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\begin{document}

\title[TANGENTIAL HILBERT PROBLEM]
{Tangential Hilbert problem for
perturbations of hyperelliptic Hamiltonian systems}

\author{D. Novikov}
\address{Laboratoire de
Topologie, Universit\'e de Bourgogne, Dijon, France}
\email{novikov@topolog.u-bourgogne.fr}

\author{S. Yakovenko}
\address{Department of Theoretical Mathematics,
The Weizmann Institute of Science, Rehovot, Israel}
\urladdr{http://www.wisdom.weizmann.ac.il/\char'176 yakov/index.html}
\email{yakov@wisdom.weizmann.ac.il}

\issueinfo{5}{08}{}{1999}
\dateposted{April 30, 1999}
\pagespan{55}{65}
\PII{S 1079-6762(99)00061-X}
\def\copyrightyear{1999}
\copyrightinfo{1999}{American Mathematical Society}
\subjclass{Primary 14K20, 34C05, 58F21; Secondary 34A20, 30C15}

\date{October 23, 1998}

\commby{Jeff Xia}


\begin{abstract}
The tangential Hilbert 16th problem is to place an upper bound for
the number of isolated ovals of algebraic level curves
$\{H(x,y)=\operatorname{const}\}$ 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.
\end{abstract}

\maketitle


\section{Tangential Hilbert problem and bounds for the number \\ of limit
cycles in perturbed Hamiltonian systems}

\subsection{Complete Abelian integrals and the tangential Hilbert
Sixteenth problem} Integrals of polynomial 1-forms over closed ovals
of real algebraic curves, called (complete) {\em 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 $\e$-close to integrable or Hamiltonian
ones. However, there is a certain hope that the ``linearized'', or
{\em tangential\/} Hilbert 16th problem can be more treatable.


Consider a polynomial perturbation of a Hamiltonian polynomial
vector field
\begin{equation}\label{pertgen}
\dot x=-\pd Hy -\e Q(x,y),\qquad \dot y=\pd Hx+\e P(x,y).
\end{equation}
An oval $\gamma$ of the level curve $H(x,y)=h$ which is a closed (but
nonisolated) periodic trajectory for $\e=0$, may generate a limit
cycle for small nonzero values of $\e$ only if the accumulated energy
dissipation is zero in the first approximation, i.e., when
\begin{equation}\label{aigen}
0=\oint_\gamma P(x,y)\,dx+Q(x,y)\,dy,
\qquad\gamma\subseteq\{H(x,y)=h\}.
\end{equation}
The expression on the right hand side of \eqref{aigen} is a {\em
complete Abelian integral\/}, and assuming the polynomials $H,P,Q$
fixed, it is a function $I=I(h)$ of the value $h\in\R$, in general
multivalued if the corresponding level curve contains several real
ovals. The value $I(h)$ is the first variation of the Poincar\'e
return map for the system \eqref{pertgen} with respect to the
parameter $\e$, computed in the chart $h$ at $\e=0$.

Thus the linearized or {\em tangential\/} Hilbert problem arises
(see \cite[problems by V.~Arnold]{problems} for a recent
reference): {\em for any collection of polynomials
$H,P,Q\in\R[x,y]$ of degree $\leqslant d$ give an upper bound for the
number of real ovals $\gamma$ over which the integral
\eqref{aigen} vanishes, but not identically\/} (in the latter case
the perturbation \eqref{pertgen} is {\em 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.

\subsection{Hyperelliptic case}
A very important particular case is the \textit{hyperelliptic\/} one,
when $H(x,y)=\tfrac12 y^2+U(x)$, $U\in\R[x]$, $\deg U=d\geqslant 5$: 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 {\em potential\/}, since such
Hamiltonian systems correspond to the Newtonian system $\ddot
x=-\pd Ux$ describing a free particle in the potential field in
one degree of freedom. The integral \eqref{aigen} in this case is
called a {\em hyperelliptic integral\/}. The tangential Hilbert
problem restricted to the hyperelliptic case, was studied in many
papers including \cite{givental,schaaf}.

