extremal polynomials in smale’s mean value conjecture

Computational Methods and Function TheoryVolume 6 (2006), No. 1, 145–163

Extremal Polynomials in Smale’s Mean Value Conjecture

Edward Crane

(Communicated by Stephan Ruscheweyh)

Abstract. Let p be a non-linear complex polynomial in one variable. Smale’smean value conjecture is a precise estimate of the derivative p′(z) in terms ofthe gradients of chords between z and a stationary point on the graph of p. Theproblem is to determine the correct constant in the estimate, but despite theapparent simplicity of the problem only a small amount of progress has beenmade since Stephen Smale first posed it in 1981. In this paper we establishthe existence of extremal polynomials for Smale’s mean value conjecture, andestablish a geometric property of the extremals.

Keywords. Complex polynomials, critical points, minimax problems.

2000 MSC. Primary 26C10; Secondary 30C15, 49J35.

1. Introduction and main results

Let p be any polynomial with coefficients in C. Then ζ ∈ C is a critical point of pwhen p′(ζ) = 0. Its image p(ζ) is the corresponding critical value. In 1981 Smaleproved the following result about critical points and critical values of polynomialsin connection with algorithms for finding roots of polynomials.

Theorem (Smale [18]). Let p be a polynomial of degree N ≥ 2 over C andsuppose that z ∈ C is not a critical point of p. Then there exists a criticalpoint ζ of p such that

(1)

∣∣∣∣ p(ζ)− p(z)

p′(z)(ζ − z)

∣∣∣∣ ≤ 4.

Thus the derivative of p at the point z can be estimated in terms of the gradientsof ‘chords’ on the graph of p from (z, p(z)) to the stationary points (ζ, p(ζ)). Inthis way it is analogous to the Mean Value Theorem for continuously differen-tiable real functions of a real variable.

Received November 21, 2005.Research carried out while the author was a Junior Research Fellow at Merton College, Oxford.

ISSN 1617-9447/$ 2.50 c© 2006 Heldermann Verlag

146 E. Crane CMFT

We will usually deal with the normalized case in which z = 0, p(0) = 0, p′(0) = 1and p is monic. Any pair (p, z) can be brought to this situation by affine changesof variable in the domain and range, and the quantity on the left-hand side ofinequality (1) is invariant under such transformations. We use the notation GN

for the space of polynomials satisfying these conditions.

The complex numbers p(ζ)/ζ, where ζ ranges over all the critical points of p,will be called the objective values of p. For p ∈ GN we define

S(p) := min

{∣∣∣∣p(ζ)

ζ

∣∣∣∣ : p′(ζ) = 0

}.

Any critical point ζ which attains this minimum will be called an essential criticalpoint of p. Smale made the following conjecture.

Smale’s Mean Value Conjecture. For any polynomial p ∈ GN ,

(2) S(p) ≤ 1− 1

N.

Furthermore the unique equality case is p = χN , where

χN(z) = zN + z.

1.1. Known cases of the conjecture. The bound S(p) ≤ 1− 1/N is knownto hold for polynomials p that satisfy any of the following conditions.

• N = 2, 3 or 4. The case N = 4 is due to J.-C. Sikorav, and was strengthenedby Tischler [19]; see also [15, Thm. 7.2.9]. In a forthcoming paper [8] wedispose of the case N = 5 by a rigorous computational method, which isunenlightening as far as the general case is concerned.

• All critical points of p lie on a line through 0. In this case the better uniformbound S(p) ≤ e− 2 is known: see [17, §10.4.11] or [15, Thm. 7.2.5].

• All zeros of p have equal modulus. This was settled by Tischler, [19],including the equality case.

• p is sufficiently close to χN . David Tischler [20] showed that χN is indeedan isolated local maximum for the function S on GN ; see also [15].

Furthermore, the uniform bound S(p) ≤ 1 is known if p satisfies any of thefollowing conditions.

• All critical points of p have equal modulus, or all critical values of p haveequal modulus. This is due to Cordova and Ruscheweyh [6], as a conse-quence of their General Subordination Theorem for polynomials; see also[17, §10.4.4].

• The sub-level set {z : |p′(z)| ≤ 1} contains a line segment joining 0 toa critical point ζ. In this case we can obtain the inequality simply byintegrating along such a line segment. A sufficient condition for this is thatthe lemniscate {z : |p′(z)| = 1} has a convex oval passing through 0, ormore generally an oval that is starlike about 0 or about a critical point.

6 (2006), No. 1 Extremal Polynomials in Smale’s Mean Value Conjecture 147

Such conditions were investigated by Kats [11]. Generically there is oneoval passing through 0, but as J. L. Walsh pointed out [21], this oval canbe made to approximate any Jordan curve. This makes it seem difficult toimprove Smale’s method of applying Koebe’s One Quarter Theorem to abranch of the inverse of p defined on a disc about 0.

• All objective values are equal. This case is due to M. Shub. Note that thepolynomial χN has this property. After rescaling in this case, we can obtaina polynomial p for which p(0) = 0 and all critical points are fixed pointsof p. Kostrikin called such a polynomial conservative. Shub pointed outthat in this case we have S(p) ≤ 1. The simplest proof is to apply Smale’sinequality (1) to the iterates of p. In fact we can show that S(p) < 1, asfollows. The universal cover of C \ {ζ : p′(ζ) = 0} is the unit disc D. Themulti-valued function p−1 lifts to a single-valued analytic map from D to D

that fixes 0 and omits the lifts of the non-critical preimages of the criticalvalues of p. Then S(p) < 1 follows from the Schwarz-Pick lemma. Tischlershowed that there are precisely

(2N−2N−1

)monic conservative polynomials of

degree N .

See [15, 19, 20, 17] for further discussion of the above results.

1.2. What is known for general degree. By considering a critical point ofminimal modulus, Schmeisser [16] showed that

S(p) ≤ 2N − (N + 1)

N(N − 1),

which gave the best known bounds at the time for degrees 5, 6, and 7. In [2],Beardon, Minda and Ng observed that Smale’s proof of K(d) ≤ 4 could besharpened to K(d) ≤ 41−1/(d−1) using an estimate of the hyperbolic density ina certain plane domain. Recently Conte, Fujikawa and Lakic [4] showed thatK(d) ≤ 4(d− 1)/(d + 1) by an ingenious repeated use of the bound |a2| ≤ 2 forthe second coefficient of a schlicht function. The two ideas have been combinedby Fujikawa and Sugawa [5] to give the bound

K(d) ≤ 4

(1 + (d− 2)41/(d−1)

d + 1

).

