ELECTRONIC RESEARCH ANNOUNCEMENTS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 5, Pages 55–65 (April 30, 1999) S 1079-6762(99)00061-X

TANGENTIAL HILBERT PROBLEM FOR PERTURBATIONS OF HYPERELLIPTIC HAMILTONIAN SYSTEMS

D. NOVIKOV AND S. YAKOVENKO

(Communicated by Jeff Xia)

Abstract. The tangential Hilbert 16th problem is to place an upper bound for the number of isolated ovals of algebraic level curves {H(x, y) = const} over which the integral of a polynomial 1-form P (x, y) dx + Q(x, y) dy (the Abelian integral) may vanish, the answer to be given in terms of the degrees n = deg H and d = max(deg P, deg Q).

We describe an algorithm producing this upper bound in the form of a primitive recursive (in fact, elementary) function of n and d for the particular case of hyperelliptic polynomials H(x, y) = y^{2} + U(x) under the additional assumption that all critical values of U are real. This is the first general result on zeros of Abelian integrals that is completely constructive (i.e., contains no existential assertions of any kind).

The paper is a research announcement preceding the forthcoming complete exposition. The main ingredients of the proof are explained and the differential algebraic generalization (that is the core result) is given.

1. Tangential Hilbert problem and bounds for the number of limit cycles in perturbed Hamiltonian systems

1.1. Complete Abelian integrals and the tangential Hilbert Sixteenth problem. Integrals of polynomial 1-forms over closed ovals of real algebraic curves, called (complete) Abelian integrals, naturally arise in many problems of geometry and analysis, but probably the most important is the link to the bifurcation of limit cycles of planar vector fields and the Hilbert Sixteenth problem. Recall that the question originally posed by Hilbert in 1900 was on the maximal number of limit cycles a polynomial vector field of degree d on the plane may have. This problem is still open even in the local version, for systems ε-close to integrable or Hamiltonian ones. However, there is a certain hope that the “linearized”, or tangential Hilbert 16th problem can be more treatable.

Consider a polynomial perturbation of a Hamiltonian polynomial vector field

ẋ = −\frac{∂H}{∂y} − εQ(x, y),  ẏ = \frac{∂H}{∂x} + εP (x, y). (1.1)

An oval γ of the level curve H(x, y) = h which is a closed (but nonisolated) periodic trajectory for ε = 0, may generate a limit cycle for small nonzero values of ε only

\begin{matrix}c \\ ◯ \end{matrix}1999 American Mathematical Society

Received by the editors October 23, 1998.

1991 Mathematics Subject Classification. Primary 14K20, 34C05, 58F21; Secondary 34A20, 30C15.

if the accumulated energy dissipation is zero in the first approximation, i.e., when

0 = ∮_{γ} P (x, y) dx + Q(x, y) dy, γ ⊆ {H(x, y) = h}. (1.2)

The expression on the right hand side of (1.2) is a complete Abelian integral, and assuming the polynomials H, P, Q fixed, it is a function I = I(h) of the value h ∈ ℝ, in general multivalued if the corresponding level curve contains several real ovals. The value I(h) is the first variation of the Poincaré return map for the system (1.1) with respect to the parameter ε, computed in the chart h at ε = 0.

Thus the linearized or tangential Hilbert problem arises (see [1, problems by V. Arnold] for a recent reference): for any collection of polynomials H, P, Q ∈ ℝ[x, y] of degree ⩽ d give an upper bound for the number of real ovals γ over which the integral (1.2) vanishes, but not identically (in the latter case the perturbation (1.1) is conservative in the first approximation, and higher variations must be considered). The bound should be given in terms of d only, in other words, it must be uniform over all combinations of polynomials of admissible degrees.

1.2. Hyperelliptic case. A very important particular case is the hyperelliptic one, when H(x, y) = \frac{1}{2}y^{2} + U(x), U ∈ ℝ[x], deg U = d ⩾ 5: in this case the level curves are hyperelliptic (rational for d = 1, 2, elliptic for d = 3, 4). The polynomial U in this case will be always referred to as the potential, since such Hamiltonian systems correspond to the Newtonian system ẍ = −\frac{∂U}{∂x} describing a free particle in the potential field in one degree of freedom. The integral (1.2) in this case is called a hyperelliptic integral. The tangential Hilbert problem restricted to the hyperelliptic case, was studied in many papers including [4, 23].

