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\def\bC{{\Bbb C}}
\def\bR{{\Bbb R}}
\def\Ho{H_{1,\ldots n}}
\def\H{\Cal H}
\def\a{\alpha}
\def\b{\beta}
\def \CA {\Cal A}
\def \CU {\Cal U}
\def \la {\lambda}
\def \al {\alpha}
\def \be {\beta}
\def \ga {\gamma}
\def \part {\partial}
\def \De {\Delta}
\def \si {\sigma}
\def \P {\Psi}
\leftline{\eightit ``I think you might do something better with the
time'', Alice said, }
\leftline{\eightit  ``than wasting it in asking riddles that have no answer''}
\leftline{\eightit \quad\quad\quad - from A Mad Tea-Party}
\vskip 1.0cm


\topmatter
\title
This strange and mysterious  Rolle's theorem
\endtitle

\author B.~Shapiro$^{\dag}$, M.~Shapiro$^\ddag$
\endauthor
\affil $^\dag$
     Department of Mathematics, University of Stockholm, S-10691, Sweden,
{\tt shapiro\@matematik.su.se}\\
$^\ddag$ Department of Mathematics, Royal Institute of Technology\\
S-10044, Sweden, {\tt mshapiro\@math.kth.se}\\
\endaffil
\leftheadtext {B \& M }
\rightheadtext {This strange and mysterious}


\abstract First year undergraduates usually learn about the classical Rolle's
theorem saying that between two consecutive
zeros of a smooth function $f$ one can always find at least one zero
of $f^\prime$. But what happens with higher derivatives or,
in more restrictive families of functions, for example,
polynomials? We describe a  Rolle's-type theorem for
smooth analogs of polynomials showing that the situation is highly
   nontrivial.
\endabstract

%\subjclass {Primary !!!!!}\endsubjclass
\keywords{A mad tea-party with hyperbolic  pseudopolynomials}
\endkeywords

\endtopmatter



\heading \S 1. Introduction. \endheading

Consider a smooth function $f$ that on a certain interval has  $n$ distinct
real zeros  which we denote by
$x_{1}^{(0)} < x_{2}^{(0)} <\ldots <x_{n}^{(0)}$ . Then $f'$ has at
least $n-1$ zeros, $f''$ has at least $n-2$ zeros, ... , $f^{(n-1)}$ 
has at least
one zero on $(x_{1}^{(0)},x_{n}^{(0)})$. If we assume that $f^{(i)}$
has exactly $n-i$ zeros
denoted by $x^{(i)}_1<x_2^{(i)}<...<x_{n-i}^{(i)},$ we get the
following system of inequalities which we  refer to as
{\it the standard Rolle's restrictions}
        $$x^{(i)}_l<x^{(j)}_l<x^{(i)}_{l+j-i}.\tag1$$


        Such a function will be called {\it degree $n$  polynomial-like}.
We can  map any degree $n$  polynomial-like $f$ to the
arrangement $\Cal A_{f}$ of all $\binom n 2$ zeros $\{x^{(i)}_{l}\}$
of $f^{(i)},\,i=0,\ldots,n-1;\,1\le l\le n-i $. Such a situation
occurs naturally if one, for example,
   takes all degree $n$ polynomials with only real zeros.
\medskip
{\smc Definition.} A real polynomial $P(x)$ in one variable is called
{\it hyperbolic} if all its roots (counted with  multiplicities) are real.
A hyperbolic $P(x)$ is called {\it strictly hyperbolic} if all its roots are
simple.

The major question we address in this note is as follows.
\medskip
{\smc Main Problem.} What additional (besides the standard Rolle's)
topological or
geometric restrictions on arrangements $\Cal A_{f}$
exist for different natural families of  polynomial-like functions?
\medskip
As an illustration of what we mean  by  topological restrictions let us
describe in more details the situation with hyperbolic polynomials.

