Zeros of Abelian Integrals and Hilbert’s XVIth problem

Susumu TANABÉ

Abstract We survey recent developments towards the solution of the (weak version of) Hilbert’s XVIth problem. The main theme is the study of number of limit cycles appearing in a perturbed Hamiltonian system which can be estimated by the number of zeros of Abelian integrals. Some methods which use Picard-Lefschetz theorem, Wronskian and Fuchsian differential equations are discussed.

0. Introduction to limit cycles

In an autonomous dynamical system,

(0.1)  \frac{d}{dt}x_{i} = v_{i}(x)  1 ≤ i ≤ n, x = (x_{1}, …, x_{n}).

whose right hand side consists of smooth functions in x ∈ ℝ^{n} (i.e. terms independent of time t), periodic régime may observed. A mathematical description of this phenomenon is furnished by the theory of ‘cycles’ (closed orbits of system (0.1)) initiated by Henri Poincaré (Mémoires sur les courbes définies par les équations différentielles I-IV, Oeuvres T.1, Gauthier-Villars, Paris). The following theorem assures us a simple geometric intuition in the study of orbits of the system (0.1) for n = 2.

The orbit (or phase curve) means the curve issued from a point (x_{1}, x_{2}) = (x_{1}(0), x_{2}(0) as a solution to the differential equation (0.1). Theorem 0.1. ([5]) An orbit (phase curve) of the dynamical system (0.1) for n = 2 i.e.

(0.1)′  \frac{d}{dt} ( \begin{matrix}x_{1} \\ x_{2} \end{matrix} ) = ( \begin{matrix}v_{1}(x) \\ v_{2}(x) \end{matrix} ) ,  x = (x_{1}, x_{2})

is either a single point or is diffeomorphic to a circle or a straight line.

The following example illustrates the theorem.

\frac{d}{dt} ( \begin{matrix}x_{1} \\ x_{2} \end{matrix} ) = ( \begin{matrix}x_{1} − x_{2} − x_{1}(x_{1}^{2} + x_{2}^{2}) \\ x_{1} + x_{2} − x_{2}(x_{1}^{2} + x_{2}^{2}) \end{matrix} )

γ_{0} = (0, 0), γ_{1}, γ_{3} : open orbits diffeomorphic to a straight line γ_{2} : closed orbit diffeomorphic to a circle

The orbit which is diffeomorphic to a circle is called ‘cycle’.

Suppose that a smooth vector field has a cycle γ. Choose a point ‘p’ on γ and draw a straight line ‘l’ passing through it and transversal to γ. An orbit starting from a point ‘q’ of the line ‘l’, sufficiently close to p shall return to a point Δ(q) on l. We call Δ(q) the monodromy transformation of q ∈ l.

It is evident that Δ(p) = p for p on a cycle γ. Especially we introduce the following notion.

Definition 1. A limit cycle of the system (0,1)’ is an isolated orbit (diffeomorphic to a circle) of the system (0,1)’. In other words, a closed orbit is a limit cycle if the fixed point of the monodromy transformation (i.e. q ∈ l such that Δ(q) = q) is isolated.

In 1900, in the occasion of the International Congress of Mathematicians, David Hilbert posed the following question among other 22 problems that shall stimulate mathematical investigations in different domains all along the XXth century (Mathematische Probleme).

Question No. 16

1. Let X(x, y) be a polynomial of degree n with real coefficients in two variables. Then how many closed ovals the level set X_{0} = {(x, y) ∈ ℝ^{2} : H(x, y) = 0} may have? Decide all possible dispositions of these ovals on ℝ^{2} (or its projectivization ℙℝ^{2}).

2. Let X_{1}(x, y), X_{2}(x, y) be two polynomials of degree n with real coefficients. Then how many limit cycles the dynamical system

\frac{dy}{dx} = \frac{X_{1}(x, y)}{X_{2}(x, y)}

may have? Let us explain the connexion between two sub-questions 1. and 2. that may seem to be, at a first

glance, alien one another.

Since more than 60 years ago [21], [10],[11], [3],[4] it has been known that the problem of the number of limit cycles of a perturbed Hamiltonian dynamical system that emerge from the original Hamiltonian system can be reduced to that of number of zeros of Abelian integrals associated to the Hamiltonian. As far as it concerns only the number of limit cycles of a perturbed Hamiltonian system (i.e. less general than the problem of Hilbert himself) , we may call it a weak version of Hilbert’s XVIth problem.

To see this let us look at a non-perturbed Hamiltonian system on the plane, (x, y) ∈ R^{2}:

(0.2)  dH(x, y) = 0.

Or in other words

ẋ = \frac{∂}{∂y} H(x, y)

(0.2)′  ẏ = −\frac{∂}{∂x}H(x, y),

where the Hamiltonian H(x, y) is a polynomial of degree n. The problem to estimate the number of limit cycles for the system (0.2) is a purely algebraic question i.e. the number of closed component of real energy surface X_{t} = {(x, y) ∈ R^{2} : H(x, y) + t = 0}.

Further we consider a perturbed dynamical system of (0.2) :

(0.3)  dH(x, y) + ϵ(P (x, y)dx + Q(x, y)dy) = 0,

or

ẋ = \frac{∂}{∂y} H(x, y) + ϵQ(x, y)

(0.3)′  ẏ = −\frac{∂}{∂x}H(x, y) − ϵP (x, y),

with polynomial perturbation of degree at most m, P (x, y) = ∑_{0≤i+j≤m} P_{i,j}x^{i}y^{j}, Q(x, y) =∑_{0≤i+j≤m} Q_{i,j} x^{i}y^{j}. Let us try to write down the monodromy transform for the system (0.3) in the neighbourhood of a closed cycle γ_{t} of (0.2). We will be interested in the limit cycles that may appear near γ_{t}. Denote such a cycle by γ_{t} + ϵδ, |ϵ| << 1, δ a curve homotopically equivalent to a closed cycle. Then the integral

T (t) = ∫_{γ_{t}+ϵδ} ẋdy − ẏdx = ∫_{γ_{t}+ϵδ} dH + ϵ(P (x, y)dx + Q(x, y)dy)

= ∫_{γ_{t}+ϵδ} dH + ϵ ∫_{γ_{t}} P (x, y)dx + Q(x, y)dy + ϵ^{2} ∫_{δ} P (x, y)dx + Q(x, y)dy,

gives the monodromy transform. This can be interpreted as an asymptotic expansion in ϵ whose first term corresponding to ϵ^{0} equals to

∮_{γ_{t}+ϵδ} H_{y}dy + H_{x}dx = H(begin) − H(end).

Thus if one choses γ_{t} + ϵδ a closed curve, then the above expression must be equal to 0. On the other hand, the term corresponding to ϵ^{1} is expressed by

(0.4)  I_{P,Q}(t) = ∫_{γ_{t}} P dx + Qdy,

which does not always vanish. If (0.4) possesses an isolated zero at t = t̃, then T (t̃) = O(ϵ^{2}). Hence we conclude that T (t) = ϵI_{P,Q}(t) + O(ϵ^{2}) must have so much zeros as I_{P,Q}(t) if | ϵ |<< 1. If the Hamiltonian system (0.3) has a closed cycle at t = t̃, I_{P,Q}(t) vanishes at point t = t̃. That is to say, the vanishing of the integral I_{P,Q}(t) is a necessary condition for the existence of a limit cycle. As a matter of fact we have the following theorem.

Theorem 0.2. If γt̃ generates a limit cycle of system (0.3) then I_{P,Q}(t) has simple zero at t = t̃. Conversely if I_{P,Q}(t) has an isolated zero at t = t̃, then γt̃ generates a limit cycle of system (0.3).

Definition 2. The integral I_{P,Q}(t) defined in (0.4) is called the Abelian integral associated to the Hamiltonian H(x, y).

This reduction permits us to apply the techniques of algebraic geometry to the study of limit cycles. In 1984, A.N. Varchenko and A.G. Khovanski proved independently the finiteness of the number of

zeros in the following form.

Theorem 0.3. For any positive integers n, m , there exists an integer c(n, m) such that the number of isolated zeros of I_{P,Q}(t) defined for deg H ≤ n, deg P, Q ≤ m (taking the multiplicity into account) must be less than c(n, m).