In all of these bounds, the upper bound on K(d) is of the form 4 − O(1/d) asd → ∞. In a forthcoming paper [9] we use the results of the present paper toprove a bound of the form

K(d) ≤ 4− C√d.

Ng [12] showed that for odd polynomials in GN we have S(p) ≤ 2, and this wasextended in [4] to apply to polynomials with vanishing coefficient of z2.

Finally, the conjecture is certainly decidable for each degree, in the followingsense. Tarski’s Theorem on the decidability of the first-order theory of the reals

148 E. Crane CMFT

says that there is an algorithm to decide whether there exists a real solutionof any finite system of polynomial equations and inequalities in finitely manyvariables with real coefficients. This is known as quantifier elimination in thereals. (See the first chapter of [3] for an explicit description of the algorithm, or[14, Thm. 3, p. 604] for a shorter exposition requiring more background in logic.)We deduce that for each value of N there exists an algebraic proof of Smale’sinequality for polynomials of degree N with the best possible constant, whateverit may be. To be absolutely clear, we are not saying here that the constant isnecessarily equal to 1 − 1/N . Furthermore, it is a consequence of the proof ofTarski’s Theorem that the best possible constant must be an algebraic number(possibly of very large degree). However, algorithms for quantifier eliminationare of exponential complexity in the number of real variables used, and in theobvious application to this problem there are 2N − 1 real variables, so it isunlikely that any further cases can be settled in this way. Moreover, quantifierelimination does not seem to lead to a proof that applies for all values of N atonce.

1.3. Main results. For each N ≥ 2 we denote by K(N) the least upper boundof S(p) over all non-linear polynomials p of degree N with p′(0) �= 0. We call apolynomial p of degree N extremal if S(p) = K(N).

Our main results are summarized in the following theorem about extremal poly-nomials.

Theorem 1. For each degree N ≥ 2, the following hold.

(i) There exists an extremal polynomial of degree N for which all critical pointsare essential, i.e. all the objective values have modulus K(N).

(ii) K(N + 1) ≥ K(N).(iii) If K(N +1) > K(N) then the set of extremal polynomials in GN is compact.(iv) If K(N +1) > K(N) then all the critical points of any extremal polynomial

in GN are essential.

The existence of an extremal polynomial is a key starting point for any approachto Smale’s conjecture that works by considering properties of extremal polyno-mials. The condition that all critical points are essential can be thought of as ageometric condition; we hope it might open new ways of attacking the conjectureusing methods of geometric function theory.

Parts (iii) and (iv) of the theorem might be useful in an inductive proof of Smale’smean value conjecture. For once one knows that K(N − 1) = 1 − 1/(N − 1),then since K(N) ≥ 1− 1/N , the hypothesis K(N) > K(N − 1) is satisfied, andthe induction step needs only to consider an extremal polynomial all of whosecritical points are essential, in the hope of proving that it must be equivalentafter rescaling to χN(z) = zN + z. Of course in trying to prove a uniform bound


S(p, x) ≤ K for polynomials of all degrees, we can already concentrate attentionon the case where all critical points are essential.

Theorem 1 is consistent with the strong form of Smale’s conjecture that says thatthe equality case is essentially unique. In a separate paper [8] we will consider thecase N = 5, giving a quantitative version of his proof. This allows us to carryout a computer proof that χ5 is the unique extremal polynomial in G5, usingverifiable interval arithmetic. In principle the same method would allow one toprove Smale’s conjecture for each degree N in turn by verifying finitely manystrict inequalities involving the values of polynomials at finitely many points inthe parameter space; however it is not clear whether it will be possible to doeven the case N = 6 in this way since the number of points to be checked growsrapidly with the dimension of the parameter space.

2. Proof that K(N + 1) ≥ K(N)

Let ε > 0 and consider any polynomial p ∈ GN such that S(p) > K(N)− ε. Wedefine another polynomial r(z) of degree N + 1, which depends on a parameterζ �= 0 as follows:

r(z) =

∫ z

0

(1− w

ζ

)p′(w) dw.

Note that r(0) = 0, r′(0) = p′(0) = 1, and the critical points of r are those of ptogether with ζ. If we write the objective value corresponding to a critical pointζi of p as λi, then its objective value as a critical point of r is

r(ζi)

ζi

=1

ζi

∫ ζi

0

(1− w

ζ

)p′(w) dw = λi −

1

ζ

(1

ζi

∫ ζi

0

wp′(w) dw

).

Observe that this converges to λi as |ζ| → ∞. The objective value correspondingto ζ is easily found to have the asymptotic behaviour ζN/(N + 1) +O(ζN−1) as|ζ| → ∞. Therefore the minimum modulus of the objective values of r canbe made to exceed K(N) − 2ε by taking ζ sufficiently large. It follows thatK(N + 1) ≥ K(N).

Notice that in this proof we neither used nor proved the existence of an extremalpolynomial of degree N or N + 1.