1.3. Background. For low degree Hamiltonians (3 or 4) there are numerous results on the number of zeros for special choices of H, many of them sharp, which will not be discussed here: we note only that the elliptic case corresponding to H(x, y) = \frac{1}{2}y^{2} + \frac{1}{3}x^{3} − x was completely investigated by G. Petrov [17], while in the case of an arbitrary cubic H a linear bound 5d + 15 for the number of zeros of I(h) was obtained by E. Horozov and I. Iliev [5]. However, the general results that would be valid for Hamiltonians of arbitrarily high degrees, are much scarcer and substantially less explicit. Perhaps the only known completely explicit general result is an upper bound for multiplicity of an isolated zero of an Abelian integral. This bound (polynomial in d) is due to P. Mardešić [13], who proved it using the approach suggested by Yu. Ilyashenko [6].

A. Khovanskiĭ in [10] and A. Varchenko in [24] proved that for any fixed d the number of isolated zeros of Abelian integrals is uniformly bounded over all Hamiltonians and forms of degree ⩽ d. The assertion of the Khovanskiĭ–Varchenko theorem is purely existential: it gives absolutely no information on how the bound may depend on d.

A simpler problem arises if we fix H and consider integrals of 1-forms ω of increasing degrees d = deg ω = max(deg P, deg Q), looking for an asymptotic bound for the number of zeros as d → ∞. In this direction a considerable progress was recently achieved: assuming the Hamiltonian H be sufficiently generic, Yu. Ilyashenko and S. Yakovenko obtained a double exponential in d upper bound for the number of isolated zeros at a positive distance from the critical values of H [7]. Almost immediately D. Novikov and S. Yakovenko improved this result, reducing the bound to a single exponent and making it uniform over all real regular values: the number of real isolated roots of the Abelian integral does not exceed exp(Cd), where C = C_{NY}(H) < +∞ is a finite constant depending only on the Hamiltonian [14]. The description of C_{NY}(H) can be done in geometric terms [7], but the bound blows up to infinity as H approaches the boundary of the open set of Morse polynomials.

Finally A. Khovanskiĭ and G. Petrov proved^{1} that the number of isolated zeros may grow at most as B(n)d + C_{KP}(n), where B(n) is an explicit expression double exponential in n = deg H, while C_{KP}(n) is a finite constant that depends on n. However, this dependence is purely existential, as before.

Summarizing this brief synopsis, we conclude that today no completely effective upper bound is known that would serve Hamiltonians of an arbitrarily high degree.

1.4. Solution of the tangential Hilbert problem for hyperelliptic Hamiltonians. In this announcement we claim a constructive upper bound for the number of zeros of hyperelliptic integrals under the additional assumption that all critical values of the potential U(x) are real. There are several broadly used classes of constructive functions, among them effective (algorithmically computable), primitive recursive (defined by a finite number of inductive rules) and elementary functions, see [12]. Our main theorem asserts the strongest form of computability of the upper bound as a function of two natural values n, d.

Theorem 1. For any real polynomial U(x) ∈ ℝ[x] of degree n+1 and any differential form ω = P dx+Q dy of degree d, the number of real ovals γ ⊂ {y^{2} +U(x) = h} yielding an isolated zero of the integral ∮_{γ} ω, is bounded by a primitive recursive (in fact, elementary) function B(n, d) of two integer variables d and n, provided that all critical values of U are real.

A closer inspection of the algorithm proving Theorem 1 suggests that the function B(n, d) grows no faster than a certain tower function (iterated exponent) of height 5 or perhaps 6. In any case, this bound is too excessive to believe that it might be realistic: this is the main reason why we never tried to write it explicitly.

2. H-fields and their polynomial-like property

The proof of Theorem 1 goes by induction on n, but the induction step requires introducing more general classes of functions than ordinary hyperelliptic integrals. In other words, Theorem 1 is obtained as a corollary to a more general Theorem 2 concerning complex zeros of analytic functions from certain Picard–Vessiot extensions [8] of the field of rational functions ℂ(t) by one or several hyperelliptic integrals.