Remark 1. Here the term “isolated zeros” has been underlined not by chance. As a matter of fact, if we set P dx + Qdy = gdH + dR for g, R ∈ ℝ[x, y] then I_{P,Q}(t) ≡ 0. One proves this property in view of the Stokes’ theorem:

\begin{matrix}I_{P,Q}(t) = ∫_{γ_{t}} gdH + dR = ∫_{γ_{t}} d(gH + R) + ∫_{γ_{t}} H dg \\ = (−t) ∫_{γ_{t}} dg = 0 \end{matrix}

It is to be remarked that the above theorem states nothing (except its existence) on the numbers c(n, m). To answer what kind of behaviour of c(n, m) can we expect, more detailed studies are demanded. Quite recently (1995), Yu. S. Ilyashenko [13], D. Novikov and S. Yakovenko [17] established the following estimate for the Hamiltonian H(x, y) with Morse type singularities. Morse type singularity means that in a neighbourhood of a critical point (x^{∗}, y^{∗}) where grad H(x, y) degenerates (i.e. (\frac{∂}{∂x}H(x^{∗}, y^{∗}), \frac{∂}{∂y} H(x^{∗}, y^{∗})) = (0, 0)), smooth coordinate change (x, y) ↦ (X(x, y), Y (x, y)) (also (X, Y) ↦ (x(X, Y), y(X, Y)) is smooth) brings the Hamiltonian into the following canonical form: H(x, y) = X^{2} + Y^{2}. For details on the Morse type singularity see S. Pnevmatikos “Anazetontas tén taxi sti perioxi ton krisimwn simeiwn”, Taxi Kai Xaos, Tomos III.

Theorem 0.4. Suppose deg H(x, y), P (x, y), Q(x, y) ≤ n in (0.3). If H(x, y) + t = 0 has only Morse type singularity for any t ∈ ℝ, then ♯{ zeros of I_{P,Q}(t)} ≤ exp(c_{0}n + c_{1}) for some constants c_{0}, c_{1} > 0 independent of n.

Theorem 0.4 is certainly a progress in comparison with Theorem 0.3, however it can not be a final version of the answer to Hilbert’s XVIth problem. As we will see later in the following sections, more precise estimates of zeros of I_{P,Q}(t) are known for lower degree Hamiltonians. The aim of this article is to give rough sketches of possible methods that must be also useful to study the Abelian integrals associated to wider class of Hamiltonians.

1. Picard-Lefschetz theorem

\underline{1.1} Vanishing cycles and an algebraic lemma. In a series of works, G.S. Petrov proposed a powerful machinery to estimate zeros of elliptic and hyperelliptic integrals. His method consists mainly in the study of ramification properties of the (hyper-) elliptic integrals by means of the Picard-Lefschetz theorem. Shortly speaking, this theorem describes the monodromy of the homology group H_{1}(X_{s}, ℤ) in terms of the intersection number of base elements of H_{1}(X_{s}, ℤ).

We will trace the argument of Petrov in chasing his central work [19].

Let us look at the following Hamiltonian

(1.1)  H(x, y; s_{0}, s_{1}) = y^{2} + x^{4} − x^{2} − s_{1}x − s_{0}.

Assume that it has 3 distinct real critical values s_{0}^{(1)} < s_{0}^{(2)} < s_{0}^{(3)}. (this is a condition on s_{1}). We fix s_{1} in our further discussions. We say that H(x^{∗}, y^{∗}; 0, s_{1}) is a critical value of H(x, y; s) (we will denote simply H(x, y; s) instead of H(x, y; s_{0}, s_{1})) if the gradient of H at a point (x, y) = (x^{∗}, y^{∗}) degenerates i.e.

grad H(x^{∗}, y^{∗}; s) = (\frac{∂}{∂x}H(x^{∗}, y^{∗}; s), \frac{∂}{∂x}H(x^{∗}, y^{∗}; s)) = (0, 0).

You can imagine it as a bottom of potential valley. Let us define a family of so called “vanishing cycles” on a family of Riemann surfaces X_{s} = {(x, y) ∈ ℂ^{2} : H(x, y; s) = 0}. The vanishing cycles form a basis in the 1st homology group of X_{s}.

Denote by x_{i}^{∗} (i = 1, 2, 3) critical points of H(x, 0, 0, s_{1}) so that s_{0}^{(i)} = −H(x_{i}^{∗}, 0, 0, s_{1}). Let x_{0}(s) < x_{1}(s) < x_{2}(s) < x_{3}(s) be the consecutive roots of the equation H(x, 0; s) = 0 for s_{0}^{(1)} < s_{0} < s_{0}^{(3)}. We denote by γ_{1}(s) (resp. γ_{2}(s)) an 1-dimensional real cycle on X_{s} whose projection on ℂ_{x}^{1}-plane goes around roots x_{0}(s), x_{1}(s) (resp. x_{2}(s), x_{3}(s)).

If one observes the deformation of cycles that happens in approaching s_{0} to s_{0}^{(1)}, then it is visually clear that γ_{1}(s) tends to a point as x_{2}(s) → x_{3}(s) in this limit process. Under this situation we call γ_{1}(s) the vanishing cycle associated with the critical value s_{0}^{(1)}. Similarly γ_{2}(s) is the vanishing cycle associated with the critical value s_{0}^{(2)}. As a matter of fact, the closed curve γ_{t} of (0.4) can be regarded as one of these cycles γ_{i}(s), we define the integral

(1.2)  I_{ω,γ_{i}}(s) = ∫_{γ_{i}(s)} ω

for a form

ω = P dx + Qdy

under the condition deg P, Q ≤ n. With these notions, Petrov’s theorem can be formulated as follows: Theorem 1.1. The elliptic integral I_{ω,γ_{i}(s)} (i = 1, 2) for ω of (1.2) has at most n + [\frac{n−1}{2}] zeros on the interval s_{0} ∈ (s_{0}^{(1)}, s_{0}^{(3)}).

We begin the proof of this theorem with the following lemma. Further we write simply I_{ω}(s) if no specification of integration cycle is demanded.

Lemma 1.2. 1. For every ω of (1.2), there exist three polynomials P_{0}(s), P_{1}(s), P_{2}(s) ∈ ℝ[s_{0}, s_{0}] such that

(1.3)  I_{ω}(s) = P_{0}(s)I_{ydx}(s) + P_{1}(s)I_{xydx}(s) + P_{2}(s)I_{x^{2}ydx}(s)
(1.4)  deg_{s_{0}} P_{0}(s), deg_{s_{0}} P_{1}(s) ≤ p − 1, deg_{s_{0}} P_{2}(s) ≤ p − 2 if n = 2p
deg_{s_{0}} P_{0}(s) ≤ p, deg_{s_{0}} P_{1}(s), deg_{s_{0}} P_{2}(s) ≤ p − 1 if n = 2p + 1.

Here deg_{s_{0}} P_{i}(s) denotes a positive integer d such that

P_{i}(s) = P_{id}s_{0}^{d} + P_{id−1}(s_{i})s_{0}^{d−1} + … + P_{i0}(s_{1})

with P_{id} ≠ 0.

2. The following estimates hold for | s_{0} |>> 1 and fixed s_{1}:

| I_{x^{i}ydx}(s) |≤ c_{0} | s_{0} |^{\frac{3+i}{4}}

for certain positive constant c_{0} > 0.

\underline{Proof} We consider the following space of 1-forms

(1.5)  H′ = \frac{A_{ℝ^{2}}^{1}}{A_{ℝ^{2}}^{0}dH + dA_{ℝ^{2}}^{0} + HA_{ℝ^{2}}^{1}} ,

where A_{ℝ^{2}}^{0} = ℝ[x, y] and A_{ℝ^{2}}^{1} is a space of 1-forms with real polynomial coefficients. It means that to every 1-form P dx + Qdy ∈ A_{ℝ^{2}}^{1} we associate a representative ψ ∈ H′ in such a way that

(1.6)  P dx + Qdy = ψ + gdH + dh + Hφ

for some g, h ∈ A_{ℝ^{2}}^{0}, φ ∈ A_{ℝ^{2}}^{1}. This decomposition is important because of the following equalities derived from Stoke’s theorem:

(1.7)  \begin{matrix}∫_{γ_{i}(s)} dh = 0 \\ ∫_{γ_{i}(s)} gdH = ∫_{γ_{i}(s)} d(gH) − ∫_{γ_{i}(s)} Hdg = 0 \\ ∫_{γ_{i}(s)} Hφ = (−s_{0}) ∫_{γ_{i}(s)} φ \end{matrix}

for a vanishing cycle γ_{i}(s) ∈ H_{1}(X_{s}, ℤ). In other words, if ψ = 0 ∈ H′ then ∫_{γ_{i}(s)} ψ = 0 After direct calculus of the quotient space H′, we see that

(1.8)  H′ ≅ ⟨ydx, xydx, x^{2}ydx⟩_{ℝ}

From this one obtains (1.3) in applying iteratively (1.7) to (1.6) (cf. Weierstrass division theorem [6]). Before proving (1.4) we shall remark the quasihomogeneous property of H(x, y; s) i.e.

