Number Real

Embed Size (px)

Citation preview

  • 7/27/2019 Number Real

    1/14

    P A C I F I C JOURNAL OF MATHEMATICSVol. 102, N o. 1, 1982

    ON TH E N U M B E R OF REAL ROOTSOF POLYNOMIALS

    T H O M A S CRAVEN AND G E O R G E CSORDAS

    A general theorem concerning the structure of a certainreal algebraic curve is proved. Consequences of this theoreminclude major extensions of classical theorems of P lya andSchur on the reality of roots of polynomials.1* Introduction and definitions* Throughout this paper, all

    polynomials are assumed to have real coefficients. In 1916, GeorgeP lya wrote a paper [6] in which he considered two polynomials,/ () = = o diX*, am 0, and h{x) ?= o M S K 0, where n ;> m,both polynomials have only real roots and the roots of h areall negative. Under these hypotheses, he proved the followingt h e o r e m .

    THEOREM 1.1 (P lya [6]). The real algebraic curve Fix, y) =hf(y) + b&f'iy) + + bmxmf{m){y) 0 has m intersection pointswith each line sx ty + u = 0, where s^ O , 0, s + > 0 andu is real*

    P lya then noted tha t this theorem gives a unified proof ofthree important special cases regarding composite polynomials:

    (1.2)(a). Setting x = 1 gives a special case of the Hermite-Poulain theorem [5, Satz 3.1].

    (1.2)(b). Setting y = 0 gives a theorem of Schur [5, Satz 7.4].(1.2)(c). Setting x y gives a theorem of Schur and P lya [7,p . 107].

    Our main theorem, proved in 2 establishes some of the prop-erties of F(x, y) = 0 when we drop all restrictions on / and requireonly that all the roots of h be real (of arbit rary sign). In thelast section, we apply this theorem to obtain an extension ofTheorem 1.1 which states that , for h restricted as in P lya's theorem,there are at least as many intersection points as the number ofreal roots of / . As corollaries, we then obtain the full strengthof the Hermite- Poulain theorem and extensions of the theorems ofP lya and Schur to arbitrary polynomials / . P lya states that histheo rem is one of the most general known theorems on the realityof roots of polynomials. We believe this statement is still true

    15

  • 7/27/2019 Number Real

    2/14

    16 THOMAS CRAVEN AND GEORGE CSORDAS

    today, more t h a n sixty years later, especially for our much moregeneral version of it . The generalizations th at this provides of(1.2)(b) and (c) are used in [2] to solve an open problem of Karlinon characterizing zero- diminishing transform ations which he says"seems to be very difficult." They also are used in [1] for manyfurther applications to polynomials and ent ire functions. For theknown results on theorems of this type, the reader is referred to[4] and [5].

    Because the subject of algebraic geometry deals almost exclu-sively with algebraically closed fields, we have found it necessaryt o develop some new terminology. As much as possible, we followWalker's book "Algebraic Curves" [8]. In particular, we use th eword component to refer to an irreducible factor of the curve F(x, y).Since we deal only with the real points of a (not necessarily irredu-cible) algebraic variety, we need a term to refer to the individualparts of the curve F(x, y) = 0, even though these parts may notbe components, or even connected components since two part s mayintersect. For th is we shall use th e word branch. Classically,branches are only defined in a neighborhood of a point on the curve[3 , p. 39] where they have the usual analytic meaning. We shalluse the word branch in the following global sense: the branch oft h e curve F containing a given point on the curve is obtained bytravelling along the curve in both directions until reaching asingularity or return ing to the starting point; at a singularitytravel out along th e other arc of th e same local branch on whichyou arrived. Thus the branches are the "pieces" into which acircuit in the protective plane [3, p. 50] is broken by removing theline at infinity. We shall see in the next section th at the branchesin which we are most interested will always go to infinity in bothdirections. In particular, if / has simple roots, the curve F(x, y)in Theorem 1.1 is nonsingular and th e branches are just th e con-nected components. As is usual in algebraic geometry, all roots,branches and components will be counted including multiplicities,with the exception of Lemma 2.1.

