On zeros of functions from Bernstein classes - Weizmann Institute of ...

Nonlinearity 13 (2000) 1087–1094. Printed in the UK

PII: S0951-7715(00)01802-8

On zeros of functions from Bernstein classes Sergei Yakovenko† Department of Theoretical Mathematics, The Weizmann Institute of Science, Rehovot 76100, Israel E-mail: [email protected]

Received 12 February 1999 Recommended by B Branner
Abstract. If a convergent Taylor series f (z) = j
0 aj zj satisfies the condition |aj |  M|ak | for some k and all j > k, then one can explicitly determine in terms of M and k the radius of a centred disc containing no more than k roots of f . This problem was solved by Yomdin and Roytwarf using the equivalence of two Bernstein classes of analytic functions and a delicate refinement of the Jensen inequality due to van der Poorten. We give two direct proofs of the above claim: one is more transparent though gives a slightly worse bound for the radius of the circle. The second proof generalizes the simple and popular differentiation–division algorithm and gives the bounds better than the original proof. AMS classification scheme numbers: 34C05, 30C15, 26C10, 34A20

1. Background and the problem 1.1. Statement of the problem for polynomials It is well known that a monic polynomial P (w) = w n + b1 w n−1 + · · · + bn−1 w + bn in one complex variable w with explicitly bounded coefficients bk ∈ C, |bk |
M, admits an explicit upper bound for the modulus of its roots (e.g. as in corollary 1 below). After the transformation w = 1/z this gives the radius of a (centred) disc free of roots of a polynomial p(z) = a0 + a1 z + a2 z2 + · · · + an zn , provided that |aj |
M|a0 | for all j  1. Clearly, this assertion should admit a generalization of the following type. If for some k between 0 and n the coefficient ak is not too small compared to the other coefficients, i.e. |aj |
M|ak |, ∀j = k and some known M, then there exists a centred disc containing no more than k roots of the polynomial p, whose radius depends only on k, n and M. The claim about zeros is completely obvious if one of the monomials is strongly dominant. For example, if nM < 1
then on the circle of radius 1 the term |ak zk | overtakes the sum of moduli of all other terms j =k aj zj , and therefore by the Rouch´e theorem there are exactly k roots of p(z) inside the unit disc and n−k roots outside it. This example can be combined with rescaling z → δz producing a slightly more general claim, but ultimately fails for polynomials having terms of comparable magnitude. † Home page: http://www.wisdom.weizmann.ac.il/∼yakov 0951-7715/00/041087+08$30.00

© 2000 IOP Publishing Ltd and LMS Publishing Ltd

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1.2. Bernstein classes and Yomdin–Roytwarf theorem The above problem might look artificial, however, this is the simplest version of a problem that constantly reappears in connection with the bifurcation of limit cycles of polynomial vector fields and Hilbert’s 16th problem. More specifically, consider a polynomial deformation unfolding an integrable planar vector field and denote by λ ∈ CN the collection of the deformation.
of all coefficients j Consider the displacement function (z, λ) = a (λ) z (the difference between the j j Poincar´e return map and the identity). Together the coefficients {aj (λ)}j
0 of this function generate the so-called Bautin ideal [6, 9] of the appropriate polynomial ring C[λ], and because of the Noetherianity there exists some finite k such that the ideal is generated by the first k coefficients a0 (λ), . . . , ak (λ). By construction, all other coefficients of the displacement as a function of parameters vanish whenever the first k of them do, moreover, one can show that |aj |
CR j max0ik |ai | for all j  k + 1 and some C, R < ∞. Hilbert’s 16th problem reduces to a determination of the maximal number and possible location of zeros of the displacement function (corresponding to limit cycles that can be born in this perturbation). Suppose that the Bautin ideal is already known, in particular, with the integer k and real constants C, R explicitly majorized. Obtaining this information is a transcendentally difficult problem which has been solved so far only in a few exceptional cases. However, we ignore this difficulty and discuss the subsequent steps, namely, how to derive the information on limit cycles from these data. The number k can be relatively easily shown to be an upper bound for the number of cycles born in the family. The problem is to estimate the region in the phase plane, where these cycles may appear, and also the domain in the parameter space for which the bound k holds, in other words, to give an explicit description of the disc in the complex z-plane, where at most k zeros of  may occur. Analysing this and similar problems, in [7, 8] Roytwarf and Yomdin introduced Bernstein classes as collections of (convergent) power series whose first several Taylor coefficients constitute in some sense the principal part of the whole infinite expression. Definition 1. The (second) Bernstein class Bk,R,M determined by one
natural parameter k ∈ N and two positive constants M, R, is the collection of power series j
0 aj zj with complex coefficients satisfying the inequality ∀j  k + 1

|aj |R j
M max |ai |R i . 0ik

(1)