H(λx, λ^{2}y; λ^{4}s_{0}, λ^{3}s_{1}) = λ^{4}H(x, y; s_{0}, s_{1})

for every λ > 0. We prescribe so called quasihomogeneous weight w(·) to each variable i.e.

w(x) = 1, w(y) = 2, w(s_{1}) = 3, w(s_{0}) = 4.

Consequently we can supply the form x^{α}y^{β}dx with the quasihomogeneous weight w(x^{α}y^{β}dx) = α+2β+1. The statement on deg_{s_{0}} P_{i}(s) can be deduced from an evident formula:

deg_{s_{0}} P_{i}(s) ≤ [\frac{w(ω) − w(x^{i}ydx)}{w(s_{0})}] .

The estimate 2) is derived from the calculus of quasihomogeneous weights of the base forms of H′ like (1.8)

q.e.d.

\underline{1.2} Picard-Lefschetz theorem In this section we review the essence of the Picard-Lefschetz theorem. Let us imagine what kind of changing happens with the Riemann surface X_{s} (and with the cycle γ_{i}(s)) if you turn the value s_{0} ∈ ℂ around s_{0}^{(3)} on the complex plane. Here we consider a turn around s_{0}^{(3)} in the following manner:

λ(t) = e^{2πit} · α + s^{(3)}, 0 ≤ t < 1, s_{0}^{(2)} − s_{0}^{(3)} < α < 0.

Thus it is possible to construct a family of diffeomorphism Γ_{t} depending continuously on t,

Γ_{t} : X_{(λ(0),s_{1})} → X_{(λ(t),s_{1})}.

Though the Riemann surface X_{(λ(1),s_{1})} coming at the end of this process is identical to X_{(λ(0),s_{1})} the automorphism of H_{1}(X_{s}, ℤ) induced by Γ_{1} is not an identical mapping.

Especially the cycle γ_{1} is transformed into another cycle Γ_{1}(γ_{1}) such that

Γ_{1}(γ_{1}) − γ_{1} ∼ −γ_{3}

here ‘∼’ denotes homotopy equivalence. In the same way,

Γ_{1}(γ_{2}) − γ_{2} ∼ −γ_{3}.

The transformation

(1.9)  γ_{i} → Γ_{1}(γ_{i}) ∼ γ_{i} − γ_{3}  i = 1, 2

is called the monodromy transform of vanishing cycles around s_{0} = s_{0}^{(3)}. (Not to confuse this termonology with Δ(q) appeared in § 0). More generally we have the following theorem.

Theorem 1.3.

Picard-Lefschetz Theorem

Let H(x, y; s) parametrized by s ∈ R^{μ} be a Hamiltonian with the property

(\frac{∂}{∂x}H(x, y; 0), \frac{∂}{∂y} H(x, y; 0)) = (0, 0) if and only if (x, y) = (0, 0).

Let γ_{i}(s) be the vanishing cycle on X_{s} = {(x, y) ∈ ℂ^{2}; H(x, y; s) = 0} that vanishes at s = s^{(i)} ∈ ℝ^{μ}. Then the monodromy transform around s^{(i)} is given by the following formula for any δ ∈ H_{1}(X_{s}, ℤ):

(1.10)  Γ_{1}(δ) = δ + (γ_{i}, δ) · γ_{i}

where (γ_{i} · δ) denotes the intersection index between γ_{i} and δ.

The Picard-Lefschetz theorem reduces the study of monodromy of cycles to the calculus of intersection index (number) between base elements of the homology group. See for the detail [6].

In his famous investigations on the Landau singularities appeared in quantum physics, F. Pham [20] made use of Picard-Lefschetz theorem to describe the monodromy of integrals

I_{ω,γ_{i}}(s) = ∫_{γ_{i}(s)} ω

in taking into account that its variation around its ramfication point s^{(j)} is given by

(1.11)  ∫_{γ_{i}} ω ⟼ ∫_{γ_{i}} ω + ∫_{⟨γ_{j},γ_{i}⟩γ_{j}} ω = ∫_{γ_{i}} ω + ⟨γ_{j}, γ_{i}⟩ ∫_{γ_{j}} ω,

according to the Picard-Lefschetz theorem.

\underline{1.3} Petrov’s proof of the theorem 1.1. In the sequel let us denote \frac{d}{ds_{0}} I_{ω}(s) = I′_{ω}(s).

Lemma 1.4. 1. The vector of integrals I(s) = (I_{ydx}(s), I_{xydx}(s), I_{x^{2}ydx}(s)) satisfy the following system of differential equations.

S(s)\frac{∂}{∂s_{0}} I = (L + V)I,

where

S(s) = [ \begin{matrix}s_{0} & \frac{3}{4}s_{1} & \frac{1}{2} \\ −\frac{1}{8}s_{1} & s_{0} − \frac{3}{4} & \frac{3}{4}s_{1} \\ −\frac{3}{16}s_{1}^{2} & −\frac{1}{2}s_{1} & s_{0} − \frac{3}{4} \end{matrix} ] ,

L + V = [ \begin{matrix}\frac{3}{4} & 0 & 0 \\ 0 & 1 & 0 \\ \frac{1}{8} & 0 & \frac{5}{4} \end{matrix} ] .

2. The differentiated integral I′_{ω}(s) defined for ω of (1.2) admits the following decomposition:

I′_{ω}(s) = Q_{0}(s)I_{y^{−1}dx}(s) + Q_{1}(s)I_{xy^{−1}dx}(s) + Q_{2}(s)I_{x^{2}y^{−1}dx}(s)

where Q_{i}(s) are polynomials with deg_{s_{0}} Q_{i}(s) satisfying the same estimates for deg_{s_{0}} P_{i}(s) as in (1.5).

3. The integral I_{y^{−1}dx,γ_{i}}(s) (i = 1, 2, 3) has no zeros on the ray s_{0} < s_{0}^{(2)}.

\underline{Proof} As for 1) see § 3.

The statement 2) is a direct consequence of 1) and lemma 1.2, 1). One shows 3) for the case γ_{i} = γ_{2}. On the interval (s_{0}^{(1)}, s_{0}^{(2)})) I_{y^{−1}dx,γ_{2}}(s) is negative. On the ray {s_{0} ≤ s_{0}^{(2)}} the roots of the equation H(x, 0; s_{0}, s_{1}) = 0 form an equilateral trapezeum, hence I_{y^{−1}dx,γ_{2}}(s) = −I_{y^{−1}dx,γ_{1}}(s) which is equal to the integrals along remaining two sides. The sides contract to the narrowest base. The sum of integrals along these sides is of sign definite hence I_{y^{−1}dx,γ_{2}}(s) must be of sign definite. For γ_{i} = γ_{1}, γ_{3} similar argument holds. q.e.d.

Proof of the Theorem 1.1 consists of several steps. \underline{Step 1} After the lemma 1.4 we pursue the following argument. By Rolle’s theorem, it is enough to estimate the number of zeros of I′_{ω,γ_{i}}(s) to estimate that of I_{ω,γ_{i}}(s). Let us investigate the number of zeros of

(\frac{I′_{ω,γ_{i}} (s)}{I_{y^{−1}dx,γ_{i}} (s)}) in the domain D = \{s_{0} ∈ ℂ_{s_{0}} \ \{s_{0} ≥ s_{0}^{(3)}\}; |s_{0}| ≤ R\} with R large enough, that is equal to

♯\{zeros (Q_{0}(s) + Q_{1}(s) (\frac{I_{xy^{−1}dx,γ_{i}}(s)}{I_{y^{−1}dx,γ_{i}}(s)}) + Q_{2}(s) (\frac{I_{x^{2}y^{−1}dx,γ_{i}}(s)}{I_{y^{−1}dx,γ_{i}}(s)}))\} in the domain D,

which makes sense by virtue of the Lemma 1.4, 3) This number is in its turn estimated by

\frac{1}{2}♯\{zeros Im(Q_{0}(s) + Q_{1}(s)\frac{I_{xy^{−1},γ_{i}}(s)}{I_{y^{−1}dx,γ_{i}}(s)} + Q_{2}(s) (\frac{I_{x^{2}y^{−1}dx,γ_{i}}(s)}{I_{y^{−1}dx,γ_{i}}(s)}))\} along ∂D.

