\tolerance=2000
\magnification=\magstep1
\parskip=6pt plus 12pt

\def\RR{{\bf R}}
\def\Rdois{\hbox{${\bf R}^2$}}
\def\Fxy{F(x) = y}
\def\FRR{\hbox{\rm $F:\Rdois \to \Rdois$} }
\def\gi{{\hbox{\rm $\Gamma_i$} }}
\def\Fgi{\hbox{\rm $F(\Gamma_i)$} }
\def\gum{\hbox{\rm $\Gamma_1$} }
\def\Fgum{\hbox{\rm $F(\Gamma_1)$} }
\def\gdois{\hbox{\rm $\Gamma_2$} }
\def\Fgdois{\hbox{\rm $F(\Gamma_2)$} }
\def\gzero{\hbox{\rm $\Gamma_0$} }
\def\Fgzero{\hbox{\rm $F(\Gamma_0)$} }
\def\Frgi{\hbox{\rm $F|_{\Gamma_i}$} }
\def\Ap{{A_p}}
\def\deg{\hbox{\rm deg} }

\pageno=1
\centerline{\bf On the problem of solving $\Fxy$ in the plane}
\vskip 6pt
\centerline{I.P. Malta, N.C. Saldanha and C. Tomei}

\centerline{Departamento de Matem\'atica, PUC-Rio
\footnote{$^1$}{Pontif\'\i cia Universidade Cat\'olica do Rio de Janeiro,
Rio de Janeiro, RJ 22453, Brasil}}
\vskip 6pt
\headline={\ifnum\pageno>1 {\hfil \it Solving $\Fxy$ in the plane \hfil}
\else{\hfil}\fi}

{\noindent \bf Abstract:}
The knowledge of a critical set together with its image provides
a wealth of information about the global geometry of mappings
from the plane to the plane.
In this note, we describe some properties satisfied by
the critical sets and indicate how they can be useful
in the explicit construction of the critical sets
and in the solution of $\Fxy$.
\vskip 12pt

In this note\footnote{$^2$}{Work supported by SCT and CNPq, Brazil},
we describe a theoretical setup to study the
problem of solving $\Fxy$ for a large class of mappings $F$ from the
plane  to the plane. The motivation to consider this problem came from
the one dimensional\  Riemann problem, that is, a conservation law
$$U_t+f(U)_x=0,\qquad f:\Rdois \rightarrow \Rdois$$
with initial condition
$$ U(0,x) = \cases{U_0 &for $x < 0$,\cr U_1 &for $x > 0$.\cr}$$

An elementary shock-wave solution of the Riemann problem is given by
$$ U(t,x) = \cases{U_0 &for $x -st< 0$,\cr U_1 &for $x-st > 0$.\cr}$$
whenever the Rankine-Hugoniot condition
$$f(U_1) - f(U_0) = s(U_1 - U_0)$$
holds. The search for elementary shock-wave solutions then leads us
to the following question: for given $s\in\RR$ and $U_0\in\Rdois$, how
many values of $U_1\in\Rdois$ are there so
that $(s,U_0,U_1)$ gives rise to
an elementary shock-wave solution, and how does this number vary as
$(s,U_0)$ vary?

Rewriting the Rankine-Hugoniot equation as
$$f(U_1) - sU_1 = f(U_0)- sU_0$$
and setting $F_s = f - sI$, one sees that $(s,U_0,U_1)$ gives rise to an
elementary shock-wave solution of the Riemann problem if and only if
$U_1$ is a solution of the non-linear equation
$$F_s(U) = F_s(U_0).$$

Thus the above question naturally induces the following problem.

{\noindent \bf Problem 1:} Given \FRR and $y\in\Rdois$, how many solutions does
the equation $\Fxy$ have, and how does the number of solutions vary with
$y$?