   {\smc Notation.} Denote by $Hyp_n$  the space of all strictly
       hyperbolic polynomials of degree $n$ and by $\widetilde{Hyp}_{n}$
       the subset of all hyperbolic polynomials with all $x^{(i)}_{l}$
       being different. Note that $Hyp_{n}$ is contractible and denote by
       $\sharp_{n}$ the number of connected components in
       $\widetilde{Hyp}_{n}$.
Let $\Cal R_{n}$ be the ${\binom n 2}$-dimensional linear space
       over $\bR$ with the coordinates $x^{(i)}_{l},$ where $0\le i \le
       n-1$ and $1\le l \le n-i$. Denote by $\Omega_{n}\subset \Cal
       R_{n}$
       the open domain
       given by the system of  inequalities $(1)$ and by
$\widetilde \Omega_{n}\subset \Omega_{n}$ its subset given by the
additional restrictions that all $x^{(i)}_{l}$ are distinct. Note
that the system $(1)$ gives a partial order on the set of variables
$x^{(i)}_{l}$. Let  $\Cal {EXT}_n$ be  the set of all linear  extensions
of this partial order and $\flat_{n}$ be the number of connected
components in $\widetilde \Omega_{n}$. Then
$\flat_n=\text{card}(\Cal {EXT}_n)=\binom{n+1}
{2}!\frac{1!2!...(n-1)!}{1!3!...(2n-1)!}$, see e.g.
\cite  {Th}. But what about $\sharp_{n}$?

Since the roots of $p,p',p'',...,p^{(n-1)}$
of any polynomial $ \widetilde {Hyp}_n$
satisfy (1) and are all distinct they form one of the linear extensions in
$\Cal {EXT}_n$.
It is therefore natural to compare
the numbers $\sharp_n$ and $\flat_n$.
But already the first nontrivial case of $\widetilde{Hyp}_4$ shows that
the situation is far from being obvious, namely
$10=\sharp_4\neq\flat_4=12$, see Fig.1. This
was apparently observed by a number of authors but the only relevant
reference we found is \cite {An} published in 1993! An explanation of this
phenomenon for $n=4$ is as follows. (See further generalizations in
\cite {KSh}.)
\medskip
{\smc Theorem,} see e.g. \cite {An}. If a degree $4$ stricly
hyperbolic polynomial satisfies the inequalities $x_{2}^{(0)}<x_{1}^{(2)},\;
x_{3}^{(0)}<x_{2}^{(2)},$ then it additionally satisfies the
inequality $x_{2}^{(1)}<x_{1}^{(3)}$.
\medskip

Easy to check when you know it! Futhermore, the results of \cite {Ko}
tell us that $\widetilde{Hyp}_{5}$ lies only  in
$116$ connected components of $\widetilde \Omega_{n}$
among totally $286$.
\medskip
\vskip 15pt
\centerline{\hbox{\epsfysize=6cm\epsfbox{fig1.eps}}}
\midspace{0.1mm} \caption{Fig.~1. Ten components of the space
$\widetilde {Hyp}_4$.}

\subheading {About this picture} The interior of the curvilinear
triangle $\triangle ABC$  is the set of all strictly hyperbolic
polynomials in the family $p=x^4-x+ux+v$ shown on the $(u,v)$-plane of
parameters. Its boundary curve is (given by the vanishing of)
the usual discriminant, i.e. the
resultant of $p$ and $p'$. Two skew lines through $A$ and $B$ form the
resultant of $p$ and $p''$. The horizontal line is the resultant of
$p$ and $p^{\prime\prime\prime}$. Finally, the vertical line through
$C$ is the resultant of $p'$ and $p^{\prime\prime\prime}$. Thus
the interior of $\triangle ABC$ is cut into ten smaller curvilinear
triangles by the union all nontrivial resultants for $n=4$.

\medskip
{\smc Open problem.} Count the number $\sharp_{n}$ of connected components in
$\widetilde{Hyp}_{n}$.