3. The algebraic geometry of Smale’s conjecture

Smale’s inequality (1) shows that the vector of objective values of a polynomialis constrained to lie in some proper subset of C

N−1. This occurs because theobjective values are invariant under two kinds of affine linear transformationof p. Let A : z → az + b and B : z → cz + d where a, b, c, d ∈ C and a, c �= 0. Wesay that the pair (A ◦ p ◦B, B−1 (x)) is a rescaling of the pair (p, x). It is easyto check that the objective values of the rescaled pair coincide with those of theoriginal pair. It therefore suffices to consider only the normalized case in which

150 E. Crane CMFT

x = 0, p(0) = 0, and p′(0) = 1. This is incidentally a convenient way of forcingthe condition that 0 is not a critical point. We will use this normalization throughthe rest of this paper, and make the abbreviation S(p) = S(p, 0). Furthermore, qwill always denote the polynomial of degree N − 1 such that p(z) = zq(z). Wehave already introduced the notation GN for the parameter space consisting ofmonic polynomials p of degree N such that p(0) = 0, p′(0) = 1. We haveGN ≡ C

N−2, taking the unconstrained coefficients of p as co-ordinates. Anypair (p, x) with deg p = N can be rescaled to a pair (p, 0) with p ∈ GN . Thisnormalized rescaled polynomial p is not in general unique, because if ω is aprimitive (N − 1)th root of unity then z → ω−1p(ωz) defines another element ofGN , and this is only equal to p in the case p = χN , defined above. The moduli ofthe objective values are also preserved by an involution of GN which sends p tothe polynomial z → p(z), where the bar denotes complex conjugation. The fixedpoints of this involution are the real polynomials in GN . Thus the function S isconstant on orbits of a real-linear action of the dihedral group of order 2(N − 1)on GN .

To each point ζ = (ζ1, . . . , ζN−1) ∈ CN−1, we associate a monic polynomial p

defined by

p(z) := N

∫ z

0

N−1∏i=1

(w − ζi) dw = zN + a1zN−1 + · · ·+ aN−1z.

Notice that p(0) = 0 and the points ζi are the critical points of p, repeatedaccording to multiplicity. We define an affine morphism (i.e. a map whose co-ordinates are polynomials) α : C

N−1 → CN−1 by

α : (ζ1, . . . , ζN−1) → (a1, . . . , aN−1).

We can express the coefficients ai explicitly in terms of the elementary symmetricfunctions of the critical points:

ak =N

N − 1− k(−1)k Symk (ζ1, . . . , ζk) .

Observe that p belongs to GN if and only if aN−1 = 1. This occurs precisely when(ζ1, . . . , ζN−1) belongs to the affine hypersurface HN that is defined by

HN :=

{(ζ1, . . . , ζN−1) :

N−1∏i=1

ζi =(−1)N−1

N

}.

The restriction α : HN → GN is surjective. Both α and its restriction to HN arefinite morphisms which are generically (N − 1)!-to-one. The symmetric groupSN−1 acts on C

N−1 by permuting co-ordinates, acting faithfully on the genericfiber of α.

Next we define

βi := ζiN−1 + a1ζi

N−2 + · · ·+ aN−1ζi0.


When ζi �= 0, we have βi = p(ζi)/ζi. Define the objective map β : CN−1 → C

N−1

byβ : (ζ1, . . . , ζN−1) → (β1, . . . , βN−1).

which maps any vector of non-zero critical points to the corresponding vectorof objective values, and it has the advantage that it is a polynomial map so itis defined even when some of the co-ordinates ζi are zero. When we permutethe labels of the critical values, the labels of the objective values are changedby the same permutation. So if L : C

N−1 → CN−1 is a linear map that acts

by permuting co-ordinates, then β ◦ L = L ◦ β. In more concise language, β isequivariant with respect to the permutation action of the symmetric group SN−1

on CN−1.

Lemma 2. The objective map β has the following properties.

(i) Each co-ordinate βi is a homogeneous polynomial of degree N − 1, and β isequivariant with respect to the permutation action of SN−1 on C

N−1.

(ii) β descends to a holomorphic map β : PN−2(C) → P

N−2(C).(iii) dim (β (HN)) = N − 2.(iv) The restriction of β to HN is a proper map.(v) β maps HN onto an irreducible hypersurface XN ⊂ C

N−1.

We call XN the objective hypersurface. The proof of Lemma 2 occupies Section3.1.

Recall our notation S(p) for the minimum modulus of the objective values of p,when p ∈ GN . We see that S ◦ α is continuous since it is the minimum of theN − 1 continuous functions |βi|. Because α is an open and surjective map, wededuce that S : GN → R is also continuous.

Smale’s inequality (1) can now be restated as

(3) K(N) = sup{min |xi| : (x1, . . . , xn−1) ∈ XN} < 4.

We will prove in Section 4 that the existence of a finite upper bound here impliesthat there exists a point of XN at which the supremum is achieved, such that|x1| = |x2| = . . . = |xN−1|. In other words, there exists an extremal polynomialof degree N all of whose objective values have modulus K(N).

3.1. Properties of β: proof of Lemma 2. Part (i) has already been ex-plained. For part (ii), note that the polynomials βi are all homogeneous of

the same degree, so β exists as a rational map; we have to show that its in-determinacy locus is empty. In other words we must show that the βi do notsimultaneously vanish except at the origin. So suppose that at some non-zerovector (ζ1, . . . , ζN−1) all the βi vanish, and let p = α(ζ1, . . . , ζN−1) be the asso-ciated polynomial. Then each root of p′ is a root of q and therefore also of p.Thus any root of p′ of multiplicity k must be a root of p of multiplicity k + 1.This can only be consistent with deg p = 1 + deg p′ if all the ζi are equal to zero.

152 E. Crane CMFT

This is a contradiction, so we are done. It follows from (ii) that β is surjectiveand is a finite projective morphism.

We now come to part (iii) of the lemma. In Section 3.2 we derive a polynomialrelation that is satisfied by the objective values. In other words, we will derive anequation for an affine hypersurface XN ⊂ C

N−1 that contains the image β (HN).

Remark. In fact one can prove part (iii) of the lemma without explicitly derivingan equation. A standard result in algebraic geometry over C is that the imageof an affine variety of dimension k under an affine morphism (i.e. a mappingwith polynomial co-ordinates) is contained in an affine variety of dimension atmost k. A subset of C

n is called constructible if it belongs to the Booleanalgebra generated by affine varieties and their complements in C

n. Chevalleyproved that the image of a constructible set under an affine morphism is againa constructible set, of no larger dimension. Chevalley’s theorem is also knownas quantifier elimination over C. It is closely related to the Theorem of Tarskimentioned in the introduction, although because C is algebraically closed, it issomewhat simpler and historically preceded it. In our application, we find thatthe image of the irreducible hypersurface HN under β is a constructible set ofdimension at most N − 2.