    2* The main theorem* We begin with a lemma which describest h e variety F(x, y) = 0 when th e polynomial h(x) has degree one.I t will be used in an inductive step in the proof of the maint h e o r e m .L E M M A 2.1. Let f(y) = ? = o be any polynomialwith real coefficients and let a be any nonzero real number. Set

    F{x, y) = af(y) + xf'(y). Assume that f has r real roots alf , arand that f has s real roots u , / 3 , listed with multiplicity if

  • 7/27/2019 Number Real

    3/14

    ON TH E N U M B E R OF REAL RO O T S OF POLYNOMIALS 17and only if they are also roots off. Then the real variety F(x, y) = 0satisfies the following;

    ( 1) The variety consists ofs + 1 one- dimensional branches(not necessarily distinct), exactly r of which cross the y- axis.

    ( 2 ) The branches are disjoint unless f has a multiple realroot. At a root at of multiplicity k, we have k 1 componentscoinciding with the line y = at and one other branch through (0, at).Other than coincidental lines y = aif branches can only intersect onthe y- axis.

    ( 3) Each branch which intersects the y- axis will intersect itonly once and will also intersect each line x = c, where c is a con-stant.

    ( 4) All branches are asymptotic to straight lines from the sety = if i 1, 2, , s and y na~ x an_xa~ .Proof. Since F(x, y) af(y) + xf\y) = 0, we can write

    (2.2) x = - U . 2M whenever f(y) 0 .

    If / has a root at of multiplicity k, then F(x, y) has a factor of(y oiY'1. Canceling this from the numerator and denominator in(2.2) shows that there is still a branch passing through (0, at).Other th an th is situat ion , it is clear from (2.2) that th e branchesare all disjoint so th at claim (2) of t he lemma holds. From (2.2)we also see th at for large values of y, th e graph is asymptotic tothe line y = na~ x an_xa~ . Thus this line together with thelines y = < (arising from f'( t ) 0) form the set of asymptotes andclaim (4) holds. Since th ere are s points if claim (1) will followfrom th e argum ents above and (3). Now consider th e interval (y19y 2) where y and y2 are two consecutive roots of / ' which are notroots of / , or y1 oo if / ' has no such root less than y2, or y2oo if / ' h as no such root greater t han yle To prove claim (3), weneed only consider the case where there exists a point y0 in (y l9 y2)such that f(y0) = 0. N ote th at th ere can be at most one such pointby the choice of yx and y2 and Rolle's theorem. Assume first thatf'(yQ) 0. Then / ' has constant sign on (yu y2), so th at / is mono-tonic on th e interval. Since y0 is a simple root of / , equation (2.2)implies that x ranges from oo to oo (or vice versa) as y rangesfrom yx to y2. Thus this branch intersects each line x c wherec is an arb itra ry constant . Now assume th at y0 is a root of / ofmultiplicity k. The k 1 components y y0 satisfy (3). The re-maining branch through (0, yQ) behaves as in the case where f'(y0)is nonzero since f\y) can be factored as (y yQ) k~ ~ times a function

  • 7/27/2019 Number Real

    4/14

    18 THOMAS CRAVEN AND GEORGE CSORDAS

    which has constant sign on (ylf y2) and f(y) factors as (y y^ )k~ times a polynomial which has a simple root at y0. Since / is mono-tonic on each subinterval (yl9 y0) and (yQ, y2), equation (2.2) impliest h a t for this branch, x ranges from - oo to ^ (or vice versa) asy ranges from yx to y2. This concludes the proof of the lemma.We now proceed to our main result. The main point of thistheorem is to guarantee that no branch of F(x, y) = 0 can pass

    through two distinct roots of f(y) on the y- axis.THEOREM 2.3. Given h{x) = ?= o hx*, n^l, bo 0, bn = 1, withonly real roots and f(y) an arbitrary polynomial, form F(x, y) =

    f ^ o W 1 ^ ) ' Each branch of the real curve F{x, y) 0 whichintersects the y- axis will intersect the y- axis in exactly one pointand will intersect each vertical line x c, where c is an arbitraryconstant. If b0 = 0, the conclusion still holds for all branchs whichdo not coincide with the y- axis. Furthermore, if two brancheswhich cross the y- axis intersect at a singular point (x0, y0) not onthe y- axis, then these branches are in fact components of the formy y0 = 0, and thus coincide as horizontal lines.

