PROJECTION OF ANALYTIC SETS AND BERNSTEIN INEQUALITIES

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BANACH CENTER PUBLICATIONS, VOLUME 44 INSTITUTE OF MATHEMATICS POLISH ACADEMY OF SCIENCES

WARSZAWA 1998

PROJECTION OF ANALYTIC SETS AND BERNSTEIN INEQUALITIES

J E A N - P I E R R E F R A N C¸ O I S E

Département de Mathématiques, Université de Paris VI, BP 172 4, Place Jussieu, tour 46-45, 5e étage, 75252 Paris, France

E-mail: jpf@ccr.jussieu.fr

Y . Y O M D I N

Department of Theoretical Mathematics, The Weizmann Institute of Science Rehovot 76100, Israel

E-mail: yomdin@wisdom.weizmann.ac.il

I. Bernstein inequality and the number of zeroes. We first give two definitions (cf. N. Roytvarf, Y. Yomdin [R-Y]). ∆R denotes, as usual, the closed disk of radius R, centred at 0.

Definition I.1. Let R > 0, 0 < α < 1 and K > 0 be given and let f be holomorphic in a neighborhood of ∆_R. We say that f belongs to the Bernstein class B_R,α,K¹ if

max{|f (z)|, z ∈ ∆R} max{|f (z)|, z ∈ ∆αR} ≤ K.

R e m a r k. The name “Bernstein class” is justified by the fact that, according to one of the classical Bernstein inequalities, any polynomial of degree d belongs to B_R,α,K¹ , K = (1/α)^d for any R and α.

Definition I.2. Let a natural N , R > 0 and C > 0 be given, and let f (z) = P∞

k=0fkz^k be an analytic function in a neighborhood of 0 ∈ C. We say that f belongs to the Bernstein class B_N,R,C² , if

|f_j|R^j ≤ C max{|f_i|Rⁱ, i = 0, . . . , N }, j ≥ N + 1.

The two classes B¹and B²essentially coincide. More precisely, we have the following

1991 Mathematics Subject Classification: 58F21, 13B21.

Received by the editors: November 5, 1996; in the revised form: July 13, 1998.

The paper is in final form and no version of it will be published elsewhere.

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Lemma I.3. Let f be an element of B²N,R,C. Then f is analytic in an open disk ˚∆R

and for any R⁰ < R, 0 < α < 1 and K = _α¹^N

1 + (1 − α^N)_1−α^α + C_1−β^β , β = R⁰/R, f belongs to B¹_R0,α,K.

P r o o f. The convergence of f (z) = P∞

k=0fkz^k on ∆R is immediate. Let m = max{|f (z)|, z ∈ ∆αR⁰}. Then by the Cauchy formula, |fi| ≤ m/(αR⁰)ⁱ for any i. In particular, |f_i|Rⁱ ≤ m/(αR⁰/R)ⁱ ≤ m/(αR⁰/R)^N for i = 0, . . . , N . Hence |f_j|R^j ≤ Cm/(αR⁰/R)^N for any j ≥ N + 1. Now we can estimate |f | on ∆R⁰ as follows:

max{|f (z)|, z ∈ ∆R⁰} ≤

N

X

k=0

|fk|R^0k+

∞

X

k=N +1

|fk|R^0k

≤ m

N

X

k=0

1 αR⁰

k

R^0k+ Cm (αR⁰/R)^N

∞

X

k=N +1

(R⁰/R)^k

= m1 α

Nh

1 + (1 − α^N) α

1 − α + C β 1 − β

i .

R e m a r k. The constant K in Lemma I.3 can be chosen as ¹_αN

1 + C_1−αβ^αβ + C_1−β^β which in some cases gives a better estimate.

Conversely, if f belongs to B_R,α,K¹ , then it belongs to B_N,R,C² with N = log_αK and C given explicitly through R, α, K (cf. N. Roytvarf, Y. Yomdin [R-Y], Hayman [Ha]).

A relevance of Bernstein classes to our purpose is explained by the following lemma (which is well known in different forms in various fields of complex analysis; we give a version, obtained by M. Waldschmidt [W] in relation to transcendent number theory).

Lemma I.4. Let R > 0, and 0 < α < 1 be given and let f be holomorphic in a neighborhood of ∆_R. Then the number of zeroes of f in ˚∆_αRdoes not exceed

Log max{|f (z)|, z ∈ ∆R}/ max{|f (z)|, z ∈ ∆αR}

Log(1 + α²)/2α .

In other words, for an element f of B¹_R,α,K,

#{f⁻¹(0) ∩ ∆αR} ≤ Log K Log(1 + α²)/2α .