The theoretical considerations in this note are also being used in the
construction of a program to solve the equation $\Fxy$ numerically,
for a rather general class of mappings $F$ described below. The key
point is that the topological arguments not only provide a satisfactory
counting of the solutions of the equation, but also show us how to
implement routines which obtain initial conditions for local iterations
(like Newton's method) which indeed converge to the desired solutions.
Details of the program will be described elsewhere [MST].

In [MT] it is proved that, for $F$ in a suitable class of $C^\infty$
``nice'' mappings of the plane into itself,
all the information needed to answer
Problem 1 is given by the action of $F$ on the set of critical points.
Nice mappings will be defined later;
over compact domains, nice mappings are generic.
Rather surprisingly, not any  set of curves in the plane, together
with a putative set of image curves, can be the critical set of a
nice mapping in the plane.
Some obvious restrictions arise from considerations of differential
topology. In the seventies, Blank and Troyer ([B],[T]) obtained
necessary and sufficient conditions for a set of curves to be the image
of the boundary under an immersion of an $n$-holed disk in the plane.
From
their work, we derive a satisfactory answer to the following problem,
which appears naturally in the computation of the critical set of a
given nice mapping $F$.

{\noindent \bf Problem 2:} Given a set $C'$ of critical points of $F$, is
there a nice mapping $G$ such that $G$ agrees with
$F$ in a neighborhood of $C'$ and $C'$ is the whole critical set of $G$?

At the end of this note, we make some comments about a delicate issue in
the project: given a nice mapping $F$ and some curves which are shown,
by making use of the appropriate tests, to be {\it all} the critical
curves
of some nice mapping $G$, how can we be sure that there are no other
critical curves?

In this text, we provide no proofs (to be found in the references)
and frequently avoid a complicated (and precise) description of the
results by making use of examples, which are supposed to convey the
spirit of the techniques employed.
We would like to thank the referee for his comments,
indicative of a careful reading.

We begin by recalling some standard definitions. A mapping $F$ from the
plane to itself is said to be proper if the inverse image of any compact
set in the plane is compact. Clearly, \FRR is proper if and only if the
mapping goes to infinity at infinity, and so $F$ can be extended
continuously from the Riemann sphere to itself by defining the value of
the extension $\tilde F$ at infinity to be infinity. Continuous proper
mappings $F$ from the plane to itself have a topological degree, which
coincides with the degree of the extension $\tilde F$ [M]. A point in
the domain of $F$ is regular if the differential $DF$ at this point is
invertible.  Points which are not regular are called critical, and their
images are the critical values of $F$. The set of critical points,
denoted by $C(F)$ or simply $C$, and its image under $F$ are called the
critical sets of $F$. Points which are not critical values are called
regular values.

To explain how all the information we need to solve the equation $\Fxy$,
for $F$ in a suitable class of $C^\infty$ mappings, is given by the
critical sets of $F$, we consider the mapping \FRR given by
$$F(u,v)=
(-6u^4 - 6u^2v^2 + uv^3 + 6v^4 - u,
{\scriptstyle 25 \over \scriptstyle 24}u^4 + u^3v + u^2v^2 +
{\scriptstyle 1 \over \scriptstyle 6} uv^3 - v^4 - v ).$$
This mapping is proper and its critical sets $C$ and $F(C)$ are given in
Figure 1, while the pre-image $F^{-1}(F(C))$ is given in Figure 2. We
use the same letter to denote a set and its image under $F$.

%Figure 1

Now, if we denote by $T_\infty$ the unbounded component of $\Rdois - F(C)$
and by $T_0$ and $T_1$ the bounded ones, we will have, from the
properness of $F$, that $F$ restricted to any connected component of
$F^{-1}(T_\alpha),\alpha = 0,1,\infty$, is a covering mapping and the action
of $F$ is described in Figure 2.  So, in particular, $F$ restricted to
any connected component of $F^{-1}(T_i), i=0,1$, is a diffeomorphism
onto $T_i$ and $F^{-1}(T_\infty)$ covers $T_\infty$ twice.  Hence the
equation $\Fxy$ has 2, 4 or 6 solutions depending on whether $y$ belongs
to $T_\infty$, $T_0$ or $T_1$ respectively.  Indeed, as proved in [MT],
for a nice $F$, the number of solutions of $\Fxy$, for $y$ varying in the
plane, can be obtained simply from knowledge of $C$ and $F(C)$.  The
additional knowledge of $F^{-1}(F(C))$ gives us information about where
to look for initial conditions for a possible iterative method to solve
the equation numerically.

