ON A THEOREM BY J. L. WALSH CONCERNING THE MODULI OF ROOTS OF ALGEBRAIC EQUATIONS

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1 ON A THEOREM BY J. L. WALSH CONCERNING THE MODULI OF ROOTS OF ALGEBRAIC EQUATIONS ALEXANDER OSTROWSKI I 1881 A. E. Pellet published 1 the followig very useful theorem: If the polyomial F(z) = 0 O + ai z + os s <IA-I s*" 1 - a* «* + fl*+i I * k+l + + \a \ z, 0 < k <, a 0 a?* 0, has two positive roots xi ad x 2 (xi <x 2 ), the the polyomial (2) ƒ(*) = a 0 + aiz + a 2 z a z has o roots i the aulus Xi<\z\ <X2 ad precisely k roots i the circle While Pellet's proof for his theorem utilizes the theorem of Rouché, J. L. Walsh published i aother more direct proof ad established i the same memoir a sort of coverse of Pellet's theorem. To formulate his result, cosider the set g of all polyomials (3) ƒ0) = ÖO + aiz + a 2 z a z which correspod to give moduli of coefficiets. All polyomials of are obtaied from oe of them, ƒ(*), if the factors o, i,, e i the expressio (4) 0 a 0 + eiaiz + 2d2Z e a z assume idepedetly all values of modulus 1. Let 2ft be the set of roots of all polyomials i fç. It is immediately see that if 9)? cotais a umber a it also cotais all umbers with the modulus \a\. It was proved by Cauchy that all roots of (4) lie o or withi the circle z =yi, where yi is the sigle positive root of the polyomial ao + tfi! z + + a -i z ~ l - a z ad that all roots of (4) lie o or are exterior to the circle z =3^2, where y2 is the sigle positive root of the polyomial 1 A. E. Pellet : Sur u mode de séparatio des racies des équatios et la formule de Lagrage, Bulleti des Scieces Mathématiques, (2), vol. 5 (1881), pp J. L. Walsh: O Pellet's theorem cocerig the roots of a polyomial, Aals of Mathematics, vol. 26 (1924), pp

2 ROOTS OF ALGEBRAIC EQUATIONS 743 a 0 1 0i z - I a j z. Sice the umbers y\ ad y% obviously belog to 2ft, it follows that 2ft lies i the closed aulus y 2^ \z\ Syi- Suppose ow that a>0 is ot cotaied i 2ft. The, if C is the circle g = a, all polyomials of g cotai the same umber k of roots withi C. Ideed, if we vary cotiuously the factors e i the represetatio (4), the roots of (4) vary also cotiuously ad their umber withi C remais the same sice oe of them is able to pass C. Now the theorem of Walsh rus as follows : Ay positive a that is ot cotaied i 2ft, ad f or which the umber of roots of (4) withi the circle \z\ =a is precisely k (0<k<), is cotaied betwee the two positive roots of the polyomial (1), that is to say: a k a k > ( a a*_i a*" 1 ) + (K + i a* a» a"). As the proof give by Walsh of this importat result is ot complete, 3 we give i what follows aother proof proceedig o differet lies. Suppose that, cotrary to (5), (6) a* a k ^ ] a v \ a v \ v=q,p?sk the, as a is ot cotaied i 2ft, we have eve (7) a k a h < a v \ a\ v=*q,v? k O the other had it follows from our hypothesis that if e ru idepedetly through all costats of modulus 1, (8) X) war 7* 0, ad therefore 3 Walsh allows i his proof the roots of/(s), which are i absolute value less tha a, to vary cotiuously ad mootoically (i absolute value) ad to approach 0. But durig this variatio the polyomial f(z) does ot ecessarily remai i the set 3 ad the correspodig sets $?o for the polyomials thus obtaied could very well cotai a i the set of the roots, so that the expressio (1) eed ot remai positive for 2 = a, as is assumed i Walsh's proof.