     \medskip

     On the other hand, if we take the set $\Cal F_{n}$ of all smooth
    degree $n$ polynomial-like functions on $\bR$ and consider the map
    $\P_{n}: \Cal F_{n}\to \Omega_{n},$ it is not too hard to show that
    $\P_{n}$ is surjective.
    \medskip


In this note we discuss a natural smooth analog of degree $n$ (hyperbolic)
polynomials. Quite unexpectedly (at least to us) there are  additional
  inequalities
besides the standard Rolle's ones on the image of the map
$\P_{n}$ restricted onto this
inifinite-dimensional family as well!
Thus, one gets a number of generalizations of
   the  Rolle's theorem involving zeros of
   higher derivatives for smooth functions.
    \medskip
    {\smc Main Definition.} A smooth real-valued function $f$ defined
    on $\bR$   is called {\it a
degree $n$ pseudopolynomial\/}  if $f^{(n)}$ never vanishes.

%We say that a pseudopolynomial $f$ on $I$ {\it has positive type\/}
%if $f^{(n)}>0$ and {\it has negative type} otherwise. (For an
%$f$ of positive type one has that $-f$ has negative type and vice
%versa.)

%In what follows we will always consider only positive type
%pseudopolynomials.
\medskip
{\smc  Some remarks.} The usual Rolle's theorem  immediately implies that any
degree $n$ pseudopolynomial $f$ has at most $n$ real zeros  (counted with
multiplicitites). The set of all degree $n$ pseudopolynomials is
not a linear space (as well as $\Cal F_{n}$) but consists of two convex
opposite cones, namely, the positive cone containing all $f$ with
$f^{(n)}> 0,$ and the negative one  with $f^{(n)}<0$.

{\smc Definition.} A degree $n$ pseudopolynomial  $f$ on
$I$ is called {\it(strictly) hyperbolic\/} if
it has exactly $n$ distinct real zeros.

{\smc Notation.} Let $PP_{n}$ (resp. $PHyp_{n}$) denote the set of all
% positive type
(resp.  strictly hyperbolic) pseudopolynomials of degree $n$. Since our main
result deals with the case $n=3$ and in order to make notations more
transparent, let us denote the 3 zeros of the function $f$ in $PHyp_{3}$ by
$x_{1}<x_{2}<x_{3}$, the 2 zeros of $f'$ by $y_{1}< y_{2}$ and the only
zero of $f''$ by $z_{1}$.
\medskip
{\smc Main Theorem. } The image $\Theta_{3}=\P_{3}(PHyp_{3})$ is given by the
following inequalities:
$$\cases \aligned
             &x_{1}<y_{1}<x_{2}<y_{2}<x_{3},\;
             y_{1}<z_{1}<y_{2}\;\text{-\;\qquad see (1)}\\
             & y_{1}-x_{1}< x_{2}-y_{1},\;
x_{3}-y_{2}<  y_{2}-x_{2}\\
	 &(y_{1}-x_{1})^2 >(z_{1}-y_{1})(2x_{2}-y_{1}-z_{1})\\
	 &(x_{3}-y_{2})^2 <
(y_{2}-z_{1})(y_{2}+z_{1}-2x_{2})\\
\endaligned \endcases\tag2
$$
\medskip
{\smc Open problem.} What is the image $\Theta_{n}=\P_{n}(PHyp_{n})$?
   \medskip

{\smc Remark.} The topic considered is so classical that
we find that it is impossible to be sure that the above result is new.
However, the authors were unable to trace anything similar in the literature
   although such facts might have been known to at least  Monsieur
   Augustin Cauchy if not to Sir Isaac Newton himself. And the last
   but not the least --- do not neglect Alice's  warning, see epigraph.

The authors are  grateful to V.~Kostov who found a mistake in a
preliminary version of \cite {KSh} which resulted in this funny
little project.
The first author wants to acknowledge
the hospitality and financial support of IHES, Paris and
Universit\'e de Nice in January 2001 where this all started.