To show that the dimension of β (HN) is at least N − 2, we check that thederivative of β at the point ζ = (1, 1, . . . , 1) is of full rank. By the homogeneityof β we find the same is true at ζt = (t, t, . . . , t), where tN = (−1)N−1/N . Sinceζt belongs to the smooth hypersurface HN , the restriction of β to HN has imageof dimension at least N − 2.

At the point ζ = (1, 1, . . . , 1) it is easy to compute the partial derivatives of β.For i �= j,

∂βi

∂ζj

= −(

N−20

)N − 1

+

(N−2

1

)N − 2

−(

N−22

)N − 3

+ · · ·+ (−1)N−1

(N−2N−2

)1

=(−1)N−1

N − 1.

For i = j,

∂βi

∂ζi

=

[−

(N−2

0

)N − 1

+

(N−2

1

)N − 2

−(

N−22

)N − 3

+ · · ·+ (−1)N−1

(N−2N−2

)1

]

+

[(1− 1

N

) (N − 1

0

)−

(1− 1

N − 1

) (N − 1

1

)+ · · ·

+(−1)N−1

(1− 1

1

) (N − 1

N − 1

)]

=(−1)N−1

N − 1+

(−1)N−1

N.


It follows that the determinant of the derivative of β at ζ is non-zero. An easyway to see this is to note that (1−N)/N is not an eigenvalue of the (N − 1) by(N − 1) matrix each of whose entries is 1.

It follows from part (iii) that β(HN) is contained in a unique irreducible hyper-surface XN . The map β is equivariant with respect to the action of SN−1, so theimage of β and hence also XN are invariant under this action. We can restatethis purely algebraically as follows:

for each N ≥ 2, there is a unique non-constant irreducible symmetriccomplex polynomial QN such that QN(λ1, . . . , λN−1) = 0 whenever(λ1, . . . , λN−1) is the vector of objective values of some p ∈ GN .

In contrast, a polynomial can be constructed with any given set of critical pointswith given multiplicities or with any given set of critical values with given mul-tiplicities of the corresponding critical points: see [1].

Part (iv) of Lemma 2, that the restriction of β to the hypersurfaceHN is proper, isa consequence of the following result, since we already know that β is continuous.

Lemma 3. Suppose that p ∈ GN and all the objective values of p satisfy |λi| ≤M .Then all the critical points satisfy

|ζi| ≤ 4N/(N−1)M1/(N−1).

Proof of Lemma 3. Consider a critical point ζ of p such that p(ζ) is a criticalvalue of maximal modulus, and let the corresponding objective value be λ; weknow λ ≤ M . Define a domain U in the Riemann sphere as follows:

U := {z ∈ C : |p(z)| > |p(ζ)|} ∪ {∞}.

There exists a single-valued branch g of p1/N defined on U , such that g(z)/z → 1as z → ∞. Observe that g is a conformal map onto the complement of a disc.The map g−1 omits all the critical points ζi. We apply the Koebe One QuarterTheorem about the point ∞ to obtain

(4) |ζi| ≤ 4|p(ζ)|1/N .

Take ζi = ζ to get

|ζ| = |p(ζ)||λ| ≤ 4|p(ζ)|1/N ,

so

|p(ζ)|(N−1)/N ≤ 4|λ| ≤ 4M.

Plugging this back into inequality (4) gives the result:

|ζi| ≤ 4(4M)1/(N−1).

This completes the proof of Lemma 3.

154 E. Crane CMFT

Finally, the surjectivity statement in part (v) of Lemma 2 follows because theimage of a proper map is closed.

A simple corollary of Lemma 3 is that (iv) =⇒ (iii) in Theorem 1. We knowthat K(N) < 4, so for any monic polynomial p whose objective values all havemodulus K(N), every critical point satisfies |ζi| < 4(N+1)/(N−1). This gives us anexplicit compact set of polynomials in which at least one extremal polynomialmust be found. If we can compute the maximum of S(p) on this compact set, wewill determine the value of K(N). This is essentially the approach to the caseN = 5 that is used in [8].

3.2. Explicit derivation of the objective hypersurface. Let us considerθ : C

N−1 → CN−1, the ‘folding map’ whose ith co-ordinate θi is the ith elementary

symmetric polynomial in the arguments. We know from part (i) of Lemma 2that the map β is equivariant with respect to the action of SN−1 on C

N−1, andhence the symmetric polynomials in the co-ordinates of β can be expanded assymmetric polynomials of the ζi. Newton’s Theorem on symmetric polynomialssays that the ring of symmetric polynomials in ζ1, . . . , ζN−1 is freely generated asa C-algebra by the elementary symmetric polynomials θ1, . . . , θN−1. Therefore βdescends to a polynomial map ϕ : GN → C

N−1 such that ϕ ◦ α = θ ◦ β. In otherwords, each elementary symmetric polynomial in the objective values of p ∈ GN

is given by a polynomial in the coefficients of p. In the rest of this section wemake this process explicit and deduce an equation for the objective hypersurfaceXN . This may be skipped at first reading since it does not contribute to the proofof Theorem 1. The reader who wishes to avoid the use of Chevalley’s Theoremmay substitute this argument in the proof of part (iii) of Lemma 2.

For p ∈ GN , we define q(z) = p(z)/z. The objective values λi = p(ζi)/ζi = q(ζi)are the values of t for which q(z)−t and p′(z) have a common root in z. Thereforethey are the roots of the resultant

R(t) = Res

(q(z)− t,

p′(z)

N

)=

N−1∏i=1

N−1∏j=1

(αj(t)− ζi),

where the αj(t) are the roots of q(z) = t as a polynomial in z, and the ζi

are the roots of p′(z) = 0, in both cases repeated according to multiplicity. Themultiplicity of roots λ of R(t) gives the appropriate multiplicity for the λi, namelythe sum of the multiplicities of the critical points whose associated objective valueis λ. One calculates the resultant R(t) from Bezout’s or Sylvester’s determinant,so the coefficients of R(t) are polynomials in the coefficients ai. One checks thatthe leading term of R(t) is (−1)N−1tN−1, independent of the choice of p. This isa useful consequence of the monic normalization of p. The remaining coefficientsof R(t), taken with appropriate signs, are the elementary symmetric functions sk

of the objective values; we see again that they are given by polynomials in theunconstrained coefficients of p. Thus for k = 1, . . . , N−1 there are polynomials σk


in N − 2 variables such that

sk = σk(a2, . . . , aN−1).