\subsection{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 $H(x,y)=\tfrac12y^2+\tfrac13 x^3-x$
was completely investigated by G.~Petrov \cite{petrov}, 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
\cite{horozov-iliev-non}. 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
{\em multiplicity\/} of an isolated zero of an Abelian integral.
This bound (polynomial in $d$) is due to P.~Marde\v si\'c
\cite{mardesic}, who proved it using the approach suggested by
Yu.~Ilyashenko \cite{il-petrovsk}.



A.~Khovanski\u\i\ in \cite{khovan-fan} and A.~Varchenko in
\cite{varchenko} proved that {\em for any fixed $d$ the number of
isolated zeros of Abelian integrals is uniformly bounded over all
Hamiltonians and forms of degree $\leqslant d$}. The assertion of the
Khovanski\u\i--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 $\omega$ of increasing degrees $d=\deg\omega=\max(\deg P,\deg
Q)$, looking for an asymptotic bound for the number of zeros as
$d\to\infty$. 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 $H$ \cite{doublexp}. 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 $C=C_{\text{NY}}(H)<+\infty$ is a finite
constant depending only on the Hamiltonian \cite{simplexp}. The
description of $C_{\text{NY}}(H)$ can be done in geometric terms
\cite{doublexp}, but the bound blows up to infinity as $H$ approaches
the boundary of the open set of Morse polynomials.

Finally A.~Khovanski\u\i\ and G.~Petrov proved\footnote{The proof of
this result was published in [17] for a hyperelliptic
polynomial $H=\tfrac12 y^2+U(x)$ 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.} that the number of isolated zeros may grow at most as
$B(n)d+C_{\text{KP}}(n)$, where $B(n)$ is an explicit expression
double exponential in $n=\deg H$, while $C_{\text{KP}}(n)$ 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.





\subsection{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 {\em effective\/}
(algorithmically computable), {\em primitive recursive\/} (defined by
a finite number of inductive rules) and {\em elementary\/} functions,
see \cite{manin}.  Our main theorem asserts the {\em strongest form of
computability\/} of the upper bound as a function of two natural
values $n,d$.

\begin{Thm}\label{aithm}
For any real polynomial $U(x)\in\R[x]$ of degree $n+1$ and any
differential form $\omega=P\,dx+Q\,dy$ of degree $d$, the number of
real ovals $\gamma\subset\{y^2+U(x)=h\}$ yielding an isolated zero of
the integral $\oint_\gamma\omega$, is bounded by a primitive recursive
\textup{(}in fact, elementary\textup{)}
 function $B(n,d)$ of two integer variables $d$
and $n$, provided that all critical values of $U$ are real.
\end{Thm}

A closer inspection of the algorithm proving Theorem \ref{aithm}
suggests that the function $B(n,d)$ grows no faster than a certain
{\em 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.

\section{H-fields and their polynomial-like property}

The proof of Theorem \ref{aithm} goes by induction on $n$, but the
induction step requires introducing more general classes of functions
than ordinary hyperelliptic integrals. In other words, Theorem \ref{aithm}
is obtained as a corollary to a more general Theorem \ref{main} concerning
{\em complex\/} zeros of analytic functions from certain {\em
Picard--Vessiot extensions\/} \cite{kaplansky} of the field of
rational functions $\C(t)$ by one or several hyperelliptic integrals.


\subsection{Analytic continuation}
Abelian integrals \eqref{aigen} admit analytic continuation as
multivalued functions of a complex argument $t$, ramified over a
finite set of points $\Sigma=\{t_1,\dots,t_\mu\}\subset\C$ and
eventually $t_0=\infty$. Generically (and always in the hyperelliptic
case), $\Sigma$ consists of {\em critical values\/} of $H$, and the
monodromy group of this extension does not depend on the integrand
$\omega$. This implies that an arbitrary Abelian integral can be
represented as a linear combination of a finite number of integrals of
some 1-forms $\omega_k$ with coefficients from $\C(t)$
\cite{nonlin-94}. These forms can be explicitly described and the
degrees of the coefficients bounded \cite{gavrilov}, but in the
hyperelliptic case the situation becomes completely transparent and
all computations explicit.


