ON THE BIRKHOFF NORMAL FORM OF A COMPLETELY

INTEGRABLE HAMILTONIAN SYSTEM NEAR A

FIXED POINT WITH RESONANCE

T. Kappeler^{∗}, Y. Kodama^{∗} and A. Némethi^{∗∗}

1. Introduction and summary of the results

In this paper we are concerned with the normal form of a completely integrable Hamiltonian system near an equilibrium point. Let H = H(z) be an analytic function, H: U → ℂ, defined on an open neighborhood U of the origin in ℂ^{2n}. Assume that H has a power series expansion near the origin z = (x, y) = 0 ∈ ℂ^{2n}

of the form H = ∑_{j=1}^{n} λ_{j}x_{j}y_{j} + O(|z|^{3}) where λ = (λ_{1}, …, λ_{n}) ∈ ℂ^{n}.

Normal form theory for Hamiltonian systems was first studied by Birkhoff (cf. [Mo]). He proved that in the case where λ_{1}, …, λ_{n} are rationally independent (i.e. in the nonresonant case), there exists a formal canonical coordinate transformation, z = φ(ζ) = ζ + O(|ζ|^{2}), so that H ∘ φ is a formal power series,

∑_{α∈ℤ_{≥0}^{n}\{0}} c_{α}ξ^{α}η^{α} (Birkhoff normal form)

with ζ = (ξ, η) ∈ ℂ^{2n}. Later Siegel [Si] showed that the power series which define the coordinate transformation φ are generically divergent. Notice that if these power series are convergent, then the Hamiltonian system is in fact integrable with I_{j} := ξ_{j}η_{j} (1 ≤ j ≤ n) being functionally independent integrals which Poisson commute. The converse is also true: first results concerning a convergent Birkhoff normal form of an integrable Hamiltonian system near a nonresonant fixed point are due to Vey [Ve] (cf. also [El] where results in the C^{∞}-case were proved) and were later substantially improved by Ito [It1].

The concept of Birkhoff normal form can be generalized to Hamiltonian systems near a resonant fixed point and one might ask again if an integrable Hamiltonian system has a Birkhoff normal form near a resonant fixed point.

The only results so far in this direction are due to Ito [It2] and concern a special case of simple resonance. In this paper we treat the general case of a simple resonance. To state our results we introduce the following notation:

Typeset by AMS-TEX

∗ Supported in part by NSF. ^{∗∗} Supported in part by an OSU Seed Grant

Let G_{j} = G_{j}(z) (1 ≤ j ≤ n) be holomorphic functions, G_{j}: U → ℂ, defined for z = (x_{k}, y_{k})_{1≤k≤n} in an open neighborhood U of the origin in ℂ^{2n} such that

G_{1}, …, G_{n} pairwise Poisson commute, i.e. \{G_{i}, G_{j}\} = ∑_{k=1}^{n} \frac{∂G_{i}}{∂x_{k}} \frac{∂G_{j}}{∂y_{k}} − \frac{∂G_{i}}{∂y_{k}} \frac{∂G_{j}}{∂x_{k}} = 0,

are elements in M^{2} and have the property that dG_{1}, …, dG_{n} are generically linearly independent. Here M^{2} denotes the vector space of germs of analytic functions f at 0, which vanish up to first order at 0 (f(0) = 0, ∂_{x_{k}}f(0) = 0, ∂_{y_{k}}f(0) = 0, 1 ≤ k ≤ n). Let A be the algebra defined by

A := {f ∈ M^{2} | {f, G_{j}} ≡ 0 , 1 ≤ j ≤ n} . (1.1)

Then A is Abelian ({f, g} = 0 for all f, g ∈ A), and has the property that h ∈ A if there exists g ∈ A with g ≠ 0 and h · g ∈ A.

Following Ito [It2], let P_{m} denote the vector space over ℂ of all homogeneous polynomials of degree m in 2n variables with complex coefficients. Then P_{2} is a Lie algebra under the Poisson bracket {·, ·}. The map P_{2} → sp(n, ℂ) asso-

ciating to f ∈ P_{2} the 2n × 2n matrix ( \begin{matrix}0 & Id \\ −Id & 0 \end{matrix} ) ( \begin{matrix}∂_{x}^{2}f & ∂_{xy}^{2}f \\ ∂_{xy}^{2}f & ∂_{y}^{2}f \end{matrix} ) is a Lie al-

gebra isomorphism. Here sp(n, ℂ) is the Lie algebra of the group Sp(n, ℂ) of 2n × 2n symplectic matrices. As sp(n, ℂ) is semisimple, P_{2} is semisimple and therefore admits a Jordan decomposition: for f ∈ P_{2}, we write f = f_{s} + f_{nil} where f_{s} = Π_{s}f is the projection of f on its semisimple part and f_{nil} = Π_{nil}f

is the projection of f on its nilpotent part, i.e. ( \begin{matrix}0 & Id \\ −Id & 0 \end{matrix} ) ( \begin{matrix}∂_{x}^{2}f_{s} & ∂_{xy}^{2}f_{s} \\ ∂_{xy}^{2}f_{s} & ∂_{y}^{2}f_{s} \end{matrix} ) is

a semisimple and ( \begin{matrix}0 & Id \\ −Id & 0 \end{matrix} ) ( \begin{matrix}∂_{x}^{2}f_{nil} & ∂_{xy}^{2}f_{nil} \\ ∂_{xy}^{2}f_{nil} & ∂_{y}^{2}f_{nil} \end{matrix} ) is a nilpotent matrix. Notice

that {f_{s}, f_{nil}} = 0 and one can find a linear symplectic change of coordinates so

that f_{s} = ∑_{j=1}^{n} λ_{j}x_{j}y_{j}. The numbers ±λ_{1}, …, ±λ_{n} coincide with the spectrum of

( \begin{matrix}0 & Id \\ −Id & 0 \end{matrix} ) ( \begin{matrix}∂_{x}^{2}f_{s} & ∂_{xy}^{2}f_{s} \\ ∂_{xy}^{2}f_{s} & ∂_{y}^{2}f_{s} \end{matrix} ) and are therefore independent of the choice of coor-

dinates. Denote by Λ_{f} the sublattice of ℤ^{n} defined by

Λ_{f} := {α ∈ ℤ^{n} | ⟨α, λ⟩ = 0}

where λ := (λ_{1}, …, λ_{n}) and ⟨α, λ⟩ = ∑_{j=1}^{n} α_{j}λ_{j}. We say that Λ_{f} is the resonance

lattice associated to f ∈ P_{2}.

For the algebra A above denote by A_{s} and A_{nil} the semisimple respectively nilpotent part of the projection A_{2} of A onto P_{2}.

Definition. (i) A is said to be nonresonant at 0 if there exists H ∈ A so that the resonance lattice Λ_{H_{s}} associated to the semisimple part H_{s} ∈ P_{2} of H^{2} (or of H, for short) is trivial, i.e. Λ_{H_{s}} = {0}.

(ii) A is said to have a simple resonance at 0 if there exists μ ∈ ℤ^{n}\{0} with |μ| := ∑_{1}^{n} |μ_{j}| ≥ 2 such that for every f in A, {kμ | k ∈ ℤ} ⊂ Λ_{f_{s}} and for some H ∈ A, Λ_{H_{s}} = {kμ | k ∈ ℤ}. The vector μ is called a prime resonance vector of A and is uniquely determined up to sign. We say that Λ_{A} := {kμ | k ∈ ℤ} is the resonance lattice of A. It is a lattice with dim Λ_{A} = 1.

Let μ ∈ ℤ^{n}\{0} be a prime resonance vector. Then we can choose a basis of ℤ^{n}, (ρ^{(j)})_{1≤j≤n}, so that ⟨ρ^{(j)}, μ⟩ = δ_{jn}. In particular, the n × n matrix whose columns are given by the ρ^{(j)}’s is unimodular (i.e. in GL(n, ℤ)) and ρ^{(1)}, ⋯, ρ^{(n−1)} is a basis (over ℤ) of the n − 1 dimensional sublattice {ξ ∈ ℤ^{n}|⟨ξ, μ⟩ = 0} of ℤ^{n}. Introduce τ_{j} := ∑_{k=1}^{n} ρ_{k}^{(j)}x_{k}y_{k} (1 ≤ j ≤ n) as well as τ_{n+1} = x^{μ^{+}}y^{μ^{−}}, τ_{n+2} = x^{μ^{−}}y^{μ^{+}} where μ^{−} := μ^{+} − μ and μ^{+} = (μ_{k}^{+})_{1≤k≤n} is given by μ_{k}^{+} := μ_{k} if μ_{k} ≥ 0 and μ_{k}^{+} := 0 if μ_{k} < 0. In section 2 we prove the following

Theorem 1.1. Assume that A has a simple resonance at 0 and let μ ∈ ℤ^{n}\{0} be a prime resonant vector of A (thus, in particular, |μ| ≥ 2). Then there exists an analytic, symplectic change of coordinates φ in a neighborhood of 0 in ℂ^{2n}, so that with respect to the new coordinates, A has the following properties:

(1) τ_{j} ∈ A (1 ≤ j ≤ n − 1);

(2) any element f in A has a convergent power series expansion in τ_{1}, …, τ_{n+2}.

Remark 1.1. The special case of Theorem 1.1 where μ = (μ_{1}, μ_{2}, 0, …, 0) ∈ ℤ^{n}\{0} is due to Ito [It2]. In the same paper he also considers the case where μ = (1, 0, …, 0) [It2, Theorem 3] which is not included in the formulation of Theorem 1.1. To prove his results Ito uses a rapidly convergent iteration procedure and it turns out that the same procedure can be applied to prove Theorem 1.1.

Remark 1.2. As in [It2, Theorem 2], there is an analogous result to Theorem 1.1 for an algebra A of germs of real analytic functions at 0, generated by real analytic integrals G_{1}, …, G_{n} which Poisson commute. We say that A is elliptic if 0 is an

elliptic fixed point (i.e. for H ∈ A arbitrary, the spectrum of ( \begin{matrix}0 & Id \\ −Id & 0 \end{matrix} ) d^{2}H_{s} is

purely imaginary where d^{2}H_{s} denotes the Hessian of H_{s}).

Theorem 1.1’. Assume that A is an algebra of germs of real analytic functions at 0, generated by functionally independent integrals G_{1}, ..., G_{n} which Poisson commute. Further assume that A_{nil} = 0 and that A is elliptic and has a simple resonance. Let μ be a prime resonance vector, |μ| ≥ 2.

Then there exists a real analytic, symplectic change of coordinates φ in a neighborhood of 0 in ℝ^{2n}, so that with respect to the new coordinates, (x̂, ŷ), A has the following properties:

(1) τ̂_{j} := ∑_{k=1}^{n} ρ_{k}^{(j)}(x̂_{k}^{2} + ŷ_{k}^{2})/2 ∈ A (1 ≤ j ≤ n − 1);

(2) any element f in A has a convergent power series expansion in τ̂_{1}, ..., τ̂_{n}, ℜτ̂_{n+1}, ℑτ̂_{n+1} where τ̂_{n} := ∑_{k=1}^{n} ρ_{k}^{(n)}(x̂_{k}^{2} + ŷ_{k}^{2})/2 and τ̂_{n+1} := ∏_{k=1}^{n}(x̂_{k} + iŷ_{k})^{μ_{k}^{+}}(x̂_{k} − iŷ_{k})^{μ_{k}^{−}}.

Notice that in view of Corollary 1.2 below, A_{nil} ≠ 0 implies that |μ| = 2. Further we remark that the case where A is elliptic with A_{nil} ≠ 0 and has a simple resonance has been treated in [It2] (cf [Ar1, Appendix 6] for a classification of quadratic Hamiltonians).

Remark 1.3. The coordinates which have the properties stated in Theorem 1.1 are not unique. One verifies that a symplectic transformation provided by a Hamiltonian flow whose Hamiltonian has a power series expansion in τ_{1}, …, τ_{n+2}, leads to new coordinates with the same properties as stated in Theorem 1.1. However, given a power series expansion in τ_{1}, …, τ_{n+2} of an element f ∈ A, one verifies that the coefficients corresponding to the monomials in τ_{1}, …, τ_{n−1} only, are independent of the choice of coordinates.

Remark 1.4. One might ask if a result similar to the one of Theorem 1.1 is true if A has multiple resonances, i.e., a resonance lattice with R = dim Λ_{A} ≥ 2. Even in the case where the resonances are decoupled (i.e. Λ_{A} has a basis μ^{(1)}, …, μ^{(R)} such that suppμ^{(i)} ∩ suppμ^{(j)} = ∅ for i ≠ j) it turns out that the method of proof used for Theorem 1.1 breaks down in general (cf. Appendix A).

As an immediate consequence of Theorem 1.1 we obtain the following

Corollary 1.2. (i) If A is nonresonant, then A_{nil} = 0 and dim A_{s} = n.

(ii) If A has a simple resonance then dim A_{s} = n − 1. If for a prime resonance vector μ, |μ| ≥ 3, then A_{nil} = {0}.

Proof. (i) By Ito’s result [It1], there exist coordinates x_{j}, y_{j} (1 ≤ j ≤ n) in a neighborhood of 0 in ℂ^{2n} so that, when expressed in these coordinates, A consists of power series in τ_{j} = x_{j}y_{j} (1 ≤ j ≤ n) which converge in a neighborhood of 0 ∈ ℂ^{2n}. Therefore, dim A_{s} = n and A_{nil} = {0}.

(ii) By Theorem 1.1, there exist coordinates x_{k}, y_{k} (1 ≤ k ≤ n) in a neighborhood of 0 in ℂ^{2n} so that, when expressed in these coordinates, any element in A has a

convergent power series expansion in τ_{1}, …, τ_{n+2} where τ_{j} := ∑_{k=1}^{n} ρ_{k}^{(j)}x_{k}y_{k} (1 ≤ j ≤

n), τ_{n+1} = x^{μ^{+}}y^{μ^{−}} and τ_{n+2} = x^{μ^{−}}y^{μ^{+}} are defined as above.

Moreover, Theorem 1.1 guarantees that the functions, τ_{1}, …, τ_{n−1}, are elements in A. This implies that dim A_{s} = n − 1, as otherwise τ_{n} would be also in A_{s} and one would conclude that A is nonresonant at 0. If, in addition, |μ| ≥ 3, then for any power series f in τ_{1}, …, τ_{n+2}, f_{nil} = 0 and thus in particular A_{nil} = {0}. □

In section 3, we make a detailed analysis of the level sets M_{c} := {(x̂, ŷ) ∈ (ℝ^{2n}, 0)| G_{j} = c_{j} (1 ≤ j ≤ n)} for real integrable systems with c = (c_{1}, ..., c_{n}), and study the fibration provided by these level sets. Here (x̂, ŷ) are the coordinates provided by Theorem 1.1’, G_{j} = τ̂_{j} (1 ≤ j ≤ n−1) and (ℝ^{2n}, 0) denotes a neighborhood of 0 invariant under the flows of the Hamiltonian vectorfields corresponding to τ̂_{j}, 1 ≤ j ≤ n − 1 (cf. section 3). G_{n} can be expressed as a power series in τ̂_{1}, ..., τ̂_{n}, ℜτ̂_{n+1}, ℑτ̂_{n+1} (which, due to the resonance assumption, does not contain a term linear in τ̂_{n}), and is such that the G_{j}’s generate A. In particular, we prove that if the prime resonance vector μ oscillates (i.e. μ has negative and positive components), then, for generic c, M_{c} is a disjoint union of tori of dimension n (cf. Proposition 3.4). If μ is nonnegative (μ_{j} ≥ 0 for 1 ≤ j ≤ n), then, for generic c sufficiently small and generic A (i.e. generic G_{n}), M_{c} has one connected component diffeomorphic to (S^{1})^{n−1} ×(0, 1) in case |μ| = 2 or 3 and is a disjoint union of tori of dimension n if |μ| ≥ 5 (cf. Proposition 3.5). In subsection 3.2, we study nongeneric level sets and in subsection 3.3 we analyze the fibration provided by the level sets. In Appendix B we analyze the level sets M_{c} for complex systems.

2. Proof of Theorem 1.1

In this section we provide a proof of Theorem 1.1. As we follow Ito’s method of proof, we present only an outline, emphasizing the parts which are different. Throughout this section we use the notation introduced in Section 1 and assume that the assumptions of Theorem 1.1 hold.