%Figure 2

In order to state the hypothesis on $F$, we recall the concept
of Whitney singularities [W]. A fold point of $F$ (resp., cusp point)
is a critical point $x$ for which there are local orientation preserving
diffeomorphisms\  around $x$ and $F(x)$ onto neighborhoods\  of the
origin
of the plane in which $F$ takes the form $(1)$ (resp., $(2)$) below.
$$F(u,v) = (u , v^2) \eqno(1)$$
$$F(u,v) = (u, av^3 - uv),\qquad a = \pm 1 \eqno(2)$$

By a celebrated theorem of Whitney [W], generically in an appropriate
topology in the space of smooth mappings from the plane  to itself,
critical points are either folds or cusps.
Mappings satisfying the three conditions below will be called nice.
\item{(a)} $F$ is a smooth proper mapping from the plane to itself.
\item{(b)} $C=C(F)$ is bounded and each critical point is a fold or a
cusp point.
\item{(c)} Images of critical curves may only intersect at a finite
number of points, and the pre-image of such a point meets
$C(F)$ at exactly two fold points.

With the techniques employed in [W], one shows that nice mappings
are generic in the class of smooth proper mappings with bounded
critical set (in a suitable $C^r$ topology).
The requirement that each critical point is a Whitney singularity
implies that zero is a regular value of $\det DF$. Thus, from condition (b),
the set of critical points $C$ is a finite disjoint union
of simple closed curves and there is only a finite number of cusp
points. Condition (a) implies that $F$ has a topological degree
$d=\deg F$. This degree is given by the number of solutions $x_j$
of the equation $\Fxy$ for any regular value $y$, counted with the
sign of $\det DF(x_j)$ [M]. Since by (b) the set of
critical points $C$ is compact, we have that $\Rdois - F(C)$ has exactly
one unbounded connected component $T_\infty$. Clearly,
$F^{-1}(T_\infty)$ is the unbounded connected component of
$\Rdois-F^{-1}(F(C))$, so the local orientation of $F$ at any solution of $\Fxy$
for $y$ in $T_\infty$ has to be the same. Thus, the number of
solutions of this equation for $y$ in $T_\infty$, which is positive,
equals the absolute value of the topological degree of $F$. In
particular, $d \ne 0$.

From (b), the restriction of $F$ to any critical curve \gi, \Fgi, is a
continuous locally injective curve and we can define the turning number,
$\tau(\Frgi)$ of \Fgi as the Brouwer degree of the mapping
$$ \vartheta \mapsto {
{F(\gamma(\vartheta + \delta)) - F(\gamma(\vartheta))} \over
|{F(\gamma(\vartheta + \delta)) - F(\gamma(\vartheta))}|},$$
where $\gamma : S^1 \to \Rdois$ is a regular orientation preserving
parametrization of \gi and $\delta$ is any small positive angle for
which $F|_{[\gamma(\vartheta),\gamma(\vartheta + \delta)]}$ is injective
for all $\vartheta \in S^1$.
Condition (c) implies that we can compute the turning number
$\tau(\Frgi)$ from the Seifert circles of \Fgi as follows. Consider the
orientation induced in the curve \Fgi by the positive (counterclockwise)
orientation of \gi. Now change slightly the curve \Fgi
near self-intersection points as in Figure 3.
Then \Fgi splits into a disjoint union of oriented simple curves -- the
turning number is the number of such simple curves counted with a sign
defined by their orientation.