2.1. Analytic continuation. Abelian integrals (1.2) admit analytic continuation as multivalued functions of a complex argument t, ramified over a finite set of points Σ = {t_{1}, …, t_{μ}} ⊂ ℂ and eventually t_{0} = ∞. Generically (and always in the hyperelliptic case), Σ consists of critical values of H, and the monodromy group of this extension does not depend on the integrand ω. This implies that an arbitrary Abelian integral can be represented as a linear combination of a finite number of integrals of some 1-forms ω_{k} with coefficients from ℂ(t) [25]. These forms can be explicitly described and the degrees of the coefficients bounded [3], but in the hyperelliptic case the situation becomes completely transparent and all computations explicit.

1The proof of this result was published in [17] for a hyperelliptic polynomial H = \frac{1}{2} y^{2} + U(x)

under the assumption that all critical points of the potential are real, but it can be generalized for all Morse Hamiltonians by several simple though nonobvious reductions.

2.2. Basic hyperelliptic integrals and Picard–Vessiot field extensions. Let U(x) = x^{n+1} + ⋯ be a monic (i.e., with leading coefficient 1) potential of degree n + 1, and ω_{k} = x^{k−1}y dx, k = 1, …, n, differential 1-forms that constitute the basis of cohomology of each nonsingular hyperelliptic complex level curve ϕ_{t} = \{\frac{1}{2}y^{2}+U(x) = t\} ⊂ ℂ^{2}, t ∈ ℂ. Define the (complete collection of) basic hyperelliptic integrals J_{kj}(t) as integrals of the forms ω_{k} over vanishing cycles δ_{j}(t) ∈ H_{1}(ϕ_{t}, ℤ), see [2], “growing” from the critical values t_{j}:

J_{kj}(t) = ∮_{δ_{j}} ω_{k}, δ_{j} = δ_{j}(t) ⊆ ϕ_{t}, diam δ_{j}(t)|_{t→t_{j}} → 0. (2.1)

Together they constitute a nondegenerate n × n-matrix J = J (t), analytically depending on t ∈ ℂ outside the critical locus Σ. This matrix function satisfies a Picard–Fuchs system [2] of first order linear differential equations (3.1) with rational coefficients, and the result of analytic continuation of J (t) along any loop in ℂ ∖ Σ is described by the Picard–Lefschetz formulas [2].

Lemma 1 (cf. [3, 25]). An arbitrary hyperelliptic Abelian integral belongs to the field 𝕜_{U} = ℂ(t)(J_{11}, …, J_{nn}), the extension of the rational functions field ℂ(t) by the basic hyperelliptic integrals J_{kj} = J_{kj}(t), j, k = 1, …, n, defined as in (2.1).

The field 𝕜_{U} completely determined by the potential U is the field of multivalued analytic functions of complex argument t, analytically continuable along any path in ℂ ∖ Σ (this construction will be further generalized in Definition 1 below). Alternatively one can describe 𝕜_{U} in purely algebraic terms as a differential field, the Picard–Vessiot extension of ℂ(t) by solutions of a linear system (3.1), adding all entries of any fundamental matrix solution to the latter. However, the system of generators J = {J_{kj}} is distinguished for many reasons.

The number n = deg U − 1 will be referred to as the gender of the field 𝕜_{U}. To define unambiguously arithmetic operations with multivalued functions, we

choose a base point t_{∗} and a collection of simple nonintersecting (except at t_{∗}) paths connecting t_{j} with t_{∗}. The integrals J_{kj}(t), originally defined only as germs at t = t_{j} [2], can be continued along these paths and define a collection of germs of analytic functions at t = t_{∗} denoted again by J_{kj}. Then the field operations in 𝕜_{U} can be identified with arithmetic operations on germs. Analytic continuation along loops attached to t_{∗}, constitute the group Mon(𝕜_{U}) of monodromy (differential) automorphisms of the field.

Each element of 𝕜_{U} can be written as a ratio of two H-polynomials, each of the form p = ∑_{k+|α|⩽d} c_{kα} t^{k}J^{α} ∈ ℂ[t, J] = ℂ[t, J_{11}, …, J_{nn}], where α = ∥α_{kj}∥ ∈ ℤ_{+}^{n^{2}} is a multiindex, J^{α} = ∏_{j,k=1}^{n} J_{kj}^{α_{kj}}, and d the degree of the H-polynomial p. The degree of an arbitrary H-function p/q ∈ 𝕜_{U} is max(deg p, deg q), as usual, and it is preserved by monodromy transformations by virtue of Picard–Lefschetz formulas.