\subsection{Basic hyperelliptic integrals and Picard--Vessiot field extensions}
Let $U(x)=x^{n+1}+\cdots$ be a {\em monic\/} (i.e., with leading
coefficient $1$) potential of degree $n+1$, and
$\omega_k=x^{k-1}y\,dx$, $k=1,\dots,n$, differential 1-forms that
constitute the basis of cohomology of each nonsingular
hyperelliptic {\em complex\/} level curve
$\phi_t=\{\tfrac12y^2+U(x)=t\}\subset\C^2$, $t\in\C$. Define the
(complete collection of) {\em basic hyperelliptic integrals\/}
$J_{kj}(t)$ as integrals of the forms $\omega_k$ over {\em
vanishing cycles\/} $\delta_j(t)\in H_1(\phi_t,{\mathbb Z})$, see
\cite{avg}, ``growing'' from the critical values $t_j$:
\begin{equation}\label{defJkj}
J_{kj}(t)=\oint_{\delta_j}\omega_k,
\qquad\delta_j=\delta_j(t)\subseteq \phi_t,\
\operatorname{diam}\delta_j(t)\big|_{t\to t_j}\to0.
\end{equation}
Together they constitute a nondegenerate $n\times n$-matrix
$\J=\J(t)$, analytically depending on $t\in\C$ outside the
critical locus $\Sigma$. This matrix function satisfies a {\em
Picard--Fuchs system\/} \cite{avg} of first order linear
differential equations \eqref{picard-fuchs} with rational
coefficients, and the result of analytic continuation of $\J(t)$
along any loop in $\C\smallsetminus\Sigma$ is described by the
{\em Picard--Lefschetz formulas\/} \cite{avg}.

\begin{Lem}[cf. \cite{gavrilov,nonlin-94}]\label{envelope}
An arbitrary hyperelliptic Abelian integral belongs to the field
$\myk_U=\C(t)(J_{11},\dots,J_{nn})$, the extension of the rational
functions field $\C(t)$ by the basic hyperelliptic integrals
$J_{kj}=J_{kj}(t)$, $j,k=1,\dots,n$, defined as in \eqref{defJkj}.
\end{Lem}


The field $\myk_U$ completely determined by the potential $U$ is the
field of multivalued analytic functions of complex argument $t$,
analytically continuable along any path in
$\C\smallsetminus\Sigma$ (this construction will be further
generalized in Definition~\ref{defHfield} below). Alternatively
one can describe $\myk_U$ in purely algebraic terms as a
differential field, the Picard--Vessiot extension of $\C(t)$ by
solutions of a linear system \eqref{picard-fuchs}, 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 {\em gender\/} of
the field $\myk_U$.

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
$t=t_j$ \cite{avg}, 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 $\myk_U$ can be identified with
arithmetic operations on germs. Analytic continuation along
loops attached to $t_*$, constitute the group $\Mon(\myk_U)$ of {\em
monodromy \textup{(differential)} automorphisms\/} of the field.

Each element of $\myk_{U}$ can be written as a ratio of two {\em
H-polynomials\/}, each of the form $p=\sum_{k+|\alpha|\leqslant
d}c_{k\alpha}\,t^k \J^\alpha\in\C[t,\J]=\C[t,J_{11},\dots,J_{nn}]$, where
$\alpha=\|\alpha_{kj}\|\in{\mathbb Z}_+^{n^2}$ is a multiindex,
$\J^{\alpha}=\prod_{j,k=1}^n J_{kj}^{\alpha_{kj}}$, and $d$ the {\em
degree\/} of the H-polynomial $p$. The degree of an arbitrary
H-function $p/q\in\myk_U$ is $\max(\deg p,\deg q)$, as usual, and it is
preserved by monodromy transformations by virtue of Picard--Lefschetz
formulas.

\subsection{General definition of H-fields}
For our purposes (mainly for Lemma \ref{innertosmaller} below) we
need a more general object, extension of $\C(t)$ by adding
hyperelliptic integrals associated with {\em several\/} different
potentials.

\begin{Def}\label{defHfield}
An {\em H-field\/}\footnote{We 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.}
$\myk_{U_1,\dots,U_\nu}$ associated with a collection of $\nu$ Morse
polynomial potentials $U_1,\dots,U_\nu\in\R[x]$ of degrees
$n_1+1,\dots,n_\nu+1$, is the extension of the field $\C(t)$ by the
complete collection of $n_1^2+\cdots+n_\nu^2$ hyperelliptic integrals
associated with each potential $U_1,\dots,U_\nu$.