    Proof The theorem follows from the following somewhatstronger statement which we shall prove:

    (2.4)(a). In the graph of F(x, y) = 0, there does not exist apair of disjoint paths from the y- axis to any point (x0, y0) off they- axis; (We say two paths are disjoint if their interiors have nopo in t in common.)

    (2.4)(b). Two coincidental paths from the y- axis to a point(x 0, y0) with x0 0 are necessarily horizontal lines y = yo;

    (2.4)(c). Multiple straight line components can intersect nobranches off the y- axis that are not coincidental straight lines.If b0 0, let br, r > 0, be the first nonzero coefficient in h andwrite g{y) = fr\y) . Then F(x, y) = xr =obi+rxg{)(y) has r compo-n e n t s which coincide with the y- axis and the new polynomials which

    replace h and / satisfy the hypotheses of the theorem. Thus wemay assume b0 0.N o t e that it suffices to prove statement (2.4) for points withx0 > 0. For if we replace h(x) by h( x), these points are reflectedabout the y- axis and the new polynomial h will still have only realroots.

  • 7/27/2019 Number Real

    5/14

    ON TH E N U M B E R OF REAL RO O T S OF POLYNOMIALS 19We now prove th e theorem by induction on n, th e degree of

    h. If n = 1, the conditions of (2.4) follow from Lemma 2.1. Nowassume n > 1 and that (2.4) holds for polynomials h of degree lessthan n and arbitrary polynomials / . Assume now that h has degreen. Since h has only real roots, we can factor h(x) = (x + aQ(x + an). Let = n and g(x) = (x + x) -( ?+ _i), so thatfc(a?)=(a? + a)g(x). Write g(x) = S=o dtxim, th e induction hypothesis saysthat th e real algebraic curve

    F^ix, y) = dof(v) + dxxf{y) + - + d^x^ ^iy) = 0satisfies statement (2.4).

    Next we define a variety in t h r e e variables by(2.5) G{x, yy t) = V , y) + xJ- F^t, y) = 0 .For any fixed value of = to> we can apply Lemma 2.1 to G(x, ?/, to)where th e polynomial in question is F^^U, y). In particular, th eintersection of each plane t t0 with th e variety G = 0 satisfiescondition (2.4) with th e i/ - axis replaced by the line x = 0 = t t0.We shall refer to this algebraic curve as Gt(x, y) = 0. F or anyfixedvalue of x x0, we obtain

    , ) = aFn_{t, y) + x^F^t, y)oy idtt = o dy

    By applying th e induction hypothesis to g(t) and the polynomial0 in th e ^/- plane with th e curveFn_x = 0. F or the remainder of this proof, we shall write 'Vaxis"in quotes to mean th e appropriate line in an y of t h e above planeswith x or t constant, for which we know that th e curve inducedby G = 0 satisfies condition (2.4).

  • 7/27/2019 Number Real

    6/14

    20 THOMAS CRAVEN AND G E O R G E CSORDASN ow we assume that condition (2.4)(a) is violated by the curve

    Fn = 0, but (2.4)(b) and (2.4)(c) both hold. Choose paths Pt and Pzin the x = t plane which give the contradiction to the theorem insuch a way that the area enclosed by the paths and th e ?/ - axis isminimal; in particular, the points yx *z y2 where the paths intersectt h e y- axis are consecutive roots of / . We also allow the possibilitythat the paths begin at a multiple root y1 = y2 of / with th e pathP x through 2/1 having greater y~values for x near zero than the pathP 2 through y2. We may also assume that (x0, y0) is a point forwhich x0 is the maximum value taken on by x in both paths.We shall consider the algebraic curve Gt = 0 for each fixed t.T he points of the paths Pi are obtained from the intersection ofth e branches of Gt with the plane x = t. Unfortunately, thesebranches do not always vary continuously with t. The followinglemma describes how discontinuities can occur.