\heading {\S 2. Proof} \endheading
\medskip
Take some $f(x)\in PHYP_{3}$.  Without loss of generality we can
assume  $f^{\prime\prime\prime}(x)>0$  implying that $f'(x)$ is
convex (otherwise multiply $f(x)$ by $-1$).  As above the 3 zeros of $f(x)$
are denoted
by $x_{1}<x_{2}<x_{3}$, the 2 zeros of $f'(x)$ by $y_{1}< y_{2}$ and the only
zero of $f''(x)$ by $z_{1}$.
Let us additionally assume that $x_{2}<z_{1}$, see Fig.2. We will
immediately derive the additional inequalities:\quad
$ y_{1}-x_{1}< x_{2}-y_{1}$ and $(x_{3}-y_{2})^2 <
(y_{2}-z_{1})(y_{2}+z_{1}-2x_{2})$.
(The case $z_{1}<x_{2}$ is completely analogous and leads to\quad
$x_{3}-y_{2}<
y_{2}-x_{2}$ and $(y_{1}-x_{1})^2 >(z_{1}-y_{1})(2x_{2}-y_{1}-z_{1})$.)
  Convexity of $f'(x)$ gives

i) on the interval $(x_{1},y_{1})$ the graph of $f'(x)$ lies above the
line segment $\{a,y_{1}\}$ tangent to the graph of $f$ at $y_{1}$;

ii) on the interval $(y_{1},x_{2})$ the graph of $f'(x)$ lies between
the $x$-axis and the  line segment $\{y_{1},b\}$ tangent to the graph
of $f$ at $y_{1}$;

iii) on the interval $(x_{2},y_{2})$ the graph of $f'(x)$ lies between the
$x$-axis and the broken line segment $\{c,d,y_{2}\}$ where $(c,d)$ is
horizontal;

iv)  on the interval $(y_{2},x_{3})$ the graph of $f'(x)$  lies
above the  line segment $\{y_{2},e\}$ which is tangent to the graph 
of $f(x)$ at
$y_{2}$.

Note that since $x_{1},x_{2},x_{3}$ are zeros of $f(x)$  one has
$\int_{x_{1}}^{y_{1}}f'(x)dx=-\int_{y_{1}}^{x_{2}}f'(x)dx$ and
$\int_{x_{2}}^{y_{2}}f'(x)dx=-\int_{y_{2}}^{x_{3}}f'(x)dx$.
   Therefore, $Ar(\triangle_{ax_{1}y_{1}}) < Ar(\triangle_{y_{1}bx_{2}})$
   and $Ar(\triangle_{ey_{2}x_{3}}) < Ar(\square_{x_{2}cdy_{2}})$ where
   $Ar$ stands for the area of the corresponding figures.  Using our
   knowledge of high-school mathematics we get,
   $Ar(\triangle_{ax_{1}y_{1}})=(y_{1}-x_{1})^2\cot \alpha/2,\;
   Ar(\triangle_{ey_{2}x_{3}})=(x_{3}-y_{2})^2\cot \beta/2,\;
   Ar(\triangle_{y_{1}bx_{2}})=(x_{2}-y_{2})^2\cot \alpha/2,\;
   Ar(\square_{x_{2}cdy_{2}})=Ar(\triangle_{z_{1}dy_{2}})
   +Ar(\square_{x_{2}cdz_{1}})=(y_{2}-z_{1})^2\cot
   \beta/2+(y_{2}-z_{1})(z_{1}-x_{2})\cot
   \beta=(y_{2}-z_{1})(y_{2}+z_{1}-2x_{2})\cot \beta$. These
   relations  immediately imply the above inequalities.

   \medskip
\vskip 15pt
\centerline{\hbox{\epsfysize=6cm\epsfbox{fig2.eps}}}
\midspace{0.1mm} \caption{Fig.~2. Derivative of $f$ and its accompaining
   elementary configuration.}