In particular, the series converges absolutely inside the disc {|z| < R}. Yomdin–Roytwarf theorem (see [7, 8]). Any function f ∈ Bk,R,M has at most k isolated zeros in the disc {|z| < R/(8k max(M, 2))}.  Clearly, application of this result to a polynomial with the unit kth coefficient and others explicitly bounded by M would yield the radius r (in general, greater than 1) such that the (intersecting) discs {|z| < r} and {|z| > 1/r} contain at most k (respectively, n − k) roots of the polynomial. The proof of Yomdin–Roytwarf theorem is based on rather delicate and indirect arguments.  The authors introduce the first Bernstein class BR,α,K as the collection of functions analytic and bounded in the disc {|z|
R} and satisfying the inequality max |f (z)|
K max |f (z)|.

|z|R

|z|αR

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It is proved then that the two Bernstein classes are essentially equivalent (after an explicit recomputing of the parameters). Then the assertion of the theorem is derived from a Jensen–  Nevanlinna-type inequality estimating the number of zeros for a function f ∈ BR,α,K in the 2 disc {|z|
R} by K/ log[(1 + α )/2α]. This result due to van der Poorten [5] is in itself a strong and delicate claim (assertions of this type are characteristic for the number-theoretic constructions). 1.3. Aims, tools and results Our primary aim was to find a direct and transparent proof of inequalities describing the location of zeros for polynomials and analytic functions in the sense explained above. We give two such direct proofs. The first one described in section 2 is based on comparing moduli of analytic functions and using the Rouch´e theorem. The polynomial or infinite series f is considered as a perturbation of its k-truncation p. If the magnitude of the perturbation f − p that begins with terms of order zk+1 is sufficiently small (and this is the case after rescaling to a sufficiently small disc), then the modulus of p on some circle (estimated from below by the Cartan lemma) is greater than that of f − p. By the Rouch´e theorem, the number of zeros of f and p inside that circle remains the same. This simple argument (actually, a modification of the illustrative example from section 1.1), allows one to prove a result similar to the Yomdin–Roytwarf theorem, but giving a weaker estimate for the radius. Another, better result can be obtained by modifying the popular differentiation–division algorithm (used in the proof of the Descartes rule) to work with complex zeros. This paradigm is described in section 3.1 before the formal proof. It turns out that this approach yields the best bound for zeros. The key argument is provided by the complex analogue of the Rolle lemma [3] asserting that the total variation of the argument of an analytic function along the boundary of a convex domain behaves much like the number of real zeros with respect to differentiations, namely, decreases at most by 2π †. One of the principal aims of this paper is to advertise this simple but powerful tool. 2. The modulus 2.1. Notation We consider the infinite-dimensional linear space
of converging Taylor expansions C{z} equipped by the weighted ∞ -norms: for p = j aj zj and any δ
1 we put pδ = sup |aj |δ j = p ◦ δ j
0

 ·  :=  · 1

where p ◦δ ∈ C{z} is a series defined as (p ◦δ)(z) = p(δz) (we will mostly use the  · 1 -norm denoted simply as  · ). Note that the operation p(z) → zν p(z) is an isometric shift in the  · 1 -norm. For convenience we introduce the classes (similar to the second Bernstein class)   aj zj ∈ C{z}: |aj |
M|ak | ∀j  k + 1 . (2) Pk,M = j
0

Each non-zero function from Pk,M after normalization f → f/ak can be considered as an M-bounded perturbation u + zk+1 q of the monic polynomial u = zk + ak−1 zk−1 + · · · + a0 with q
M. † Reference [3] is available on the author’s internet home page.