Frequently, in the theory of differential equations, we deal with analytic developments f (z) = P∞

k=0fkz^k where fk is defined inductively by an expression which involves the preceding coefficients. So it can often be shown that f belongs to a certain Bernstein class B². Combining the above results, we can estimate the number of zeroes of the functions in B² as follows:

Proposition I.5. Let f be an element of B²N,R,C. Then for any R⁰⁰< R, the number of zeroes of f in ˚∆R⁰⁰ does not exceed

N · min

{α,(R⁰⁰/R)<α<1}

1 + Log 1 + (1 − α^N)_1−α^α + C_1−γ^γ / Log(1/α) 1 + Log (1 + α²)/2/ Log(1/α) , where γ = R⁰⁰/αR < 1.

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P r o o f. For any α, R⁰⁰/R < α < 1, let R⁰= R⁰⁰/α. Then by Lemma I.3, f belongs to B_R¹0,α,K, with K = _α¹^N

1 + (1 − α^N)_1−α^α + C_1−γ^γ , where γ = R⁰⁰/αR = R⁰/R. Hence, by Lemma I.4 the number of zeroes of f on ˚∆R⁰⁰= ˚∆αR⁰ is bounded by

Log(1/α)^N 1 + (1 − α^N)_1−α^α + C_1−γ^γ Log (1 + α²)/2α

= N ·1 + Log 1 + (1 − α^N)_1−α^α + C_1−γ^γ / Log(1/α) 1 + Log((1 + α²)/2)/ Log(1/α) . Since the value of α between (R⁰⁰/R) and 1 or, equivalently, the value of R⁰, R > R⁰>

R⁰⁰ can be chosen arbitrarily, the proposition follows.

Corollary I.6. Let f be an element of B²N,R,C. Then

1) For R⁰⁰= R/4, the number of zeroes of f on ˚∆_R⁰⁰does not exceed N log_5/4(4+2C).

2) For R⁰⁰= R/2 max(C, 2), the number of zeroes of f on ˚∆_R⁰⁰ is at most 20N . 3) For R⁰⁰= Re^{−(10N +2)}/ max(C, 2), this number is at most N .

P r o o f. To prove 1), take α = ¹₂. Then γ = ¹₂ and

#{f⁻¹(0) ∩ ∆_R⁰⁰} ≤ N ·1 + Log(1 + (1 − ¹₂^N

) + C)/ Log 2

1 + Log⁵₈/ Log 2 ≤ N log_5/4(4 + 2C).

In 2) we also choose α = ¹₂. Then γ = 1/ max(C, 2), and we get

#{f⁻¹(0) ∩ ∆_R⁰⁰} ≤ N ·1 + Log 1 + (1 − ¹₂N

) + 2/ Log 2

1 + Log⁵₈/ Log 2 ≤ 20N.

Finally, for R⁰⁰= Re^{−(10N +2)}/ max(C, 2), we put α = e^−10N; then γ = 1/e²max(C, 2), and #{f⁻¹(0) ∩ ∆_R⁰⁰} ≤ N · (1 + (2/3N )). Since the number of zeroes is an integer, this yields #{f⁻¹(0) ∩ ∆R⁰⁰} ≤ N .

R e m a r k. By taking into account the Bernstein inequality, the last conclusion is strong enough to prove that the number of zeroes of a polynomial of degree d does not exceed d.

II. Projection of analytic sets. We introduce now the algebra C(R) as follows:

f (x, z) = fx(z) = f (x1, . . . , xn; z) =

∞

X

k=0

z^kfk(x1, . . . , xn)

belongs to C(R) if there are (α, β) so that the coefficients f^k(x1, . . . , xn) (k = 0, 1, . . .) are polynomials in x = (x1, . . . , xn) of degree less than αk + b and if P∞

k=0R^k|fk| < ∞.

The norm |fk| of the polynomial fk is (for instance) the sum of the absolute value of its coefficients.

Definition II.1. We define the Bautin ideal I of f (x, z) as the ideal generated by the coefficients f_k(x₁, . . . , x_n). The Bautin index is the minimal integer N so that f0(x1, . . . , xn), . . . , fN(x1, . . . , xn) generate this ideal I.

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We fix a total ordering < on Nⁿ so that:

α ∈ Nⁿ, β ∈ Nⁿ, β 6= 0, then α < α + β;

α ∈ Nⁿ, β ∈ Nⁿ, γ ∈ Nⁿ, α < γ then α + β < γ + β.

There are several possible choices of such an ordering. For instance, we can choose

< as follows: Let C(α) =Pn

i=1αi, α < β if C(α) < C(β) or if C(α) = C(β) and there exists k, 1 ≤ k ≤ n, such that α_j= b_j for j < k and α_k< β_k.

Given a polynomial f = P

A∈Nⁿf_Ax^A, and the total ordering on Nⁿ, we denote by exp f the largest exponent A such that fA6= 0. Let exp I = {A ∈ Nⁿ, A = exp f, f ∈ I}.

There is a unique minimal set E = {E₁, E₂, . . . , E_d} such that for every element β of exp I, there is an element ε in E and an element α in Nⁿ such that β = ε + α.

If the ordering of the multi-indices is given, we call a set of elements {h1, . . . , hd} of I such that exp h_k= E_k a standard basis (or Gr¨obner basis) of the ideal I.