\medskip


   Now in order to prove that $\Theta_{3}$ coincides with
   the whole domain given by (2) pick a $6$-tuple of real numbers
   $x_{1},x_{2},x_{3},y_{1},y_{2},z_{1}$ satisfying $(2)$. Let us
   again assume that $x_{2}< z_{1}$. (Cases $z_{1}\le x_{2}$ are completely
   analogous.) Draw a piecewise linear function $g(x)$ shown on Fig. 3.
   (The only difference with Fig.2 is that we force $c=b$.)
   The inequalities $(2)$ imply that $Ar(\triangle_{ax_{1}y_{1}}) < 
Ar(\triangle_{y_{1}bx_{2}})$
   and $Ar(\triangle_{ey_{2}x_{3}}) < Ar(\square_{x_{2}bdy_{2}})$.
   It is easy to approximate $g(x)$ by a convex function $h(x)$ so that

   i) $(y_{1},y_{2})$ are the zeros of $h(x)$ and $z_{1}$ is the zero
   of $h'(x)$;

   ii) $\int_{x_{1}}^{y_{1}}h(x)dx<-\int_{y_{1}}^{x_{2}}h(x)dx$ and
$\int_{x_{2}}^{y_{2}}h(x)dx<-\int_{y_{2}}^{x_{3}}h(x)dx$.

Keeping the function $h(x)$ as it is on the intervals $(x_{1},y_{1})$
and $(y_{2},x_{3})$ and descreasing its absolute value  on
$(y_{1},y_{2})$ we can construct  a new convex function $\tilde h(x)$
satisfying

   i) $(y_{1},y_{2})$ are the zeros of $\tilde h(x)$ and $z_{1}$ is the zero
   of $\tilde h'(x)$;

    ii) $\int_{x_{1}}^{y_{1}}\tilde h(x)dx=-\int_{y_{1}}^{x_{2}}\tilde 
h(x)dx$ and
$\int_{x_{2}}^{y_{2}}\tilde h(x)dx=-\int_{y_{2}}^{x_{3}}\tilde h(x)dx$.
The function $f(x)\in PHYP_{3}$ we were looking for is now obtained
as $f(x)=-\int_{y_{1}}^{x}\tilde h(t)dt$.
\smallskip
Q.E.D.



   \medskip
\vskip 15pt
\centerline{\hbox{\epsfysize=6cm\epsfbox{fig3.eps}}}
\midspace{0.1mm} \caption{Fig.~3. Constructing an appropriate function.}


\bigskip

\Refs
\widestnumber \key{ShSh}

\ref \key An \by B.~Anderson
\paper Polynomial root dragging
\jour Amer. Math. Monthly
\vol 100
\yr 1993
\pages 864--866
\endref

%\ref \key Ko1 \by V.~Kostov
%\paper On the geometric properties of Vandermonde's mapping and on
%the problem of moments
%\jour Proc. R.Soc. Edinb
%\vol 112
%\issue 3-4
%\pages 203--211
%\yr 1989
%\endref

\ref \key Ko \by V.Kostov
\paper Discriminant sets of families of hyperbolic polynomials of
degree $4$ and $5$
\finalinfo Preprint \# 620, Univ. de Nice-Sophia Antipolis, submitted
to Serdica Math. J
\yr 2001
\endref

\ref \key KSh \by V.~Kostov and B.~Shapiro
\paper On arrangements of roots for a real hyperbolic polynomial and
its derivatives
\finalinfo Preprint \# 619, Univ. de Nice-Sophia Antipolis, submitted
to Bulletin des sciences mathematiques
\yr 2001\endref

\ref \key Th \by M.~Thrall
\paper A combinatorial problem
\jour Michigan Math. J
\vol 1
\yr 1952
\pages 81--88
\endref