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The Yomdin–Roytwarf theorem applied to functions from the class Pk,M asserts that each such function may have no more than k isolated roots in the open disc of radius  −1 Y–R = 8k max(M, 2) . (3) Rk,M Conversely, any result concerning zeros of functions from Pk,M can be easily reformulated for Bernstein classes. 2.2. Bounds for the class P0,M First we quote a simple lower bound for roots of functions from P0,M which will also serve as a template for further constructions. Lemma 1 (cf [1]). Let p ∈ P0,M , i.e. p = 1 + zq and all coefficients of q are bounded by M. Then p has no roots inside the disc {|z| < (1 + M)−1 }. Proof. On the circumference of radius r < 1 we have for any p ∈ P0,M |p(z)||z|=r
p(1 + r + r 2 + r 3 + · · ·)
p/(1 − r)

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and hence on this circumference |zq(z)|
Mr/(1 − r) under our assumptions. For r < (1 + M)−1 this is less than 1, so by the Rouch´e theorem 1 + zq with q = zq = M has no zeros in the disc {|z|
(1 + M)−1 }. Note that this bound is sharp (see the example at the end of the paper).  Corollary 1. The monic polynomial p(z) = zn + q(z), deg q
n − 1, has all roots inside the centred disc of radius 1 + q.  2.3. Claim with shortest demonstration The simplest result in the spirit of the Yomdin–Roytwarf theorem is as follows. Proposition 1. A function from the class Pk,M has at most k roots in the disc of radius  −1 mod = 21 (8e)k (M + 1) e ≈ 2.718 28 . . . . Rk,M

(5)

Proof. Consider an arbitrary normalized series p ∈ Pk,M represented as p = u + zk+1 q with a monic polynomial u of degree k and q
M. Let 1 > r > 0 be a positive number to be chosen later. As the polynomial u is monic, by Cartan’s inequality [4] there exists a finite number of exceptional discs with the sum of their diameters less than r such that on the complement to their union u admits the lower bound |u(z)|  (r/4e)k . Consider the annulus {r
|z|
2r} foliated by circumferences {|z| = ρ}, r
ρ
2r. As the sum of diameters of the exceptional discs is less than r, at least one such circumference is disjoint with their union and hence u is bounded from below on it by (r/4e)k . On the other hand, on any such circumference the term zk+1 q(z) admits an explicit upper bound using (4): |zk+1 q(z)||z|=ρ
M

ρ k+1 (2r)k+1
M . 1−ρ 1 − 2r

The domination inequality (r/4e)k > (2r)k+1 M/(1 − 2r) ensures that the Rouch´e theorem applies to the circumference {|z| = ρ} and guarantees that the number of roots of p and u (the

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latter being at most k) in the disc {|z|
ρ} coincide. Resolving the domination inequality mod with respect to r gives r < Rk,M .  The constant 8e in this estimate is worse than its counterpart 8 occurring in (3), though the proof of (5) appears to be much more transparent than the original proof of (3). However, using a different set of arguments, we show in section 3 that both bounds can be improved. 2.4. Remark on the Rouch´e theorem and Cartan lemma In [2] the authors suggest a construction similar to that discussed in this section, but without using the Cartan lemma. Instead for a polynomial p of degree k, following the idea of Douady, they find explicitly the radius r < 1 of a circumference on which one of the monomials overtakes the rest. This allows one to apply the Rouch´e theorem to this leading monomial and consider the rest as its perturbation. The estimate based on using the Cartan lemma appears to be more transparent and gives somewhat better bounds. 3. The argument 3.1. The paradigm