We claim there must exist a non-trivial polynomial relation RN(σ1, . . . , σN−1) = 0among the polynomials σk. In fact, any N − 1 elements of the polynomial ringC[a2, . . . , aN−1] must satisfy a non-trivial polynomial relation. Although this isa standard result in commutative algebra, we give the elementary proof here tomake our exposition self-contained. We simply estimate in two different ways thedimension of the complex vector space V generated by the monomials of degreeless than or equal to L in each of the variables σ1, . . . , σN−1. Suppose there wereno linear relation among these monomials; then dim(V ) = (L + 1)N−1. On theother hand, the degree of each of these monomials (expanded as a polynomialin the variables ai) is at most Lm in each variable ai, where m is the maximumexponent of any variable appearing in any of the σi. So they are all containedin the vector space W of polynomials of degree at most Lm in each of thevariables ai. We have

dim(W ) = (1 + Lm)N−2 ≤ (1 + L)N−2mN−2.

Since V ⊂ W we should have (L + 1)N−1 ≤ (1 + L)N−2mN−2. But this is falsefor L ≥ mN−2, so we have the required contradiction to prove the claim.

We may choose the polynomial RN to be of minimal degree and therefore irre-ducible. (Incidentally, the argument of the preceding paragraph places a boundon the degree of RN). Substitute into RN the expansions of the sk in terms ofthe variables λi. This yields an algebraic relation between the λi, say

QN(λ1, . . . λN−1) = 0.

This relation is non-trivial because there is no non-trivial algebraic relation be-tween the elementary symmetric polynomials si in the ring C[λ1, . . . , λN−1].

4. The Max-Min problem on hypersurfaces in CN

Our discussion of the objective hypersurface XN suggests a metric problem aboutgeneral complex affine hypersurfaces. Define

Z(Q) := {(x1, . . . , xk) ∈ Ck : Q(x1, . . . , xk) = 0},

where Q ∈ C[x1, . . . , xk] is a non-constant polynomial. Then define

(5) f(Q) := supx∈Z(Q)

(min

i|xi|

).

The problem is to characterize those polynomials Q for which f(Q) is finite, andfor such a polynomial to determine whether the bound is attained by some pointx ∈ C

k. We will say that a monomial term of the polynomial Q is the dominantmonomial of Q if it of maximal degree in each variable separately; equivalentlyit is the leading term in the lexicographic ordering of monomials correspondingto every permutation of {1, . . . , k}.

156 E. Crane CMFT

Lemma 4. Let Q be a non-constant complex polynomial in k variables. Thenf(Q) <∞ if and only if Q has a dominant monomial.

Proof. Firstly suppose that Q has the dominant monomial λx1m1x2

m2 . . . xkmk .

Consider any point x = (x1, . . . , xk) ∈ Z(Q) such that min |xi| = t ≥ 1; if thereis no such point then f(Q) ≤ 1 so we are done. There at most m1m2 . . . mk

monomial terms in Q. For any monomial term of Q other than the dominantterm, say cx1

n1 · · ·xknk , we have ni ≤ mi for each 1 ≤ i ≤ k and nj < mj for

some particular index j, and therefore

|cx1n1 · · ·xk

nk | ≤ |c|∣∣x1

m1 · · ·xj−1mj−1xj

mj−1xj+1mj+1 · · ·xk

mk∣∣ .

Suppose that the maximum modulus of the coefficients of Q is M . Then therelation Q(x1, . . . , xk) = 0 implies, by the triangle inequality,

|λx1m1x2

m2 · · ·xkmk |

≤ M

(k∏

i=1

mi

)(k∑

j=1

|x1m1 · · ·xj−1

mj−1xjmj−1xj+1

mj+1 · · ·xkmk |

).

Since the xi are all non-zero we may divide the above inequality through by thedominant monomial and use 1/|xi| ≤ 1/t to obtain the inequality

t ≤ kM

λ

k∏i=1

mi.

We conclude that

f(Q) ≤ max

(1,

lM

λ

k∏i=1

mi

).

The converse implication is trivial in the case k = 1, so suppose that k ≥ 2 andthat Q does not have a dominant monomial. We wish to show that Q = 0 hassolutions with minimum modulus greater than an arbitrary constant K. Thelack of a dominant monomial implies that we can find two different lexicographicorderings on monomials that give different leading monomial terms for Q. Af-ter re-labeling the variables if necessary, we may take these monomials to beax1

m1 · · ·xkmk and bx1

n1 · · ·xknk where a, b �= 0, m1 < n1, and m2 > n2.

Now fix a generic choice of x3, . . . , xk such that |xi| > K for i = 3, . . . , k, anddefine a two-variable polynomial R as follows:

R(x, y) := Q(x, y, x3, . . . , xk) = cxn1yn2 + dxm1ym2 +O(xn1−1ym2−1

),

where c, d �= 0. Let s(y) be the (n1−m1)th elementary symmetric function of the

roots with respect to x of R(x, y). Then s(y) is a rational function of y whoseLaurent expansion at infinity has leading term ±(d/c)ym2−n2 . So s(y) exceeds(

n1

n1−m1

)Kn1−m1 when |y| is sufficiently large, and therefore there is a solution of

R(x, y) = 0 with |x|, |y| > K, as required.


Lemma 5. Let T be a non-constant k-variable polynomial with a dominantmonomial. Suppose that (x1, . . . , xk) ∈ Z(T ) and |x1| = . . . = |xd| = f(T )but |xi| > f(T ) for i > d, where d < k. Then T (x1, . . . , xd, yd+1, . . . , yk) isidentically zero as a polynomial in the (k − d) variables yd+1, . . . , yk.

Proof. Suppose for a contradiction that yd+1, . . . , yk are some values for whichT (x1, . . . , xd, yd+1, . . . , yk) �= 0. Consider the two-variable polynomial S definedby

S(v, w) := T (vx1, vx2, . . . , vxd, wxd+1 + (1− w)yd+1, . . . , wxk + (1− w)yk).