2.1. Preliminaries.

Choose H in A so that Λ_{H_{s}} = Λ_{A} where H_{s} denotes the semisimple part of H,

H_{s} := ∑_{j=1}^{n} λ_{j}x_{j}y_{j} . (2.1)

For a power series f = ∑_{α,β∈ℤ_{≥0}^{n}} c_{αβ}x^{α}y^{β} at the origin we use the notation

(2.2)  f = f^{d} + f^{d+1} + f^{d+2} + ⋯

where f^{j} (j ≥ d) is a homogeneous polynomial of degree j with d = degree(f^{d}) ≥ 0. We refer to f^{d} as the lowest order part of f. A power series f is said to be in H_{s}- normal form (or Birkhoff normal form) if

\{H_{s}, f\} = ∑_{k=1}^{n} \frac{∂H_{s}}{∂x_{k}} \frac{∂f}{∂y_{k}} − \frac{∂H_{s}}{∂y_{k}} \frac{∂f}{∂x_{k}} ≡ 0 .

It is said to be in H_{s}-normal form up to order d + d_{1} if f^{d} + ⋯ + f^{d+d_{1}} is in H_{s}-normal form. Notice that a power series f which is in H_{s}-normal form can be considered as a power series in (n + 2) variables τ_{1}, …, τ_{n+2}. Moreover, as

(2.3)  τ_{n+1}τ_{n+2} = x^{μ^{+}+μ^{−}}y^{μ^{+}+μ^{−}} = ∏_{j=1}^{n}(x_{j}y_{j})^{|μ_{j}|} ,

is a function of τ = (τ_{1}, …, τ_{n}), f is of the form

(2.4)  f(z) = f_{1}(τ, τ_{n+1}) + f_{2}(τ, τ_{n+2})

where f_{j}(τ, τ_{n+j}) are power series in τ_{1}, …, τ_{n} and τ_{n+j} (j = 1, 2). Alternatively, f can be considered as a Laurent series in τ_{1}, …, τ_{n+1}, eliminating τ_{n+2} in f_{2}(τ, τ_{n+2}) by using (2.3).

For the remainder of all of section 2, given a power series f in H_{s}-normal form, we denote by \frac{∂f}{∂τ_{j}} (1 ≤ j ≤ n+1) the partial derivative of f with respect to τ_{j} when f is considered as a Laurent series in τ_{1}, …, τ_{n+1}.

Using that

(2.5)  {τ_{i}, τ_{j}} = 0 (1 ≤ i, j ≤ n) , {τ_{i}, τ_{n+1}} = 0  (1 ≤ i ≤ n − 1) ,
{τ_{n}, τ_{n+1}} = −τ_{n+1}

the Poisson bracket {f, g} of power series f, g in H_{s}-normal form can be computed as

(2.6)  \{f, g\} = −τ_{n+1} (\frac{∂f}{∂τ_{n}} \frac{∂g}{∂τ_{n+1}} − \frac{∂f}{∂τ_{n+1}} \frac{∂g}{∂τ_{n}} ) .

2.2. Construction of a formal coordinate transformation .

In this subsection, we construct the transformation φ formally. Introduce the projection Π_{N}f of a power series f onto its H_{s}-normal part and define Π_{R}f := f − Π_{N}f. For any convergent power series f denote by X_{f}^{t} := exp(tX_{f}) the flow corresponding to the Hamiltonian vector field X_{f}. The coordinate transformation φ of Theorem 1.1 is constructed from a sequence of transformations each of which is a Hamiltonian flow obtained in a well known fashion:

Proposition 2.1. Let K = K^{2} + K^{3} + ⋯ be a power series with K_{s} = H_{s} ≡

∑_{j=1}^{n} λ_{j}x_{j}y_{j}. Assume that K is in H_{s}-normal form up to order 1 + d (d ≥ 1). Then

there exists a unique polynomial W of the form

(2.7)  W = W^{d+2} + ⋯ + W^{2d+1}

with Π_{N}W = 0 such that K ∘ X_{W}^{t=1} is in H_{s}-normal form up to order 1 + 2d. (The flow X_{W}^{t} (ζ) exists for |t| ≤ 1 for ζ in a sufficiently small neighborhood of 0.)

Applying Proposition 2.1 successively for H = H^{2} + H^{3} + ⋯, one obtains

Corollary 2.2. There exists a sequence of symplectic coordinate transformations φ_{j} (j ≥ 0), φ_{j} = X_{W_{j}}^{t=1}, where W_{j} is the polynomial provided by Proposition 2.1 with d = 2^{j}, so that the coordinate transformation φ^{(j)} := φ_{0} ∘ ⋯ ∘ φ_{j} takes the Hamiltonian H into H_{s}-normal form up to order 1 + 2^{j+1}. Consequently, φ := lim_{j→∞} φ^{(j)} is a formal symplectic transformation such that H ∘ φ is in H_{s}-normal

form.

One verifies by a straightforward inductive argument that the following Lemma holds:

Lemma 2.3. Assume that H is in H_{s}-normal form up to order 2 + d and G is an integral of H, i.e. {H, G} = 0, then G = G^{ℓ} + G^{ℓ+1} + ⋯ is in H_{s}-normal form up to order ℓ + d.

In view of Lemma 2.3 and Corollary 2.2, one concludes that, for a proof of Theorem 1.1, it remains to establish that the formal coordinate transformation φ of Corollary 2.2 is given by a convergent power series. For this purpose we need to estimate the function W obtained in Proposition 2.1. Recall from the introduction that ρ^{(j)} ∈ ℤ^{n}\{0} (1 ≤ j ≤ n) is a basis of ℤ^{n} with ⟨ρ^{(j)}, μ⟩ = δ_{jn}.

Lemma 2.4. Let f be a convergent power series. Then the projection Π_{N}f of f on its H_{s}-normal part is given by

(2.8)  Π_{N}f(x, y) = ∫_{0}^{1} dθ_{1} ⋯ ∫_{0}^{1} dθ_{n−1}f(e^{2πiθ}x, e^{−2πiθ}y)

where e^{2πiθ}x is defined by

(2.9)  e^{2πiθ}x = (e^{2πi ∑_{j=1}^{n−1} θ_{j}ρ_{1}^{(j)}}x_{1}, …, e^{2πi ∑_{j=1}^{n−1} θ_{j}ρ_{n}^{(j)}}x_{n})

and where e^{−2πiθ}y is defined similarly.

Proof. It suffices to consider the case where f is a monomial, f(x, y) = x^{α}y^{β}. Then

f(e^{2πiθ}x, e^{−2πiθ}y) = x^{α}y^{β} exp(2πi ∑_{j=1}^{n−1} θ_{j}⟨ρ^{(j)}, α − β⟩) .

In order for such a term not to get averaged when integrated over θ_{1}, …, θ_{n−1}, it is necessary and sufficient that ⟨ρ^{(j)}, α − β⟩ = 0 for 1 ≤ j ≤ n − 1, i.e. α, β in ℤ_{≥0}^{n} have to be of the form

α = γ + ℓ_{1}μ^{+} + ℓ_{2}μ^{−} , β = γ + ℓ_{1}μ^{−} + ℓ_{2}μ^{+}

where γ ∈ ℤ_{≥0}^{n} and ℓ_{1}, ℓ_{2} ∈ ℤ_{≥0}. □

Consider the disc Ω_{r} := {z ∈ ℂ^{2n} | |z_{j}| < δ_{j}r (1 ≤ j ≤ 2n)} where r > 0 and δ_{j} > 0 (1 ≤ j ≤ n).

For a polynomial W introduce

(2.10)  |W |_{r}^{sup} := sup_{z∈Ω_{r}} |W (z)| .

Lemma 2.5. Let W be a polynomial with Π_{N}W = 0. Then

(2.11)  |W |_{r}^{sup} ≤ 2π ∑_{k=1}^{n−1} |{τ_{k}, W }|_{r}^{sup} .

Proof. Introduce, for z = (x, y) ∈ Ω_{r} fixed,

(2.12)  Ŵ(θ) := W (e^{2πiθ}x, e^{−2πiθ}y)

where e^{2πiθ}x and e^{−2πiθ}y are defined as above with θ = (θ_{1}, …, θ_{n−1}), 0 ≤ θ_{j} ≤ 1. Notice that (x̃(θ), ỹ(θ)) := (e^{2πiθ}x, e^{−2πiθ}y) ∈ Ω_{r} and Ŵ(θ = 0) = W (z). By the mean value theorem,

|Ŵ(θ) − Ŵ(0)| ≤ ∑_{j=1}^{n−1} max_{θ∈[0,1]^{n−1}} |\frac{∂Ŵ(θ)}{∂θ_{j}} |

where

\frac{∂Ŵ(θ)}{∂θ_{j}} = ∑_{k=1}^{n} \frac{∂W}{∂x_{k}} |_{x̃,ỹ} 2πiρ_{k}^{(j)}x̃_{k} − ∑_{k=1}^{n} \frac{∂W}{∂y_{k}} |_{x̃,ỹ} 2πiρ_{k}^{(j)}ỹ_{k} .

As τ_{j} = ∑_{ℓ=1}^{n} ρ_{ℓ}^{(j)}x_{ℓ}y_{ℓ} one sees that

\frac{∂τ_{j}}{∂x_{k}} = ρ_{k}^{(j)}y_{k}  and  \frac{∂τ_{j}}{∂y_{k}} = ρ_{k}^{(j)}x_{k} .

This leads to

|\frac{∂W}{∂θ_{j}} |_{r}^{sup} ≤ 2π |∑_{k=1}^{n} (\frac{∂W}{∂x_{k}} \frac{∂τ_{j}}{∂y_{k}} − \frac{∂W}{∂y_{k}} \frac{∂τ_{j}}{∂x_{k}} ) |_{x̃,ỹ}|_{r}^{sup}
≤ 2π|{W, τ_{j}}|_{r}^{sup}

and, therefore,

|Ŵ(θ) − Ŵ(0)|_{r}^{sup} ≤ 2π ∑_{j=1}^{n−1} |{W, τ_{j}}|_{r}^{sup} .

Using the assumption Π_{N}W = 0 one concludes from Lemma 2.4 that

∫_{0}^{1} dθ_{1} ⋯ ∫_{0}^{1} dθ_{n−1}Ŵ(θ) = 0 .

Therefore, W (z) = ∫_{0}^{1} dθ_{1} ⋯ ∫_{0}^{1} dθ_{n−1}(Ŵ(0) − Ŵ(θ)) and

|W |_{r}^{sup} ≤ 2π ∑_{j=1}^{n−1} |{W, τ_{j}}|_{r}^{sup} .

According to Lemma 2.5, we obtain an estimate of W (as in Proposition 2.1) from an estimate of the Poisson brackets {τ_{k}, W } for 1 ≤ k ≤ n − 1.

2.3. Estimate for {τ_{k}, W }.

To estimate {τ_{k}, W } for 1 ≤ k ≤ n − 1 we make use of the assumption that G_{1}, …, G_{n} are integrals of H and that dG_{1}, …, dG_{n} are generically independent.

Assume that the lowest order part of G_{j} is of degree s_{j} ≥ 1 so that G_{j} is of the form

(2.13)  G_{j} = G_{j}^{s_{j}} + G_{j}^{s_{j}+1} + G_{j}^{s_{j}+2} + ⋯

where G_{j}^{s_{j}+d} are polynomials homogeneous of degree s_{j} + d. As pointed out by Ito [It2, Lemma 4.4, p. 422] we may assume that dG_{1}^{s_{1}}, …, dG_{n}^{s_{n}} are generically independent. Assume that H is in H_{s}-normal form up to order 1 + d (d ≥ 1). By Lemma 2.3, G_{j} is in H_{s}-normal form up to order s_{j} + d − 1. Write

(2.14)  G_{j}(z) = g_{j}(z) + Ĝ_{j}(z)

where g_{j} is in H_{s}-normal form, i.e. g_{j} = Π_{N}g_{j} and Ĝ_{j}(z) = O(|z|^{s_{j}+d}). Notice that

(2.15)  rank (\frac{∂(g_{1}^{s_{1}}, …, g_{n}^{s_{n}})}{∂(τ_{1}, …, τ_{n+1})}) = n

on an open dense subset near z = 0, where we recall that we view g_{1}^{s_{1}}, …, g_{n}^{s_{n}} as Laurent series in τ_{1}, …, τ_{n+1}.

Let z := φ(ζ) = X_{W}^{t=1}(ζ) be the transformation described in Proposition 2.1. Then, again by Lemma 2.3,

(2.16)  G_{j}(φ(ζ)) = g_{j}(ζ) + {g_{j}(ζ), W (ζ)} + Ĝ_{j}(ζ) + O(|ζ|^{s_{j}+2d})

is in H_{s}-normal form up to order s_{j} − 1 + 2d. Therefore (1 ≤ j ≤ n)

(2.17)  {g_{j}(ζ), W (ζ)} + Π_{R}Ĝ_{j}(ζ) = O(|ζ|^{s_{j}+2d})

which can be written as (1 ≤ i ≤ n)

(2.18)_{i}  ∑_{j=1}^{n+1} \frac{∂g_{i}}{∂τ_{j}} · \{τ_{j}, W \} = −Π_{R}Ĝ_{i}(ζ) + O(|ζ|^{s_{i}+2d}) .

This is a linear system of n equations for {τ_{1}, W }, …, {τ_{n+1}, W }, from which we would like to derive estimates for {τ_{1}, W }, …, {τ_{n−1}, W }. Notice that {G_{i}, G_{j}} = 0 implies {g_{i}^{s_{i}}, g_{j}^{s_{j}}} = 0, which, by (2.6), implies (1 ≤ i, j ≤ n)

\frac{∂g_{i}^{s_{i}}}{∂τ_{n}} \frac{∂g_{j}^{s_{j}}}{∂τ_{n+1}} − \frac{∂g_{i}^{s_{i}}}{∂τ_{n+1}} \frac{∂g_{j}^{s_{j}}}{∂τ_{n}} = 0 .

This means that the n×2 matrix (\frac{∂g^{0}}{∂τ_{n}} , \frac{∂g^{0}}{∂τ_{n+1}} ) with g^{0} = (g_{1}^{s_{1}}, …, g_{n}^{s_{n}}) has rank at

most 1, i.e. \frac{∂g^{0}}{∂τ_{n}} and \frac{∂g^{0}}{∂τ_{n+1}} are linearly dependent. But due to (2.15) either \frac{∂g^{0}}{∂τ_{n}} ≠ 0

(case 1) or \frac{∂g^{0}}{∂τ_{n+1}} ≠ 0 (case 2). By reordering the G_{j}’s, if necessary, we have that

either \frac{∂g_{n}^{sn}}{∂τ_{n}} ≠ 0 (case 1) or \frac{∂g_{n}^{sn}}{∂τ_{n+1}} ≠ 0 (case 2). Let us outline how one proceeds in case 1 to obtain estimates for {τ_{1}, W }, …, {τ_{n−1}, W } from the system (2.18). One uses equation (2.18)_{n} to eliminate {τ_{n}, W } and obtains a system of n − 1 equations for {τ_{1}, W }, …, {τ_{n−1}, W } and {τ_{n+1}, W }. Using that {G_{n}, G_{i}} = 0, one concludes that the terms involving {τ_{n+1}, W } are of sufficiently high order and, therefore, can be included in the error term. In more detail, we eliminate {τ_{n}, W } from the system (2.18) by forming (2.18)_{i} \frac{∂g_{n}}{∂τ_{n}} − (2.18)_{n} \frac{∂g_{i}}{∂τ_{n}} to obtain (1 ≤ i ≤ n − 1)

(2.19)_{i} ∑_{j=1}^{n−1} α_{ij}(ζ){τ_{j}, W } = − α_{i,n+1}(ζ){τ_{n+1}, W }

− \frac{∂g_{n}}{∂τ_{n}} Π_{R}Ĝ_{i} + \frac{∂g_{i}}{∂τ_{n}} Π_{R}Ĝ_{n} + \frac{∂g_{n}}{∂τ_{n}} O(|ζ|^{s_{i}+2d})

− \frac{∂g_{i}}{∂τ_{n}} O(|ζ|^{s_{n}+2d})

where α_{ij} are defined as

(2.20)  α_{ij}(ζ) = \frac{∂g_{n}}{∂τ_{n}} \frac{∂g_{i}}{∂τ_{j}} − \frac{∂g_{n}}{∂τ_{j}} \frac{∂g_{i}}{∂τ_{n}} .