Suppose that a real polynomial p(z) = n0 aj zj is normalized to satisfy ak = 1, |aj |
M for j = k. Then its kth derivative p(k) has the free term k! and all coefficients bounded by n(n − 1) · · · (n − k + 1)M. Lemma 1 allows us to find explicitly a disc D free of roots of p(k) . By the Rolle lemma the number of real zeros of p on D ∩ R can be at most k. Unfortunately, the reference to the Rolle theorem does not allow one to count non-real zeros. However, there is a close substitute suitable for our purposes. 3.2. The Rolle–Voorhoeve lemma and argument-based approach Let % ⊂ C be a piecewise-smooth curve parametrized by t ∈ [0, 1], t → z(t) ∈ C and f a function analytic in a neighbourhood of %. If f has no zeros on %, then f (z(t)) = r(t) exp iφ(t) for some piecewise-smooth branches of r(t) > 0 and φ(t). Definition 2 (cf [3]). The Voorhoeve index of f on % is the total variation of the argument of f along %    1  1  1  d f       ˙ V (f ; %) = |φ(t)| dt = (6)  dt Im ln f (z(t)) dt
 f (z(t)) |˙z(t)| dt 0 0 0 (here a dot denotes the derivative in the real variable t, whereas a prime denotes derivation with respect to the complex variable z). This definition will be applied to closed non-self-intersecting curves % with z(0) = z(1). 1 However, unlike the topological index 0 φ˙ dt equal to 2π times the number of zeros of f inside % if f extends analytically there, the Voorhoeve index changes continuously when zeros of f cross the curve %. Besides, also in contrast with the topological index, the Voorhoeve index behaves nicely with respect to differentiation. Since for closed convex curves the integral curvature is 2π, the following claim can be considered as a generalization of the Rolle theorem (one of several possible) for analytic functions. Lemma 2 (Rolle–Voorhoeve inequality [3]). For a closed curve % ⊂ C with integral curvature ((%), V (f ; %)
V (f  ; %) + ((%).

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An analogue (actually, an improvement) of lemma 1 is the following claim. Let Cr be the circle Cr = {|z| = r} ⊂ C. Lemma 3. For p ∈ P0,M and r < 21 (1 + M)−1 , the Voorhoeve index of p on the circumference Cr satisfies V (p; Cr ) < 2π. Somewhat more generally, if zqδ = Mδ for some 0 < δ < 1, then for p = 1 + zq and r < 21 δ(1 + Mδ )−1 the same inequality V (p; Cr ) < 2π holds. Proof. Consider first the base case δ = 1. Since zq = M, the majorant (4) implies max |p(z) − 1|
z∈Cr

Mr 1−r

max |p  (z)|
z∈Cr

M . (1 − r)2

As Cr is parametrized by z = r exp 2π it, t ∈ [0, 1], so that |˙z(t)| ≡ 2π r, we see that the integrand |˙z||p  |/|p| in the right-hand side of (6) is majorized by 2π and hence V (p; Cr ) < 2π , provided that r max Cr

|p  | 2(M + 1): then (1 − r)2 > 1 − (M + 1)−1 = M/(M + 1) as required. The general case of δ < 1 is reduced to the previous one by the substitution p ◦ δ instead of p.  3.3. Derivation in the ∞ -norm In order to apply the real paradigm from section 3.1, one has to majorize p (k) /k! in terms of known M = p. This can be easily done for polynomials, but without a priori bounds for the degree the differentiation operator is unbounded in the ∞ -norm. This is the reason why the weighted norms  · δ with δ  1 appear. Lemma 4. If p = M, then p (k) /k! δ
M/(1 − δ)k+1 for any 0 < δ < 1. Proof. If p = p1 = M, then p(z) has M/(1 − z) as the majorizing series (i.e. all Taylor coefficients of the latter are positive and greater than or equal to the absolute values of the respective coefficients of p). This relationship survives the formal differentiation, therefore p (k) /k! has M/(1 − z)k+1 as the majorizing series. The value of the latter at z = δ majorizes p (k) /k! δ since the sum of a series with non-negative terms majorizes the maximum of absolute values of these terms.  3.4. Assertion with the strongest claim Combining this result with the bound for the Voorhoeve index (lemma 3), we arrive at the following result. Theorem 1. For an arbitrary function p ∈ Pk,M and any λ > 1 the disc of radius   arg Rk,M,λ = 21 1 − λ1 (λk+1 M + 1)−1 contains no more than k roots of p.