    L E M M A 2.7. Set ft(y) = Fn_x(t, y), so that for fixed t equation(2.5) becomes(2.8) Gt(x, y) = aft(y) + xft(y) .I n Gt(x, y) 0, the 'partition of the points of the curve into branchesfor fixed t varies continuously with t except possibly at a pointt = s such that fs or / / has a multiple real root.(a) / / fs has a double real root at y0, then for values of tnear s, the curve Gt(x9 y) may have two branches crossing the y- axisnear y0 or may have two branches coming close to (0, yQ) and con-tained entirely on opposite sides of the y- axis. If f8 has a root yQof multiplicity m > 2, then for values of t near s, the curve Gthas r ^ m branches crossing the y- axis near yQ; the number m ris even; and there are at most (m r)/ 2 branches contained entirelyin the half plane x > 0.

    (b) If fs has a real root of multiplicity m > 1 at y0 withfs(Vo) = 0, then the curve G8 has a (multiple) horizontal asymptotey = y0. For nearby values of t, the asymptotes can separate, creat-ing new branches (not crossing the y- axis) or disappear in pairsfcausing adjacent branches which were separated by an even numberof asymptotes to join into a single branch. (We shall refer to thisas

  • 7/27/2019 Number Real

    7/14

    ON TH E N U M B E R OF REAL ROOTS OF POLYNOMIALS 21at th e i/- axis, corresponding to a multiple root of ft, or at infinity,corresponding to a multiple asymptote of even multiplicity (i.e., amultiple real root of / / ). Thus discontinuities can only occur incases (a) and (b). Theanalysis of the individual cases is clear fromLemma 2.1 and a consideration of how multiple roots of a poly-nomial and its derivative may vary as the polynomial variescon-tinuously.

    We now continue th e proof of the theorem. Assume first thatV = 22 and, as t increases, th e branches of Gt which generate th epaths in Fn trace out straight lines y = y1 = y2 in Fn_. For thetwo paths P1 an d P2 to meet at (x0, y0,x0), the corresponding branchesof Gt must vary to become a pair of straight lines y = yQ= y inG Q(%, V) = 0; but then Lemma 2.1 implies that there is at least onemore branch of GH passing through (0, y0,x0). Thus th e straightlines in Fn_x meet another branch a t (0, y0,xQ), in contradiction to(2.4)(c). Therefore, using (2.4)(b) and (c), we can conclude that ast increases from zero, th e branches of Go through y1 = y2 mustseparate, continuing to cross th e 2/-axis, for small values of t, att h e points where th e branches of Fn_x through yx = y2 meet th econstant t planes.

    As we analyze th e behavior of the branches of Gt, we shall belooking for a contradiction in one of th e curves Fn_1>cwhere c isa constant with 0 c

  • 7/27/2019 Number Real

    8/14

    22 THOMAS CRAVEN AND G E O R G E CSORDASaxis." Henceforth, we shall assume a > 0. Theanalysis for a < 0is similar andwill be omitted. Since a > 0, branches can only bejoined at infinity if the lower onedoes not cross the ^/ - axis. Ifsuch a branch lies between thebranch through yx and the branchthrough y2, then joining it to thebranch through y has essentiallyno effect, since thebranch through yx could have been continuouslyvaried to obtain the same result. So let (0, y, t0) be thepoint onthe "y- axis" at which twobranches of Gt come to the "?/ - a is," orappear as a pair of straight lines through a root of ft as inLemma2.7(a), prior to at least one of them joining with one of our originalbranches. Let yt(t) denote the^/ - coordinate at x = 0 of thebranchof Gt continuously obtained from the original branch through ' yt\this is well defined for t > 0 until the first time y^t) is a multipleroot of ft. Then we have three possibilities:(A) y,{Q> y >y2(t0)(B) y ^ y,{Q(C ) y y2(Q.

    We consider theeffect of each of these possibilities andshowthat no combination of them can allow the paths in Fn to occurwithout giving rise to a contradiction in one of the curves Fn_Uc,where 0 c^ x