This is seen from the fact that a complex valued function f(s) has a zero in an domain if its imaginary part Im f(s) has two zeros on the boundary of the domain. If Im f(s) makes 2N turns around zero on ∂D, it implies that f(s) has N zeros in D.

To apply the argument principle of complex functions we estimate the zeros of Im (\frac{I′_{ω,γ_{i}} (s)}{I_{y^{−1}dx,γ_{i}} (s)}) along

a circle of large radius {s_{0} ∈ C; |s_{0}| = R}. The lemma 1.2, 2) and lemma 1.4, 2) give the estimate

(1.12)  \frac{1}{2}♯\{zeros of Im (\frac{I′_{ω,γ_{i}}}{I_{y^{−1}dx,γ_{i}}} (s)) along \{s_{0} ∈ C; |s_{0}| = R\}\} ≤ [\frac{n + 1}{2}] .

It remains to estimate zeros of Im (\frac{I′_{ω,γ_{i}} (s)}{I_{y^{−1}dx,γ_{i}} (s)}) along the cut \{s_{0} ≥ s_{0}^{(3)}\}.

\underline{Step 2} The following equality follows easily from the fact Im Q_{0}(s) ≡ 0 on {s_{0} ≥ s_{0}^{(3)}} :

(1.13) \frac{1}{2}♯\{zeros Im(Q_{0}(s) + Q_{1}(s) (\frac{I_{xy^{−1}dx}(s)}{I_{y^{−1}dx}(s)}) + Q_{2}(s) (\frac{I_{x^{2}y^{−1}dx}(s)}{I_{y^{−1}dx}(s)}))\} along \{s_{0} ≥ s^{(3)}\}

= \frac{1}{2}♯\{zeros Im(Q_{1}(s) (\frac{I_{xy^{−1}dx}(s)}{I_{y^{−1}dx}(s)}) + Q_{2}(s) (\frac{I_{x^{2}y^{−1}dx}(s)}{I_{y^{−1}dx}(s)}))\} along \{s_{0} ≥ s^{(3)}\}

= \frac{1}{2}♯\{zeros of\{\frac{1}{|I_{y^{−1}dx,γ_{i}(s)} |^{2}} ((Im I_{xy^{−1}dx}Re I_{y^{−1}dx} − Im I_{y^{−1}dx}Re I_{xy^{−1}dx}) Q_{1}(s)

+ (Im I_{x^{2}y^{−1}dx}Re I_{y^{−1}dx} − Im I_{y^{−1}dx}Re I_{x^{2}y^{−1}dx}) Q_{2}(s))} along {s_{0} ≥ s^{(3)}}.

Here we have fixed γ_{i} = γ_{1} and we omitted suffices indicating the cycle of integration.

\underline{Step 3} Let us show that on the cut \{s_{0} ≥ s_{0}^{(3)}\}, Im I_{ω,γ_{1}}(s) = ±\frac{i}{2}I_{ω,γ_{3}}(s) for any form ω ∈ A_{ℝ^{2}}^{1}. As I_{ω,γ_{1}}(s) is a complex conjugate function,

(1.14)  I_{ω,γ_{1}}(s) − I_{ω,γ_{1}}(s̄) = 2i Im I_{ω,γ_{1}}(s)

On the other hand this difference is nothing but the variation of the integral I_{ω,γ_{1}}(s) after its turn around the point s_{0} = s_{0}^{(3)} i.e. the monodromy transform around s_{0} = s_{0}^{(3)} minus identity. Here apply Picard-Lefschetz theorem (cf. (1.9) or (1.11)) in order to express the monodromy transformation by means of cycles:

(1.15)  I_{ω,γ_{1}}(s̄) = I_{ω,γ_{1}±γ_{3}}(s).

In comparing (1.14) and (1.15) we get

(1.16)  Im I_{ω,γ_{1}}(s) = ±\frac{i}{2}I_{ω,γ_{3}}(s).

\underline{Step 4} Let us examine the term

(1.17)  Im I_{xy^{−1}dx,γ_{1}}(s)Re I_{y^{−1}dx,γ_{1}}(s) − Im I_{y^{−1}dx,γ_{1}}(s) Re I_{xy^{−1}dx,γ_{1}}(s)

that appeared in (1.13). By using (1.16) we see

(1.17) = ±\frac{i}{2} det ( \begin{matrix}I_{y^{−1}dx,γ_{1}}(s) I_{y^{−1}dx,γ_{3}}(s) \\ I_{xy^{−1}dx,γ_{1}}(s) I_{xy^{−1}dx,γ_{3}}(s) \end{matrix} ) .

Let us denote this determinant iω(γ_{1}, γ_{3}). Similarly

(1.18)  Im I_{x^{2}y^{−1}dx,γ_{1}}Re I_{y^{−1}dx,γ_{1}} − Im I_{y^{−1}dx,γ_{1}} Re I_{x^{2}y^{−1}dx,γ_{1}}

= ±\frac{i}{2} det ( \begin{matrix}I_{y^{−1}dx,γ_{1}}(s) I_{y^{−1}dx,γ_{3}}(s) \\ I_{x^{2}y^{−1}dx,γ_{1}}(s) I_{x^{2}y^{−1}dx,γ_{3}}(s) \end{matrix} ) .

The Wronskian method explained in the following §2 and the lemma 1.2.2) yield that (1.15) is a holomorphic function bounded at ∞. That is to say (1.18) is a real constant.

\underline{Step 5} In combining (1.17) and (1.18), we see that the number (1.13) is nothing but

(1.19)  \frac{1}{2}♯ zeros of \{iω(γ_{1}, γ_{3})Q_{1}(s) + real const. Q_{2}(s)\} .

By the Wronskian method we see that ω(γ_{1}, γ_{3}) is also holomorphic at s_{0} = s_{0}^{(3)}, s_{0}^{(1)}. Thus the fuction iω(γ_{1}, γ_{3})Q_{1}(s) + real const. Q_{2}(s) is holomorphically extended to D_{1} = ℂ \ {s_{0} < s_{0}^{(2)}}. Let us estimate (1.19) near s_{0} = +∞. After the argument principle like in Step 1 and lemma 1.2. 2), the function iω(γ_{1}, γ_{3})Q_{1}(s) + real const. Q_{2}(s) goes around the origin less than [\frac{n−1}{2}] times if s_{0} turns around a large circle {|s_{0}| = R, R >> 1}. The cut along {s_{0} < s_{0}^{(2)}} gives the contribution to (1.19) estimated by

(1.20)  \frac{1}{2}♯\{zeros of Im iω(γ_{1}, γ_{3})Q_{1}(s)\}.

The lemma 1.4. 3), and the following lemma entail that (1.20) must be less than deg Q_{1}(s) ≤ \frac{n}{2} − 1.

Lemma 1.5. The real valued function iω(γ_{1}, γ_{3}) admits a holomorphic extension to D_{1} and its imaginary part is proportional to I_{y^{−1}dx,γ_{1}(s)} on the ray s_{0} < s_{0}^{(2)}.

\underline{Proof} The holomorphicity follows from the Wronskian argument. As for the second we observe

(1.21)  Im iω(γ_{1}, γ_{3}) = \frac{1}{2} (ω(γ_{1}, γ_{3}) + ω̄(γ_{1}, γ_{3})) = \frac{1}{2}ω(γ_{1}, γ_{3} + γ̄_{3}),

because γ_{1} is a real cycle on the ray s_{0}^{(1)} < s_{0} < s_{0}^{(3)}.The sum of cycles γ_{3} +γ̄_{3} is homotopically equivalent to ±γ_{2} on the interval (s_{0}^{(1)}, s_{0}^{(2)}) and to the cycle ±(γ_{2} − γ_{1}) on the ray {s_{0} > s_{0}^{(1)}}. Thus (1.21) is proportional to

det ( \begin{matrix}I_{y^{−1}dx,γ_{1}} I_{y^{−1}dx,γ_{1}+γ_{2}} \\ I_{xy^{−1}dx,γ_{1}} I_{xy^{−1}dx,γ_{1}+γ_{2}} \end{matrix} ) = det ( \begin{matrix}I_{y^{−1}dx,γ_{1}} 0 \\ I_{xy^{−1}dx,γ_{1}} const. \end{matrix} ) = const. I_{y^{−1}dx,γ_{1}}

The last equality results from the fact that γ_{1} + γ_{2} is homotopic to a cycle encircling the infinity and the elementary calculus of residues around x = ∞.

q.e.d.