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Proof. We represent p = u + zk+1 q with unitary u and bounded zk+1 q = M. Lemma 4 then implies that p(k) /k! = u(k) /k! + w = 1 + w, where w is a series divisible by z and wδ
M/(1 − δ)k+1 for any δ ∈ (0, 1). By lemma 3, the Voorhoeve index V (1 + w; Cr ) is strictly less than 2π provided that r arg is small enough to satisfy r −1  2(1 + M/(1 − δ)k+1 )/δ that resolves to r < Rk,M,λ as in 1 . Finally, lemma 2 guarantees that in this case (8) after the change of the parameters λ = 1−δ V (p; Cr ) < 2π + 2πk and therefore at most k isolated zeros of p can occur in {|z|
r} for so small r.  The choice of λ > 1 in (8) can be optimized for each given combination of parameters k and M. However, simply taking λ = 2 already gives a bound which is better than (3). Corollary 2. The disc of radius Rk,M,2 = (4 + 2k+3 M)−1 arg

contains at most k roots of a function p ∈ Pk,M .



Corollary 3. A function from the second Bernstein class Bk,R,M may have at most k isolated roots in the disc of radius 

1 1 (λk+1 M + 1)−1 R 1 − 2 λ for any choice of λ > 1. Proof. Indeed, f ∈ Bk,R,M if and only if f ◦ R ∈ Bk,1,M , and it remains only to note that Bk,1,M ⊆ 1j N Pj,M (and conversely Pk,M ⊆ Bk,1,M ).  Remark. The series p(z) = zk − Mzk+1 /(1 − z) from Pk,M has k + 1 roots in the closed disc of radius (1 + M)−1 . This gives an insight into the degree of accuracy of theorem 1. 4. Concluding remark: quasi-Bernstein classes of entire functions The bound for zeros obtained for Bernstein classes (or their equivalent counterparts Pk,M ) has a somewhat awkward appearance because of the unboundedness of differentiation in any ∞ -norm weighted by a geometric progression. A more elegant result can be obtained if instead of the classes Pk,M with finite radius of convergence we consider the classes of entire functions (cf with (2))   aj zj ∈ C{z} : j ! |aj |
Mk! |ak | ∀j  k + 1 . (9) Qk,M =


j
0

Indeed, a series j
0 aj zj with coefficients aj satisfying |aj |
M/j ! for all j , is majorized by the series M exp z. Hence the same arguments as in lemma 4 prove the implication p ∈ Qk,M ⇒ p(k) ∈ Q0,M . Combining this observation with the obvious remark that Q0,M ⊂ P0,M and hence lemma 3 applies to the derivative p (k) , we arrive at the following analogue of theorem 1. Theorem 2. An arbitrary function from the class Qk,M has at most k isolated zeros in the disc of radius fact = 21 (1 + M)−1 . RM

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Note that (unlike the bound (8)), the radius of the disc containing at most k roots, is independent of k. This feature cannot be achieved by methods described in section 2. However, for the moment we are unaware of possible applications of results pertinent to these classes, therefore do not explore this direction further. Acknowledgments I am grateful to Yosef Yomdin for stimulating discussions. The idea of the first proof appeared in conversations with Dmitri Novikov, to whom my sincere gratitude goes. References [1] Borwein P and Erd´elyi T 1995 Polynomials and Polynomial Inequalities (New York: Springer) (MR 97e:41001) [2] Hauser H, Risler J-J and Teissier B 1999 The reduced Bautin index of planar vector fields Duke Math. J. 100 425–45 [3] Khovanski˘ı A and Yakovenko S 1996 Generalized Rolle theorem in Rn and C J. Dynam. Control Syst. 2 103–23 (MR 97f:26016) [4] Levin B Ja 1980 Distribution of Zeros of Entire Functions revised edn (Providence, RI: American Mathematical Society) (MR 81k:30011) [5] van der Poorten A J 1977 On the number of zeros of functions Enseignement Math. 23 19–38 (MR 57 #16221) [6] Roussarie R 1998 Bifurcation of Planar Vector Fields and Hilbert’s Sixteenth Problem (Progress in Mathematics vol 164) (Boston, MA: Birkh¨auser) [7] Roytwarf N and Yomdin Y 1997 Bernstein classes Ann. Inst. Fourier (Grenoble) 47 825–58 (MR 98h:34009a) [8] Yomdin Y 1998 Oscillation of analytic curves Proc. Am. Math. Soc. 126 357–64 (MR 98d:32033) [9] Yakovenko S 1995 A geometric proof of the Bautin theorem Concerning the Hilbert 16th Problem (AMS Transl. Ser. 2 165) (Providence, RI: American Mathematical Society) pp 203–19