    0, which, as we have seen, must satisfy theinduc-

    tion hypothesis.Assume first that situation (A) occurs alone. Then the only

    way to have thepaths Px andP2 meet each other is to have oneof the new real roots of ft vary to meet yt(t) and the other tomeet y2(t), and then subsequently joining these branches. (Notethat connections at infinity cannot be used since they require thatat least one branch being joined not cross the i/ - axis. Also thepolynomial // has an oddnumber of roots between y^t) and j/ 2(t) )But we obtain a contradiction as soon as both of the newbrancheshave joined to theoriginal ones since this generates a path in Fn_xfrom (0, yx) to (0, y2) passing through thepoint (0 (see equation (2.2)). Now we consider the effectof situation (B). The newbranch must meet yt(t) andpull off the2/- axis as t increases. The newbranch thus formed cannot joinat infinity with the branch coming from y2 (again because a > 0),hence to obtain the connection of P and P2, the branches must

  • 7/27/2019 Number Real

    9/14

    ON TH E N U M B E R OF REAL ROOTS OF POLYNOMIALS 23rejoin th e " / - axis" and part again. The lower one must then joineither with th e branch through y2(t)or with the upper branch fromsituation (A), while th e lower branch joins with the branch throughy2(t). (Asnoted above, applications of (A) have no substantial effecton th e argument.) This use of (B) prevents a contradiction fromoccuring in Fn^ by breaking th e path which would otherwise begenerated from yx to y2. But, if we let xm be th e maximum valueof x to which th e branch pulls away from the / - axis, say att tm, and note that xm 0, there exists a pair of straightlines in Gt which generate the path. These pairs of straight linesgenerate coincidental paths in Fn_ lf so by th e induction hypothesisth e paths in Fn^ are horizontal straight lines. But then we musthave straight lines in Fn also. Finally, for (2.4)(c), we assume that

  • 7/27/2019 Number Real

    10/14

    24 THOMAS CRAVEN AND GEORGE CSORDAS

    Fn has m multiple straight lines y = yQ which intersect some otherbranch of Fn at t = tQ > 0. Then, as above, we obtain m straightlines y = y0 in each Gt, 0 and at least m + 1 lines in G vBy Lemma 2.1(2), these generate m + 1 straight lines y = y0 in. F V L which intersect another branch of F ^ at (0, yo)f contradictingth e induction hypothesis. This completes the proof of the theorem.REMARK 2.9. All of the complications in the preceding proof

    are caused by the fact that it is possible to have branches in Fn_twhich cross the y- axis and have more t h a n one point with the samec-coordinate. This sometimes happens when the degree of h is lesst h a n the degree of / . We conjecture that this does not happen ift h e degree of h is greater t h a n or equal to the degree of / ; i tcertainly cannot happen if / has only real roots, for then a verticalline would intersect the curve in more points t h a n the degree. Aninteresting example in this regard is obtained by taking h(x) to be apower of x + 1 and f(y) = 11 - 16y + 8y2 - Ayz + y*(see Figure 2.1).Then using the notation of the proof above, the curve F2(x, y) has abranch which intersects some vertical lines in three points, but noo t h e r Fn has such a branch. The curve F 4 is also interesting; itcomes very close to having a singularity near (.6, -1.5) causing oneof the branches to intersect some horizontal lines in th ree points.

    V

    Fj (X. T) = 0

    F4tx , ) = o

    F5(X,Y)FIGURE 2.1.

  • 7/27/2019 Number Real

    11/14

    ON TH E N U M B E R OF REAL ROOTS OF POLYNOMIALS 25Also of interest is the fact that, in going from F2 to F5, the numberof branches increases from two to four.

    3* Applications* We begin with a direct extension of Theorem1.1 of P lya to arbitrary polynomials / .

    THEOREM 3.1. Let h(x) = ?= o M* be a polynomial of degree nwith only real negative roots and f{y) an arbitrary polynomialwith r real roots and degree at most n. The real algebraic curveF(x, y) = ?= o &i / )(l/) = 0 has at least r intersection points witheach line sx ty + u = 0 where s ^ 0, t ^ 0, s + t > 0 and u isreal.

    Proof. Write f(y) =o (&

  • 7/27/2019 Number Real

    12/14

    26 THOMAS CRAVEN AND G E O R G E CSORDASProof. Assume first that zero is no t a root of h. Then the

    only difference in hypotheses from the previous theorem is that thedegree of / may be greater than th e degree of h. The proof isthus th e same as the previous proof, except that the polynomial inequation (3.2) may have zero as a root. That is, the branches ofF(xf y) may have horizontal asymptotes. Hence lines of zero slopemay miss the branches, but lines of positive slope will still intersectany branch which crosses the #- axis.