2.3. General definition of H-fields. For our purposes (mainly for Lemma 3 below) we need a more general object, extension of ℂ(t) by adding hyperelliptic integrals associated with several different potentials.

Definition 1. An H-field^{2} 𝕜_{U_{1},...,U_{ν}} associated with a collection of ν Morse polynomial potentials U_{1}, …, U_{ν} ∈ ℝ[x] of degrees n_{1} + 1, …, n_{ν} + 1, is the extension of the field ℂ(t) by the complete collection of n_{1}^{2} + ⋯ + n_{ν}^{2} hyperelliptic integrals associated with each potential U_{1}, …, U_{ν}.

The gender of the H-field 𝕜_{U_{1},...,U_{ν}} is the sum n = n_{1} + ⋯ + n_{ν}. The critical locus Σ ⊂ ℂ, generically consisting of n points, is the union {t_{1}, …, t_{n}} of critical values of all respective potentials U_{s}.

As in the case of a single potential, one can fix settings (the base point, system of paths, etc.) so that each element of the H-field will be associated with a unique algebraic expression and the degree of elements is well defined.

2.4. Theorem on zeros for H-fields. We prove that under the additional assumption that Σ ⊂ ℝ, i.e., that all critical values of all potentials are real, the H-fields possess the property that makes them similar to the field of rational functions: the number of complex isolated zeros of any H-function admits an upper bound in terms of its degree and gender.

If Σ ⊂ ℝ, then we can assume that the critical points are ordered, −∞ < t_{1} < ⋯ < t_{n} < +∞. They subdivide the real axis into n + 1 finite or semiinfinite intervals ℓ_{j}. Theorem 2 places an upper bound for the number of real (i.e., on ℓ_{j}) and complex (in the upper or lower half-planes) zeros of any H-function: a simply connected domain when zeros are counted, needs to be specified because of the multivaluedness of H-functions.

Theorem 2 (main). There exists a primitive recursive (in fact, elementary) function B(n, d) such that the number of complex isolated zeros of any H-function f ∈ 𝕜_{U_{1},...,U_{ν}} of gender n and degree d in the upper or lower half-planes {± Im t > 0} and on each real interval ℓ_{j} can be at most B(n, d), provided that all critical values of all potentials are real.

Reduction from Theorem 2 to Theorem 1 is provided by Lemma 1. We believe that the assumption Σ ⊂ ℝ is technical, but for the moment it cannot be dropped.

The function B(n, d) is determined by the algorithm given in the proof. In principle and if necessary, its growth rate for large n, d can be estimated by a closer inspection of the algorithm. Note that the bound is uniform over all combinations of potentials generating H-fields with the same gender, and over all values of coefficients of H-functions of a given degree.

3. The structure and main ingredients of the proof

3.1. Preliminary normalization. For any given gender n the H-fields of this gender are parameterized by a combinatorial invariant (a partition of n describing how many different potentials of each degree were used) and, as soon as the partition is fixed, by the strings of coefficients of all potentials U_{s}(x). The latter can be to a certain extent resized: using affine transformations x ↦ λ_{s}x+λ′_{s} with λ_{s}, λ_{s} ∈ ℂ, it is possible to normalize the string of collections of each potential U_{s} independently. Besides, one can make a change of the “independent variable” t ↦ μt + μ′ with μ, μ′ ∈ ℂ, common for all potentials. Using these transformations, one can achieve the following: (a) the coefficients of all potentials are explicitly bounded; (b) the roots of all potentials are in the unit disk; (c) the overall critical locus Σ is centered at 0, so that ∑_{j=1}^{n} t_{j} = 0, and does not shrink too much or stretch to infinity: max_{i≠j} |t_{i} − t_{j}| = 1. Clearly, these transformations cannot affect any bound on the number of zeros.

2We would like to use the expression “hyperelliptic field” instead of the abbreviation “H-field”, but the former term is already in use (though not very common). On the other hand, it would be certainly inadmissible to call elements of an H-field “hyperelliptic functions”, since the latter name is firmly attached to functions of a different class. Thus we had to decide between at least three-word-long term and an abbreviation.

We refer to such H-fields as (properly) resized, introducing at the same time the notion of resized H-polynomials, H-functions, etc.