%Figure 3

Since critical points are fold or cusp points, we can orient the image
\Fgi of each critical curve \gi so that a small disk around any fold
point in \gi is sent by $F$ to the left side of the oriented curve \Fgi.
In particular, each image $F(x)$ of a cusp point $x \in \gi$ points to
the right side of the oriented curve \Fgi (see Figure 4). This
orientation will be said to be given by the sense of folding.
The turning $w(\gi)$ of the curve \Fgi oriented by the sense of folding is
defined by $w(\gi) = \pm \tau(\Frgi)$.
This sign is positive if this
orientation agrees with the one induced by $F$ on \Fgi from the positive
orientation on \gi.

%Figure 4

The three results below are stringent relations among
the numbers defined so far (see [MT] for proofs).

{\bf \noindent Theorem A:} Let $F$ be a nice mapping. Then
$$ |\deg F | = k - 2w + 1  >  0,$$
where $k$ is the number of cusp points and
$w = \sum_{{\Gamma_i} \subset C} {w(\gi)}$.

Let \FRR be a nice mapping and $C' = \bigcup_{i=1}^{n}{\gi}$
a union of critical
curves of $F$. Let \gzero be a circle around $C'$ and suppose that
$\gzero\cap C(F)= \emptyset$.
Let $D_0$ be the open disk bounded by \gzero.
Let $w(\gzero)$ be the turning number of
$F(\gzero)$
oriented so that a small neighborhood in $\overline D_0$ of a point in
\gzero is sent by $F$ to the left side of $F(\gzero)$, and
$w(\gi)$ for $i=1,\ldots,n$ be the turning of \Fgi oriented by the
sense of folding.

{\bf \noindent Proposition {\rm [MT]}:} If $D_0 \cap C(F) = C'$ then
$$w(\gzero) = k - 2w(C') + 1,$$
where $w(C')= \sum_{i=1}^{n}{w(\gi)}$ and $k$ is the number of cusp
points in $C'$.

{\bf \noindent Corollary:} If $C'=C(F)$ then $|\deg F|=w(\gzero)$.

We show by an example how to solve Problem 2.
Suppose we have found a critical curve \gum of a nice mapping $F$ such
that the
image curve oriented by the sense of folding and the
behavior of $F$ in a neighborhood of \gum are given in Figures 5 and 6
respectively.

%Figures 5 and 6

Since $w(\gum) = 1$ and \gum has one cusp point, we conclude from
Theorem A that \gum cannot be the whole critical set of $F$, so we have
to look for more critical curves.

Suppose we find another critical curve \gdois as in Figure 7, close to
which the local behavior of $F$ is given in Figure 8.

%Figures 7 and 8

From Theorem A we learn that if $\gum \cup \gdois$ is a critical set of
a nice mapping $G$ then $| \deg\,G| = 1$. So, if we take a circle
around the
curves in the range, its pre-image is a simple, closed regular curve
\gzero around the critical curves \gum and \gdois as in Figure 9.
By the proposition and corollary above, $w(\gzero) = 1$.
The orientation in \Fgzero means that any point in the open disk bounded
by \gzero is sent by $F$ to the left side of \Fgzero.

%Figure 9

Clearly the problem of existence of a nice mapping $G$ as required is then
reduced to the following question.
Is there an extension $G$ of
$F|_{\gzero \cup \gum \cup \gdois}$ to the closed disk $\overline{D_0}$
bounded by \gzero such that $G$ is an immersion outside the curves \gum
and \gdois and preserves the behavior of $F$ near
$\gum \cup \gdois$ ?
To answer this question, first observe that this problem decouples in
similar problems. Let $C' = \gzero \cup \gum \cup \gdois$ and $g =
F|_{C'}$.

\item{(i)} For $i = 1,2$, can $g|_{\gi}$ be extended
to the closed disk $\overline{D_i}$
bounded by \gi, as an immersion in $D_i$ and such that the extension
preserves the behavior of $F|_{\overline{D_i}}$ near \gi?