Both sides of (2.19)_{i} are meromorphic functions of ξ_{j}, η_{j} with a possible pole at points ξ, η with ξ_{j} = η_{j} = 0 for some 1 ≤ j ≤ n. To remove the poles in (2.19)_{i}

we multiply this equation by the polynomial P (ζ) := τ_{n+1} ∏_{k=1}^{n} ξ_{k}η_{k}. To see that P

serves its purpose, write (1 ≤ j ≤ n + 1)

(2.21)  \frac{∂}{∂τ_{j}} = D_{τ_{j}} + \frac{∂τ_{n+2}}{∂τ_{j}} D_{τ_{n+2}}

where D_{τ_{j}} denotes the partial derivative of a function f = f(τ_{1}, …, τ_{n+1}, τ_{n+2}) (when not considering it as a function of τ_{1}, …, τ_{n+1}). Recall that τ_{n+1}τ_{n+2} = ξ^{μ^{+}+μ^{−}}η^{μ^{+}+μ^{−}} and, therefore,

(2.22)  \frac{∂τ_{n+2}}{∂τ_{n+1}} = − (\frac{1}{τ_{n+1}} )^{2} ξ^{μ^{+}+μ^{−}}η^{μ^{+}+μ^{−}} = −\frac{τ_{n+2}}{τ_{n+1}} .

Further (1 ≤ j ≤ n)

\frac{∂τ_{n+2}}{∂τ_{j}} = ∑_{k=1}^{n} \frac{∂τ_{n+2}}{∂(ξ_{k}η_{k})} \frac{∂(ξ_{k}η_{k})}{∂τ_{j}} .

To compute this derivative introduce the inverse B = (b_{kj}) of A where A is the unimodular n × n matrix whose j’th row is given by ρ^{(j)}. Then

\frac{∂(ξ_{k}η_{k})}{∂τ_{j}} = b_{kj} .

Moreover,

\frac{∂τ_{n+2}}{∂(ξ_{k}η_{k})} = \frac{1}{τ_{n+1}} \frac{∂(ξ^{μ^{+}+μ^{−}}η^{μ^{+}+μ^{−}})}{∂(ξ_{k}η_{k})} = \frac{|μ_{k}|}{τ_{n+1}} \frac{ξ^{μ^{+}+μ^{−}}η^{μ^{+}+μ^{−}}}{ξ_{k}η_{k}} .

Altogether, we obtain (1 ≤ j ≤ n)

\frac{∂τ_{n+2}}{∂τ_{j}} = (∑_{k=1}^{n} b_{kj} \frac{|μ_{k}|}{ξ_{k}η_{k}} ) τ_{n+2} . (2.23)

From (2.22) and (2.23) we, therefore, conclude that

P (ζ)\frac{∂τ_{n+2}}{∂τ_{n+1}} = − (∏_{k=1}^{n} ξ_{k}η_{k}) τ_{n+2}

and

P (ζ)\frac{∂τ_{n+2}}{∂τ_{j}} = τ_{n+1}τ_{n+2} ∑_{k=1}^{n} b_{kj}|μ_{k}| ( ∏_{{1≤j≤n \atop j≠k}} ξ_{j}η_{j}) .

These computations are now used to estimate the term −α_{i,n+1}(ζ){τ_{n+1}, W } on the right hand side of (2.19)_{i}. Using (2.6) we see that

(2.24) P (ζ)α_{i,n+1}(ζ) = (∏_{1}^{n} ξ_{k}η_{k}) τ_{n+1} (\frac{∂g_{n}}{∂τ_{n}} \frac{∂g_{i}}{∂τ_{n+1}} − \frac{∂g_{n}}{∂τ_{n+1}} \frac{∂g_{i}}{∂τ_{n}} )

= − (∏_{k=1}^{n} ξ_{k}η_{k}) {g_{n}, g_{i}} .

Now we make use of the assumption that 0 = {G_{n}, G_{i}} (integrability) to conclude that

(2.25)  {g_{n}, g_{i}} = −{g_{n}, Ĝ_{i}} − {Ĝ_{n}, g_{i}} − {Ĝ_{n}, Ĝ_{i}}
= O(|ζ|^{s_{n}+s_{i}+d−2}) .

Combining (2.24) and (2.25) we conclude that the order of P (ζ)α_{i,n+1}(ζ) is given by 2n + s_{n} + s_{i} + d − 2, i.e.

(2.26)  −P (ζ)α_{i,n+1}(ζ){τ_{n+1}, W } = O(|ζ|^{m_{i}+2d−2})

where m_{i} = 2n + s_{n} + s_{i} + |μ| − 2. This is the key point in this proof. It uses the integrability of the system to insure that {τ_{n+1}, W } can be treated as an error term in (2.19).

Further, one verifies that

P (ζ) (\frac{∂g_{n}}{∂τ_{n}} O(|ζ|^{s_{i}+2d}) − \frac{∂g_{i}}{∂τ_{n}} O(|ζ|^{s_{n}+2d})) = O(|ζ|^{m_{i}+2d}) .

The system of equations (2.19)_{i} is, therefore, reduced to (1 ≤ i ≤ n − 1)

(2.27)_{i}  ∑_{j=1}^{n}P (ζ)α_{ij}(ζ){τ_{j}, W } =

=P (ζ) (\frac{∂g_{i}}{∂τ_{n}} Π_{R}Ĝ_{n} − \frac{∂g_{n}}{∂τ_{n}} Π_{R}Ĝ_{i}) + O(|ζ|^{m_{i}+2d}) .

By compairing homogeneous parts of (2.27), we can obtain equations to be solved for W^{d+2}, …, W^{2d+1} inductively, which we describe below.

In case 2, we argue similarly to obtain (1 ≤ i ≤ n − 1)

(2.28)_{i}  ∑_{j=1}^{n−1}P (ζ)α̃_{ij}(ζ){τ_{j}, W } =

=P (ζ) (\frac{∂g_{i}}{∂τ_{n+1}} Π_{R}Ĝ_{n} − \frac{∂g_{n}}{∂τ_{n+1}} Π_{R}Ĝ_{i}) + O(|ζ|^{2n+s_{n}+s_{i}+2d})

where α̃_{ij}(ζ) := \frac{∂g_{n}}{∂τ_{n+1}} \frac{∂g_{i}}{∂τ_{j}} − \frac{∂g_{n}}{∂τ_{j}} \frac{∂g_{i}}{∂τ_{n+1}} (1 ≤ i, j ≤ n).

To treat case 1 and case 2 simultaneously, introduce the polynomials a_{ij}(ζ)

(2.29) a_{ij}(ζ) = \{ \begin{matrix}P (ζ) (\frac{∂g_{n}}{∂τ_{n}} \frac{∂g_{i}}{∂τ_{j}} − \frac{∂g_{n}}{∂τ_{j}} \frac{∂g_{i}}{∂τ_{n}} ) & (case 1) \\ P (ζ) (\frac{∂g_{n}}{∂τ_{n+1}} \frac{∂g_{i}}{∂τ_{j}} − \frac{∂g_{n}}{∂τ_{j}} \frac{∂g_{i}}{∂τ_{n+1}} ) & (case 2) . \end{matrix}

Denote by a_{ij}^{m_{i}}(ζ) the leading order part of a_{ij}(ζ),

a_{ij}^{m_{i}}(ζ) = \{ \begin{matrix}P (ζ) (\frac{∂g_{n}^{sn}}{∂τ_{n}} \frac{∂g_{i}^{s_{i}}}{∂τ_{j}} − \frac{∂g_{n}^{sn}}{∂τ_{j}} \frac{∂g_{i}^{s_{i}}}{∂τ_{n}} ) & (case 1) \\ P (ζ) (\frac{∂g_{n}^{sn}}{∂τ_{n+1}} \frac{∂g_{i}^{s_{i}}}{∂τ_{j}} − \frac{∂g_{n}^{sn}}{∂τ_{j}} \frac{∂g_{i}^{s_{i}}}{∂τ_{n+1}} ) & (case 2) \end{matrix}

with m_{i} = s_{n} + s_{i} + 2n + |μ| − 2 in case 1 and m_{i} = s_{n} + s_{i} + 2n in case 2. We summarize the results obtained above in the following.

Lemma 2.6. Assume that there exists a polynomial W = W^{d+2} + ⋯ + W^{2d+1} with Π_{N}W = 0 so that, with z = φ(ζ) = X_{W}^{t=1}(ζ), G_{j} ∘ φ are in H_{s}-normal form up to order s_{j} + 2d − 1. Then each homogeneous polynomial W^{ℓ+2} (d ≤ ℓ ≤ 2d − 1) satisfies the following system of (n − 1) equations (1 ≤ i ≤ n − 1)

(2.30)  ∑_{j=1}^{n−1} a_{ij}^{m_{i}}(ζ){τ_{j}, W^{ℓ+2}} = F_{i}^{m_{i}+ℓ+2}(ζ)

where F_{i}^{m_{i}+ℓ+2}(ζ) is given by

P (ζ) ∑_{j=0}^{ℓ−d} (\frac{∂g_{i}^{s_{i}+j}}{∂τ_{n}} · Π_{R}Ĝ_{n}^{s_{n}+ℓ−j} − \frac{∂g_{n}^{sn+j}}{∂τ_{n}} Π_{R}Ĝ_{i}^{s_{i}+ℓ−j})

− ∑_{j=1}^{n−1} ∑_{k=1}^{ℓ−d} a_{ij}^{m_{i}+k}{τ_{j}, W^{ℓ+2−k}}  (case 1)

P (ζ) ∑_{j=0}^{ℓ−d} (\frac{∂g_{i}^{s_{i}+j}}{∂τ_{n+1}} Π_{R}Ĝ_{n}^{s_{n}+ℓ−j} − \frac{∂g_{n}^{sn+j}}{∂τ_{n+1}} Π_{R}Ĝ_{i}^{s_{i}+ℓ−ν})

− ∑_{j=1}^{n−1} ∑_{k=1}^{ℓ−d} a_{ij}^{m_{i}+k}{τ_{j}, W^{ℓ+2−k}}  (case 2)

(2.31)

Moreover,

(2.32)  p(ζ) := det[a_{ij}^{m_{i}}(ζ)_{1≤i,j≤n−1}] ≠ 0 .

The system (2.30) can be solved for {τ_{j}, W^{ℓ+2}},

(2.33)  \{τ_{k}, W^{ℓ+2}\} = \frac{q_{k}^{s+ℓ+2}(ζ)}{p(ζ)}

where q_{k}^{s+ℓ+2}(ζ) is also a determinant and given by Cramer’s rule.

We point out that, {τ_{k}, W^{ℓ+2}} being a polynomial, formula (2.33) shows that the numerator q_{k}^{ℓ}(ζ) is divisible by p(ζ).

2.4. Estimate of W.

We now combine Lemma 2.5 and Lemma 2.6 to obtain an estimate for W.

For convenience, we assume that the Hamiltonian H is normalized so that ∥μ∥ = ∥λ∥.

According to Ito [It2, Lemma 5.1], for a small but otherwise arbitrary positive number r > 0, there exist constants 0 < δ_{i} < 1 (1 ≤ i ≤ 2n) such that, for ζ ∈ Δ_{r} := {ζ = (ζ_{1}, …, ζ_{2n}) ∈ ℂ^{2n} | |ζ_{j}| = δ_{j}r (1 ≤ j ≤ 2n)},

(2.34)  |p(ζ)| ≥ c_{1}r^{s}

where s is the degree of p(ζ) := det ((a_{ij}^{m_{i}}(ζ))_{1≤i,j≤n−1}) (cf. (2.32)) and c_{1} > 0 is independent of r. Introduce the polydiscs Ω_{r} := {ζ ∈ ℂ^{2n} | |ζ_{j}| < δ_{j}r (1 ≤ j ≤ 2n)} and denote by A(Ω_{r}) the space of power series in ζ which are absolutely convergent in \overline{Ω}_{r}. Further, introduce the subspaces A_{m}(Ω_{r}) := \{f ∈ A(Ω_{r}) | f = f^{m} + f^{m+1} + ⋯ } and the following norms for f ∈ A(Ω_{r}),

|f|_{r}^{sup} := max_{\overline{Ω}_{r}} |f(ζ)|; ∥f∥_{r} := ∑_{j=0}^{∞} |f^{j}|_{r}^{sup}

and, for f ∈ A_{m}(Ω_{r}),

∥f∥_{r,m} := \frac{∥f∥_{r}}{r^{m}} .

It follows from (2.32), (2.33) and the maximum principle that

(2.35)  |\{τ_{k}, W^{ℓ+2}\}|_{r}^{sup} ≤ \frac{max_{ζ∈Δ_{r}} |q_{k}^{s+ℓ+2}(ζ)|}{min_{ζ∈Δ_{r}} |p(ζ)|} ≤ \frac{1}{c_{1}} \frac{|q_{k}^{s+ℓ+2}|_{r}^{sup}}{r^{s}}

with W = W^{d+2} + ⋯ + W^{2d+1}. According to Cramer’s rule, q_{k}^{s+ℓ+2} is given by

q_{k}^{s+ℓ+2} = det ( \begin{matrix}a_{11}^{m_{1}} & ⋯ & F_{1}^{m_{1}+ℓ+2} & ⋯ & a_{1n−1}^{m_{1}} \\ ⋮ & & ⋮ & & ⋮ \\ a_{n−1 1}^{m_{n−1}} & ⋯ & F_{n−1}^{m_{n−1}+ℓ+2} & ⋯ & a_{n−1 n−1}^{m_{n−1}} \end{matrix} )

(cf. (2.31) for the definition of F_{i}^{m_{i}+ℓ+2}).

In a straightforward way one then verifies that (1 ≤ k ≤ n − 1; d ≤ ℓ ≤ 2d − 1)

(2.36)  ∥\{τ_{k}, W^{ℓ+2}\}∥_{r} ≤ c_{2} ∑_{i=1}^{n−1} \frac{∥F^{m_{i}+ℓ+2}∥_{r}}{r^{m_{i}}}

for some c_{2} > 0.

We need to introduce some more notation: for a power series f = ∑_{α,β} c_{αβ}ξ^{α}η^{β}

denote by f̃ the power series with coefficients |c_{αβ}|

f̃ = ∑_{α,β} |c_{αβ}|ξ^{α}η^{β} .

Let c_{3} > 0 be a constant such that

∑_{1≤i,j≤n} \frac{∥D_{τ_{j}}g_{i}^{s_{i}}∥_{r}}{r^{s_{i}−2}} + ∑_{{1≤i≤n \atop k=n+1,n+2}} \frac{∥D_{k}g_{i}^{s_{i}}∥_{r}}{r^{s_{i}−|μ|}} ≤ c_{3}

and define c_{4} := max(c_{2}, 1 + c_{2}c_{3}). Using (2.36) one then proves (cf. [It2, Lemma 5.3]).

Lemma 2.7. If

∑_{i,j=1}^{n} ∥\frac{∂(g̃_{i} − g_{i}^{s_{i}})}{∂τ_{j}} ∥_{r} \frac{1}{r^{s_{i}−2}} + ∑_{i=1}^{n} ∑_{j=n+1}^{n+2} ∥\frac{∂(g̃_{i} − g_{i}^{s_{i}})}{∂τ_{j}} ∥_{r} \frac{1}{r^{s_{i}−|μ|}} < \frac{1}{2c_{4}} (2.37)

then the polynomials {τ_{k}, W } (1 ≤ k ≤ n − 1) satisfy

(2.38)  ∥\{τ_{k}, W \}∥_{r} ≤ 2c_{4} ∑_{i=1}^{n} \frac{∥˜Ĝ_{i}∥_{r}}{r^{s_{i}−2}} .

Set δ := min_{1≤j≤n} δ_{j} and c_{5} := 4πc_{4}.