3.2. Picard–Fuchs system for hyperelliptic integrals and variation of argument along arcs distant from the singular locus. The multivalued matrixvalued function J (t) formed by basic hyperelliptic integrals (2.1) satisfies a linear system of ordinary differential equations of the form

(t + A)J̇(t) = BJ (t), A, B ∈ Mat_{n×n}(ℂ), (3.1)

where A, B are constant matrices depending only on the potential U. The general form of (3.1) was established in [4] from geometric considerations, and in [19] the system (3.1) was derived by elementary arguments allowing for explicit description of the matrices in terms of the potential.^{3} In particular, det(t + A) = ∏_{j=1}^{n}(t − t_{j}) (the product is taken over all critical values of the potential U), and the norms ∥A∥, ∥B∥ are bounded in terms of n if the potential is properly resized.

For the case of a general H-field associated with a collection of potentials {U_{s}} a similar system can be written for each potential and, after passing to symmetric products, for the collection of all H-monomials of degree ⩽ d for any particular d:

Δ(t) · \frac{d}{dt} (t^{k}J^{α}) = ∑_{|β|⩽d} A_{k,α,β}(t) J^{β}, ∀ k + |α| ⩽ d, (3.2)

where A_{k,α,β} ∈ ℂ[t] are polynomials of explicitly bounded (in terms of n and d) degrees with bounded coefficients, and Δ(t) = ∏_{t∈Σ}(t − t_{j}) is the product taken over the union of all critical values of all potentials U_{s}.

In other words, any H-polynomial of a known degree d and gender n can be written as a linear combination of coordinate functions restricted to a certain trajectory of the polynomial (more precisely, rational) vector field (3.2) in the affine space of appropriate dimension (depending on d and n). The main result of [15, 16] applied to the system (3.2) in combination with [26, Corollary 2.7] yields the following property of resized H-fields.

Lemma 2. Variation of argument of any resized H-function of degree d and gender n along any arc γ ⊂ ℂ ∖ Σ admits an explicit upper bound in terms of n, d, and geometry of the arc γ (its length |γ| and the distance from γ to Σ, measured by inf_{t∈γ} |Δ(t)| > 0).

Variation of argument of any such function along any sufficiently small circular arc around any singular point t_{j} ∈ Σ or t_{0} = ∞ is bounded from above by an explicit expression involving only d and n.

The bound provided by Lemma 2 is already given (assuming γ for simplicity at the distance 1 from Σ) by a tower function of height 4 in the variables n and d [15]. This explains why any bound based on using the lemma, must be very large.

3Note added in proof : the demonstration from the Thesis [19] recently appeared in the book [22, pp. 83–84].

3.3. Clusterization. The general principle established in Lemma 2, immediately implies an upper bound for the number of isolated zeros on any compact simply connected subset of ℂ∖Σ, by virtue of the argument principle. To extend this result for zeros arbitrarily close to ramification points, additional efforts are required.

However, the assumption that the H-field is already resized, makes it possible to break the critical locus into at least two parts with a sufficient spacing between them and also distant from infinity, which is another ramification point. Covering each part by a convex simply connected domain D_{j}, called cluster, with the boundary at a controlled distance from all other singularities, splits the problem of zeros into that for each cluster separately, for a neighborhood D_{∞} of infinity and for the complement C = ℂ ∖ (D_{∞} ∪ D_{1} ∪ D_{2}). The bound for zeros in C follows from Lemma 2 and the argument principle (being multiply connected, C should be further split into simply connected pieces without singularities inside). Thus it is the problem for a single separate cluster that has to be considered.

Zeros near infinity (inside the cluster D_{∞}) can be counted combining the main result of [20] with that from [15]: this works in fact for any cluster with only one singularity inside. The arguments briefly described below, show how the ideas of [20] can be generalized to cover the case of several ramification points. Very roughly, one has to find a system of functions that after restriction to a cluster would have the same monodromy as the hyperelliptic integrals, but be in some sense simpler. The solution is to consider a full collection of hyperelliptic integrals associated with an appropriate potential of inferior degree (i.e., smaller gender): then one can proceed by induction using the construction from [17]. The latter is briefly explained in the following section.