    If zero is a root of h, let bs, s > 0, be the first nonzero coeffi-cient of h an d write g(y) = f (y). Note that g has at least r sreal roots by Rollers theorem. Then F(x, y) - ? ?= o bt+tfg y)has s components coinciding with th e y- B.xi& and th e remainder ofth e curve has at least r s intersections with each line of positiveslope. Since every line of positive slope must cross the y- axia, thetheorem holds.

    We can now obtain thecorollaries mentioned in th e introduction.The first is a corollary of Theorem 2.3.

    COROLLARY 3.4 (Hermite- Poulain). Let f{x) be a polynomialwith real coefficients and h{x) = b0 + b x + + bnxn a polynomialwith only real roots. Then the polynomial g(x) = ?= o &*/ (i)( ) hasat least as many real roots as f. I f f has only real roots, thenevery multiple root of g is also a multiple root of f.

    Proof. The real curve F(x, y) = ?= o MV{i> (2/) = 0 has a branchcrossing the ^/-axis for each real root of / . By Theorem 2.3, thesebranches intersect the line x = 1. The polynomial ?= o i/ (

  • 7/27/2019 Number Real

    13/14

    ON TH E N U M B E R OF REAL ROOTS OF POLYNOMIALS 27fs)(y) and apply Theorem 3.1 to b^x'g^iy), setting y = 0. ByRolle's theorem, the number of real roots of / is at most equal tos plus the number of real roots of g. Thus the conclusion againholds.

    A slight change in the hypotheses of this corollary leads us toCorollary 3.6 in which we obtain the result separately for thepositive and negative roots. In [2], Corollary 3.6 is used to solvean open problem of Karlin.

    COROLLARY 3.6. Let h(x) ?= o bxx*be a polynomial of degreen with only real negative roots and let f(x)==^T=oaixif m n.Then the polynomial U Aff* has at least as many positive rootsas f has positive roots and at least as many negative roots as fhas negative roots. The multiplicity of zero as a root is the same.

    Proof. This follows from the proof of Theorem 3.1; since theasymptotes all have negative slopes, any branch which crosses the /-axis at a positive root of / will cross the cc-axis at a positiver o o t of ^ iilaibixif and similarly for negative roots.

    COROLLARY 3.7. Let h(x) = ?= o * be a polynomial with onlyreal nonpositive roots and let f be an arbitrary polynomial withreal coefficients. Then the polynomial ?= o b^f^ x) has at least asmany real roots as f.

    Proof. Set y = x in Theorem 3.3.This corollary is the starting point for our work in [1], where

    it is used to obtain many results concerning polynomials and entirefunctions.

    REFERENCES1. T. Craven and G. Csordas, An inequality for the distribution of zeros of polyno-mials and entire functions, Pacific J. M a t h . , 95 (1981), 263-280.2. , Zero- diminishing linear transformations, Proc. Amer. Math. Soc, 80(1980), 544- 546.3. J. Coolidge, A Treatise on Algebraic Plane Curves, Dover, New York, 1959.4. B. Ja . Levin, Distribution of zeros of entire functions, Amer. M a t h . Soc. TransL,P ro v ide nce , R.L, 1964.5. N . Obreschkoff, Verteilung und Berechnung der Nullstellsn Reeler Polynome, VebD e uts e he r Verlag der Wissenschaften , Berlin, 1963.6. G. P lya, Uber algebraische Gleichungen mit nur reelen Wurzeln, Viertelj chr.N a t u r fo r s c h . Ges. Zurich, 61 (1916), 546-548.7. G. P lya and J. Schur. Uber zwei Arten von Faktorenfolgen in der Theorie deralgebraischen Gleichungen, J. Reine Angew. M a t h . , 14 4 (1914), 89- 113.

  • 7/27/2019 Number Real

    14/14

    28 T H O M A S C R A V E N A N D G E O R G E C S O RDA S8. R. Walker, Algebraic Curves, Dover, New York, 1950.

    Received May 19, 1981. The first auth er was pa rtially supported by NSF gra ntMCS79-00318.UNIVERSITY OP HAWAII AT MANOAHONOLULU, HI 96822