We have S(1, 1) = 0 and S(1, 0) �= 0, so S(v, w) is not identically zero on the linev = 1 and it follows that Z(S) is an affine curve which projects locally near thepoint (1, 1) onto the v co-ordinate as a branched cover. Therefore there is a point(v, w) near to (1, 1) such that S(v, w) = 0, |v| > 1 and |wxj + (1−w)yj| > f(T )for all d + 1 ≤ j ≤ k. This contradicts the definition of f(T ), so we are done.

Proposition 6. Let T be a non-constant k-variable polynomial with a dominantmonomial. Then there exists (x1, . . . , xk) ∈ Z(T ) with |x1| = . . . = |xk| = f(T ).

Proof. For k ≥ 1, let the induction hypothesis IH(k) consist of the followingtwo statements.

(i) Let T (x1, . . . , xk) be a polynomial with a dominant monomial. Then thereexists (x1, . . . , xk) ∈ Z(T ) such that |x1| = |x2| = . . . = |xk| = f(T ).

(ii) Let R(x0, x1, x2, . . . , xk) be a complex polynomial such that for each win some open disc D ⊂ C, the k-variable polynomial Q(w) defined byQ(w)(x1, . . . , xk) = R(w, x1, . . . , xk) has the non-constant dominant termλ(w)x1

m1 . . . xkmk . (This means that some mi > 0 and λ(w) �= 0 for each

w ∈ D.) Then log(f(Q(w)) is subharmonic in w, and therefore f(Q(w))has no isolated local maximum in D.

For k = 1 statement (i) is true since T (x1) has only finitely many roots, whilestatement (ii) is true because the roots of Q(w) are algebraic functions of w.

To begin the induction step, suppose that k ≥ 2 and assume part (ii) of IH(k−1).Let T (x1, . . . , xk) be a polynomial with the dominant monomial λx1

m1 . . . xkmk .

The proof of Lemma 4 provides an explicit upper bound f(T ) ≤ C. If |xk| > Cand T (x1, . . . , xk) = 0 then mink

i=1 |xi| < |xk|, so minki=1 |xi| = mink−1

i=1 |xi|. Nowwe divide the equation T (x1, . . . , xk) = 0 by xk

mk and make the substitutionxk = 1/w to obtain a polynomial equation of the form

R(w, x1, . . . , xk−1) = 0.

Now for a given value of w set Q(w)(x1, . . . , xk−1) = R(w, x1, . . . , xk−1). Thepolynomial Q(w) has a term λ(w)x1

m1 . . . xk−1mk−1 which is non-zero and domi-

nant so long as λ(w) �= 0. But λ(w) is a polynomial in w and λ(0) = λ �= 0, sothere is some disc D(0, ε) of values of w for which Q(w) has this dominant term.Hence statement (ii) of IH(k− 1) applies. We deduce that there exists w1 �= 0 in

158 E. Crane CMFT

D(0, ε) and ε′ < |w1| such that for all w ∈ D(0, ε′) we have f(Q(w1)) ≥ f(Q(w)).We may also stipulate that |1/w1| > C. Re-phrasing this in terms of the vari-able xk, we have shown that for the purposes of computing f(T ) we may ignoreall points in C

k for which |xk| > 1/ε′. The same argument applies to each co-ordinate in place of xk, which shows that in fact there is a compact polydiscD = D1 × · · · ×Dk ⊂ C

k and a point x = (x1, . . . , xk) ∈ Z(T ) ∩D such that

min |xi| ≥ supy∈Z(T )\D

(min |yi|).

Since the intersection of the polydisc D with the hypersurface Z(T ) is a compactset and the function min |xi| is a continuous function on C

k, we find that thesupremum is attained. Finally we use Lemma 5 to find an extremal point suchthat |xi| = f(T ) for all i. This completes the proof of statement (i) of IH(k).

We now prove statement (ii) of IH(k). It suffices to assume that 0 ∈ D andshow that log(f(Q(w))) has a subharmonic envelope at w = 0. Let us writeQ(w) = Q0 + wQ1 + · · ·+ wmQm, where the coefficients belong to C[x1, . . . , xk],so that

R(w, x1, . . . , xk) =m∑

j=0

wjQj(x1, . . . , xk).

We are given that for w ∈ D, the k-variable polynomial Q(w) has a dominantmonomial λ(w)x1

m1 . . . xkmk . Here λ(w) is a polynomial with constant term

λ0 �= 0. From statement (i) of IH(k) applied with T = Q0 = Q(0), we know thatthere exists x = (x1, . . . , xk) ∈ Z(Q0) such that min |xi| = f(Q(0)). If x is a zeroof all of Q0, . . . , Qm then x belongs to Z(Q(w)) for all w; then f(Q(w)) ≥ f(Q(0))for all w, so we are done. Otherwise Q(w)(x) �= 0 for sufficiently small w �= 0.Consider the affine curve defined by

Y := {(w, t) ∈ C2 : Q(w)(tx) = 0}.

We know (0, 1) ∈ Y and that Y is defined by a single polynomial equation in thetwo variables w and t. We also know that (w, 1) �∈ Y for sufficiently small w �= 0.It follows that the projection of Y to the t-plane is locally a branched coveringmap near (0, 1). For any point (w, t) ∈ Y we have f(Q(w)) ≥ |t|f(Q(0)). Butthe function g(w) = max{log |t| : (w, t) ∈ Y } is subharmonic with g(0) ≥ 0. Solog(f(Q(w))) is bounded below by a subharmonic function with which it agreesat 0, as required.

We now apply the above results to the polynomial T = QN . Recall that QN isan irreducible symmetric polynomial that relates the objective values appearingin Smale’s conjecture. Smale’s inequality says that f(QN) < 4. In particularf(QN) is finite, so QN has a dominant monomial, and therefore Z(QN) has anextremal point with all |xi| equal to K(N). Since the map β is surjective, thispoint is the vector of objective values of some polynomial in GN , and we haveproved part (i) of Theorem 1. Lemma 5 also shows that if the set of extremal


polynomials in GN is compact, so that its image under β is compact, then theextremal points of Z(QN) consist only of points with all |xi| equal to K(N). Inother words, (iii) =⇒ (iv) in Theorem 1. We will prove part (iii) of the maintheorem in the next section.