3.4. Argument principle after Petrov. Suppose we have an H-function f with ramification locus Σ on the real axis. Consider a symmetric (with respect to ℝ) domain Ω ⊂ ℂ ∖ Σ formed by cutting the cluster (disk) D = \overline{Ω} along two rays emanating from two adjacent singular points in the opposite directions; see Figure 1. Since Ω is simply connected, f extends as a single-valued function analytic in Ω. To majorize the number of zeros of f in this domain, we apply the argument principle. Assuming H-field to be resized and the exterior arc γ_{∗} distant from Σ, the variation of argument along the arcs γ_{j}, γ_{∗} is explicitly bounded by Lemma 2. It remains to majorize the variation of argument of f along the upper and lower edges of the real intervals σ_{j}^{±} between the singular points t_{j} and t_{j+1}.

Let σ be one such edge. It was an observation made by G. Petrov in [18], that variation of argument of an analytic function f along a connected curve is at most π times the number of roots of Im f on that curve plus 1, since between any two consecutive roots of Im f there, variation of argument of f can be at most π. In other words, to majorize the number of zeros of f in Ω, it is sufficient to majorize the number of zeros of Im_{σ} f = Im(f|_{σ}) on every interval σ that is a part of ∂Ω.

To apply this argument recursively, one has to restore the settings and extend the imaginary parts analytically from their respective edges of definition to obtain germs at t_{∗}. One can easily show that for an H-function from the field 𝕜 = 𝕜_{U_{1},...,U_{ν}} all these extensions will again belong to the same field, i.e. there are well-defined maps Im_{σ} 𝕜 → 𝕜 for all σ = σ_{j}^{±}. This follows, e.g., from the fact that the matrices A, B occurring in the system (3.1), are real.

γ_{∗} Γ t_{j}  γ_{j} σ Ω  ℝ

Figure 1. The cluster and its boundary.

The above construction reduces the problem of zeros for one H-function f to that for several other functions Im_{σ_{j}^{±}} f associated with all real edges σ_{j}^{±} ⊂ ∂Ω. The gain occurs if these new H-functions are simpler than the original one.

3.5. D-inner subfield and D-restricted monodromy. The choice of a cluster D introduces an asymmetry between the critical values t_{j}, the respective vanishing cycles and hence between the basic integrals J_{kj} generating the H-field.

Assume that the base point t_{∗} used to identify elements of the Picard–Vessiot extension with germs, belongs to D together with the paths connecting it with all “interior” singularities t_{j} ∈ D (the paths connecting t_{∗} with the “outer” singular points outside D, can be arbitrary).

Definition 2. The D-restricted monodromy (sub)group Mon^{D}(𝕜), 𝕜 = 𝕜_{U_{1},...,U_{ν}}, is a subgroup of the full monodromy group Mon(𝕜) formed by analytic continuation over loops entirely belonging to the cluster D. The dual object is the D-inner subfield 𝕜^{D} = 𝕜_{U_{1},...,U_{ν}}^{D} invariant under all transformations from Mon^{D}(𝕜). Alternatively, this subfield can be described as the extension of ℂ(t) by the integrals J_{kj} over the cycles vanishing only at the inner points t_{j} ∈ D.

Lemma 3. There exists an H-field 𝕜_{V_{1},...,V_{μ}} of gender m < n and a collection of germs r_{1}, …, r_{ρ}, ρ = nm, each invariant under the restricted monodromy (i.e., extendable to single-valued meromorphic functions in D), such that

𝕜_{U_{1},...,U_{ν}}^{D} = 𝕜_{V_{1},...,V_{μ}}(r_{1}, …, r_{ρ}).

The collection (r_{1}, …, r_{ρ}) satisfies a system of first order linear ordinary differential equations with bounded rational coefficients, similar to (3.2).

This lemma is an analytic corollary of the fact that the restricted monodromy of the inner subfield is the same as the (unrestricted) monodromy of an appropriate H-field of a smaller gender, associated with a collection of potentials V_{1}, …, V_{μ}. The proof of this corollary uses Thom’s theorem on versal deformations and the Lyashko–Looijenga theorem; see [11]. Without loss of generality the functions r_{k} can be assumed real on the real axis, being replaced by r_{k}(t)+\overline{r_{k}(\overline{t})} and i^{−1}(r_{k}(t)− \overline{r_{k}(\overline{t})}).