At last the contribution to (1.19) from the little circle around s_{0} = s_{0}^{(2)} can be estimated as zero, as the following asymptotic expansion holds,

iω(γ_{1}, γ_{3}) = a_{0} log(s_{0} − s_{0}^{(2)}) + a_{1}(s_{0} − s_{0}^{(2)}) log(s_{0} − s_{0}^{(2)}) + …

after the Wronskian argument and Picard-Lefschetz theorem. In summary, estimates (1.12), (1.19), (1.20) plus Rolle’s theorem give the desired estimate of the Theorem 1.1. q.e.d.

2. Wronskian method

In a work appeared in 1978 Yu.S. Ilyashenko [12] proposed a new method to study the multiplicity of zeros of Abelian integrals by means of their Wronskians. His main result obtained after this method is the following.

Theorem 2.1. Let us look at the Hamiltonian

H(x, y) = x^{3} − 3x − y^{2}

and a perturbed Hamiltonian system:

(2.1)  dH + ϵ(P (x, y)dx + Q(x, y)dy) = 0

with deg P (x, y) ≤ 2, deg Q(x, y) ≤ 1. Then the multiplicity of limit cycles that can appear in (2.1) is less than 2.

We remark that the above theorem covers results established before and by F. Takens and R.I. Bogdanov (Versal deformations of a singularity of a vector field on the plane in the case of zero eigenvalues. Petrovski Seminar Proceedings (1976) vol. 2. English translation: Seclecta Math. Sovietica 1. (1981) 389-421).

To prove the Theorem 2.1, Ilyashenko studied the Abelian integrals

(2.2)  I_{ydx,γ}(s_{0}) = ∫_{γ(s_{0})} ydx, I_{xydx,γ}(s_{0}) = ∫_{γ(s_{0})} xydx

where γ(s_{0}) a vanishing cycle of the Riemann surface X_{s_{0}} = {(x, y) ∈ ℂ^{2} : H(x, y) + s_{0} = 0}. It is well known that X_{s_{0}} is topologically equivalent to a torus in this case and dim H_{1}(X_{s_{0}}, ℤ) = 2. So we choose a basis {γ_{1}, γ_{2}} of H_{1}(X_{s_{0}}, ℤ) such that γ_{1} is the vanishing cycle associated with the critical value s_{0} = 2. To I_{ω,γ_{i}}(s_{0}) defined in the same manner as in § 1 (see (1.2)), we associate the Wronskian as follows:

(2.3)  W (s_{0}, ω) := det ( \begin{matrix}I′_{ω,γ_{1}}(s_{0}) I′_{ω,γ_{2}}(s_{0}) \\ I′′_{ω,γ_{1}}(s_{0}) I′′_{ω,γ_{2}}(s_{0}) \end{matrix} )

where I′_{ω,γ}(s_{0}) = \frac{d}{ds_{0}} I_{ω,γ}(s_{0}). We shall be interested in the case ω = (a + bx)ydx. Remark that

\frac{d}{ds_{0}} ∫_{γ_{i}}(a + bx)ydx = ∫_{γ_{i}}(a + bx)\frac{dx}{y}.

In principle the integrals I_{ω,γ_{i}}(s_{0}) ramify around the critical values of the Hamiltonian H(x, y) (in this case s_{0} = ±2, ∞). The great advantage to make use of the Wronskian W (s_{0}, ω) is, however the univalence of W (s_{0}, ω) (not a multivalued function). In fact we will see that W (s_{0}, ω) is a rational function in s_{0}.

Let us observe the action induced by monodromy transformation on W (s_{0}, ω). Let us denote a turn of s_{0} ∈ ℂ around s_{0} = −2 by l. Then by the aid of the Picard-Lefschetz formula (1.10), (1.11):

l_{∗}W (s_{0}, ω) = det ( \begin{matrix}I′_{γ_{1}} + ⟨γ_{1}, γ_{2}⟩I′_{γ_{2}} I′_{γ_{2}} + ⟨γ_{2}, γ_{2}⟩I′_{γ_{2}} \\ I′′_{γ_{1}} + ⟨γ_{1}, γ_{2}⟩I′′_{γ_{2}} I′′_{γ_{2}} + ⟨γ_{2}, γ_{2}⟩I′′_{γ_{2}} \end{matrix} )

It is known by E. Cartan’s formula that ⟨γ_{2}, γ_{2}⟩ = 0 ([20]). Therefore the above expression is equal to

det ( \begin{matrix}I′_{γ_{1}} + ⟨γ_{1}, γ_{2}⟩I′_{γ_{2}} I′_{γ_{2}} \\ I′′_{γ_{1}} + ⟨γ_{1}, γ_{2}⟩I′′_{γ_{2}} I′′_{γ_{2}} \end{matrix} )

= det ( \begin{matrix}I′_{γ_{1}} I′_{γ_{2}} \\ I′′_{γ_{1}} I′′_{γ_{2}} \end{matrix} ) · det ( \begin{matrix}1 0 \\ ⟨γ_{1}, γ_{2}⟩ 1 \end{matrix} )

= det ( \begin{matrix}I′_{γ_{1}} I′_{γ_{2}} \\ I′′_{γ_{1}} I′′_{γ_{2}} \end{matrix} ) = W (s_{0}, ω).

The same argument works for a turn around s_{0} = 2. Hence W (s_{0}, ω) does not ramify neither at s_{0} = 2 nor s_{0} = −2, it is an univalent function on ℂ_{s_{0}}. On the other hand, by computing the growth by the size |γ_{i}(s_{0})| and the integrand ω as s_{0} approaches to a critical value of H(x, y), one sees

(2.4)  W (s_{0}, w) = O(s_{0}^{−2/3}) as s_{0} → ∞

(2.5)  W (s_{0}, ω) = O((s_{0} ± 2)^{−1}) as s_{0} → ±2.

This is verified by the same argument as that of lemma 1.2, 2). From the above estimates (2.4) and (2.5), one concludes that W (s_{0}, ω) admits the following simple expression:

(2.6)  W (s_{0}, ω) = \frac{A}{s_{0} − 2} + \frac{B}{s_{0} + 2} = \frac{(A + B)s_{0} + 2(A − B)}{(s_{0}^{2} − 4)}

for A, B ∈ ℂ. All rational functions satisfying (2.4) and (2.5) can be written down like this. On proving A + B ≠ 0, Ilyashenko asserts that the multiplicity of zeros of I_{ω,γ_{2}}(s_{0}) must be less than 2 because

{the multiplicity of zeros I′_{ω,γ_{1}}(s_{0})} + { the multipliciy of zeros of I′′_{ω,γ_{2}}(s_{0})} ≤ 1,

This is an evident conclusion from (2.3) and (2.6). q.e.d.

Several authors exploited the above method to obtain estimates on the muliplicities of zeros of abelian integrals defined for other Hamiltonians. For example, P. Mardešic obtained the following “best estimate” ever known till now for the Hamiltonians with minimally restrictive conditions. Namely he considered the situation below. Let H(x, y) = ∑_{i=0}^{n+1} H_{i}(x, y) ∈ ℝ[x, y] be a polynomial of degree n + 1. Here H_{i}(x, y) is a polynomial of homogeneous degree i. Assume that every critical value (singularity) of H(x, y) is of Morse type (see the commentary before Theorem 0.1). Secondly suppose that the algebraic variety {(x, y, z) ∈ ℂℙ^{2} : H_{n+1}(x, y) − z^{n+1} = 0} is a smooth submanifold of ℂℙ^{2}. Then the following estimate holds.

Theorem 2.2. Let γ(s_{0}) be a vanishing cycle of X_{s_{0}} = {(x, y) ∈ ℂ^{2} : H(x, y) + s_{0} = 0}.

Suppose deg P (x, y), Q(x, y) ≤ n and the conditions above are satisfied then the multiplicity of isolated zeros of an Abelian integral I_{P dx+Qdy,γ}(s_{0}) is less than \frac{n^{4}+n^{2}−2}{2}.