5. Analysis of extremal polynomials using schlichtfunctions

In this section we describe a compact parameter space for Smale’s conjecture,consisting of polynomials of degrees between 2 and N . We use this parameterspace to show that if the extremal subset of GN were not compact then we wouldhave K(N) = K(N − 1). This is part (iii) of Theorem 1, and as we have seen itimplies part (iv). Therefore if we are to have K(N) = 1−1/N , as is conjectured,then the extremal subset of GN will have to be compact.

The class S of schlicht functions consists of all univalent functions f : D → C

such that f(0) = 0 and f ′(0) = 1. It is well known that S is compact whenequipped with the topology of locally uniform convergence (see [13, Thm. 1.7]).The following lemma is also classical.

Lemma 7. Fix N ≥ 2 and let T ⊂ S be the subset consisting of inverse branchesof polynomials of degree at most N . Then T is closed, therefore compact, in thetopology of locally uniform convergence. The coefficients of the correspondingpolynomials are continuous functions on T .

Proof. A function f ∈ S is a branch of the inverse of a given polynomialp(z) = aNzN + · · ·+ a1z + a0 if and only if a0 = 0, a1 = 1 and p(f(z)) ≡ z.Suppose we have a sequence (fn) of elements of S such that fn → f locallyuniformly, and each fn is a branch of the inverse of a polynomial pn. The KoebeOne Quarter Theorem says that each image fn(D) contains the disc D(0, 1/4).We claim that the restrictions of the polynomials pn to the disc D(0, 1/4) con-verge locally uniformly there to the restriction of f−1. To show this, take any0 < ε < 1/2. By Koebe’s One Quarter Theorem again, if |w| = 1 − ε then|fn(w)| > (1− ε)/4. Now for |z| ≤ (1− 2ε)/4 we have

pn(z) = f−1n (z) =

1

2πi

∫C(0,1−ε)

wf ′n(w) dw

fn(w)− z.

Since fn converges uniformly on this circular contour and is uniformly boundedaway from z, we see that pn(z) converges uniformly for |z| ≤ (1 − 2ε)/4. Theuniform limit on this disc of a sequence of polynomials of degree at most Nis again a polynomial of degree at most N . In fact, by evaluating the Taylorcoefficients of p = f−1 and of pn using Cauchy’s integral formula on the samecircular contour, we find that the coefficients of the polynomials pn converge tothose of p. So T is closed, and the coefficients of the polynomial f−1 dependcontinuously on f as f ranges over T .

160 E. Crane CMFT

We call a non-linear polynomial p schlicht-normalized when p(0) = 0, p′(0) = 1,and p has an inverse branch f ∈ T such that C(f) = limz→1 f(z) is a criticalpoint of p. Of course p(C(f)) = 1. Every polynomial in GN can be rescaled toobtain a schlicht-normalized polynomial. To see this, consider the largest discD(0, r) on which a branch f of p−1 can be defined that sends 0 to 0. There isat least one critical point ζ of p which forms an obstruction to increasing r, bywhich we mean that |p(ζ)| = r and that ζ is the limit of f(z) as z → p(ζ) fromwithin D(0, r). Then we can define a schlicht-normalized polynomial p by

p(z) :=p(p(ζ)z)

p(ζ).

We recall that affine rescaling does not change the quantity S(p) that appears inSmale’s inequality. Note that we cannot usually rescale a polynomial so that itbecomes both monic and schlicht-normalized. Since every polynomial in GN canbe rescaled to become schlicht-normalized, we have

K(N) = supf∈T

S(f−1).

Let pn be a sequence of schlicht-normalized polynomials of degree at most N ,such that S(pn) → K(N) as n → ∞. Denote the inverse branch of pn whichbelongs to T by fn, and let Cn be the critical point C(fn). After passing to asubsequence we can assume that fn converges to a univalent function f . Lemma 7shows that f ∈ T . Denote by p the polynomial which coincides with f−1 wherethey are both defined. By Lemma 7, we have pn → p as n →∞, locally uniformlyand in each coefficient. Since K(N) ≥ 1 − 1/N we may pass to a subsequenceso that S(pn) ≥ 1/2 for all n. It follows that |Cn| ≤ 2 for all n, and thereforewe can pass to a further subsequence so that Cn → ζ as n →∞. We then havep′(ζ) = 0 and p(ζ) = 1. The key point here is that the limit polynomial p isnon-linear.

Now S(p) = |p(η)/η| for some η �= 0 such that p′(η) = 0. Choose ε > 0 so smallthat η is the only critical point of p in D(η, 2ε). Then the number of zeros of p′nin D(η, ε) is

1

2πi

∫C(η,ε)

p′′n(w) dw

p′n(w).

As n → ∞, the integrand converges uniformly on the contour to p′′(w)/p′(w).Therefore for sufficiently large n we can choose a critical point ηn of pn, in sucha way that ηn → η as n → ∞, and therefore pn(ηn)/ηn → p(η)/η as n → ∞.But S(pn) ≤ |pn(ηn)/ηn|, so lim sup S(pn) ≤ S(p). Therefore S(p) = K(N).

Now suppose that K(N) > K(N − 1). Then the limit polynomial p must havedegree exactly N . Define the extremal set

Ex(N) := {f ∈ T : S(f−1) = K(N)}.Then, Ex(N) consists of inverse branches of polynomials of degree N . We can ap-ply the above argument to show that any sequence fn ∈ Ex(N) has a convergent


subsequence whose limit is again in Ex(N); hence Ex(N) is a compact subsetof T consisting only of inverse branches of polynomials of degree exactly N .Therefore each polynomial coefficient of f−1 is bounded as f ranges over Ex(N),and the coefficient of zN is bounded away from zero. It follows that the monicrescalings of the polynomials f−1 for f ∈ Ex(N) form a bounded subset of GN .We have proved the following result.

Proposition 8. Suppose that K(N) > K(N − 1). Then there exists a poly-nomial p of degree N such that S(p) = K(N), i.e. an extremal polynomial fordegree N . Furthermore the set Ex(N) of extremal polynomials forms a compactsubset of GN .