3.6. Depression of gender for D-inner subfield. The Petrov construction allows us to reduce the problem of zeros for the inner subfield 𝕜^{D} to that for the “model” H-field 𝕜_{V} = 𝕜_{V_{1},...,V_{μ}}. Assume that the number of complex zeros (and poles, if necessary) of any function g from the latter field is already known to be computable in terms of deg g. By Lemma 3, any H-function f from 𝕜^{D} can be written as a combination f = ∑_{j} h_{j}g_{j} with h_{j} ∈ ℂ(r_{1}, …, r_{ρ}) real on D ∩ ℝ and g_{j} ∈ 𝕜_{V}, involving a finite (controllable in terms of d = deg f and n) number of terms.

Applying the Petrov construction to the function f/g_{1} = h_{1} + ∑_{j⩾2} h_{j}g_{j}/g_{1}, we see that for any real interval σ ⊂ ∂Ω the imaginary part Im_{σ} f/g_{1} is the sum ∑_{j⩾2} h_{j}g̃_{j,σ} with g̃_{j,σ} = Im_{σ}(g_{j}/g_{1}) ∈ 𝕜_{V}, which contains fewer terms than h. Note that each division by an element from 𝕜_{V} affects the number of zeros in a controllable way by the induction assumption. Iterating this step, one can reduce the question of zeros for f to that for finitely many functions from the H-field 𝕜_{V} of strictly inferior gender m.

3.7. Reduction from outer to inner polynomials. A similar slightly more elaborate construction is used to make the last step and reduce the problem from the entire H-field 𝕜 = 𝕜_{U_{1},...,U_{ν}} to its D-inner subfield 𝕜′ = 𝕜_{U_{1},...,U_{ν}}^{D}; in fact, it is sufficient to count zeros inside the cluster D for elements of 𝕜′[Z], where by Z = (Z_{1}, …, Z_{τ}), τ = n^{2} − nm, we denoted the collection of outer (noninner) basic hyperelliptic integrals J_{kj} (2.1). Moreover, changing if necessary the system of generators Z, one can always assume that they are real on some inner interval σ_{∗} ⊂ D. The assertion below follows from the triangular form of the Picard– Lefschetz formulas and the fact that the real and imaginary parts of each monomial Z^{α} can be obtained by linear combinations of analytic continuations along loops inside the cluster.

Lemma 4. For any real segment σ ⊂ ∂Ω and any monomial Z^{α} in outer variables,

(3.3) Re_{σ} Z^{α} = Z^{α} + p_{1}, Im_{σ} Z^{α} = p_{2}, where p_{1,2} ∈ 𝕜′[Z], deg_{Z} p_{j} < |α|.

Now we can easily explain the last reduction. For a polynomial p = ∑_{|α|⩽d} c_{α}Z^{α}, c_{α} ∈ 𝕜′, we consecutively divide p by the coefficients c_{α} with |α| = d (which makes a controllable change in the number of zeros) and then apply the Petrov construction amounting to taking imaginary parts. After a finite number of steps all leading terms of degree d in the variables Z_{k} will be eliminated by virtue of (3.3). The construction provides the inductive step for induction in d = deg_{Z} p.

3.8. Concluding remarks. The alternating division by a function and subsequent differentiation with application of the Rolle lemma is a common tool in bounding the number of zeros of real functions; see [21]. The Petrov principle may be thus seen as an analogue of the Rolle lemma for the operators Im_{σ} rather than for \frac{d}{dt}; see [20].

The only place where we do not know how to get rid of the assumption that all critical points of the potentials are real, is Lemma 4: all other steps of the proof can be easily modified to cover the general case.

However, it would be interesting to notice that using completely different methods, A. Givental in [4] established a certain Lagrangian nonoscillation for the system (3.1), while R. Schaaf proved that the “hyperelliptic” integral ∮ dx/y^{3} has no real zeros under the same assumption on the critical points. For the moment it seems a pure coincidence that the same condition reappeared in Theorem 1.

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The papers [7, 14, 15, 16, 19, 20, 25, 26] are available starting from the URL indicated below.

Laboratoire de Topologie, Université de Bourgogne, Dijon, France

E-mail address: novikov@topolog.u-bourgogne.fr

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

URL: http://www.wisdom.weizmann.ac.il/~yakov/index.html
E-mail address: yakov@wisdom.weizmann.ac.il