To prove this theorem, Madešic studied the Wronskian defined as the determinant of a n^{2} × n^{2} matrix

W (s_{0}, ω) = det ((\frac{d}{ds_{0}} )^{j} I_{γ_{i}}(s_{0})) \begin{matrix}1 ≤ i ≤ n^{2} \\ 0 ≤ j ≤ n^{2} − 1 \end{matrix} ,

where {γ_{1}, …, γ_{n^{2}}} a basis of the homology group H_{1}(X_{s_{0}}, ℤ). He shows the univalence hence the rationality of W (s_{0}, ω) by an similar argument to the above one, on showing

\begin{matrix}W (s_{0}, ω) = O (s0^{\frac{3n^{2}−n^{4}}{2}} ) as s_{0} → ∞ \\ W (s_{0}, ω) = O ((s_{0} − s_{0}^{(i)})^{2−n^{2}}) as s_{0} → s_{0}^{(i)} i = 1, …, n^{2} \end{matrix}

where s_{0}^{(i)} is a critical value of the Hamiltonian H(x, y). This implies that the Wronskian is a rational function of the form,

W (s_{0}, ω) = \frac{A(s_{0})}{B(s_{0})},

where deg A(s_{0}) ≤ \frac{n^{4}−n^{2}}{2} , deg B(s_{0}) ≤ (n^{2} − 1)n^{2}. Hence we see the multiplicity of isolated zeros of an Abelian integral (\frac{∂}{∂s_{0}} )^{n^{2}−1}I_{P dx+Qdy,γ_{i}}(s_{0}) is less than \frac{n^{4}−n^{2}}{2} which gives the estimate of the theorem q.e.d.

3. Fuchsian differential equation

3.1 Statement of the result

We remember that in the lemma 1.2, 1) we observed a system of differential equations satisfied by I_{x^{i}y^{ℓ}dx}(s), 0 ≤ i ≤ 2 defined for the Hamiltonian (1.1). As a matter of fact analogous system of differential equations appears in the investigation of Abelian integrals defined for more general type of Hamiltonians. Namely this method allows one to obtain estimates on the multiplicity of zeros of the Abelian integrals.

Let us try to get a system for the case of a Hamiltonian of the following form:

(3.1)  H(x, y) = F (x, s′) − y^{ν},

where the polynomial F (x, s′) is assumed to be as follows:

(3.2)  F (x, s′) = x^{μ+1} + s_{μ−1}x^{μ−1} + … + s_{1}x.

that depends on μ − 1 real parameters (s_{1}, …, s_{μ−1}).

Remark that every polynomial G(X) of degree μ + 1 in one variable can be reduced to the form F (x, s′) + s_{0} after a simple change of coordinate X → x = X + c, for certain c ∈ R. If you put μ = 2 and ν = 2 you reproduce the situation of §1.

In effect, every notion that has been introduced in §1 can be defined and studied in a similar manner for the Hamiltonian (3.1). That is to say we shall look at a family of Riemann surface

(3.3)  X_{s}^{(μ,ν)} = {(x, y) ∈ P^{2}C; F (x, s′) − y^{ν} + s_{0} = 0}

and the vanishing cycles of the homology group H_{1}(X_{s}^{(μ,ν)}). It is known that the rank of this group is equal to μ(ν − 1). Further we shall look at the quotient space like (1.5) for our Hamiltonian (3.1) and obtain from it a basis of the cohomology group H^{1}(X_{s}^{(μ,ν)}) in the following manner:

(3.4)  x^{k}y^{ℓ}dx, (0 ≤ k ≤ μ − 1, 1 ≤ ℓ ≤ ν − 1).

It justifies the duality between H_{1}(X_{s}^{(μ,ν)}) and H^{1}(X_{s}^{(μ,ν)}). That is to say, the ranks of two groups are equal.

In order to introduce the Abelian integrals, we fix a vanishing cycle γ_{s} ∈ H_{1}(X_{s}^{(μ,ν)}). We define the Abelian integral

(3.5)  K_{i}^{λ}(s) = ∫_{γ_{s}} x^{i}(F (x, s′) + s_{0})^{λ}dx,  i = 0, …, μ + 1.

for λ = \frac{ℓ}{ν} . Further we shall use the notation λ in this sense for the sake of simpicity. You imagine γ_{s} as a sum of cycles which turn around two roots of the equation F (z, s′) + s_{0} = 0.

This is a (multivalued) function in μ variables. Leray’s residue theorem ([7], [6]) assures that one needs not to fix concretely a cycle γ_{s} in (3.5) during the search for differential equations satisfied by (3.5). Namely Leray’s residue theorem tells us,

(3.6)  \frac{∂}{∂s_{0}} K_{ℓ}^{λ} = ∫_{γ_{s}} z^{i} \frac{∂}{∂s_{0}} (F (z, s′) + s_{0})^{λ}dz.

Every integral defined by (3.5) satisfies the same differential equation independent of the cyle γ_{s}.

Further we shall impose a condition on the Hamiltonian (3.1) that its critical values s_{0}^{(i)} never coincides (i.e. s_{0}^{(i)} ≠ s_{0}^{(j)} if i ≠ j) and it has singularity of Morse type i.e. the Hessian is non-degenerate at each critical point (cf. §1).

As we intend to establish an estimate of the multiplicity of zeros of a ramifying function, we will precise the definition of this notion:

Definition 3. If a convergent series in the neighbourhood of t = t_{0} defines a multivalued function in a sector Δ_{θ} = {t : |t − t_{0}| < θ} ⊂ C,

f(t) = ∑_{ρ∈Q,k∈N} f_{ρ,k}(t − t_{0})^{ρ}(log(t − t_{0}))^{k}

this series is called a Dulac function.

If the above series has the following characteristic exponents:

ρ_{1} = {min ρ; f_{ρ,k} ≠ 0},
k_{1} = {max k; f_{ρ_{1},k} ≠ 0},

we say that a Dulac function has t_{0} as a zero of multiplicity (k_{1} + 1)([ρ_{1}] + 1).

Dulac [8] remarked that the monodromy transform Δ(q) of a dynamical system defined in §0 is a Dulac function in q.

Further we shall understand the zero of the Abelian integral as that of a Dulac function in variable s_{0}. After Picard-Lefschetz theorem, the integral Kk^{\frac{m}{ν}} (s) has the critical values \{s_{0}^{(1)}, ⋯, s_{0}^{(μ)}\} of F (x, s′)+

y^{ν} as its ramifiation points.

Theorem 3.1. Under the above situation let us consider the Abelian integral Kk^{\frac{m}{ν}} (s) defined along a cycle γ_{s} = {(x, y) ∈ R^{2} : H(x, y) + s_{0} = 0},

Kk^{\frac{m}{ν}} (s) = ∫_{γ_{s}} x^{k}y^{m}(x, y)dx,

with K ∈ N, m ∈ Z. Suppose Kk^{\frac{m}{ν}} (s) ≢ 0. Then we have the following estimates on the multiplicity N of zeros of the Abelian integral Kk^{\frac{m}{ν}} (s).

i) At one of its ramification points s̃_{0} ∈ {s_{0}^{(1)}, ⋯, s_{0}^{(μ)}}, N satisfies:

N ≤ 2[\frac{m}{ν} + \frac{3}{2}].

ii) At a point where it is smooth i.e. s_{0} = s̃_{0} ∉ {0, s_{0}^{(1)}, ⋯, s_{0}^{(μ)}}, N does not surpass μ.

iii) At the orign s_{0} = 0, N satisfies:

N ≤ max\{μ − 1, [\frac{m}{ν}] + k + 2\}.

3.2 Differential operator annihilating the Abelian integral.

To get concrete expression of the Abelian integrals in the neighbourhood of ramification points, we start from the following proposition.

Proposition 3.2 ([15], [18]). The Abelian integrals K_{0}^{λ}(s), …, K_{μ+1}^{λ}(s) satisfy the following overdetermined system of differential equations:

(3.7)_{i}  ∑_{ℓ=0}^{μ−1} s_{ℓ} \frac{∂}{∂s_{0}} K_{ℓ+i}^{λ} + \frac{∂}{∂s_{0}} K_{μ+1+i}^{λ} = λK_{i}^{λ}, 0 ≤ i ≤ μ − 1,

(3.8)_{j}  ∑_{ℓ=1}^{μ−1} ℓs_{ℓ} \frac{∂}{∂s_{0}} K_{ℓ+j}^{λ} + (μ + 1)\frac{∂}{∂s_{0}} K_{μ+1+j}^{λ} = −(j + 1)K_{j}^{λ}, −1 ≤ j ≤ μ − 1.