\item{(ii)} Can $g$ be extended to $A= \overline{D_0} - (D_1 \cup D_2)$
as an immersion outside of the boundary of $A$ preserving the
behavior of $F|_A$ near $\partial A$?

Clearly, for $i= 1,2$, $g|_\gi$ has an extension as a diffeomorphism
from $D_i$ onto the open disk bounded by \Fgi. To give an answer to (ii),
we use a slight adaptation of a criterion introduced by Blank [B] and
Troyer [T]. To explain how to do this, we consider first a simpler
problem. Suppose that $\Gamma$ is a critical curve of a nice mapping $H$ and
that $h = H|_\Gamma$ acts as in Figure 10.

%Figure 10

The problem is to decide if there exists an extension of $h$ to the
closed disk $\overline D$ bounded by $\Gamma$ as an immersion in $D$,
having $x$ and $y$ as cusp points, and such that a small neighborhood in
$\overline D$ of a point in $\Gamma - \{x,y\}$ is sent  to the left side
of the oriented curve $h(\Gamma)$. Proceed as follows. Let
$T_{\infty}$ be the unbounded connected component of $\Rdois  -  h(\Gamma)$.
Choose a point $p_i$ in each bounded connected
component of $\Rdois -  h(\Gamma)$. Let $r_i$ be a proper embedding of
the non-negative real axis into the plane with $r_i(0)=p_i$ (called a
ray from $p_i$) such that the rays $r_1,\ldots,r_m$ do not contain
intersection points of $h(\Gamma)$, cut $h(\Gamma)$ transverswally and
are pairwise disjoint. Now assign to each intersection of a ray with the
curve a positive or a negative sign, depending on whether the curve
crosses the ray from right to left (that is, $p_i$ is on the left side
of the curve) or the curve crosses the ray from left to right. Complete
the set of rays by choosing for each cusp point $x$ a ray from $h(x)$
assigning to this point a negative sign (see Figure 11).

%Figure 11

Now construct a word (the Blank word for the oriented curve $h(\Gamma)$)
by following the orientation of the curve $h(\Gamma)$ and collecting the
intersection points with the rays, keeping track of signs and rays.
Clearly the word is defined up to cyclic permutation. In this example, a
Blank word is given by
$$BW = r_1^+r_2^+r_3^+r_4^+r_5^+r_6^+r_7^+r_3^+r_4^+r_5^+
r_2^+r_3^+r_5^-r_6^+r_7^-.$$

We say that a Blank word admits a simplification if there exists a pair
$r_i^+, r_i^-$ such that (after a cyclic permutation if necessary) there
are no letters with negative exponent between $r_i^+$ and $r_i^-$ .
If this is the case, a simplified Blank word is obtained by eliminating
the subword $r_i^+\ldots r_i^-$ (or $r_i^-\ldots r_i^+$). We say that a Blank
word groups (or has a grouping) if there are successive simplifications
such that the final Blank word has no letters with negative exponents.
In the example, a possible choice of simplification is
$$BW \to r_1^+r_2^+r_3^+r_5^+r_6^+r_7^+r_3^+r_4^+r_6^+r_7^- \to
r_1^+r_2^+r_3^+r_5^+r_6^+$$
and so the Blank word of the example groups.
The fact that the
Blank word groups, together with the fact that the turning number
minus the number of cusps equals one, is sufficient (and necessary!)
to guarantee the existence of the desired extension. The complete
description of Blank's criterion can be found in [B], while the
mild extension being used is in [MST].

