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\begin{document}

\title{ First steps of local contact algebra}

\author{ V.I.Arnold
\thanks{This work was partially supported by the Russian
 Basic Research Foundation,
project 99-01-01109 and by the Institute Universitaire de France. 
}}

\date{}
\maketitle

\rightline{Dedicated to H.S.M. Coxeter}

\section{  Introduction } %s1

The belief that all simple (having no continuous moduli)
objects in the nature are controlled by the Coxeter groups
is a kind of religion. The corresponding theorem
in singularity theory is due to A.B.Givental \cite{1}. It
classifies simple singularities of caustics and wave fronts,
 defined by the projections of Lagrange and Legendre subvarieties of
 symplectic and contact manifolds, in terms of the Coxeter
 euclidean reflections groups, extending
 to the case of singular varieties my
 previous $A-D-E$ -- classification \cite{2}
  (corresponding to smooth submanifolds).

The present work is an attempt to start the classification of singular
simple curves in contact manifolds.

The idea that every reasonable mathematical theory should
 have a symplectic and contact versions is also based
 on the success of Coxeter's extension of linear algebra
 (considered as the theory of the root systems $A$)
 to other mirrors configurations. The application of this
 idea to calculus has led to the foundation of
 symplectic and contact topologies (see \cite{3}).

 In the present article the same idea is applied to a modest local problem.
 It is astonishing that this problem -- the classification
 of simple curves in a contact space -- is rather difficult and
 leads to interesting new
 interactions of the local commutative algebra
 with symplectic and contact structures.

A contact structure on  an odd-dimensional manifold is a
field of hyperplanes in the tangent spaces,
 which is completely nonintegrable.
The classical Darboux-Givental theorem claims that a germ of a smooth
submanifold of a contact manifold is well defined (up to a contactomorphism,
preserving the contact structure) by the induced structure on the submanifold
(see \cite{4}).

The present paper shows that at a singular point of a curve there exist
more invariants -- some ghost of the contact structure persists. It
would be interesting to describe this ghost algebraically, in terms of the
local algebra of the singularity and of the Poisson brackets. Such a formula
is missing in the paper. I just calculate the normal forms, showing the
existence of the ghost.

{\it A singularity of a curve} is a germ  of
 a smooth (holomorphic, formal,...)  mapping of the
 line into a smooth (holomorphic, formal,...) manifold at a singular
point (where the derivative of the mapping vanishes). The singularities
are considered   up to
the diffeomorphisms   of the images in the target manifold (that is, up
to the right-left equivalence in terms of the singularity theory).

A singularity is called {\it simple} if all the singularities of
the neighbouring mappings
belong to a finite set of equivalence classes. The classification of
simple
singularities of curves is described in \cite{5}.

The codimension of the set of nonsimple singularities of
mappings of a line to $\C^N$ equals to $6N-6$ if $N$ is large
(nonsimple singularities first occur at some points of some curve of generic
families,
if the number of parameters reaches $6N-7$).

The list of simple singularities of curves starts with the series
$$
A_2\quad \leftarrow\quad A_4\quad \leftarrow\quad \dots, \qquad A_{2k} =
(s^2,\, s^{2k+1},\, 0,\, \dots,\, 0).
$$

These are all the singularities, whose Taylor series contain nonzero quadratic
terms (neglecting those which are infinitely degenerate).
The list of the simple singularities contains also the (finitely degenerate)
singularities, whose Taylor expansions start from nonzero cubic terms
(they form two 3-indices series, related to $E_6$ and to $E_8$).
There are also seven 1-index series of finitely degenerate singularities,
whose 6-jet has the form $(s^4,\, s^6,\, 0,\, \dots ,\, 0).$

The remaining 32  {\it sporadic} simple  singularities have the Taylor
series starting 
with $(s^4,\, s^5,\, 0,\, \dots ,\, 0)$ (7 curves), 
with $(s^4,\, s^7,\, 0,\, \dots ,\, 0)$ (15 curves),
with $(s^4,\, s^9,\, 0,\, \dots ,\, 0)$ (1 curve), 
with $(s^5,\, 0,\, \dots ,\, 0)$ (6 curves), and 
with $(s^6,\, 0,\, \dots ,\, 0)$  (3 curves).

They  live in spaces of dimension at least 2 (5 curves),
at least 3 (13 curves),
at least 4 (9 curves),
at least 5 (4 curves),
and at least 6 (one curve).

I mention here this information, hoping that it might help someone
to find the missing relation of this list of simple curves to other
simple objects. The special role of the number 6 suggests that the
affine version of $E_6$ might be responsible to the first
nonsimple singularities generation  (like in the $A - D - E$
theory) and the cardinality 32 suggests a possibility of a relation to the
complex reflection groups.

The main results of the present paper  are the classifications of the simple
singularities of integral curves of series $A_{2k}$ in contact manifolds
and  of the  nonintegral curves with a semicubical singularity
$A_2.$ The solution of the last problem is based on the study of the simplest
objects of the symplectic and contact  local algebra -- of the Diraciens
of the ideals in the algebras of functions on symplectic manifolds
and of their contact versions.



\section{Integral curves of series A}

The $A_{2k}$ -- singularity is the singularity diffeomorphic to that of the
plane curve $ x^{2k+1} = y^2$ at the origin.