3 744 ALEXANDER OSTROWSKI [October (9) \ak\a k 5* ]C "^ö v I pssqtvj) k But from (7) ad (9) it follows that for all values of e i questio (10) \a k \a k < For, if we have for a particular set of e -values, e v : (ID a k \ a > E 0 v e v a v a p*=0, V9e k we see from (7) ad (11), that the right-had side of (10) becomes equal to a k \ a k for some other set of e, cotrary to (9). Now it follows from (10) that the polyomials (12) have o roots at a ad therefore o roots o the circle C. O the other had, we have obviously by (10) everywhere o C: \f*(z)\ >\a k z k \. Hece, by the theorem of Rouché, sice f*(z) a k z k has exactly k roots iside C, the same is true for ay polyomial ƒ*(z). The result arrived at may be aouced i the followig form: If o,, &_!, 6fc.fi, -,, ru idepedetly through all values of modulus 1, let fc~l <K*0 = X e v a v z v, f(z) = X e v a v z'; the the differece <j>(z) \p(z) does ot vaish o C ad has exactly k roots iside C. It follows, that i particular k-l S ^fl ' j^o ^ w 22 e»^a v=fr+l But the it is impossible that we have simultaeously for oe particular set e ' of e ad for aother, e ", 22 x<w > 22 ^fl*

4 I94i] ROOTS OF ALGEBRAIC EQUATIONS 745 ^e v f a v a v < X) e"a v a v Hece oly two cases are possible : A. We have always fc-l z *va v a v K=0 > X) " a " a y-jfc+1 ad therefore everywhere o C: </>(s) > yp{z) \. But the, by the theorem of Rouché, 4>{z) has iside C the same umber of roots as <f>(z) \l/(z), that is k, ad this is impossible, <j>(z) beig of degree k 1. B. We have always X) e v a v a v < 23» a v av ad therefore everywhere o C: \ 4>{z) \ < J \f/(z). But the x[/(z) would have iside C exactly k roots, while \p(z) vaishes at 2 = 0 with the multiplicity k + 1 at least. We see that (7) ad (6) are impossible ad the proof of (5) is completed. The theorems of Pellet ad Walsh dealt with i the precedig paragraphs allow us to describe immediately the set 50Î. Cosider the + \ equatios: (13) (14) (15) X) a z' - v=k+l v=0 v=0 X) U, 12 v - a J z v = 0, z v = 0, k = 1,,» 1, flo = 0. If, as we will assume, a 0 a ^0, each of the equatios (13), (15) possesses oe positive root p resp. p 0, ad we have po^p sice p is the exact upper ad p 0 the exact lower limit for the moduli of 9JÎ. As to the equatios (14), every oe of them possesses either two positive roots or exactly oe double positive root or o positive roots at all. Strike out the equatios (14) correspodig to the two last cases; each of the remaiig equatios (14) possesses two positive roots, Xk, x

5 746 RICHARD COHN [October (xk<xk). The we obtai 9ft by removig from the plae of z the two circular domais \z\ <p 0 ad z\ >p ad all auli Xu< \z\ <x{, correspodig to the equatios (14) with two differet positive roots. It is the a cosequece of Walsh's theorem that the differet (ope) itervals (x&, xl) have o poits i commo ad lie i the iterval (po, p); moreover, if for two of these itervals (#&, Xk), (x m, Xm) we have k<m, the we have certaily x{ tkx mi ad it is easily see that we have eve x{ <x m. As Walsh remarks, his proof of Pellet's theorem remais valid also i the case of a power-series ad of its roots iside the circle of covergece. It is hardly ecessary to remark that our proof of Walsh's theorem also applies mutatis mutadis to a power series, if we oly cosider its roots withi the circle of covergece. UNIVERSITY OF BASEL SOME EXCEPTIONAL VALUES OF THE LIMIT OF THE RATIO OF ARC TO CHORD RICHARD COHN It was observed by E. Kaser 1 that i the complex euclidea plae the limitig value of the ratio of the arc of a curve to its chord, while oe ed poit of the arc is fixed, ad the other approaches it alog the curve, is ot always uity; but assumes for aalytic curves taget to a miimal lie, a sequece of real values,.94,.86,.80,. These values are fuctios of the order of cotact oly, ad approach zero as the latter icreases. I this ote we shall describe two similar situatios which occur i real spaces. The problem i the case of the K plae 2 has bee worked out i Professor Kaser's Semiar i Geometry. 3 I this plae the legth of the curve y =f(x) passig betwee poits of abscissae xi, X2, i that order, is give by X1 \dx/ 1 E. Kaser, The ratio of the arc to the chord of a aalytic curve eed ot approach uity, this Bulleti, vol. 21 (1914), pp Similar questios for three dimesios are discussed i E. Kaser, Complex geometry ad relativity, theory of the "rac" curvature, Proceedigs of the Natioal Academy of Scieces, 1932, p Kaser, Trihorometry, a ew chapter of coformai geometry, Proceedigs of the Natioal Academy of Scieces, vol. 23, p R. Colema, S. Jablo ad D. Mittlema obtaied the results for the K plae give below.