Lemma 2.8. Let 0 < ρ < r. Under the assumption of Lemma 2.7, the following estimates hold:

(i) ∥W ∥_{r} ≤ c_{5} ∑_{i=1}^{n} \frac{∥˜Ĝ_{i}∥_{r}}{r^{s_{i}−2}}

(ii) ∥\frac{∂W}{∂ζ_{i}} ∥_{ρ} ≤ \frac{c_{5}}{δ(r−ρ)} ∑_{k=1}^{n} \frac{∥˜Ĝ_{k}∥_{r}}{r^{s_{k}−2}} (1 ≤ i ≤ 2n)

(iii) ∥\frac{∂^{2}W}{∂ζ_{i}∂ζ_{j}} ∥_{ρ} ≤ \frac{4c_{5}}{δ^{2}(r−ρ)^{2}} ∑_{k=1}^{n} \frac{∥˜Ĝ_{k}∥_{r}}{r^{s_{k}−2}} (1 ≤ i, j ≤ 2n).

Proof. (i) By Lemma 2.5,

|W^{ℓ+2}|_{r}^{sup} ≤ 2π ∑_{k=1}^{n−1} |{τ_{k}, W^{ℓ+2}}|_{r}^{sup}

which, combined with Lemma 2.7 implies (i). Estimate (ii) and (iii) follow from (i) by applying Cauchy’s integral formula. □

2.5. Proof of Theorem 1.1.

The proof of Theorem 1.1 is now completed as in Ito [It2]. One first provides, using Lemma 2.8, estimates at the k’th iteration step for the flow X_{W_{k}}^{t} and G_{i}^{(k)} ·φ_{k} where φ_{k} = X_{W_{k}}^{t=1} and then uses them to prove that the limit φ = lim_{k→∞} φ_{0} ⋯ φ_{k} (cf.

Corollary 2.2) defines a holomorphic coordinate transformation in a neighborhood of the origin.

3. On the level sets of an integrable system near a fixed point with a simple resonance

In this section, we consider real integrable systems of the type described in Theorem 1.1′. Such a system can be viewed as a complex Hamiltonian system by introducing the symplectic coordinates

(x, iy) = (\frac{x̂ + iŷ}{\sqrt{2}}, i\frac{x̂ − iŷ}{\sqrt{2}}) (3.1)

where ẑ := (x̂, ŷ) are coordinates as provided by Theorem 1.1′. In order to keep the exposition simple, we will often use the complex notation x, y with the understanding that (3.1) always holds and use z = (x, y) rather than (x, iy) as we are interested in the geometry of the the level sets rather than the dynamics of the system. Choose G_{j} = τ_{j} (1 ≤ j ≤ n − 1) and denote by

M_{c} := {ẑ:= (x̂, ŷ) ∈ (ℝ^{2n}, 0) | G_{j}(z) = c_{j} (1 ≤ j ≤ n)} (3.2)

a level set of the integrable system where (ℝ^{2n}, 0) is a neighborhood of 0 in ℝ^{2n} which is invariant under the Hamiltonian flows corresponding the the Hamiltonians G_{j}, 1 ≤ j ≤ n−1, (cf. paragraph before Proposition 3.2) and by M_{c;z} the connected component of M_{c} containing z.

3.1 Generic level sets.

Notice that for generic ẑ := (x̂, ŷ) ∈ (ℝ^{2n}, 0), d_{z}G_{1}, …, d_{z}G_{n} are linearly independent. By Sard’s theorem (cf. e.g. [Hi]) the following result holds:

Proposition 3.1. For generic c, d_{z}G_{1}, …, d_{z}G_{n} are linearly independent for arbitrary ẑ in M_{c}.

For the sequel assume that the connected component M_{c,z} of M_{c} with z ∈ M_{c,z} has the property that d_{z′}G_{1}, …, d_{z′}G_{n} are linearly independent for arbitrary z′ in M_{c,z}. Following a standard argument one uses the flows φ_{j}^{t_{j}}(z) corresponding to the Hamiltonian vector fields X_{G_{j}} with initial condition φ_{j}^{0}(z) = z (1 ≤ j ≤ n) to construct a local diffeomorphism

\begin{matrix}Φ(·, z) : ℝ^{n−1} × I(z) & → M_{c,z} \\ (t_{1}, …, t_{n}) & ↦ φ_{1}^{t_{1}} ∘ ⋯ ∘ φ_{n}^{t_{n}}(z) \end{matrix}

where I(z) denotes the maximal interval of existence of the flow φ_{n}^{t_{n}}(z). As the flows corresponding to X_{G_{j}}(1 ≤ j ≤ n − 1) are periodic ( the integrals G_{j} (1 ≤ j ≤ n − 1) are an incomplete set of action variables) one concludes that M_{c,z} is either diffeomorphic to (S^{1})^{n} or (S^{1})^{n−1} × I(z). (Flows can also be used to study the level sets of complex systems. In that case the flow variables t_{j} are complex and the orbits φ_{j}^{t_{j}}(z) are Riemann surfaces whose properties however are not easily analyzed (cf. e.g. [Sb]).)

Our aim is to obtain additional information, depending on properties of μ and G_{n}, on M_{c} and the fibration they induce. Without loss of generality, we may assume that μ and ϱ^{(j)} satisfy the following (normalization) conditions

μ_{j} = 0 (1 ≤ j ≤ ℓ); μ_{j} ≠ 0 (ℓ + 1 ≤ j ≤ n); μ_{n} > 0; (3.3)

ϱ^{(j)} := e_{j} (1 ≤ j ≤ ℓ); ϱ_{k}^{(j)} = 0 (l + 1 ≤ j ≤ n; 1 ≤ k ≤ l)

where ℓ is some integer, 0 ≤ ℓ ≤ n − 2, and e_{1}, …, e_{n} is the standard basis in ℝ^{n}. Introduce

Ω := \{ω = (ω_{1}, …, ω_{n}, ω_{n+1}) ∈ \overline{ℝ}_{+}^{n} × ℂ | ∏_{j=ℓ+1}^{n} ω_{j}^{|μ_{j}|} = |ω_{n+1}|^{2}\}

where \overline{ℝ}_{+} is the closure of ℝ_{+} := \{x ∈ ℝ | x > 0\} and define

ψ : \{(x, y) ∈ ℂ^{2n} | x_{j} = \overline{y}_{j}(1 ≤ j ≤ n)\} → Ω

given by

ω_{k} ≡ ψ_{k}(x, y) := x_{k}y_{k} (1 ≤ k ≤ n); ω_{n+1} ≡ ψ_{n+1}(x, y) := x^{μ^{+}}y^{μ^{−}}.

For (Ω, 0) we use the standard neighborhood system {U_{ϵ}}_{ϵ} with U_{ϵ} := {ω ∈ Ω : ∑_{j=1}^{n+1} |ω_{j}|^{2} < ϵ}. For c = (c_{1}, …, c_{n}) with |c| << ϵ, introduce

B_{c} := {ω ∈ U_{ϵ} | G_{j}(ω) = c_{j} (1 ≤ j ≤ n)}

where here we view the G_{j}’s as functions of ω_{1}, …, ω_{n+1} which are real analytic in ω_{1}, …, ω_{n}, ℜω_{n+1}, ℑω_{n+1}. By a standard argument (cf. e.g. [Lo]), for generic c (i.e. for c ∉ Δ for some proper analytic subspace (Δ, 0) ⊂ (ℝ^{n+1}, 0)) and for |c| << ϵ, B_{c} is a one dimensional real smooth C^{∞}−manifold, and its C^{∞}−type does not depend on the choice of ϵ and c. Since ψ^{−1}(0) = 0 and ψ is proper, V_{ϵ} = ψ^{−1}(U_{ϵ}) defines a neighborhood system {V_{ϵ}}_{ϵ} of (ℝ^{2n}, 0). Define for c = (c_{1}, ⋯, c_{n}) with |c| << ϵ

M_{c} := {z ∈ ψ^{−1}(U_{ϵ}) | G_{j}(z) = c_{j} (1 ≤ j ≤ n)}.

Notice that ψ induces a map M_{c} → B_{c} which we again denote by ψ.

The following result reduces the analysis of M_{c} to the one of B_{c}, at least for generic c.

Proposition 3.2. For generic c ∈ (ℝ^{n}, 0) with |c| << ϵ, the following statements hold:

(i) ψ : M_{c} → B_{c} is a fiber bundle with fiber (S^{1})^{n−1}; (ii) ψ : M_{c} → B_{c} admits a real analytic trivialization Ψ,

B_{c} × (S^{1})^{n−1} ⟶^{Ψ} M_{c}
pr_{1} ↘  ↙ ψ
B_{c}

where pr_{1} : B_{c} × (S^{1})^{n−1} → B_{c} denotes the canonical projection.

Remark 3.1. In subsection 3.3, we present a different trivialization which is independent of c, obtained from using the Hamiltonian flows corresponding to τ_{1}, ⋯, τ_{n}. The one constructed in the proof of Proposition 3.2 will be convenient in subsection 3.2.

To prove Proposition 3.2 we need the following auxilary result. Define Ω′ := {ω ∈ Ω | ω_{j} ≠ 0 for 1 ≤ j ≤ n}.

Lemma 3.3. For generic c ∈ (ℝ^{n}, 0), B_{c} ⊂ Ω′.

Proof. For a generic c ∈ (ℝ^{n}, 0) we may assume that c_{j} ≠ 0 (1 ≤ j ≤ n). Thus, for any ω ∈ B_{c}, ω_{j} = τ_{j} = c_{j} ≠ 0 for 1 ≤ j ≤ ℓ. Assume that there exist ω ∈ B_{c} and j_{0}, ℓ + 1 ≤ j_{0} ≤ n, with ω_{j_{0}} = 0 or, equivalently, that ω_{n+1} = 0. Then, the system of equations

∑_{k≠j_{0}} ρ_{k}^{(j)}ω_{k} = c_{j} (1 ≤ j ≤ n − 1); G_{n}(ω_{1}, …, ω_{n}, 0) = c_{n} (3.4)

has a solution ω^{0} := (ω_{1}, …, ω_{j_{0}−1}, 0, ω_{j_{0}+1}, …, ω_{n}, 0). Introduce

Σ := {ω ∈ (Ω, 0) | ω_{j_{0}} = 0} (= (ℝ^{n−1}, 0)) .

The closure of G(Σ) is an analytic variety of dimension not greater than n − 1, and c ∈ G(Σ). We conclude that c is not generic. This shows that for generic c, ω_{n+1} ≠ 0 for any (ω_{1}, …, ω_{n+1}) ∈ B_{c}. □

Proof of Proposition 3.2. Notice that the base space Ω′ is a product (ℝ_{+})^{ℓ} × Ω̃′ where

Ω̃′ := {(ω_{ℓ+1}, …, ω_{n+1}) ∈ (ℝ_{+})^{n−ℓ} × ℂ^{∗} | ∏_{j=ℓ+1}^{n} ω_{j}^{|μ_{j}|} = |ω_{n+1}|^{2}}

with ℂ^{∗} := ℂ\0. Correspondingly, the map ψ can be written as ψ_{1} × ψ_{2} where

ψ_{1} : \{(x, y) ∈ ℂ^{2ℓ} | x_{j} = \overline{y}_{j} ≠ 0 (1 ≤ j ≤ ℓ)\} → (ℝ_{+})^{ℓ}

is given by ψ_{1}(x, y) := (x_{1}y_{1}, …, x_{ℓ}y_{ℓ}) and

ψ_{2} : \{(x, y) ∈ ℂ^{2n−2ℓ} | x_{j} = \overline{y}_{j} ≠ 0(ℓ + 1 ≤ j ≤ n)\} → Ω̃′

is given by ψ_{2}(x, y) := (x_{ℓ+1}y_{ℓ+1}, …, x_{n}y_{n}, ∏_{j=ℓ+1}^{n} x_{j}^{μ_{j}^{+}}y_{j}^{μ_{j}^{−}}).

The map ψ_{1} is a trivial fiber bundle

(ℝ_{+})^{ℓ} × ψ_{1}^{−1}(1, …, 1) ⟶^{Ψ_{1}} ψ_{1}^{−1}((ℝ_{+})^{ℓ})
pr_{1} ↘  ↙ ψ_{1}
(ℝ_{+})^{ℓ}

where Ψ_{1} is given by

Ψ_{1}(ω_{1}, …, ω_{ℓ}, x_{1}, …, x_{ℓ}, \overline{x}_{1}, …, \overline{x}_{ℓ}) := (\sqrt{ω_{j}}x_{j}, \sqrt{ω_{j}}\overline{x}_{j})_{1≤j≤ℓ}

and the fiber ψ_{1}^{−1}(1, …, 1) is

ψ_{1}^{−1}(1, …, 1) = \{(x_{1}, …, x_{ℓ}, \overline{x}_{1}, …, \overline{x}_{ℓ}) | x_{j}\overline{x}_{j} = 1 (1 ≤ j ≤ ℓ)\} = (S^{1})^{ℓ}.

The map ψ_{2} is also a trivial fiber bundle

Ω̃′ × ψ_{2}^{−1}(1, …, 1) ⟶^{Ψ_{2}} ψ_{2}^{−1} ˜ (Ω′)
pr_{1} ↘  ↙ ψ_{2}
Ω̃′

where

Ψ_{2}((ω_{j})_{ℓ+1≤j≤n+1}, (x_{j}, \overline{x}_{j})_{ℓ+1≤j≤n}) :=

(\sqrt{ω_{j}}(\frac{ω_{n+1}}{|ω_{n+1}|})^{ρ_{j}^{(n)}}x_{j}, \sqrt{ω_{j}}(\overline{\frac{ω_{n+1}}{|ω_{n+1}|}})^{ρ_{j}^{(n)}} \overline{x}_{j})_{ℓ+1≤j≤n}.

Notice that ψ_{2}^{−1}(1, …, 1) ≃ (S^{1})^{n−l−1}. Indeed, fix a point (x, x̄) ∈ ψ_{2}^{−1}(1, ⋯, 1). Then R : (S^{1})^{n−l−1} → ψ_{2}^{−1}(1, ⋯, 1), defined by

R(z_{l+1}, ⋯, z_{n−1}) = (( ∏_{j=l+1}^{n−1} z_{j}^{ρ_{k}^{(j)}} ) x_{k}, ( ∏_{j=l+1}^{n−1} z̄_{j}^{ρ_{k}^{(j)}} ) x̄_{k})

is an isomorphism where S^{1} = {z ∈ ℂ | |z| = 1}. (Recall that by the normalization (3.3), ρ_{k}^{(j)} = 0 if l + 1 ≤ j ≤ n and 1 ≤ k ≤ l.) □

The following two propositions relate the nature of the prime resonance vector μ with the topology of B_{c}.

Proposition 3.4. Assume that μ is oscillating, i.e. that there exists j_{0} with ℓ+1 ≤ j_{0} ≤ n − 1 such that μ_{j_{0}} < 0. (Recall that μ_{n} > 0.) Then, for c ∈ ℝ^{n} sufficiently small, B_{c} is compact. If B_{c} is smooth (which is the case for generic c), then each connected component of B_{c} is diffeomorphic to S^{1}.

Remark 3.2. If μ is oscillating, there exists H ∈ A, so that the Hessian of H at 0 is positive definite. Thus Proposition 3.4 is related to a result due to Weinstein [We]. This result asserts that for a Hamiltonian system on a symplectic manifold of dimension 2n with Hamiltonian H, there exist at least n periodic orbits on each energy surface which is sufficiently close to an energy surface {H(z) = H(z_{0})} where z_{0} is a critical point of H with positive definite Hessian. (The Hamiltonian system needs not to be completely integrable and no resonance assumptions are made.)