We get (3.7)_{i} in applying directly (3.6) to the definition of the integral (3.5). The left hand side of (3.8)_{j} turns out to be equal to

∫_{Γ_{s}} z^{j+1} \frac{∂}{∂z} (F (z, s′) + s_{0})^{λ}dz,

which in its turn must be equal to the right hand side of (3.8)_{j} by integration by parts.

The relations (3.7)_{i} (3.8)_{j} can be rewritten in a matrix representation as follows.

Σ · b⃗ = a⃗,

where

Σ = [ \begin{matrix}s_{0} & s_{1} & ⋯ & s_{μ−2} & s_{μ−1} & 0 & 1 & ⋯ & 0 & 0 & 0 \\ 0 & s_{0} & ⋯ & s_{μ−3} & s_{μ−2} & s_{μ−1} & 0 & ⋯ & 0 & 0 & 0 \\ 0 & 0 & ⋯ & s_{μ−4} & s_{μ−3} & s_{μ−2} & s_{μ−1} & ^{^{.}^{.} .} & 0 & 0 & 0 \\ ⋮ & ⋮ & ^{^{.}^{.} .} & ⋮ & ⋮ & ⋮ & ⋮ & ⋯ & ⋮ & ⋮ & ⋮ \\ 0 & 0 & ⋯ & s_{1} & s_{2} & s_{3} & s_{4} & ⋯ & 1 & 0 & 0 \\ 0 & 0 & ⋯ & s_{0} & s_{1} & s_{2} & s_{3} & ⋯ & 0 & 1 & 0 \\ 0 & 0 & ⋯ & 0 & s_{0} & s_{1} & s_{2} & ⋯ & s_{μ} & 0 & 1 \\ s_{1} & 2s_{2} & ⋯ & (μ − 1)s_{μ−1} & 0 & μ + 1 & 0 & ⋯ & 0 & 0 & 0 \\ 0 & s_{1} & ⋯ & (μ − 2)s_{μ−2} & (μ − 1)s_{μ−1} & 0 & μ + 1 & ^{^{.}^{.} .} & 0 & 0 & 0 \\ ⋮ & ⋮ & ⋯ & ⋮ & ⋮ & ⋮ & ^{^{.}^{.} .} & ⋯ & ⋮ & ⋮ & ⋮ \\ 0 & 0 & ⋯ & s_{1} & 2s_{2} & 3s_{3} & ⋯ & ⋯ & μ + 1 & 0 & 0 \\ 0 & 0 & ⋯ & 0 & s_{1} & 2s_{2} & ⋯ & ⋯ & 0 & μ + 1 & 0 \\ 0 & 0 & ⋯ & 0 & 0 & s_{1} & ⋯ & ⋯ & (μ − 1)s_{μ−1} & 0 & μ + 1 \end{matrix} ],

a⃗ =^{t} (λK_{0}^{λ}, λK_{1}^{λ}, ⋯, λK_{μ−1}^{λ}, 0, −K_{0}^{λ}, −2K_{1}^{λ}, ⋯, −μK_{μ−1}^{λ})

b⃗ =^{t} (\frac{∂}{∂s_{0}} K_{0}^{λ}, \frac{∂}{∂s_{0}} K_{1}^{λ}, ⋯, \frac{∂}{∂s_{0}} K_{2μ}^{λ}).

Here one shall remark the following simple relation, \frac{∂}{∂s_{i}} K_{j}^{λ}(s) = \frac{∂}{∂s_{j}} K_{i}^{λ}(s).

In fact 2μ Abelian integrals take part in the equations (3.7)_{i}, (3.8)_{j} instead of μ integrals. You may wonder why I do not speak about μ(ν − 1) integrals that can be obtained from the froms (3.4). The reason is that for fixed λ = \frac{ℓ}{ν} with 1 ≤ ℓ ≤ ν − 1 the integrals have non-trivial relations only among μ integrals K_{j}^{λ}(s) 0 ≤ j ≤ μ − 1 after Proposition 3.2.

Proposition 3.3. The integrals K(s) = (K_{0}(s)^{λ}, ⋯, K_{μ−1}^{λ}(s)), s = (s_{0}, ⋯, s_{μ−1}) satisfy the following system of differential equations:

(3.9)  S \frac{∂}{∂s_{0}} K = (L + V (s_{2}, ⋯, s_{μ−1}))K,

where V (s_{2}, ⋯, s_{μ−1}) is a rank μ − 2 matrix with polynomial entries depending only on s_{2}, ⋯, s_{μ−1}. The matrix S admits the following representation

S = s_{0}id_{μ} + C(s_{1}, …, s_{μ−1})

with a polynomial matrix C(s_{1}, …, s_{μ−1}) and

L = diag(λ + \frac{1}{μ + 1}, …, λ + \frac{μ}{μ + 1}).

We will make use of the notation Δ_{μ}(s) designating a monic polynomial in variable s_{0} that is the discriminant of the polynomial z^{μ+1} + s_{μ−1}z^{μ−1} + … + s_{1}z + s_{0}. It means that for s ∈ R^{μ} the equation F (x, s′) + s_{0} = 0 has multiple (double, triple etc.) roots. The discriminant can be calculated as determinant of the matrix S(s) of the Proposition 3.3,

Δ_{μ}(s_{0}, s′) = det S(s) = det Σ.

Specifically the matrix Σ is often called Sylvester’s resultant (cf. van der Waerden, Modern Algebra).

Definition 4. One says that s′ ∈ R^{μ−1} belongs to the bifurcation set of the Hamiltonian (3.1) B ⊂ R^{μ−1} if and only if there exists s_{0} ∈ C such that

Δ_{μ}(s_{0}, s′) = \frac{∂}{∂s_{0}} Δ_{μ}(s_{0}, s′) = 0.

Remark 2. The bifurcation set classifies the Hamiltonians (3.2). The Hamiltonians outside of B are those in generic position i.e. the case that you encounter most often in the nature as they compose a subset of codimension 0 in the space of all Hamiltonians. And our condition that the Hamiltonian has only Morse type singularities and different critical values is equivalent to say that the Hamiltonian is outside of B. The next typical case will be those on the regular part of B. After that comes those on codimension 1 subset of B, etc.

After Proposition 3.3, our strategy shall be developed in the following manner. We seek for a differential operator that annihilates K_{i}^{λ}(s) (Proposition 3.4). It turns out that these operators are Fuchsian differential operators with its singular locus along D = {s ∈ C^{μ}; Δ_{μ}(s) = 0} possibly plus an affine plane. We shall determine the characteristic exponents (i.e. how fast the function annihilated by these operators can vanish at its singular locus) by reducing the general situation to a special example by means of isomonodromy technique.The maximum of the characteristic exponents shall estimate the multiplicity of the zeros of the integrals K_{i}^{λ}(s).

In order to get the Fuchsian differential operator annihilating K_{i}^{λ}(s), one shall introduce determinant of a matrix with operator elements like the following.

P(s, \frac{∂}{∂s_{0}} ) = [ \begin{matrix}p_{0,0} & p_{0,1} & ⋯ & p_{0,μ−1} \\ p_{1,0} & ⋮ & ⋯ & p_{1,μ−1} \\ ⋮ & ⋮ & ⋯ & ⋮ \\ p_{μ−1,0} & p_{μ−1,1} & ⋯ & p_{μ−1,μ−1} \end{matrix} ] ,

with operator elements like the following,

detP(s, \frac{∂}{∂s_{0}} ) :=

(3.10)  = ∑_{i_{0},i_{1},··· ,i_{μ−1}} sign(i_{0}, i_{1}, ⋯, i_{μ−1})p_{i_{μ−1},μ−1} ⋯ p_{i_{1},1}p_{i_{0},0},

for p_{i,j} = p_{i,j}^{1}(s)\frac{∂}{∂s_{0}} + p_{i,j}^{0}(s), with p_{i,j}^{1}(s), p_{i,j}^{0}(s) ∈ R[s]. In (3.10), the indices (i_{0}, i_{1}, ⋯, i_{μ−1}) run through all the permutations of (0, 1, ⋯, μ − 1).

Proposition 3.4. Differential operator P(s, \frac{∂}{∂s}) of degree μ defined below annihilates the integral K_{0}^{λ}(s),

P(s, \frac{∂}{∂s})K_{0}^{λ}(s) = 0,

where

P(s, \frac{∂}{∂s}) := det(S \frac{∂}{∂s_{0}} − L − V (s′)),

the matrices S, L, V (s′) are those which are defined in the Proposition 3.3.