\endRefs
\enddocument

We now introduce the closure of the space $PHyp_{n}$ in the
appropriate topology.
\medskip
{\smc Notation.} Let $\phi_{1}$ and $\phi_{2}$ be two nondecreasing functions
on $I$. It is well-known that any monotone
function on an interval is continuous  and moreover is differentiable
in the complement of at most
countable set of points, see e.g. \cite {FK}.
We say that $\phi_{1}$ and $\phi_{2}$ are {\it equivalent} if
they coincide at every point where they both are continuous. Let
$\frak Mon$ denote the space of all equivalence classes of monotone
nondecreasing functions and  let $\frak M$ denote
the space of all locally finite  nonnegative measures on $I$
with the topology of weak convergency??? (Ne nuzhno li chto-nibud'
dobavit' pro svoistvo mer, naprimer, absolytno nepreryvnye
otnositel'no mery Lebega ili chto nositel' singulyarnoi chasti mery
ne bolee chem scheten???)
The space $\frak {Mon}$ is
a closed subset of is a closed subset of
the space of $\Cal L_{1}$ of all $L^1$-functions on $I$??? and the
space of derivatives of functions in $\frak {Mon}$ coincides with
$\frak M$ (in what sense derivatives???).

{\smc Main definition. \/}  Denote by
$GP_{n}$  the space of all functions on $I$ having

i)\quad  continuous derivatives of order up to $n-2$;

ii)\quad piecewise continuous and nondecreasing
    derivatives of order $n-1$.

The above 2 conditions are equivalent to the fact that the
$n$-th derivative (understood in the sense of generalized functions)
belongs to $\Cal M$.
Obviously, $GP_{n}\subset PP_{n}$. We call functions
from $GP_{n}$ {\it generalized degree $n$ positive type pseudopolynomials}.
For a $C^1$-function $\phi$ defined on $I$ we call by its {\it pseudozero}
the maximal interval where $f$ vanishes identically.  (The same
definition works
even for continuous functions having finite
right and left derivatives at every point.) A {\it generalized zero} of $\phi$
is either a usual zero or a pseudozero.
\medskip
{\smc Lemma.} Any $f\in GP_{n}$ has on $I$ at most $n$ generalized zeros of
which at most $1$ is a pseudozero.

\demo {Proof} Obvious. \qed

\enddemo

{\smc Definition.} A generalized pseudopolynomial $f\in GP_{n}$ is called
{\it hyperbolic} if it has exactly $n$ generalized zeros counted with
multipllicities. Kak schitat' kratnost' psevdo nulya??? Let us
denote by $GHyp_{n}\subset GP_{n}$ the space of all generalized hyperbolic
degree $n$ pseudopolynomials.

{\smc Definition.} One can define the correspondence $G\Psi:
GHyp_{n}\to \Omega_{n}???$ by sending each $f\in GHyp_{n}$ to the set
of all $\binom {n} 2$ (generalized) zeros of
$f,f',f'',\ldots,f^{(n-1)}$.


{\smc More notation.}  Let $\overline \Theta_{n}$
denote the closure of $\Theta_{n}$. Obviously, $\overline \Theta_{n}$
lies in the closure $ \overline \Omega_{n}$ of $\Omega_{n}$.
\medskip
{\smc Lemma.} The image $G\Psi(GHyp_{n})$ coincides with $\overline
\Theta_{n}$.

\demo {Proof} Obvious. \qed

\enddemo
\medskip
Instead of the infinite-dimensional $GHyp_{n}$  we now choose its
$\binom n 2$-dimensional subfamily which will be homeomorphically
mapped by $G\Psi$ onto $\overline \Theta_{n}$.

{\smc Notation.} Let $\Cal {MM_{n}}\subset \Cal M_{n}$ be the set of all
finite measures whose support consists of !!! points. The inverse
image of $\Cal {MM_{n}}$ under the map of taking the $n$-th
derivative is called $\frak {Ess}_{n}$.
\medskip
{\smc Lemma.} $\frak {Ess}$ is it !

We start with some general statements about $PHyp_{n}$ and $\Psi$.

\medskip
{\smc Lemma 2.1.} The map $\Psi_{n} :PHyp_{n}\to \Omega_{n}$ is a surjection
on the image $\Theta_{n}$ with contractible fibers.