Proof. It is to prove that for c sufficiently small, B_{c} is compact. Consider the linear system

∑_{k=1}^{n} ρ_{k}^{(j)}ω_{k} = c_{j} (1 ≤ j ≤ n − 1)

which admits a solution ω_{k} = \frac{μ_{k}}{μ_{n}} ω_{n} − b_{k} where

b_{k} ≡ b_{k}(c′) := ∑_{j=1}^{n−1} (−b_{kj} + \frac{μ_{k}}{μ_{n}} b_{nj}) c_{j} (1 ≤ k ≤ n − 1); b_{n} ≡ b_{n}(c′) = 0

with c′ = (c_{1}, …, c_{n−1}) and, as in subsection 2.3, (b_{kj}) is the inverse of the unimodular n×n matrix whose rows are given by ρ^{(1)}, …, ρ^{(n)}. Introduce J^{±} := {j | ℓ+1 ≤ j ≤ n; ±μ_{j} > 0\}. For j ∈ J^{+} we write ω_{j} = \frac{μ_{j}}{μ_{n}} (ω_{n} − \frac{μ_{n}}{μ_{j}} b_{j}) to conclude from

ω_{j} ≥ 0 that ω_{n} ≥ \frac{μ_{n}}{μ_{j}} b_{j}. Similarly, for j ∈ J^{−}, we write ω_{j} = (−\frac{μ_{j}}{μ_{n}} ) (\frac{μ_{n}}{μ_{j}} b_{j} − ω_{n}) and conclude that \frac{μ_{n}}{μ_{j}} b_{j} ≥ ω_{n}. Therefore, if ω ∈ B_{c},

0 ≤ M^{+} ≤ ω_{n} ≤ M_{−} < ∞

where M^{+} ≡ M^{+}(c′) := max\{\frac{μ_{n}}{μ_{j}} b_{j} | j ∈ J^{+}\} and M_{−} ≡ M_{−}(c′) := min\{\frac{μ_{n}}{μ_{j}} b_{j} | j ∈ J^{−}}. Notice that M_{−} < ∞ as J^{−} ≠ ∅ and 0 ≤ M^{+} as n ∈ J^{+} and b_{n} = 0. In particular, we observe that B_{c} = ∅ if M_{−} < M^{+}. For the case M^{+} ≤ M_{−} recall that ω ∈ B_{c} implies that ∏_{ℓ+1}^{n} ω_{j}^{|μ_{j}|} = |ω_{n+1}|^{2}. To rewrite this equation in a convenient way let

f_{+}(t) := ∏_{j∈J^{+}} (t − \frac{μ_{n}}{μ_{j}} b_{j})^{μ_{j}} , f_{−}(t) := ∏_{j∈J^{−}} (\frac{μ_{n}b_{j}}{μ_{j}} − t)^{|μ_{j}|}

and

f(t) := Cf_{+}(t)f_{−}(t)

where

C := ( ∏_{j=ℓ+1}^{n} |\frac{μ_{j}}{μ_{n}} |^{|μ_{j}|}) > 0.

One verifies that, with ω_{j} = \frac{μ_{j}}{μ_{n}} ω_{n} − b_{j}, |ω_{n+1}|^{2} = ∏_{j=ℓ+1}^{n} ω_{j}^{|μ_{j}|} = f(ω_{n}), that f(t) is a polynomial in t with f(t) > 0 for M^{+} < t < M_{−} and f(M^{+}) = f(M_{−}) = 0. Moreover, if M^{+} < M_{−}, there exists a unique element t_{∗}, M^{+} < t_{∗} < M_{−}, so that Max_{f} = f(t_{∗}) where

Max_{f} ≡ Max_{f}(c′) := max{f(t) | M^{+} < t < M_{−}}.

Notice that lim_{c′→0} b_{j}(c′) = 0 and therefore

lim_{c′→0} M^{+}(c′) = lim_{c′→0} M_{−}(c′) = lim_{c′→0} Max_{f}(c′) = 0.

Further, if ω ∈ B_{c}, then

0 ≤ ω_{n} ≤ M_{−}(c′), and |ω_{n+1}|^{2} ≤ Max_{f}(c′).

We conclude that for any ε-disc D_{ε} in \overline{ℝ_{+}}^{n} × ℂ there exists δ ≡ δ(ε) > 0 so that if |c| < δ, then B_{c} ∩ D_{ε} ⊂ D_{ε/2} i.e. for c sufficiently small, B_{c} is compact. If B_{c} is smooth, it then follows that it is a disjoint union of circles. □

Proposition 3.5. Assume that μ_{j} ≥ 0 (1 ≤ j ≤ n). Then for generic c in (ℝ^{n}, 0) with |c| << 1 and generic G_{n} (generic in the sense that A is a generic Poissonalgebra with simple resonance) the number #_{open} of open connected components of B_{c} (i.e. components which are diffeomorphic to the unit interval (0,1)) is given as follows:

(i) if |μ| = 2 or 3, then #_{open} = 1; (ii) if |μ| = 4, then either #_{open} = 0 or #_{open} = 1 (iii) if |μ| ≥ 5, then #_{open} = 0.

Proof. In a first step we show that for generic c, B_{c} can be identified with the Milnor fiber of a hypersurface, obtained from G_{n}. Using the same notation as in the proof of Proposition 3.4, f(t) = ∏_{j=ℓ+1}^{n} (\frac{μ_{j}}{μ_{n}} )^{μ_{j}} (t − \frac{μ_{n}}{μ_{j}} b_{j})^{μ_{j}} as J^{−} = ∅. Notice that

f : (M^{+}, ∞) ⟶ (0, ∞)

is a real analytic isomorphism and admits a square root, f^{\frac{1}{2}} : (M^{+}, ∞) ⟶ (0, ∞), which is again a real analytic isomorphism. Denote by g ≡ g_{c′} the inverse of f^{\frac{1}{2}} where c′ = (c_{1}, ⋯, c_{n−1}). Substituting ω_{j} = \frac{μ_{j}}{μ_{n}} g(r) − b_{j} (ℓ + 1 ≤ j ≤ n − 1), and taking into account that f(ω_{n}) = r^{2} and ω_{n+1} = re^{iθ}, B_{c} can be identified with the Milnor fiber

\{(r, θ) | r ∈ (\overline{ℝ}_{+}, 0); H(g(r), r, θ) = c_{n}\}

where H(ω_{n}, r, θ) ≡ H_{c′}(ω_{n}, r, θ) := G_{n}(ω).

In a next step consider the space B_{0} corresponding to c_{j} = 0 (1 ≤ j ≤ n), i.e.

B_{0} := {(t, r, θ) | t ∈ (ℝ_{+}, 0); t^{|μ|} = C^{|μ|}r^{2}; H(t, r, θ) = 0}

where C ≡ C(μ) := ∏_{k=ℓ+1}^{n−1}(\frac{μ_{k}}{μ_{n}})^{−μ_{k}/|μ|}. H admits an expansion of the form

H(t, r, θ) := ∑_{ℓ,k≥0} t^{ℓ}r^{k}( ∑_{j+j′=k} a_{kℓj}(cos θ)^{j}(sin θ)^{j′}).

Substituting t = C(μ)r^{2/|μ|}, one obtains

H_{1}(r, θ) ≡ H(C(μ)r^{2/|μ|}, r, θ) = ∑_{q=k+2ℓ/|μ|} P_{q}(θ)r^{q}

where P_{q}(θ) := ∑_{q=k+2ℓ/|μ|} C^{ℓ}( ∑_{j+j′=k} a_{kℓj}(cos θ)^{j}(sin θ)^{j′}).

As P_{q}(θ) is a polynomial in cos θ and sin θ (and thus, in particular, periodic of period 2π) the number #_{q} of roots of P_{q}(θ) in the interval [0, 2π), counted with multiplicity, is even. Let

q_{0} := min \{q = k + \frac{2ℓ}{|μ|} | P_{q} ≢ 0; k, ℓ ≥ 0\}.

If P_{q_{0}}(θ) has only simple roots, we claim that, for generic c in (ℝ^{n}, 0), the number #_{open} of noncompact components of B_{c} is given by \frac{1}{2}#_{q_{0}}. This can be proved as follows: consider a sufficiently small, fixed disc D_{ε} := {(r, θ) | |r| ≤ ε}. The equation defining B_{c} is a perturbation of r^{q_{0}}P_{q_{0}}(θ) = 0. The solution set {(r, θ) | P_{q_{0}}(θ) = 0} are rays which intersect ∂D_{ε} transversely (in fact, orthogonally). Since intersecting transversally is an open property, the number of noncompact components of B_{c}, for |c| sufficiently small, is given by

\frac{1}{2}#(B_{c} ∩ ∂D_{ε}) = \frac{1}{2}#\{θ ∈ [0, 2π) | P_{q_{0}}(θ) = 0\}

and the above claim follows.

It remains to analyze P_{q_{0}}. For that purpose, expand G_{n}(ω),

G_{n}(ω) = ∑_{1}^{n} a_{j}ω_{j} + ∑_{j,k} a_{jk}ω_{j}ω_{k} + β_{1}Re(ω_{n+1}) + β_{2}Im(ω_{n+1}) + ...

where the dots stand for terms of higher order, and substitute, for 1 ≤ j ≤ n, ω_{j} = \frac{μ_{j}}{μ_{n}} ω_{n} = \frac{μ_{j}}{μ_{n}} Cr^{\frac{2}{|μ|}} to obtain, as |μ| ≥ 2,

\begin{aligned}H(Cr^{\frac{2}{|μ|}} , r, θ) & = (∑_{1}^{n} a_{j}μ_{j}) \frac{C}{μ_{n}} r^{\frac{2}{|μ|}} + αr^{\frac{4}{|μ|}} + O(r^{\frac{6}{|μ|}}) \\ & + β_{1}r cos θ + β_{2}r sin θ + O(r^{1+ \frac{2}{|μ|}}) \end{aligned}

where α := (\frac{C}{μ_{n}} )^{2} ∑_{j,k} a_{jk}μ_{j}μ_{k}.

As G_{n} ∈ A and μ is a prime resonant vector for A, ∑_{j=1}^{n} a_{j}μ_{j} = 0. Further, for G_{n} generic, α ≠ 0 as well as (β_{1}, β_{2}) ≠ (0, 0). Therefore, if |μ| ≥ 5, q_{0} = \frac{4}{|μ|}, P_{q_{0}} ≡ α and thus #_{q_{0}} = 0. If |μ| = 2 or 3, then q_{0} = 1, P_{q_{0}}(θ) = β_{1} cos θ + β_{2} sin θ has two simple roots in [0, 2π) and therefore #_{q_{0}} = 2. If |μ| = 4, P_{q_{0}}(θ) = α + β_{1} cos θ + β_{2} sin θ, then P_{q_{0}}(θ) has generically only simple roots and, depending of the size of α, β_{1}, β_{2}, either #_{q_{0}} = 0 or #_{q_{0}} = 2. □

To illustrate the above results, we present a number of examples.

Example 1. (cf. Proposition 3.5) For a first set of examples, let n = 2, μ = (2, 1), choose ρ^{(1)} = (−1, 2), ρ^{(2)} = (1, −1), G_{1} := τ_{1} = −ω_{1} + 2ω_{2} and let G_{2} be a power series in ω_{1}, ω_{2}, ℜω_{3} and ℑω_{3} where ω_{3} = x_{1}^{2}x_{2}. Then, with c = (c_{1}, c_{2}),

B_{c} = \{(ω_{1}, ω_{2}, ω_{3}) ∈ (\overline{ℝ}_{+}^{2} × ℂ, 0) | ω_{1}ω_{2} = |ω_{3}|^{2}; G_{1} = c_{1}; G_{2} = c_{2}\} .

Writing t := ω_{2}, ω_{3} = re^{iθ} and ω_{1} = 2t − c_{1}, B_{c} can be described, with some abuse of notation,

B_{c} = {(t, r, θ) | (t, r) ∈ R_{c_{1}}; θ ∈ ℝ; G_{2}(t, r, θ) = c_{2}}

where R_{c_{1}} := {(t, r) ∈ (ℝ^{2}, 0) | t ≥ 0; r ≥ 0; 2t − c_{1} ≥ 0; (2t − c_{1})t = r^{2}}. The set R_{c_{1}} is the graph in the positive (t, r)-quadrant of a curve which is strictly increasing. We consider B_{c} as a fiber space above R_{c_{1}}, where B_{c} → R_{c_{1}}, (t, r, θ) ⟼ (t, r) is the projection and the fiber above (t, r) is given by the solution set F_{c_{2}}(t, r) := {θ ∈ ℝ | G_{2}(t, r, θ) = c_{2}}.

Let A = (a_{ij}) be the unimodular 2 × 2 matrix whose rows are given by ρ^{(1)}, ρ^{(2)}. The rows of the transpose of the inverse, (A^{−1})^{T} = (β_{ij}), are then given by ν^{(1)} = (1, 1), ν^{(2)} = (2, 1). Let x_{j} = \sqrt{ω}_{j}e^{iθ_{j}} (j = 1, 2) and notice that (ω_{j}, θ_{j}) are symplectic polar coordinates, ω_{j} = (x̂_{j}^{2} + ŷ_{j}^{2})/2, θ_{j} = arg(x̂_{j} + iŷ_{j}). The coordinate transformation (τ_{1} := ∑ a_{1j}ω_{j} = −ω_{1} + 2ω_{2}, s_{1} := ∑ β_{1j}θ_{j} = θ_{1} + θ_{2}; τ_{2} := ∑ a_{2j}ω_{j} = ω_{1} − ω_{2}, s_{2} := ∑ β_{2j}θ_{j} = 2θ_{1} + θ_{2}) is canoncial. Note that s_{2} = θ and the variable s_{1} is a new time variable for the reduced Hamiltonian system under the reduction G_{1} = c_{1} (cf. [Ar2, p 259]). Instead of using the coordinates t = ω_{2}, r = ω_{1}\sqrt{ω}_{2}, θ = s_{2} to describe the level sets B_{c}, we could equally well use the coordinates τ_{1}, τ_{2}, s_{2}.

Ex. 1.1. Let G_{2} = ℜω_{3} = r cos θ. Then F_{c_{2}}(t, r) = ∅ if r < |c_{2}|. The equation r cos θ = c_{2} has one solution in θ( mod 2π) if r = |c_{2}| and two solutions if r > |c_{2}|. One concludes that for |c_{2}| > 0, B_{c} is diffeomorphic to the unit interval (0, 1).

As shown in [Ar2, p. 259], the intersection of M_{c} ∩ {θ_{2} = constant} with the (ω_{1}, θ_{1}) plane (Poincaré section) consists of two disconnected components of one dimension which are, generically, parametrized by ω_{2}. These components are however connected when considered in the space Ω.

Ex. 1.2. Let G_{2} = ℜω_{3}^{k} = r^{k} cos kθ. By a similar argument, one concludes that for c_{2} ≠ 0, B_{c} = ⨆_{k}(0, 1), i.e. B_{c} is a disjoint union of k copies of the unit interval.

Ex. 1.3. Let G_{2} = (ℜω_{3})^{4} +(ℑω_{3})^{4}. Notice that, for c_{2} < 0, B_{c} = ∅ and for c = 0, B_{0} consists of the origin only. Following the arguments of the proof of Proposition 3.5, we conclude that for c_{2} > 0, B_{c} is diffeomorphic to S^{1}. □

Example 2. (cf. Proposition 3.4) For this second set of examples, let n = 2, μ = (1, −1) and G_{1} := τ_{1} = ω_{1} + ω_{2}. Then

B_{c} = \{(ω_{1}, ω_{2}, ω_{3}) ∈ (\overline{ℝ}_{+}^{2} × ℂ, 0) | ω_{1}ω_{2} = |ω_{3}|^{2}; ω_{1} + ω_{2} = c_{1}; G_{2} = c_{2}\}.

Writing again t := ω_{2}, ω_{3} = re^{iθ}, one has ω_{1} = c_{1} − t and obtains

B_{c} = {(t, r, θ) | (t, r) ∈ R′_{c_{1}}; θ ∈ ℝ; G_{2} = c_{2}}

where R′_{c_{1}} := \{(t, r) ∈ (\overline{ℝ}_{+}^{2}, 0) | c_{1} − t ≥ 0; (c_{1} − t)t = r^{2}\}. If c_{1} < 0, then B_{c} = ∅ and if c_{1} = 0, B_{c} consists only of (ω_{1}, ω_{2}, ω_{3}) = (0, 0, 0).

Ex. 2.1. Let c_{1} > 0 and G_{2} := ℜω_{3} = r cos θ. Then B_{c} = S^{1} if |c_{2}| < \frac{c_{1}}{2} and B_{c} = ∅ for |c_{2}| > \frac{c_{1}}{2} .

Ex. 2.2. Let c_{1} > 0 and G_{2} := ℜω_{3}^{k} = r^{k} cos kθ. Then B_{c} = ⨆_{k} S^{1} (disjoint union

of k copies of S^{1}) if |c_{2}|^{\frac{1}{k}} < \frac{c_{1}}{2} and B_{c} = ∅ if |c_{2}|^{\frac{1}{k}} > \frac{c_{1}}{2} .

We remark that Ex 2.1-2.2 fit into the following more general situation: According to the proof of Proposition 3.4 and in view of the equations ω_{j} = \frac{μ_{j}}{μ_{n}} ω_{n} − b_{j} (notation as in the proof of Proposition 3.4), B_{c} is determined by the following equations (t := ω_{n}; ω_{n+1} = re^{iθ})

(i) r^{2} = f(t) where M^{+}(c′) < t < M_{−}(c′); (ii) G_{n}(ω) ≡ H(t, r, θ) = c_{n}.