Let ρ = max{|h_i− x^Eⁱ|, i = 1, . . . , d}.

Theorem II.2. There exists C > 0, depending on f , such that the function

∞

P

k=0

|fk|z^k belongs to B_N,R/(1+ρ)² nα,C.

P r o o f. Let h1(x1, . . . , xn), . . . , hd(x1, . . . , xn) be a Gr¨obner basis of the ideal I.

Moreover, since f₀, . . . , f_N generate I, we have h_j =PN

i=0φ^j_if_i. Let C₁= max_i,j|φ^j_i|.

From classical estimates on division of polynomials by an ideal, we obtain that there is C2> 0 such that for any element h of the ideal I,

h =

d

X

j=1

gjhj, with |gj| ≤ C2|h|(1 + ρ)^{n deg h}.

Several generalizations of these crucial estimates have been produced in the setting of analytic coefficients (cf. [Br], [H], [Ga]). Hence,

h =

N

X

i=0

g_i⁰fi, with |g_i⁰| ≤ dC1C2|h|(1 + ρ)^{n deg h}.

In particular, for any j ≥ N + 1 we have

fj =

N

X

i=0

g^0j_i fi, with |g_i^0j| ≤ dC1C2|fj|(1 + ρ)^n(αj+β)≤ dC1C2C3((1 + ρ)^nα/R)^j,

since for an element f (x, z) = P∞

k=0z^kf_k(x₁, . . . , x_n) of C(R), there is a constant C3

such that |fj| ≤ C3. Denote dC1C2C3(1 + ρ)^nβ by C4. This yields

|fj| ≤

N

X

i=0

|g_i^0j| |fi| ≤ C4((1 + ρ)^nα/R)^j

N

X

i=0

|fi|.

We obtain:

|fj|R⁰j ≤ C4(N + 1) max{|fi|, i = 0, . . . , N } ≤ C5max{|fi|R⁰ⁱ, i = 0, . . . , N }, where C5= C4max(1/R^0N, 1), R⁰= R/(1 + ρ)^nα.

This proves Theorem II.2 with C = C5.

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Theorem II.3. Let f be an element of C(R). Assume that there exist α, β so that the coefficients fk(x1, . . . , xn) are homogeneous of degree αk + β. Then for all x, the series f (x; z) =P∞

k=0z^kf_k(x) belongs to B_N,R² 0/|x|^α,C|x|^β. Write

fj(x) =

N

X

i=0

g_i^0j(x)fi(x).

Denote by homi(p(x)) the homogeneous component of degree i of a polynomial p(x). We obtain

f_j(x) =

N

X

i=0

hom_α(j−i)+β(g^0j_i (x))f_i(x).

Then we get

|homα(j−i)+β(g_i^0j(x))| ≤ |g^0j_i | |x|^α(j−i)+b≤ dC1C2C3(1/R⁰)^j|x|^α(j−i)+β, and this yields

|fj(x)| ≤

N

X

i=0

dC1C2C3(1/R⁰)^j|x|^α(j−i)+β|fi(x)| ≤ C4|x|^β(|x|^α/R⁰)^j

N

X

i=0

|fi(x)|/|x|^αi. Hence,

|fj(x)|(R⁰/|x|^α)^j ≤ C|x|^βmax{|fi(x)|(R⁰/|x|^α)ⁱ, i = 0, . . . , N }.

Theorem II.4. Let f be an element of C(R), N, C be as above. Let R⁰(x) = ¹₄R⁰/|x|^α, R⁰⁰(x) = (R/|x|^α)/2 max(C|x|^β, 2), R^∗(x) = (R/|x|^α)e^{−(10N +2)}/ max(C|x|^β, 2). Then for any x, the function fx(z) can have on the disks ˚∆R⁰(x), ˚∆R⁰⁰(x), ˚∆R^∗(x), at most N log_5/4(4 + 2C|x|^β), 20N and N zeroes, respectively.

In the article [F-Y], we followed a different presentation based on the use of the norm

“maximum on a polydisc”. This allows to handle more general data (f may have analytic coefficients) but it is necessary to use priviliged neighborhoods.

The Lojasiewicz inequality appears closely related to the subject. We take the oppor- tunity of this Symposium to mention briefly the connection and postpone its developments to further studies.

The Lojasiewicz inequality entails a constant K and an exponent δ so that

KX^∞

k=0

f_k²(x)δ

≤

N

X

i=0

f_i²(x).

It yields the inequality

|fj(x)| ≤ C max{|f_i(x)|^1/δ, i = 0, . . . , N }, j ≥ N + 1.

Going back to the Jensen inequality, we obtain the same type of bound for the number of zeroes (cf. Lemma I.4). The only change is that this bound gets multiplied by 1/δ.

The authors thank warmly E. Bierstone and P. Milman for several discussions which improved the presentation of their results. We like to add the reference [ L-T-Z], which recently appeared, where the most general case (codimension ≥ 1) has been considered with different techniques.

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