\demo {Proof}  We first prove contractability of the fibers of $\Psi$.
Let $\bar X=\{(x_{1}^{(0)}<x_{2}^{(0)}<\ldots<x_{n}^{(0)});\;$
$(x_{1}^{(1)}<x_{2}^{(1)}<\ldots<x_{n-1}^{(1)});\;\ldots;\;(x_{1}^{(n-1)})\}$
be some point of $\Theta_{n}$. By definition, the
necessary and sufficient conditions that a function $f$ belongs to
$PHyp_n$ and $\Psi(f)=\bar X$ are
$$
\left\{\aligned &f= (x-x_{1}^{(0)})\ldots(x-x_{n}^{(0)})\phi_{0}\\
                     &f'=(x-x_{1}^{(1)})\ldots(x-x_{n-1}^{(1)})\phi_{1}\\
                      &\vdots \\
                   &f^{(n-1)}=(x-x_{1}^{(n-1)})\phi_{n-1}\endaligned \right. ,
$$
where $\phi_{0},\ldots,\phi_{n-1}$ are some smooth strictly positive
functions defined on $I$. For a fixed $\bar X$ every condition
$f^{(i)}=(x-x_{1}^{(i)})\ldots(x-x_{n-i}^{(i)})\phi_{i}$ where
$\phi_{i}$ runs over the set of all strictly positive functions
determines a convex cone (of codimension $n-i$) in the space of all
smooth functions. Thus the set of solutions to the above system is
convex and therefore contractible if nonempty.
It remains to prove that
$\Psi$ is open and surjective.
The openness is obvious at least for any $f\in PHyp_{n}$ since
any sufficiently small deformation of $f$ in the
$\bold {C}^{n-1}$-topology still belongs to
$PHyp_{n}$, i.e. we can independently
and arbitrarily deform all zeros in $\bar X$ if they are all distinct
by a $\bold {C}^{n-1}$-small deformation of $f$. Thus we have to
treat the case when some
zeros of derivatives of $f$ coincide. (Note that by definition of
$PHyp_{n}$ no zeros of
consecutive derivatives of $f$ can coincide since this implies the existence
of a double zero of $f$.) The local deformation argument is then as
follows.
Choose $g:(\bR,0)\to\bR$ a germ of a smooth function
satisfying the following properties. Its derivatives up to order $n-1$ have
at most simple zeros at the origin, i.e. no pair of two consecutive Taylor
coefficients in its $(n-1)$-jet at the origin vanish.
Let $0\le j_1<j_{2}<\dots <j_k\le n-1$ be the orders of vanishing coefficients
in the $(n-1)$-jet of $g$ at the origin or, equivalently,
the orders of vanishing derivatives.
We show that there exist small perturbations of $g$ with the above
properties and with an arbitrary given arrangement of simple zeros
of the derivatives of orders $j_1,\dots , j_k$  near the origin.
       Add to $g(x)=\sum_{i=0}^{n-1}a_ix^i+\dots$
       where $a_{j_1}=\dots = a_{j_k}=0$
a germ $h(x)=b_{j_1}x^{j_1}+\dots +b_{j_k}x^{j_k}$. The zero of the $j_{l}$-th
derivative $(l=1,\ldots,r)$ of $g+h$ near the origin is up to infinitesimals
of higher order given by the equation
$x\approx -\frac {b_{j_{l}}}{(j_{l}+1)a_{j_{l}+1}}.$  Recall
that $a_{j_{l}+1}\neq 0$ by the assumption on $g$.
Therefore by adding appropriate $b_{j_{l}}x^{j_{l}}$ one can easily get any
chosen arrangement of zeros of $g$'s derivatives of orders
$j_{1},\ldots,j_{k}$ near the origin. Using this argument
for some $f\in PHyp_{n}$ with several coinciding zeros of its
derivatives we can make its finite supported perturbation with the
estimated modulus of its derivatives up to order $n$ producing an
arbitrary small deformation of all coinciding zeros. (Independent
deformation of several coinciding zeros is the step which can not be
in general performed within the space of polynomial of some bounded
degree.) \qed
\enddemo

\medskip
{\smc Corollary 2.2.} The space $PHyp_{n}$ is contractible and therefore the
image $\Theta_{n}$ is contractible as well.


\medskip





\heading {\S 3. Final remarks.} \endheading