Assume that G_{n} depends only on ω_{n+1}. Then H is independent of t and, for generic c, the set of solutions (r, θ), r ∈ (\overline{ℝ}_{+}, 0), satisfying (i) and (ii) is given by F ⨆_{∂F} F (the disjoint union of two copies of F which are identified along the

boundaries) where F is the closed fiber of H in the disc of radius \sqrt{Max_{f}(c′)}, centered at 0,

F ≡ F_{H}(c) := \{(r, θ) : r ≤ \sqrt{Max_{f}(c′)}; H(r, θ) = c_{n}\}.

To verify this, notice that for given r > 0, the equation f(t) = r^{2} has no solution if r^{2} > Max_{f}(c′), has exactly one solution if r^{2} = Max_{f}(c′) and has two solutions if r^{2} < Max_{f}(c′). In particular, if #_{o} and #_{c} are the number of open, respectively compact connected components of F_{H}(c), then B_{c} has #_{o} + 2#_{c} connected components, each analytically diffeomorphic to S^{1}.(If |c_{n}| << Max_{f}(c′), then F is the Milnor fiber of G_{n}.) □

3.2 Nongeneric level sets.

Let us make a few remarks concerning the level sets M_{c} for nongeneric c. In this case, B_{c}\Ω′ ≠ ∅ and, for ω ∈ B_{c}\Ω′, the fiber ψ^{−1}(ω) might be different from (S^{1})^{n−1} (cf. Proposition 3.2). To analyze M_{c} for nongeneric c, we stratify the set Ω. Taking into account the normalization conditions (3.3), one sees that Ω has a product decomposition, Ω = (\overline{ℝ_{+}})^{l} × Ω̃ where

Ω̃ := \{(ω_{j})_{l+1≤j≤n+1} ∈ (\overline{ℝ_{+}})^{n−l} × ℂ | ∏_{j=l+1}^{n} ω_{j}^{|μ_{j}|} = |ω_{n+1}|^{2}\}.

For arbitrary subsets I ⊆ {1, ..., l} and J ⊆ {l + 1, ..., n}, define the stratum

Ω_{I,J} := {ω ∈ Ω| for 1 ≤ j ≤ n : j ∈ I ∪ J iff ω_{j} = 0}.

Notice that for ω ∈ Ω_{I,J} with J ≠ ∅, ω_{n+1} = 0 and that Ω′ = Ω_{∅,∅} as well as Ω = ⋃_{I,J} Ω_{I,J}. It turns out the toplogical type of the fiber ψ^{−1}(ω) depends only on the stratum Ω_{I,J} which contains ω. Indeed, argueing as in the proof of Proposition 3.2 we see that

ψ^{−1}(ω) = ψ_{1}^{−1}(ω_{1}, ..., ω_{l}) × ψ_{2}^{−1}(ω_{l+1}, ..., ω_{n+1})

where here ψ_{1}(x, y) := (x_{1}y_{1}, ..., x_{l}y_{l}) is defined on {(x, y) ∈ ℂ^{2l} | x_{j} = ȳ_{j} (1 ≤

j ≤ l)} and ψ_{2}(x, y) := (x_{l+1}y_{l+1}, ..., x_{n}y_{n}, ∏_{j=l+1}^{n} x_{j}^{μ_{j}^{+}}y_{j}^{μ_{j}^{−}}) is defined on {(x, y) ∈ ℂ^{2n−2l} | x_{j} = ȳ_{j} (l + 1 ≤ j ≤ n)}. Identifiying S^{1} with {z ∈ ℂ | |z| = 1}, one has, for ω ∈ Ω_{I,J}, ψ_{1}^{−1}(ω_{1}, ..., ω_{l}) = (S^{1})^{l−|I|} and, if |J| ≠ ∅, ψ_{2}^{−1}(ω_{l+1}, ..., ω_{n+1}) = (S^{1})^{n−l−|J|}. By Proposition 3.2, for J = ∅, ψ_{2}^{−1}(ω_{l+1}, ..., ω_{n+1}) = (S^{1})^{n−l−1}. To study the degeneration of the fibers for ω ∈ Ω_{I,J}, the cases J ≠ ∅ and J = ∅ are treated in the same way and thus we consider the case J ≠ ∅ only. Choose a continuous path ω(t) ∈ Ω (0 ≤ 0 ≤ 1) without self-intersections, so that ω(0) = ω (∈ Ω_{I,J}) and ω(t) ∈ Ω′ ≡ Ω_{∅,∅} for 0 < t ≤ 1. Choose a continuous lift (x_{k}(t), x̄_{k}(t))_{1≤k≤n} of the path ω(t), i.e. ψ((x_{k}(t), x̄_{k}(t))_{1≤k≤n}) = ω(t). Introduce Y := ψ^{−1}(ω(0)) and X := ψ^{−1}({ω(t) | 0 ≤ t ≤ 1}) and define (with S^{1} = {z ∈ ℂ | |z| = 1}) R_{I,J} : (S^{1})^{n−1} × [0, 1] → X by

R_{I,J} (s_{1}, ..., s_{n−1}, t) := ((∏_{j=1}^{n−1} s_{j}^{ρ_{k}^{(j)}})x_{k}(t), (∏_{j=1}^{n−1} s̄_{j}^{ρ_{k}^{(j)}})\overline{x_{k}}(t))_{1≤k≤n} .

Notice that R_{I,J} is onto and R_{I,J} |_{(S^{1})^{n−1}×(0,1]} is 1 − 1. Therefore, R_{I,J} induces a strong deformation retract r_{I,J} : X → Y given by ( with s = (s_{1}, ..., s_{n−1}))

r_{I,J} (R_{I,J}(s, t)) := R_{I,J} ((s, 0)) .

Further, as J ≠ ∅, R_{I,J} ((S^{1})^{n−1} × {0}) can be identified with

{ ((∏_{j=1}^{n−1} s_{j}^{ρ_{k}^{(j)}})x_{k}(0), (∏_{j=1}^{n−1} s̄_{j}^{ρ_{k}^{(j)}})x̄_{k}(0))_{k∈{1,...,n}\I∪J} | (s_{1}, ..., s_{n−1}) ∈ (S^{1})^{n−1}}

which is diffeomorphic to (S^{1})^{n−|I|−|J|} and R_{I,J} : (S^{1})^{n−1}×{0} → (S^{1})^{n−|I|−|J|} can be identified to the projection pr_{I,J} : (S^{1})^{n−1} → (S^{1})^{n−|I|−|J|} on the corresponding factors.

The topological type of M_{c} for nongeneric c can now be determined as follows: To make the exposition simpler, assume that c is such that B_{c,ω} \ Ω′ consists of a finite set of points, B_{c,ω} \ Ω′ = {ω^{(1)}, ..., ω^{(d)}} where B_{c,ω} denotes the connected component of B_{c} which contains ω. (If B_{c,ω} \ Ω′ is not finite, there exists (I, J) ≠ (∅, ∅) so that B_{c,ω} ⊆ Ω_{I,J}. This case is treated similarly as the case (I, J) = (∅, ∅) and is left to the reader.)

For 1 ≤ j ≤ d, there exists (I_{j}, J_{j}) ≠ (∅, ∅), so that ω^{(j)} ∈ Ω_{I_{j},J_{j}}. Let z ∈ M_{c} be a lift of ω, i.e. ψ(z) = ω. Then the connected component M_{c,z} of M_{c} containing z can be identified with the quotient space B_{c,ω} × (S^{1})^{n−1}/ ∼ where for two elements (ω′, z′), (ω′′, z′′) ∈ B_{c,ω} × (S^{1})^{n−1}, (ω′, z′) ∼ (ω′′, z′′), if there exists 1 ≤ j ≤ d, so that ω′ = ω′′ = ω^{(j)} and pr_{I_{j},J_{j}}(z′) = pr_{I_{j},J_{j}}(z′′). The identification of M_{c,z} with B_{c,ω} × (S^{1})^{n−1}/ ∼ can be obtained as follows: Since all the fibers of ψ are connected, B_{c,ω} can be lifted to M_{c,z} by a continuous, injective map B_{c,ω} → M_{c,z}, ω′ ↦ (x(ω′), x̄(ω′)). This lift is used to define a continuous map Λ : B_{c,ω} × (S^{1})^{n−1} → M_{c,z} given by

Λ(ω′, s_{1}, ..., s_{n−1}) := ((∏_{j=1}^{n−1} s_{j}^{ρ_{k}^{(j)}} ) x_{k}(ω′), (∏_{j=1}^{n−1} s̄_{j}^{ρ_{k}^{(j)}} ) x̄_{k}(ω′))_{1≤k≤n} .

3.3 Fibration by the level sets.

Consider a deformation (M_{c})_{c∈C} of level sets, with C denoting the parameter space of the deformation. We claim that (M_{c})_{c∈C} can be obtained from the corresponding deformation (B_{c})_{c∈C}. In particular, if, for I ⊂ {1, ⋯, l} and J ⊂ {l + 1, ⋯, n} arbitrary, the deformations (B_{c} ∩Ω_{I,J})_{c∈C} (cf. notation of subsection 3.2) are topologically trivial, then (M_{c})_{c∈C} is topologically trivial as well. First notice that this statement does not follow from the identification M_{c,z} ≃ B_{c,ω}×(S^{1})^{n−1}/ ∼ provided in subsection 3.2, as the identification is constructed by choosing a section of ψ over B_{c,ω} and therefore depends on c. An identification which is independent of c can be obtained by using the Hamiltonian flows corresponding to τ_{1}, ⋯, τ_{n} with initial conditions x_{j} = x̄_{j} := r_{j} ≥ 0 (1 ≤ j ≤ n). Define ϕ : (S^{1})^{n} × (\overline{ℝ_{+}})^{n} → ℝ^{2n}, given by ϕ(s_{1}, ⋯, s_{n}, r_{1}, ⋯, r_{n}) = (ϕ_{k}, \overline{ϕ_{k}})_{1≤k≤n} where

ϕ_{k}(s_{1}, ⋯, s_{n}, r_{1}, ⋯, r_{n}) := (∏_{j=1}^{n} s_{j}^{ρ_{k}^{(j)}} ) r_{k}.

Then ϕ induces an isomorphism between (S^{1})^{n} × ℝ_{+}^{n} and {(x, y) ∈ ℂ^{2n} | x_{k} = \overline{y_{k}}; ∏_{k=1}^{n} x_{k}y_{k} ≠ 0\}. Further ϕ collapses some closed subgroups of (S^{1})^{n} over {(x, y) ∈ ℝ^{2n} | ∏_{k=1}^{n} x_{k}y_{k} = 0}. One verifies that the composition α := ψ · ϕ : (S^{1})^{n} × \overline{ℝ_{+}}^{n} → Ω is given by α(s_{1}, ⋯, s_{n}, r_{1}, ⋯, r_{n}) = (r_{1}^{2}, ⋯ r_{n}^{2}, r^{μ^{+}+μ^{−}}s_{n}) and is therefore universal, i.e. does not depend on the choice of B_{c} ⊂ Ω or c. Then M_{c} can be obtained from B_{c} by taking the inverse image α^{−1}(B_{c}) and collapsing this space by ϕ. Of course, for generic c, this construction gives an alternative proof of Proposition 3.2 and for arbitrary c, this construction leads to the same result as the one given in subsection 3.2.

As an application we provide a basis for the fundamental group π_{1}(ψ^{−1}(ω)) of ψ^{−1}(ω) where ω ∈ B_{c}. Let η := (\sqrt{ω}_{j})_{1≤j≤n} and ξ := (1, ⋯, 1, s_{n}), where s_{n} is chosen to be 1 if ω_{n+1} = 0,

and define pathes α^{(1)}, …, α^{(n−1)} in the fiber of M_{c} ⟶^{ψ} B_{c} above ω by

α^{(j)}(t_{j}) := (e^{−iρ_{k}^{(j)}t_{j}}ϕ_{k}(ξ, η), e^{iρ_{k}^{(j)}t_{j}} \overline{ϕ_{k}(ξ, η)})_{1≤k≤n}

where 0 ≤ t_{j} ≤ 2π. Notice that on ψ^{−1}(ω), x_{k}dy_{k} is a closed one form (1 ≤ k ≤ n). Thus, by Stokes’ theorem, ∫_{α^{(j)}} x_{k}dy_{k} (1 ≤ k ≤ n) depends only on the homotopy class [α^{(j)}]_{π_{1}(ψ^{−1}(ω))} of α^{(j)} in the fundamental group π_{1}(ψ^{−1}(ω)) of ψ^{−1}(ω).

It follows that [α^{(j)}]_{π_{1}(ψ^{−1}(ω))}(1 ≤ j ≤ n − 1) spans π_{1}(ψ^{−1}(ω)) and, for generic c, is a basis (over ℤ) of π_{1}(ψ^{−1}(ω)).

Appendix A: Decoupled Resonances

The aim of this appendix is to indicate why the method used to prove Theorem 1.1 seems only to work in the case of a simple resonance.

We continue to use the notation introduced in Sections 1 and 2. In addition, we introduce the notion of decoupled resonances – they are the most elementary type of resonances one can think of. Let A be an algebra as defined in (1.1) and denote by Λ_{A} its resonance lattice. This is a lattice in ℤ^{n} such that Λ_{A} = ⋂_{f∈A} Λ_{f}. We

consider the case of multiple resonances, i.e., R ≡ dim Λ_{A} ≥ 2.

Definition. A is said to be resonant with decoupled resonances at z = 0 if Λ_{A} admits a basis μ^{(1)}, …, μ^{(R)} such that

(i) suppμ^{(j)} ∩ suppμ^{(k)} = ∅ (j ≠ k)

(ii) |μ^{(j)}| := ∑_{k=1}^{n} |μ_{k}^{(j)}| ≥ 2.

Notice that R ≤ \frac{n}{2}, if A is resonant with decoupled resonances.

In the following, we want to indicate where the proof of Theorem 2.1 as presented in Section 2 seems to break down for an algebra A, which is resonant with R ≥ 2 decoupled resonances.

Assume that there is an element H in A so that H_{nil} = 0 and Λ_{H_{s}} = Λ_{A}. Let

H_{s} be given by H_{s} = ∑_{j=1}^{n} λ_{j}x_{j}y_{j} with λ = (λ_{1}, …, λ_{n}) ∈ ℂ^{n}.

It is convenient to introduce the following notation slightly differing from the one used in the previous sections: Denote by ρ^{(j)} (R + 1 ≤ j ≤ n), ρ^{(j)} ∈ ℤ^{n}\{0}, a basis (over ℤ) of the sublattice of ℤ^{n}, orthogonal to Λ_{A}. Further choose ρ^{(j)} (1 ≤ j ≤ R) so that ρ^{(j)} (1 ≤ j ≤ n) is a basis of ℤ^{n}. Let

τ_{j} := ∑_{k=1}^{n} ρ_{k}^{(j)}x_{k}y_{k} (1 ≤ j ≤ n)

and

σ_{j}^{+} := x^{μ^{(j)+}}y^{μ^{(j)−}} , σ_{j}^{−} := x^{μ^{(j)−}}y^{μ^{(j)+}} (1 ≤ j ≤ R)

where μ_{k}^{(j)+} := μ_{k}^{(j)} (if μ_{k}^{(j)} ≥ 0), μ_{k}^{(j)+} := 0 (if μ_{k}^{(j)} < 0) and μ^{(j)−} := μ^{(j)+} − μ^{(j)}.

Let f = f(z) ∈ P be a formal power series in H_{s}-normal form considered as a power series in τ_{j}, σ_{k}^{±} (1 ≤ j ≤ n; 1 ≤ k ≤ R). Note that

(A.1)  σ_{k}^{+}σ_{k}^{−} = ∏_{j=1}^{n}(x_{j}y_{j})^{μ_{j}^{(k)}} .

Therefore, f can be represented uniquely as

(A.2)  f(z) = ∑_{sign} f_{sign}(τ, σ_{sign})

where τ := (τ_{1}, …, τ_{n}), sign := (γ_{1}, …, γ_{R}) with γ_{i} ∈ {± } and σ_{sign} = (σ_{1}^{γ_{1}}, …, σ_{R}^{γ_{R}}). Here f_{sign}(τ, σ_{sign}) are power series in τ_{1}, …, τ_{n}, σ_{1}^{γ_{1}}, …, σ_{R}^{γ_{R}}.