6. A special case

Tischler [19, §2] used an algebraic method to deal with the special case of themean value conjecture in which all the non-zero roots of p have equal modulus. Inconjunction with Theorem 1, the same piece of algebra yields bounds on the dis-criminant of an extremal polynomial, under the assumption K(N) > K(N − 1).In the following, Δ(p) is the discriminant of p, and the polynomial q is definedby p(z) = zq(z).

Lemma 9. For any p in GN , we have

N−1∏i=1

|λi| =|Δ(p)|NN−1

.

Thus

mini|λi| ≤

|Δ(p)|1/(N−1)

N=|Δ(q)|1/(N−1)

N.

Proof. Let z0 = 0, z1, . . . , zN−1 be the roots of p and let ζ1, . . . , ζN−1 be theroots of p′, repeated according to multiplicity. Recall from Section 3.1 that theobjective values λi = p(ζi)/ζi are the roots of the resultant polynomial

R(t) = Res

(q(z)− t,

p′(z)

N

).

The constant coefficient of R(t) is

Res

(q,

p′

N

)=

N−1∏j=1

N−1∏i=1

(zj − ζi) =N−1∏j=1

p′(zj)

N

= N

N−1∏j=0

p′(zj)

N= N

N−1∏j=0

1

N

∏i�=j

(zj − zi)

=(−1)N(N−1)/2

NN−1Δ(p).

162 E. Crane CMFT

The condition p′(0) = 1 implies Δ(p) = ±Δ(q). The leading term of R(t) is(−1)N−1tN−1. Since the λi are the roots of R(t), we find that

N−1∏i=1

|λi| =|Δ(p)|NN−1

.

Theorem 10 (Tischler). Among polynomials p in GN such that the roots ofq(z) = p(z)/z all lie on the unit circle, p(z) = zN + z is extremal for Smale’smean value conjecture.

Proof. Here is Tischler’s proof. It follows from Hadamard’s inequality for deter-minants [15, Thm 1.7.3] that among all monic polynomials q whose roots all lieon the unit circle, the modulus of the discriminant is maximized uniquely whenthose roots are equally spaced around the unit circle. Observe that the minimumof the |λi| is at most their geometric mean, with equality only if they are all equal.The normalizing condition q(0) = p′(0) = 1 leaves a unique polynomial maximalamong those whose roots lie on the unit circle, namely q(z) = zN−1 + 1.

In the light of Theorem 1 we can obtain another corollary of Lemma 9.

Corollary 11. If p ∈ GN is an extremal polynomial, and K(N) > K(N − 1),then p satisfies

(N − 1)N−1 ≤ Δ(p) ≤ (4N)N−1.

Proof. We know in this case from Theorem 1 that the |λi| are all equal to K(N).Since 4 ≥ K(N) ≥ (N − 1)/N , the first formula of Lemma 9 gives the result.

Acknowledgement. Some of the results of this paper appeared in the author’sPh.D. thesis [7]. The author would like to thank Keith Carne and Alan Beardonfor many interesting discussions on this topic, and the referee, who suggested anumber of improvements to the exposition.

References

1. A. F. Beardon, T. K. Carne and T. W. Ng, The critical values of a polynomial, Constr.Approx., 18 no.3 (2002), 343–354.

2. A. F. Beardon, D. Minda and T. W. Ng, Smale’s mean value conjecture and the hyperbolicmetric, Math. Ann. 322 no.4 (2002), 623–632.

3. S. Basu, R. Pollack and M.-F. Roy, Algorithms in Real Algebraic Geometry, Algorithmsand Computation in Mathematics, Vol. 10, Springer, 2003.

4. A. Conte, E. Fujikawa and E. Lakic, Smale’s mean value conjecture and the coefficients ofunivalent functions, to appear in Proc. Amer. Math. Soc.

5. E. Fujikawa and T. Sugawa, Geometric function theory and Smale’s mean value conjecture,preprint.

6. A. Cordova and S. Ruscheweyh, Subordination of polynomials, Rocky Mountain J. Math21 (1) (1991), 159–70.


7. E. T. Crane, Topics in conformal geometry and dynamics, PhD thesis, University of Cam-bridge, 2004; download from http://www.maths.ox.ac.uk/∼crane

8. , A computational proof of the degree 5 case of Smale’s mean value conjecture, toappear.

9. , A bound for Smale’s mean value conjecture, to appear.10. P. Griffiths and J. Harris, Principles of Algebraic Geometry, John Wiley and Sons, 1978.11. B. A. Kats, Convexity and star-shapedness of the level curves of polynomials. Math. Notes

15 (1974), 419–424.12. T. W. Ng, Smale’s mean value conjecture for odd polynomials, J. Aust. Math. Soc. 75

no.3 (2003), 409–411.13. C. Pommerenke, Univalent Functions, Vandenhoeck and Ruprecht, Gottingen, 1975.14. M. O. Rabin, Decidable Theories, Chapter C.3 of Handbook of Mathematical Logic, in:

Kreisler et al (eds.), Studies in Logic and the Foundations of Mathematics, Vol. 90, North-Holland, 1977.

15. Q. Rahman and G. Schmeisser, Analytic Theory of Polynomials, LMS Monographs NewSeries, no. 26, Oxford University Press, 2002.

16. G. Schmeisser, The conjectures of Sendov and Smale, Approximation Theory, 353–369,DARBA, Sofia 2002.

17. T. Sheil-Small, Complex Polynomials, Cambridge Studies in Advanced Mathematics, no.75, Cambridge University Press, 2002.

18. S. Smale, The fundamental theorem of algebra and complexity theory, Bull. Amer. Math.Soc. (N. S.) 4 (1981), 1–36.

19. D. Tischler, Critical points and values of complex polynomials, J. Complexity 5 (1989),438–56.

20. , Perturbations of critical fixed points of analytic maps, Asterisque 222 (1994),407–422.

21. J. L. Walsh, On the convexity of the ovals of lemniscates, Studies in mathematical analysisand related topics, 419–423, Stanford Univ. Press, 1962.

Edward Crane E-mail: [email protected]: Mathematics Department, University of Bristol, University Walk, Bristol BS8 1TW,United Kingdom.

extremal polynomials in smale’s mean value conjecture

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