Returning to the example given in Figure 9, we follow
(a slight modification of) a procedure due to Troyer ([T]) which
is the analogue of Blank's criterion to the case where the domain
of the extension being sought consists of more than one curve. We
begin by constructing rays as before (see Figure 12), and collecting one
word for each curve $g(\gi), i = 0,1,2$. A Blank word for $g(C')$ is a
single word obtained as the adjunction of a cyclic permutation of each
word. In the example, denoting by $BW(\gi)$ the Blank word of
$g(\gi), i = 0,1,2$, we have

%Figure 12

$$BW(\gzero) = a^+b^+c^+d^+e^+f^+q^+r^+,$$
$$BW(\gum) = d^+e^+f^+c^-,$$
$$BW(\gdois) = c^+d^+f^-q^+r^-a^-,$$
so a possible Blank word for $g(\gzero \cup \gum \cup \gdois)$ is
$$BW(\gzero \cup \gum \cup \gdois) =
d^+e^+f^+c^-c^+d^+f^-q^+r^-a^-a^+b^+c^+d^+e^+f^+q^+r^+ $$
which has a grouping. As before, since $w(\gzero) + w(\gum) + w(\gdois)
- (\hbox{number of cusp points}) = -1$, and a Blank word groups, we can
guarantee the existence of the desired extension.

Let $A_1,\ldots,A_m$ be the connected components of $D_0-C'$,
with the notation defined above.
Then for each $p=1,\ldots,m$, the boundary of $\Ap$ is
$\partial \Ap = \Gamma_{p_0} \cup ({\bigcup_{i=1}^{n_p}{\Gamma_{p_j}}})$
where $\Gamma_{p_0}$ is the exterior boundary of $\Ap$, that is,
$\Ap$ is contained in the disk bounded by $\Gamma_{p_0}$.
As in the example,
the problem of existence of a nice mapping $G$ which agrees with
$F$ in $C'$, has the same behavior as $F$ in a neighborhood of $C'$ and
such that $C(G)=C'$ can be reduced to the existence of extensions of
$F|{\partial \Ap}$
to $\Ap$ as an immersion in $\Ap$ preserving the behavior of
$F$ in a neighborhood of $\partial \Ap$ in $\Ap$. So we have to make a
distinction among the cusps in $\partial \Ap$ in such a way as to select
the ones which are detected by the action of $F$ in $\Ap$.
Let $D_i$ be the open disk bounded by
the critical curve \gi. A cusp point $x \in \gi$ will be called
inward
if, for each sufficiently small neighborhood $U$ of $x$, $F$ maps
$U \cap \overline D_i$ onto a neighborhood of $F(x)$. It is clear from
the local form (2) that if a cusp point $x$ is not inward,
then $F$ maps
$(\Rdois - D_i) \cap U$ onto a neighborhood of $F(x)$. In this case, $x$
is said to be an outward cusp point.
In other words, the fact that $x$ is a
cusp point is detected by the action of $F$ in $D_i$ or in
$\Rdois - \overline D_i$ if $x$ is an inward or outward cusp respectively.
These cases correspond to Figures 13 (a) and (b).

%Figure 13

Clearly, for the problem of existence of the desired extension of
$F|{\partial \Ap}$, we have to consider as cusp points only the inward
cusp points of $\Gamma_{p_0}$, and the outward cusp points of
$\Gamma_{p_j}, j=1,\ldots,n_p$. So, for each $p$, construct a set of
rays for the curves $F(\partial \Ap)$ oriented by the sense of folding
considering only the cusps described above. In Figures 14 (a) and (b),
we draw the rays associated to the components $A_1$ and $A_2$ for $F$
given in Figure 1.

%Figure 14

For each $p$, construct the Blank word of
$F(\Gamma_{p_j}),j=0,\ldots,n_p$
and consider the set of Blank words for $F(\partial \Ap)$.
For each $p$, let $k_p=k_{\ast {p_0}}+
{\sum_{j=1}^{n_p}{k_{p_j}^\ast}}$,
where $k_{\ast {p_0}}$ and $k_{p_j}^\ast$ are, respectively, the number
of inward and outward cusps in the corresponding curve.

{\bf \noindent Theorem B:} ([MST])
$C'$ is the set of critical points of a nice
mapping $G$ which agrees with $F$
in a neighborhood of $C'$ if and only if

\item{(i)} $k_p-({\sum_{j=0}^{n_p}w(\Gamma _{p_j})}) - n_p + 1 = 0$\
for $p=1,\ldots,m$, and
\item{(ii)} for each $p=1,\ldots,m$, there exists a Blank word for
$F(\partial \Ap)$ which has a grouping.