The {\em gender\/} of the H-field $\myk_{U_1,\dots,U_\nu}$ is the sum
$n=n_1+\cdots+n_\nu$. The critical locus $\Sigma\subset\C$, generically
consisting of $n$ points, is the union $\{t_1,\dots,t_n\}$ of critical
values of all respective potentials $U_s$.
\end{Def}


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.


\subsection{Theorem on zeros for H-fields}
We prove that under the additional assumption that $\Sigma\subset\R$,
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 {\em complex\/} isolated zeros of
any H-function admits an upper bound in terms of its degree and gender.

If $\Sigma\subset\R$, then we can assume that the critical points are
ordered, $-\infty0\}$ and on each real
interval $\ell_j$ can be at most $B(n,d)$, provided that all critical
values of all potentials are real.
\end{Thm}

Reduction from Theorem \ref{main} to Theorem \ref{aithm} is provided by
Lemma \ref{envelope}. We believe that the assumption $\Sigma\subset\R$ 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.








\section{The structure and main ingredients of the proof}


\subsection{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 $x\mapsto
\lambda_s x+\lambda'_s$ with $\lambda_s,\lambda_s\in\C$, 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'' $t\mapsto \mu t+\mu'$ with
$\mu,\mu'\in\C$, 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 $\Sigma$ is centered at $0$, so that $\sum_{j=1}^n
t_j=0$, and does not shrink too much or stretch to
infinity: $\max_{i\ne j}|t_i-t_j|=1$. Clearly, these
transformations cannot affect any bound on the number of zeros.

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

\subsection{Picard--Fuchs system for hyperelliptic integrals and
variation of argument along arcs distant from the singular locus}
The multivalued 
matrix-valued function $\J(t)$ formed by basic hyperelliptic integrals
\eqref{defJkj} satisfies a linear system of ordinary differential
equations of the form
\begin{equation}\label{picard-fuchs}
(t+A)\dot\J(t)=B\J(t),
\qquad A,B\in\operatorname{Mat}_{n\times n}(\C),
\end{equation}
where $A,B$ are constant matrices depending only on the potential
$U$. The general form of \eqref{picard-fuchs} was established in
\cite{givental} from geometric considerations, and in
\cite{roitman} the system \eqref{picard-fuchs} was derived by
elementary arguments allowing for explicit description of the
matrices in terms of the potential.\footnote{{\em Note added in
proof\/}: the demonstration from the Thesis [19]
recently appeared in the book [22, pp.~83--84].} 
In particular, $\det(t+A)=\prod_{j=1}^n(t-t_j)$ (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 $\leqslant d$ for any particular
$d$:
\begin{equation}\label{bigsys}
\Delta(t)\cdot\tfrac d{dt}\,(t^k\J^\alpha)=\sum\nolimits_{|\beta|\leqslant
d}{\mathcal A}_{k,\alpha,\beta}(t)\,\J^\beta, \qquad \forall\
k+|\alpha|\leqslant d,
\end{equation}
where ${\mathcal A}_{k,\alpha,\beta}\in\C[t]$ are polynomials of
explicitly bounded (in terms of $n$ and $d$) degrees with bounded
coefficients, and $\Delta(t)=\prod_{t\in\Sigma}(t-t_j)$ is the
product taken over the union of {\em all\/} critical values of
{\em 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 \eqref{bigsys} in the affine
space of appropriate dimension (depending on $d$ and $n$). The
main result of \cite{mean,annalif-99} applied to the system
\eqref{bigsys} in combination with \cite[Corollary 2.7]{fields}
yields the following property of resized H-fields.

\begin{Lem}\label{mean}
Variation of argument of any resized H-function of degree $d$ and
gender $n$ along any arc $\gamma\subset\C\smallsetminus\Sigma$
admits an explicit upper bound in terms of  $n$, $d$, and geometry
of the arc $\gamma$ \textup{(}its length $|\gamma|$ and the distance from
$\gamma$ to $\Sigma$, measured by $\inf_{t\in\gamma}|\Delta(t)|>0$\textup{)}.