We can get by a similar method a differential operator P_{x^{0}}^{(k)}(s, \frac{∂}{∂s}) of degree μ + 1 annihilating K_{k}^{λ} :

P_{0}^{(k)}(s, \frac{∂}{∂s_{0}} )K_{k}^{λ}(s) = 0.

These operators have the following “principal terms” (the terms with highest order differentials):

(3.11)  P(s, \frac{∂}{∂s}) = Δ_{μ}(s)(\frac{∂}{∂s_{0}} )^{μ} + ∑_{1≤j≤μ} P_{μ−j}(s)(\frac{∂}{∂s_{0}} )^{μ−j},

and

P_{0}^{(k)}(s, \frac{∂}{∂s})

(3.12)  = s_{0}Δ_{μ}(s)(\frac{∂}{∂s_{0}} )^{μ+1} + ∑_{1≤j≤μ+1} P_{μ−j+1}^{(k)}(s)(\frac{∂}{∂s_{0}} )^{μ+1−j},

for k ≥ 1. And these operators are Fuchsian differential operators, that is to say, deg_{s_{0}}P_{μ−j}(s) = μ − j, deg_{s_{0}}P_{μ+1−j}^{(k)}(s) = μ + 1 − j. The notation deg_{s_{0}} is that of (1.4). Remark that this Proposition already proves the statement ii) of Theorem 3.1, because as it is easy to see, a solution to a differential equation of order μ + 1 cannot have a zero of multiplicity strictly larger than μ. Now the proof of the statement i), iii) remained.

3.3 Calculus of the characteristic exponents

In order to explain the notion of the characteristic exponents, let us remember Frobenius method for a second order Fuchsian operator. Let P (t, \frac{∂}{∂t}) be an ordinary differential operator defined as follows:

P (t, \frac{d}{dt}) = t^{2}b_{2}(t)(\frac{d}{dt})^{2} + tb_{1}(t)\frac{d}{dt} + b_{0}(t),

where b_{2}(0)b_{1}(0)b_{0}(0) ≠ 0. Then the solutions of the equation P (t, \frac{d}{dt})u(t) = 0 are known to be linear combination of functions of the form v(t) = t^{λ}(log t)^{κ}α(t) with α(t) a holomorphic function at t = 0 i.e. a Dulac function. To determine numbers λ and permissible values of κ, we shall solve so called indicial equation (see (3.13) below) for P (t, \frac{d}{dt}). Substituting u(t) = t^{λ} one sees that a function like v(t) can be solution of the equation if and only if

(3.13)  b_{2}(0)λ(λ − 1) + b_{1}(0)λ + b_{0}(0) = 0.

If the equation (3.13) has distinct roots λ_{1}, λ_{2} called characteristic exponents, then we have two linearly independent solutions:

u(t) = t^{λ_{i}}α_{i}(t), i = 1, 2

i.e. κ = 0. While if the (3.13) has a double root λ_{0}, then we have two linearly independent solutions:

u(t) = t^{λ_{0}}α_{0}(t) or = t^{λ_{0}}(log t)α̃_{0}(t),

i.e. κ = 0, 1. See the book by Coddington-Levinson, Ince or Whittaker-Watson “Modern Analysis” to review more detailed and generalized discussion.

We can apply the same method to a higher order Fuchsian differential operator. As a matter of fact, the indicial equation associated to the equation (3.11) (resp. (3.12)) has always a set of μ (resp. μ + 1) rational numbers independent of the choice of s′ ∈ R^{μ−1} \ B. One can see it from the fact that the characteristic exponents of (3.12) must be rational numbers λ_{1}(s′), ⋯, λ_{μ+1}(s′), while the indicial equation itself changes continuously depending on s′ ∈ R^{μ−1} \ B. On B the multiplicity of the indicial equation can change and we cannot extend it beyond B.

After Picard-Lefschetz theorem explained in §1, the characteristic exponents of (3.12) must satisfy

e^{2πiλ_{1}(s′)} = −e^{2πi \frac{m}{ν}} , e^{2πiλ_{2}(s′)} = ⋯ = e^{2πiλ_{μ+1}(s′)} = 1.

This means that

λ_{i}(s′_{1}) − λ_{i}(s′_{2}) ∈ Z, 1 ≤ i ≤ μ + 1,

for s′_{1} ≠ s′_{2} ∈ R^{μ−1} \ B. In taking into account the continuous dependence of the characteristic exponents on s′ ∈ R^{μ−1} we conclude that λ_{i}(s′) must be constant on s′ ∈ R^{μ−1} \ B (isomonodromy technique). Now we can choose any point s′ ∈ R^{μ−1} \ B to calculate concretely the characteristic exponents of (3.11), (3.12). Let us choose for the sake of simplicity s′ = (s_{1}, 0, ⋯, 0). Then the equations under question will be as follows:

(3.13).  [∏_{j=0}^{μ−1}(s_{0} \frac{∂}{∂s_{0}} − \frac{j + 1}{μ + 1} + j − \frac{m}{ν}) + \frac{s_{1}}{μ + 1}(−\frac{μ}{μ + 1}s_{1} \frac{∂}{∂s_{0}} )^{μ}]K0^{\frac{m}{ν}} (s_{0}, s_{1}) = 0

On the other hand for Kk^{\frac{m}{ν}} (s_{0}, s_{1}), we have

[(s_{0} \frac{∂}{∂s_{0}} − \frac{k}{μ + 1} + μ − 1 − \frac{m}{ν}) ∏_{j=0}^{μ−1}(s_{0} \frac{∂}{∂s_{0}} +

(3.14)  +\frac{μj − k}{μ + 1} − \frac{m}{ν}) + \frac{s_{1}}{μ + 1}(−\frac{μ}{μ + 1}s_{1} \frac{∂}{∂s_{0}} )^{μ}(s_{0} \frac{∂}{∂s_{0}} − k − 1 − \frac{m}{ν})]Kk^{\frac{m}{ν}} (s_{0}, s_{1}) = 0.

If you calculate the characteristic exponents at singular points of the equations (3.13), (3.14) we get the following result.

Proposition 3.5. The characteristic exponents at a singular point s_{0} = μ(−\frac{s_{1}}{μ+1})^{\frac{μ+1}{μ}} of the equations (3.13) are the following:

ρ = 0, 1, ⋯, μ − 2, \frac{1}{2} + \frac{m}{ν}.

At the same point, the characteristic exponents of (3.14) are the follwoing:

ρ = 0, 1, ⋯, μ − 1, \frac{1}{2} + \frac{m}{ν}.

At the point s_{0} = 0 its characteristic exponents are as follows:

ρ = 0, 1, ⋯, μ − 1, \frac{m}{ν} + k + 1.

As it has been explained on the Frobenius method, if the indicial equation has double roots ρ then there exists a solution to (3.14) in the form

Kk^{\frac{m}{ν}} (s_{0}, s_{1}) = (s_{0} − s_{0}(s_{1}))^{ρ} log(s_{0} − s_{0}(s′))α′_{0}(s_{0} − s_{0}(s′)),

with a holomorphic function α′_{0}(•) in the neighbourhood of s_{0} = s_{0}(s′). This situation can happen only with ρ = \frac{1}{2} + \frac{m}{ν}. After definition 3, the multiplicity of zeros N of Kk^{\frac{m}{ν}} (s_{0}, s_{1}) must satisfy the inequality of the statement Theorem 3.1, i).

The statement iii) follows from the fact that \frac{m}{ν} ∉ N. If \frac{m}{ν} = ℓ ∈ N then the integral K_{k}^{ℓ}(s) has an exact form as its integrand. Hence Kk^{\frac{m}{ν}} (s) ≡ 0. Q.E.D.

The detailed proof of Theorem 3.1 will appear in [22]

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[23] A.N.Varchenko , An estimate of the number of zeros of an Abelian integral depending on a parameter and limit cycles , Func.Anal.Appl. 18, no.2 (1984) , pp. 14-25.

Max Planck Institut für Mathematik, Vivatsgasse7, D-53111, Bonn, Germany

Moscow Independent University, Bol’sdhoj Vlasievskij perulok 11,

Moscow, 121002, Russia E-mails: tanabe@mpim-bonn.mpg.de,

tanabe@ipu.rssi.ru