\noindent Moreover, if (i) and (ii) are satisfied, then
$|\deg\, G| = w(\gzero)$.

{\bf \noindent Remark:} Condition (i) simply says that for each $p$,
$\chi(\overline A_p) = w(\partial \Ap)-k_p$, where $\chi(\overline A_p)$
is the Euler characteristic of $\overline A_p$, which is $1-n_p$.

We are still left with the following intriguing issue: once we found a
(sub)set of critical curves of the mapping $F$ which is indeed the
critical set of a mapping $G$ (since it satisfies the tests described
above), how can we be sure that $F$ does not have other critical curves?
The answer is obvious: this cannot be guaranteed by mere topological
arguments.

The problem is already present in the one dimensional case in two
different versions: how can we be sure, given a mapping from the line
to itself that we know all its zeros, or all the zeros of its
derivative? There are two obvious difficulties: a computer program will
search for zeros in a bounded set and not beyond a certain level of
refinement within this set. Implicitly, we assume the knowledge of a
priori estimates which guarantee that there are no zeros outside the
set being scanned and that, within this set, the mapping does
not oscillate so much as to generate additional zeros. Thus, unless the
program itself generates the required estimates, it cannot certify that
all zeros will be found.

We provide two classes of examples in which the set of critical curves
is not found completely. The reader will have no trouble in convincing
himself that the examples are typical. In Figure 15, the knowledge of
the critical curve $\Gamma$ together with the behavior of the
mapping $F$ at infinity (its degree) is not enough to decide whether
the critical set of $F$ is (a) or (b). The problem is that one can draw
a curve $\gamma$ around the remaining critical curves in (b) for which
$F(\gamma)$ is a simple closed curve: if you never enter the disk $D$
bounded by $\gamma$ you will never distinguish between the complicated
behavior of $F$ in $D$ described in (b) and the simple one (a
diffeomorphism) shown in (a). So, unless the original search for
critical curves spontaneously browsed through $D$, the topological
arguments presented in this note would not indicate that $D$ should be
searched for critical points.

%Figure 15

The other example is shown in Figure 16. Knowing curve $\Gamma$ and
the
degree of $F$ would not induce the program to search for the (possibly
missing) two critical curves. The reason is that the image of a simple
curve tightly surrounding $\Gamma$ has the same topological behavior as
the
image of a large circle: both are taken to (isotopic) curves of
equal turnings.
So, again, from the topological information, both
situations are undistinguishable.

%Figure 16

\vskip 16pt plus 16pt

\centerline{\bf Bibliography}
\item{[B]} Blank, S.J.: {\it Extending immersions of the circle},
Expos\'e 342, S\'eminaire Bourbaki 1967-68, Benjamin, NY, 1969.
\item{[M]} Milnor, J.W.: {\it Topology from the differentiable
viewpoint}, University Press of Virginia, 1969.
\item{[MST]} Malta, I., Saldanha, N.C., Tomei, C.: {\it Critical sets
and the numerical inversion of mappings in the plane}, in preparation.
\item{[MT]} Malta, I., Tomei, C.: {\it Singularities of vector fields
arising from one dimensional Riemann problems}, J.Diff.Eq., 93 (1991).
\item{[T]} Troyer, S.F.: {\it Extending a boundary immersion to the disk
with $n$ holes}, Ph.~D. Dissertation, Northeastern U., Boston,
Mass., 1973.
\item{[W]} Whitney, H.: {\it On singularities of mappings of Euclidean
spaces, I: mappings of the plane into the plane}, Ann. of Math. (1955),
374-410.

\bigskip
\obeylines
\parskip=0pt
\parindent=0pt

malta@mat.puc-rio.br
nicolau@mat.puc-rio.br, http://www.mat.puc-rio.br/$\sim$nicolau/
tomei@mat.puc-rio.br

\bye