It is useful to consider functions in H_{s}-normal form as Laurent series in n + R variables τ_{1}, …, τ_{n}, σ_{1}^{+}, …, σ_{R}^{+} by eliminating σ_{j}^{−} using (A.1). For an element f ∈ P in H_{s}-normal form we denote by \frac{∂f}{∂τ_{j}} (1 ≤ j ≤ n) and \frac{∂f}{∂σ_{j}^{+}} (1 ≤ j ≤ R) the

derivatives of f with respect to τ_{j}, σ_{j}^{+} when f is considered as a (formal) Laurent series in τ_{1}, …, τ_{n}, σ_{1}^{+}, …, σ_{R}^{+}.

Using that

{τ_{j}, τ_{k}} = 0 (1 ≤ j, k ≤ n) , {σ_{j}^{+}, τ_{k}} = 0 (1 ≤ j ≤ R , 1 ≤ k ≤ n; j ≠ k) , (A.3)
{σ_{j}^{+}, τ_{j}} = σ_{j}^{+} ; {σ_{j}^{+}, σ_{k}^{+}} = 0 (1 ≤ j, k ≤ R) ,

one verifies that for f, g ∈ P in H_{s}-normal form

\{f, g\} = − ∑_{j=1}^{R} (\frac{∂f}{∂τ_{j}} \frac{∂g}{∂σ_{j}^{+}} − \frac{∂f}{∂σ_{j}^{+}} \frac{∂g}{∂τ_{j}} ) σ_{j}^{+} . (A.4)

Recall that we have denoted by Π_{N}f the projection of a power series f onto its H_{s}-normal form part. Π_{N}f can be computed by an averaging procedure

(A.5)  Π_{N}f(x, y) = ∫_{0}^{1} dθ_{R+1} ⋯ ∫_{0}^{1} dθ_{n}f(e^{2πiθ}x, e^{−2πiθ}y)

where e^{2πiθ}x is defined by

(A.6)  e^{2πiθ}x = (e^{2πi ∑_{R+1}^{n} θ_{j}ρ_{1}^{(j)}}x_{1}, …, e^{2πi ∑_{R+1}^{n} θ_{j}ρ_{n}^{(j)}}x_{n})

and e^{−2πiθ}y is defined similarly. Formula (A.5) implies that, given a polynomial W with Π_{N}W = 0, the supremum norm |W |^{sup} on a neighborhood of the origin can be computed by

(A.7)  |W |^{sup} ≤ 2π ∑_{k=R+1}^{n} |{τ_{k}, W }|^{sup} .

One of the main points of the proof of Theorem 1.1, as presented in Section 2, is to estimate |W |^{sup}. Let z = φ(ζ) ≡ X_{W}^{t=1}(ζ) be the transformation described in Proposition 2.1 with W being a polynomial, W = W^{d+2} + ⋯ + W^{2d+1}. Then, by Lemma 2.3, with G_{j} = g_{j} + Ĝ_{j} and Ĝ_{j}(ζ) = O(ζ^{s_{j}+d}),

(A.8)  G_{j} ∘ φ(ζ) = g_{j}(ζ) + {g_{j}(ζ), W } + Ĝ_{j}(ζ) + O(|ζ|^{s_{j}+2d})

is in H_{s}-normal form up to order s_{j} − 1 + 2d. Therefore, (1 ≤ j ≤ n)

(A.9)  {g_{j}(ζ), W (ζ)} + Π_{R}Ĝ_{j}(ζ) = O(|ζ|^{s_{j}+2d})

which can be written as (1 ≤ i ≤ n)

(A.10)_{i}  ∑_{j=1}^{n} \frac{∂g_{i}}{∂τ_{j}} \{τ_{j}, W \} + ∑_{j=1}^{R} \frac{∂g_{i}}{∂σ_{j}^{+}} \{σ_{j}^{+}, W \}
= −Π_{R}Ĝ_{i}(ζ) + O(|ζ|^{s_{i}+2d}) .

Notice that (A.10) is a linear system of n equations for n + R quantities {τ_{1}, W }, …, {τ_{n}, W } and {σ_{1}^{+}, W }, …, {σ_{R}^{+}, W }.

We then try to eliminate the 2R variables {τ_{j}, W }, {σ_{j}^{+}, W } (1 ≤ j ≤ R) from (A.10) up to terms of higher order, and reduce these n equations to n−R equations for {τ_{j}, W } with j = R + 1, …, n. To illustrate this procedure, we recall the case of a simple resonance R = 1 (cf. Section 2): we first write (A.10) in matrix form,

( \begin{matrix}\frac{∂g_{1}}{∂τ_{1}} & \frac{∂g_{1}}{∂σ_{1}^{+}} \\ \frac{∂g_{i}}{∂τ_{1}} & \frac{∂g_{i}}{∂σ_{1}^{+}} \end{matrix} ) \binom{\{τ_{1}, W \}}{\{σ_{1}^{+}, W \}} = \binom{F_{1}}{F_{i}} (2 ≤ i ≤ n) (A.11)

where F_{1} and F_{i} include the variables {τ_{j}, W } with j = 2, …, n. Let us consider, for example, the case where (\frac{∂g_{i}}{∂σ_{1}^{+}} , −\frac{∂g_{1}}{∂σ_{1}^{+}} ) ≢ (0, 0) (cf. case 2 in Section 2). Then

eliminating {σ_{1}^{+}, W } from (A.11) (Cramer’s rule), we obtain, after multiplication with τ_{1}σ_{1}^{+},

τ_{1}σ_{1}^{+} det ( \begin{matrix}\frac{∂g_{1}}{∂τ_{1}} & \frac{∂g_{1}}{∂σ_{1}^{+}} \\ \frac{∂g_{i}}{∂τ_{1}} & \frac{∂g_{i}}{∂σ_{1}^{+}} \end{matrix} ) \{τ_{1}, W \} = τ_{1}σ_{1}^{+} \frac{∂g_{i}}{∂σ_{1}^{+}} F_{1} − τ_{1}σ_{1}^{+} \frac{∂g_{1}}{∂σ_{1}^{+}} F_{i} . (A.12)

Notice that the determinant in (A.12) is related to the Poisson bracket {g_{1}, g_{i}} (cf (A.4)). Using {G_{1}, G_{i}} = 0 (integrability), we see that, for 2 ≤ i ≤ n, {g_{1}, g_{i}}= O(ζ^{s_{1}+s_{i}+d−2}). Therefore, the left hand side of (A.12) is O(ζ^{s_{1}+s_{i}+d+d+2}), so that we obtain from (A.10), up to error terms O(ζ^{s_{1}+s_{i}+2d}), a linear system of n − 1 equations for {τ_{i}, W } (2 ≤ i ≤ n).

Now we consider the case R = 2. The equation corresponding to (A.11) is given by (3 ≤ i, j ≤ n)

A(1, 2; i, j) ( \begin{matrix}\{τ_{1}, W \} \\ \{σ_{1}^{+}, W \} \\ \{τ_{2}, W \} \\ \{σ_{2}^{+}, W \} \end{matrix} ) = ( \begin{matrix}F_{1} \\ F_{2} \\ F_{i} \\ F_{j} \end{matrix} )

where F_{1}, F_{2}, F_{i} and F_{j} include the variables {τ_{k}, W } with 3 ≤ k ≤ n, and A(1, 2; i, j) is a 4 × 4 matix given by

A(1, 2; i, j) := ( \begin{matrix}\frac{∂g_{1}}{∂τ_{1}} & \frac{∂g_{1}}{∂σ_{1}^{+}} & \frac{∂g_{1}}{∂τ_{2}} & \frac{∂g_{1}}{∂σ_{2}^{+}} \\ \frac{∂g_{2}}{∂τ_{1}} & \frac{∂g_{2}}{∂σ_{1}^{+}} & \frac{∂g_{2}}{∂τ_{2}} & \frac{∂g_{2}}{∂σ_{2}^{+}} \\ \frac{∂g_{i}}{∂τ_{1}} & \frac{∂g_{i}}{∂σ_{1}^{+}} & \frac{∂g_{i}}{∂τ_{2}} & \frac{∂g_{i}}{∂σ_{2}^{+}} \\ \frac{∂g_{j}}{∂τ_{1}} & \frac{∂g_{j}}{∂σ_{1}^{+}} & \frac{∂g_{j}}{∂τ_{2}} & \frac{∂g_{j}}{∂σ_{2}^{+}} \end{matrix} ) (A.13)

for 3 ≤ i, j ≤ n and i ≠ j. Analogous to the case R = 1, we would like to show that (A.10) leads to a linear system (similar to (A.12)), where the terms involving the four variables {τ_{1}, W }, …, {σ_{2}^{+}, W } are of the order of error terms. A necessary condition is that, for some 3 ≤ i < j ≤ n, τ_{1}τ_{2}σ_{1}^{+}σ_{2}^{+} det A(1, 2; i, j) = O(ζ^{s_{1}+s_{2}+s_{i}+s_{j}+d}). However, this is not true in general as can be seen as follows: By the integrability {G_{i}, G_{j}} = 0, we obtain by (A.4) relations among the functions, \frac{∂g_{i}}{∂τ_{k}}, \frac{∂g_{j}}{∂σ_{ℓ}^{+}} (1 ≤ i, j, k ≤ n, 1 ≤ ℓ ≤ 2),

∑_{k=1}^{2} (\frac{∂g_{i}}{∂τ_{k}} \frac{∂g_{j}}{∂σ_{k}^{+}} − \frac{∂g_{i}}{∂σ_{k}^{+}} \frac{∂g_{j}}{∂τ_{k}} ) τ_{k}σ_{k}^{+} = O(ζ^{s_{i}+s_{j}+d}) . (A.14)_{ij}

It is convenient to introduce the following notations,

a_{k}^{i} := τ_{k} \frac{∂g_{i}}{∂τ_{k}} , b_{k}^{i} := σ_{k}^{+} \frac{∂g_{i}}{∂σ_{k}^{+}}

and to denote the coefficients τ_{k}σ_{k}^{+} (\frac{∂g_{i}}{∂τ_{k}} \frac{∂g_{j}}{∂σ_{k}^{+}} − \frac{∂g_{i}}{∂σ_{k}^{+}} \frac{∂g_{j}}{∂τ_{k}} ) in (A.14)_{ij} by [i, j]_{k}. Then

[i, j]_{k} = det ( \begin{matrix}a_{k}^{i} & b_{k}^{i} \\ a_{k}^{j} & b_{k}^{j} \end{matrix} ) = O(ζ^{s_{i}+s_{j}}) .

Equations (A.14)_{ij} and (A.14)_{kℓ} lead to the 2 × 2 system

( \begin{matrix}[i, j]_{1} & [i, j]_{2} \\ [k, ℓ]_{1} & [k, ℓ]_{2} \end{matrix} ) ( \begin{matrix}1 \\ 1 \end{matrix} ) = ( \begin{matrix}O(ζ^{s_{i}+s_{j}+d}) \\ O(ζ^{s_{k}+s_{l}+d}) \end{matrix} )

and therefore, by Cramer’s rule, (1 ≤ i, j, k, ℓ ≤ n)

[i, j]_{1}[k, ℓ]_{2} − [i, j]_{2}[k, ℓ]_{1} = O(ζ^{s_{i}+s_{j}+s_{k}+s_{l}+d}) . (A.15)

The identities (A.15) give, in general, a complete set of bilinear relations for the expressions of the form [i, j]_{1}[i′, j′]_{2} for 1 ≤ i, j, i′, j′ ≤ n.

Analogous to the case R = 1, we would like to show that, at least for some 3 ≤ i < j ≤ n, the term τ_{1}τ_{2}σ_{1}^{+}σ_{2}^{+} det A(1, 2; i, j) is factorized by the relation (A.15). Let us express such a term as a product of terms [i, j]_{k}. This can be done, by using the Laplace expansion for a determinant [Ma 1, p 189], as follows

τ_{1}τ_{2}σ_{1}^{+}σ_{2}^{+} det A(1, 2; i, j) = τ_{1}τ_{2}σ_{1}^{+}σ_{2}^{+} det(A(1, 2; i, j)^{T}) =

= det ( \begin{matrix}a_{1}^{1} & a_{1}^{2} & a_{1}^{i} & a_{1}^{j} \\ b_{1}^{1} & b_{1}^{2} & b_{1}^{i} & b_{1}^{j} \\ a_{2}^{1} & a_{2}^{2} & a_{2}^{i} & a_{2}^{j} \\ b_{2}^{1} & b_{2}^{2} & b_{2}^{i} & b_{2}^{j} \end{matrix} ) = I − II + III

where I, II and III are given by

\begin{aligned}I & = [1, 2]_{1}[i, j]_{2} + [1, 2]_{2}[i, j]_{1} , \\ II & = [1, i]_{1}[2, j]_{2} + [1, i]_{2}[2, j]_{1} , \\ III & = [1, j]_{1}[2, i]_{2} + [1, j]_{2}[2, i]_{1} . \end{aligned}

Using (A.15) and [i, j]_{k} = O(ζ^{s_{i}+s_{j}}), we obtain

(A.16)  τ_{1}τ_{2}σ_{1}^{+}σ_{2}^{+} det A(1, 2; i, j) = 2{[1, 2]_{1}[i, j]_{2} − [1, i]_{1}[2, j]_{2}
+ [1, j]_{1}[2, i]_{2}} = O(ζ^{s_{1}+s_{2}+s_{i}+s_{j}}) .

Notice that the terms in (A.16) are similar to the Plücker relations [cf Ma 2, p 4], applied to v_{k} := (a_{k}^{1}, a_{k}^{2}, a_{k}^{i}, a_{k}^{j}) (= τ_{k}(\frac{∂g_{1}}{∂τ_{k}} , \frac{∂g_{2}}{∂τ_{k}} , \frac{∂g_{i}}{∂τ_{k}} , \frac{∂g_{j}}{∂τ_{k}})) and w_{k} := (b_{k}^{1}, b_{k}^{2}, b_{k}^{i}, b_{k}^{j}) (k = 1, 2). The Plücker relations are obtained from (v_{k} ∧ w_{k}) ∧ (v_{k} ∧ w_{k}) = 0,

[1, 2]_{k}[i, j]_{k} − [1, i]_{k}[2, j]_{k} + [1, j]_{k}[2, i]_{k} = 0.

However these relations cannot be used to improve the estimate (A.16) further, as in (A.16) both subindices, 1 and 2, appear.

Since the relations (A.15) are the only bilinear relations among the terms [k, l]_{1}· [k′, l′]_{2}, we conclude that in a generic situation, with no further assumptions,

τ_{1}τ_{2}σ_{1}^{+}σ_{2}^{+} det A(1, 2; i, j) = O(ζ^{s_{1}+s_{2}+s_{i}+s_{j}})

(and not better) for {1, 2} ∩ {i, j} = ∅ and i ≠ j.

Appendix B: Generic Level Sets of Complex Systems

In this appendix, we want to present results corresponding to various results of section 3 for complex systems.

Assume that (x, y) ∈ (ℂ^{2n}, 0) are coordinates so that the conclusions of Theorem 1.1 hold. To analyze the level sets M_{c} := {z = (x, y) ∈ (ℂ^{2n}, 0) | G_{j}(z) = c_{j} (1 ≤ j ≤ n)} we argue similarly as in section 3. We again assume the normalization conditions (3.3) for μ and introduce the following subsets of ℂ^{n+2}

\begin{aligned}Ω & := \{ω ∈ ℂ^{n+2}| ∏_{j=ℓ+1}^{n} ω_{j}^{|μ_{j}|} = ω_{n+1}ω_{n+2}\}, \\ Ω′ & := \{ω ∈ ℂ^{n+2}| ω_{j} ≠ 0(1 ≤ j ≤ ℓ); ω_{n+1} ≠ 0\}, \\ Ω′′ & := \{ω ∈ ℂ^{n+2}| ω_{j} ≠ 0(1 ≤ j ≤ ℓ); ω_{n+2} ≠ 0\} \end{aligned}

and set Ω^{0} := Ω′ ∪ Ω′′.