Variation of argument of any such function along any sufficiently
small circular arc around any singular point $t_j\in\Sigma$ or
$t_0=\infty$ is bounded from above by an explicit expression involving
only $d$ and $n$.
\end{Lem}

The bound provided by Lemma \ref{mean} is already given (assuming
$\gamma$ for simplicity at the distance $1$ from $\Sigma$) by a
tower function of height $4$ in the variables $n$ and $d$
\cite{mean}. This explains why any bound based on using
the lemma, must be {\em very\/} large.

\subsection{Clusterization}
The general principle established in Lemma \ref{mean}, immediately
implies an upper bound for the number of isolated zeros on any compact
simply connected subset of $\C\smallsetminus\Sigma$, 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 {\em 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 $D_\infty$ of infinity and for the complement
$C=\C\smallsetminus(D_\infty\cup D_1\cup D_2)$.  The bound for
zeros in $C$ follows from Lemma \ref{mean} 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 $D_\infty$) can be counted
combining the main result of \cite{mrl} with that from \cite{mean}:
this works in fact for any cluster with only one singularity
inside. The arguments briefly described below, show how the ideas of
\cite{mrl} 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 \cite{petrov}. The latter is
briefly explained in the following section.







\subsection{Argument principle after Petrov}
Suppose we have an H-function $f$ with ramification locus $\Sigma$ on
the real axis. Consider a symmetric (with respect to $\R$) domain
$\Omega\subset\C\smallsetminus\Sigma$ formed by cutting the cluster
(disk) $D=\ol\Omega$ along two rays emanating from two adjacent
singular points in the opposite directions; see Figure~1. Since
$\Omega$ is simply connected, $f$ extends as a single-valued function
analytic in $\Omega$. 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 $\gamma_*$ distant from $\Sigma$, the
variation of argument along the arcs $\gamma_j,\gamma_*$ is
explicitly bounded by Lemma \ref{mean}. It remains to majorize the
variation of argument of $f$ along the upper and lower edges of the
real intervals $\sigma_j^{\pm}$ between the singular points $t_j$ and
$t_{j+1}$.
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\caption{The cluster and its boundary.}
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Let $\sigma$ be one such edge. It was an observation made by G.~Petrov
in \cite{petrov-elliptic}, that {\em variation of argument of an
analytic function $f$ along  a connected curve is at most $\pi$  times
the number of roots of $\myIm f$ on that curve plus $1$}, since between any
two consecutive roots of $\myIm f$ there, variation of argument of $f$
can be at most $\pi$. In other words, to majorize the number of zeros
of $f$ in $\Omega$, it is sufficient to majorize the number of zeros
of $\myIm_\sigma f=\myIm(f\big|_\sigma)$ on every interval $\sigma$ that
is a part of $\partial\Omega$.

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 {\em for an H-function from the field
$\myk=\myk_{U_1,\dots,U_\nu}$ all these extensions will again belong
to the same field\/}, i.e.~there are well-defined maps
$\myIm_{\sigma}\:\myk\to\myk$ for all $\sigma=\sigma_j^\pm$. This
follows, e.g., from the fact that the matrices $A,B$ occurring in
the system \eqref{picard-fuchs}, are real.


The above construction reduces the problem of zeros for {\em one\/}
H-function $f$ to that for {\em several\/} other functions
$\myIm_{\sigma_j^\pm}f$ associated with all real edges
$\sigma_j^\pm\subset\partial\Omega$. The gain occurs if these new
H-functions are simpler than the original one.


\subsection{$D$-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
$t_j\in D$ (the paths connecting $t_*$ with the ``outer'' singular
points outside $D$, can be arbitrary).

\begin{Def}
The {\em $D$-restricted monodromy\/} (sub)group $\Mon^D(\myk)$,
$\myk=\myk_{U_1,\dots,U_\nu}$, is a subgroup of the full monodromy
group $\Mon(\myk)$ formed by analytic continuation over loops
entirely belonging to the cluster $D$.
The dual object is the {\em $D$-inner\/} subfield
$\myk^D=\myk^D_{U_1,\dots,U_\nu}$ invariant under all transformations
from $\Mon^D(\myk)$. 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 $t_j\in D$.
\end{Def}



\begin{Lem}\label{innertosmaller}
There exists an H-field $\myk_{V_1,\dots,V_\mu}$ of gender $m