fixed number p (k) which depends only on k.

This result was proved for the case p = 1, k = 1, 2 by Landau,t using Pellet's Theorem. The result was proved for p = 1 and all values of k by Fejdr4 although by a method not involving Pellet's Theorem.

The following theorem is a generalization of these results, and can be established by the method indicated in the note by the author to which reference has just been made:

There eUist a number p(a, k) such that every polynomial

1 +alxY1+asx+ **+ aii-le+h1 I

+ a x& A+ a*+, axna + a16+2 x'b+ . + *+ ak X

has at least Pi roots whose moduli are not greater than ,u(a, k), no matter what may be the numbers h, a,, a2, * , awl-i, a,+i, **. ak, VI, V2, .*., Vj, provided merely that

0 < VI < V2 < ... < Vky

We add the remark that Pellet's Theorem seems to have important applications to the theory of Graeffe's method for the computation of the roots of an algebraic equation. Those applications have not yet been made in detail.?

* Walsh, Comptes Bndus, voL 176 (1928)? pp. 1861-1868. See alo Bienski, Comptes Rendu, voL 177 (1928), pp. 1198-1194.

t Annale de I'?ole Normale (8), voL 24 (1907), pp. 198-201. + Math. Antnale, voL 65 (1907-08), pp. 418-428. ? Se, howar, X. EL Carvallo, AMUodpraTiq powr la resolution it compe

ii.quatiotu algibriqus o f trasceadaue (Pauis, 1896).



Documents

On Pellet's Theorem Concerning the Roots of a Polynomial