Define ψ : ℂ^{2n} → Ω by setting

ψ(x, y) := ((x_{j}y_{j})_{1≤j≤n}, ∏_{j=ℓ+1}^{n} x_{j}^{μ_{j}^{+}}y_{j}^{μ_{j}^{−}}, ∏_{j=ℓ+1}^{n} x_{j}^{μ_{j}^{−}}y_{j}^{μ_{j}^{+}} )

and introduce for c = (c_{1}, ⋯, c_{n})

B_{c} := {ω ∈ (Ω, 0) | G_{j}(ω) = c_{j} (1 ≤ j ≤ n)}.

Notice that ψ induces a map M_{c} → B_{c} which we again denote by ψ. For the remainder of this subsection it is convenient to denote by M_{c} the inverse image ψ^{−1}(B_{c}). The level set M_{c} is then semilocal as the generic fiber of ψ is equal to (ℂ^{∗})^{n−1}. (If we would consider ψ_{local} : (ℂ^{2n}, 0) → (Ω, 0) and define M_{c} by ψ_{local}^{−1}(B_{c}), the generic fiber ψ_{local}^{−1}(ω), with ω ∈ (Ω, 0), is an annular domain {z = (z_{1}, ⋯, z_{n−1}) ∈ ℂ^{n−1} | a_{j} < |z_{j}| < b_{j}, (1 ≤ j ≤ n − 1)} where 0 < a_{j} < b_{j} < ∞ might depend on the choice of ω. Notice that for different choices of a_{j}’s and b_{j}’s, these domains are not bianalytically isomorphic.)

Analoguous to Proposition 3.2 we have

Proposition B.1. For generic c ∈ (ℂ^{n}, 0)

(i) ψ : M_{c} → B_{c} is a fiber bundle with fiber (ℂ^{∗})^{n−1}; (ii) ψ : M_{c} → B_{c} admits an analytic trivialization

Ψ

B_{c} × (ℂ^{∗})^{n−1} ⟶ M_{c}
pr_{1} ↘  ↙ ψ

B_{c}

where pr_{1} : B_{c} × (ℂ^{∗})^{n−1} → B_{c} denotes the canonical projection.

To prove Proposition B.1 we need the following auxilary result, which is proved in a similar way as Lemma 3.3.

Lemma B.2. For generic c ∈ (ℂ^{n}, 0), B_{c} ⊂ Ω^{0}.

Proof of Proposition B.1. Notice that the inverse image ψ^{−1}(Ω^{0}) is a product E_{1} ×_{Ω^{0}} E_{2} where

E_{1} := {((x_{k}, y_{k})_{1≤k≤ℓ}, ω) ∈ (ℂ^{∗})^{2ℓ} × Ω^{0} | x_{j}y_{j} = ω_{j}(1 ≤ j ≤ ℓ)}

and

\begin{aligned}E_{2} & := \{((x_{k}, y_{k})_{ℓ+1≤k≤n}, ω) ∈ (ℂ^{∗})^{2n−2ℓ} × Ω^{0} | x_{j}y_{j} = ω_{j}(ℓ + 1 ≤ j ≤ n); \\ & ∏_{j=ℓ+1}^{n} x_{j}^{μ_{j}^{+}}y_{j}^{μ_{j}^{−}} = ω_{n+1}; ∏_{j=ℓ+1}^{n} x_{j}^{μ_{j}^{−}}y_{j}^{μ_{j}^{+}} = ω_{n+2}\}. \end{aligned}

Let ψ_{j} : E_{j} → Ω^{0} denote the projection (j = 1, 2) and observe that

≃
ψ^{−1}(Ω^{0}) ⟶ E_{1} ×_{Ω^{0}} E_{2}
ψ ↘  ↙ ψ_{1} × ψ_{2}
Ω^{0}

For ψ_{1} : E_{1} → Ω^{0}, a holomorphic trivialization Ψ_{1} : (ℂ^{∗})^{ℓ} × Ω^{0} → E_{1} is given by

Ψ_{1}(t_{1}, ⋯, t_{ℓ}, ω) = ((t_{j}ω_{j}, \frac{1}{t_{j}} )_{1≤j≤ℓ} , ω) .

It remains to show that the bundle induced by E_{2} → Ω^{0} over B_{c} is holomorphically trivial for generic c. This is done in three steps: (S1) To define local trivializations of E_{2} → Ω^{0} introduce the fiber

𝔽 = {(z_{ℓ+1}, ⋯, z_{n}) ∈ (ℂ^{∗})^{n−ℓ} | ∏_{j=ℓ+1}^{n} z_{j}^{μ_{j}} = 1}

and recall that ρ^{(n)} = (0, ⋯, 0, ρ_{ℓ+1}^{(n)}, ⋯, ρ_{n}^{(n)}) ∈ ℤ^{n} satisfies ⟨ρ^{(n)}, μ⟩ = 1. Then Ψ′_{2} : 𝔽 × Ω′ → ψ_{2}^{−1}(Ω′) given by

Ψ′_{2}(z_{ℓ+1}, ⋯, z_{n}, ω) := ((ωk^{\frac{μ{− \atop k}}{|μ_{k}|}} z_{k}ω_{n+1}^{ρ_{k}^{(n)}}, ωk^{\frac{μ{+ \atop k}}{|μ_{k}|}} z_{k}^{−1}ω_{n+1}^{−ρ_{k}^{(n)}})_{ℓ+1≤k≤n} , ω)

is a trivialization of ψ_{2} : E_{2} → Ω^{0} above Ω′ and Ψ′′_{2} : 𝔽 × Ω′′ → ψ_{2}^{−1}(Ω′′) given by

Ψ′′_{2}(z_{ℓ+1}, ⋯, z_{n}, ω) := ((ωk^{\frac{μ{+ \atop k}}{|μ_{k}|}} z_{k}ω_{n+2}^{−ρ_{k}^{(n)}}, ωk^{\frac{μ{− \atop k}}{|μ_{k}|}} z_{k}^{−1}ω_{n+2}^{+ρ_{k}^{(n)}})_{ℓ+1≤k≤n} , ω)

is a trivialization of ψ_{2} : E_{2} → Ω^{0} above Ω′′. (Notice that Ψ′_{2} and Ψ′′_{2} are not symmetric.) The gluing map of the two trivializations, G : Ω′ ∩ Ω′′ → Aut(𝔽) is given by

G(ω) := diag (\frac{(ω_{n+1}ω_{n+2})^{ρ_{ℓ+1}^{(n)}}}{ω_{ℓ+1}} , …, \frac{(ω_{n+1}ω_{n+2})^{ρ_{n}^{(n)}}}{ω_{n}} ) (B.1)

i.e. G(ω) is a diagonal matrix. Thus ψ_{2}^{−1}(Ω^{0}) = 𝔽×Ω′ ⊔𝔽×Ω′′/ ∼, where (z′, ω) ∼ (z′′, ω) if ω ∈ Ω′∩Ω′′, (z′, ω) ∈ Ω′, (z′′, ω) ∈ 𝔽×Ω′′ and z′′_{j} = (ω_{n+1}ω_{n+2})^{ρ_{j}^{(n)}}ω_{j}^{−1}z′_{j} (ℓ+ 1 ≤ j ≤ n). (S2) In view of Lemma 3.7, we may assume that B_{c} ⊂ Ω^{0}. Consider the pullback

M_{c}^{(1)} ×_{B_{c}} M_{c}^{(2)} → B of E_{1} ×_{Ω^{0}} E_{2} → Ω^{0}. Then M_{c}^{(1)} →^{ψ_{1}} B_{c} is a trivial fibration

whose fiber is (ℂ^{∗})^{ℓ}. To investigate the fibration M_{c}^{(2)} →^{ψ_{2}} B_{c} we consider the line bundles induced by the diagonal elements of the gluing map G, introduced above. Denote by L_{j} (ℓ + 1 ≤ j ≤ n) the line bundle (ℂ × Ω′ ⊔ ℂ × Ω′′)/ ∼ above Ω^{0} where the equivalence relation is defined as follows: for (u′, ω′) ∈ ℂ × Ω′ and (u′′, ω′′) ∈ ℂ × Ω′′, (u′, ω′) ∼ (u′, ω′′) if ω′ = ω′′ ∈ Ω′ ∩ Ω′′ and u′′ = (ω_{n+1}ω_{n+2})^{ρ_{j}^{(n)}}ω_{j}^{−1}u′. We claim that the pullback L_{j};_{B_{c}} → B_{c} of the line bundles L_{j} → Ω^{0} over B_{c} are all trivial. This follows from the following three observations:

(01) The exact sequence of sheaves of holomorphic functions

O → ℤ → O ⟶^{exp} O^{∗} → O

induces a long exact sequence in cohomology

⋯ → H^{1}(B_{c}, O) → H^{1}(B_{c}, O^{∗}) → H^{2}(B_{c}, ℤ) → ⋯

(Recall that O^{∗} denotes the sheaf of holomorphic functions on B_{c} which vanish nowhere on B_{c}.)

(02) Since B_{c} is a non compact analytic variety of dimension 1, B_{c} is a Stein space and, as a consequence, H^{1}(B_{c}, O) = 0 (cf. e.g., [KK,p.224]).

(03) Since B_{c} is a Stein space and of dimension 1, it is homotopy equivalent to a real one dimensional CW-complex, hence H^{2}(B_{c}, ℤ) = 0 (cf.e.g., [GR,p.156]).

Thus we conclude that H^{1}(B_{c}, O^{∗}) = 0, or, equivalently, that any line bundle over B_{c} is trivial. Denote by Θ_{j} a trivialization of L_{j};_{B_{c}}

Θ_{j}
ℂ × B_{c} ⟶ L_{j};_{B_{c}}
pr_{2} ↘  ↙
B_{c}

and introduce B′_{c} := B_{c} ∩ Ω′, B′′_{c} := B_{c} ∩ Ω′′ and the restrictions Θ′_{j} := Θ_{j} ↾_{ℂ×B′_{c}}, Θ′′_{j} := Θ_{j} ↾_{ℂ×B′′_{c}}. Recall that L_{j} = ℂ × Ω^{0} ∩ ℂ × Ω^{0}/ ∼ and therefore there exist u′_{j} ∈ O^{∗}(B′_{c}) and u′′_{j} ∈ O^{∗}(B′′_{c}) so that

Θ′_{j}(v, ω) = (u′_{j}(ω)v, ω); Θ′′_{j}(v, ω) = (u′′_{j}(ω)v, ω).

For ω ∈ B′_{c} ∩ B′′_{c},

u′′_{j}(ω) = (ω_{n+1}ω_{n+2})^{ρ_{j}^{(n)}}ω_{j}^{−1}u′_{j}(ω). (B.2)

In the next step, the functions u′_{j}, u′′_{j} (ℓ + 1 ≤ j ≤ n) are used to construct a trivialization of M_{c} → B_{c}. (S3) The pull back M_{c} = ψ_{2}^{−1}(B_{c}) → B_{c} of ψ_{2} : E_{2} → Ω^{0} over B_{c} has a presentation of the form

ψ_{2}^{−1}(B_{c}) = 𝔽 × B′_{c} ⊔ 𝔽 × B′′_{c}/ ∼

where (z′, ω) ∈ 𝔽 × B′_{c} and (z′′, ω) ∈ 𝔽 × B′′_{c} are equivalent, (z′, ω) ∼ (z′′, ω), if ω ∈ B′_{c} ∩ B′′_{c} and z′′_{j} = (ω_{n+1}ω_{n+2})^{ρ_{j}^{(n)}}ω_{j}^{−1}z′_{j} (ℓ + 1 ≤ j ≤ n). Define Δ_{2} : ψ_{2}^{−1}(B_{c}) → (ℂ^{∗})^{n−l} × B_{c} as follows: Δ′_{2} := Δ_{2} ↾_{𝔽×B′_{c}} and Δ′′_{2} := Δ_{2} ↾_{𝔽×B′′_{c}} are given by

Δ′_{2}(z′, ω) := ((\frac{z′_{j}}{u′_{j}(ω)})_{ℓ+1≤j≤n} , ω)

Δ′′_{2}(z′′, ω) := ((\frac{z′′_{j}}{u′′_{j}(ω)})_{ℓ+1≤j≤n} , ω) .

Notice that for (z′, ω) ∼ (z′′, ω), one has ω ∈ B′_{c} ∩ B′′_{c} and

z′′_{j} = (ω_{n+1}ω_{n+2})^{ρ_{j}^{(n)}}ω_{j}^{−1}z′_{j}

and therefore,

\frac{z′′_{j}}{u′′_{j}(ω)} = \frac{(ω_{n+1}ω_{n+2})^{ρ_{j}^{(n)}}ω_{j}^{−1}u′_{j}(ω)}{u′′_{j}(ω)} \frac{z′_{j}}{u′_{j}(ω)} = \frac{z′_{j}}{u′_{j}(ω)}

where for the last equality we have used (B.2). This shows that Δ_{2} is well defined. Notice that Δ_{2} is an embedding and that the following diagram is commutative

Δ_{2}
ψ_{2}^{−1}(B_{c}) ⟶ (ℂ^{∗})^{n−ℓ} × B_{c}
↘  ↙ pr_{2}
B_{c}

In order to identify the range of Δ_{2}, range (Δ_{2}), we argue over B′_{c} and B′′_{c} separately. Let (v_{j})_{ℓ+1≤j≤n} ∈ (ℂ^{∗})^{n−ℓ} so that there exists ((z′_{j})_{ℓ+1≤j≤n}, ω) ∈

ψ_{2}^{−1}(B′_{c}) with \frac{z′_{j}}{u′_{j}(ω)} = v_{j}, or z′_{j} = v_{j}u′_{j}(ω) (ℓ + 1 ≤ j ≤ n). Substituting these

identities into ∏_{ℓ+1}^{n}(z′_{j})^{μ_{j}} = 1 leads to

(∏_{ℓ+1}^{n} v_{j}^{μ_{j}}) (∏_{ℓ+1}^{n} u′_{j}(ω)^{μ_{j}}) = 1. (B.3)

The same argument can be used over B′′_{c} to conclude that

(∏_{ℓ+1}^{n} v_{j}^{μ_{j}}) (∏_{ℓ+1}^{n} u′′_{j}(ω)^{μ_{j}}) = 1. (B.4)

Further, it follows from (B.2) that

∏_{ℓ+1}^{n} u′_{j}(ω)^{μ_{j}} = ∏_{ℓ+1}^{n} u′′_{j}(ω)^{μ_{j}}. (B.5)

and therefore we can define δ ∈ O^{∗}(B_{c}) by

δ(ω) = \{ \begin{matrix}∏_{j=ℓ+1}^{n} u′_{j}(ω)^{μ_{j}} & if ω ∈ B′_{c} \\ ∏_{j=ℓ+1}^{n} u′′_{j}(ω)^{μ_{j}} & if ω ∈ B′′_{c}. \end{matrix}

Conditions (B.3) and (B.4) can thus be expressed over all of B_{c} by

δ(ω) ∏_{ℓ+1}^{n} v_{j}^{μ_{j}} = 1

and Δ_{2} : M_{c} = ψ_{2}^{−1}(B_{c}) → {((v_{j})_{ℓ+1≤j≤n}, ω) ∈ (ℂ^{∗})^{n−ℓ} × B_{c} | δ(ω) ∏_{ℓ+1}^{n} v_{j}^{μ_{j}} = 1} is a bianalytic isomorphism. (S4) To find a local trivialization of Δ_{2}(M_{c}), define

T_{2} : Δ_{2}(M_{c}) → { ((ṽ_{j})_{ℓ+1≤j≤n}, ω) ∈ (ℂ^{∗})^{n−ℓ} × B_{c} | ∏_{ℓ+1}^{n} ṽ_{j}^{μ_{j}} = 1}

by setting T_{2}((v_{j})_{ℓ+1≤j≤n}, ω) = ((δ(ω)^{−ρ_{j}^{(n)}}v_{j})_{ℓ+1≤j≤n}, ω). Then

∏_{ℓ+1}^{n} ṽ_{j}^{μ_{j}} = (∏_{ℓ+1}^{n} v_{j}^{μ_{j}}) δ(ω)^{−Σμ_{j}ρ_{j}^{(n)}} = δ(ω)\frac{1}{δ(ω)} = 1.

We have thus shown that T_{2} · Δ_{2} : M_{c} = ψ_{2}^{−1}(B_{c}) → B_{c} × 𝔽 is a bianalytic isomorphism. □

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