Consider the contact space $\C^{2n+1}$ equipped with the Darboux coordinate
system and with the contact structure $ dz - \sum p_i dq_i.$  Consider
the integral curves $ A_{2k,r}$ whose projections to the symplectic
$2n$--space are
$$
A_{2k,0}:\quad (q_1 = s^2,\ p_1 = s^a,\  p_i = q_i =0\quad {\rm for}\quad i>1);
$$
$$
A_{2k,r}:\quad (q_1 = s^2,\, q_2 = s^a,\, p_1 = s^b,\, p_2 =0,\,
 p_i = q_i =0\quad {\rm for}\quad i>2),
$$
where $a =2k+1,\quad b = a+2r,\quad r>0$;  of course
$ z =(2/b +2)s^{b+2}$ along such integral curves.


\begin{theorem}\label{T1} %1
Almost every  integral curve in a contact space, whose quadratic term is
non\-zero, is simple and contact equivalent to one of the curves
$ A_{2k,r} \quad (0\leq r \leq 2k) $ above.
\end{theorem}

 The exceptional
curves are nonsimple and form a
subset of codimension infinity  consisting of infinitely
degenerate curves.

One might replace $s^b$ by $0$ in the normal form $A_{2k,2k}$  --
the corresponding integral curves are diffeomorphic.

The stratum $ A_{2k,r}$ has codimension $ (2n-1)(k-1)+r$
in the space of integral curves in $\C^{2n+1}$ having a singular
point at the origin.
In 3-space only the series $ A_{2,0}\leftarrow A_{4,0} \leftarrow \dots $
occurs,\ ${\rm codim}(A_{2k,0}) = k-1.$


\section{Semicubical singular curves in contact spaces}

A semicubical singularity is the singularity diffeomorphic to
that of the curve  $ x^3 = y^2$ at the origin.

This plane  curve is the image of the maping $ x=s^2,\ y=s^3.$
A singularity of a mapping into a contact space is {\it simple},
if the singularities of the images of the neighbouring mappings are
contactomorphic to those of a finite set of curves.

\begin{theorem} %2
 There are exactly 5 simple semicubical singularity types
   in the contact space
$\C^{2n+1}.$ They are contactomorphic to the singularities
$$
\begin{array}{ccccccccc}
a^0 &\leftarrow&  b^1&\leftarrow& c^2 &\leftarrow& e^3&\leftarrow & f^4\\
&&&&          \uparrow           &    &       \uparrow && \uparrow     \\
&&&&   (d^3)    &\leftarrow&  (h^4) &\leftarrow& (j^5),               \\
\end{array}
$$
where
$$
\begin{array}{l}
a^0: \quad z=s^2,\quad q_1 =s^3,\quad p=0,\quad q_{>1} =0;    \\
b^1: \quad z=s^3,\quad q_1 =s^2,\quad p=0,\quad q_{>1} =0;    \\
c^2: \quad z=s^4,\quad q_1 =s^2,\quad p_1=s^3,\quad p_{>1}=q_{>1} =0;    \\
e^3: \quad z=s^4,\quad q_1 =s^2,\quad q_2=s^3,\quad p_1=s^5,\quad p_{>1} = q_{>2} =0;    \\
f^4: \quad z=s^4,\quad q_1 =s^2,\quad q_2=s^3,\quad p=0,\quad q_{>2} =0.    \\
\end{array}
$$
\end{theorem}

Here the superscript denotes the codimension in the space of singular
curves in $\C^{2n+1}.$ The classes in brackets are not simple. For $n=1$
(that is,in the contact 3-space) only 3 simple singularities
$ a^0\leftarrow b^1 \leftarrow c^2 \leftarrow (d^3) $ exist.

The nonsimple singularities of semicubical curves in a contact space
form a set of codimension 3 in the space of singular curves.
Such singularities occur in $m$-parametric families of generic curves
in the contact $2n+1$-space if $m\geq 2n+3.$
 In the contact 3-space the nonsimple
semicubical singularities first occur when the number of parameters equals 5.

\section{Classification of integral curves  A }

Consider the projection $(z,p,q)\mapsto (p,q)$ of the contact space
$\C^{2n+1}$ onto the symplectic space $\C^{2n}.$
We may suppose, without loss of generality, that the $z$-axis at the
singular point of our curve having the $ A_{2k}$ -- singularity,
does not belong to the tangent 2-plane of the curve at the singular point.

The projected curve has then also the $A_{2k}$-singularity. The symplectic
classification of such singularities is given in \cite{6}. It is just the
list of $A_{2k,r}$ in Theorem \ref{T1} above $ (0\leq r\leq 2k).$

The symplectomorphism $ h(p,q) = (P,Q),$ reducing the
projection to the symplectic normal form, can be lifted to a
 contactomorphism $g$ of the $(z,p,q)$-space by the formula
 $$
 g(z;p,q) = \left( z+S(p,q); P(p,q),Q(p,q)\right).
 $$
 The generating function $S$ is defined here by the relation
 $ dS =PdQ-pdq.$

 The contactomorphism $g$ sends the integral curves onto the
 integral curves. It reduces the initial integral
 curve $\gamma$ to the normal form of Theorem \ref{T1}.

 The integral curves $A_{2k,r}$ are pairwise contact nonequivalent.
 Indeed, $k$ is a diffeomorphism invariant. The contact invariance
 of
 the index $r>0$ follows from its description as of {\it the order of the
 closeness of $\gamma$ to the closest smooth integral 2-surface.}

 To prove this description, consider the isotropic 2-surface, providing the
 closest approximation of the projected curve in the symplectic space. The
 distance to this surface is vanishing along the projected curve as
 $s^b$ if $r<2k$ and is vanishing for $r =2k$ (according to \cite{6}).
 Integrating $pdq$ we lift this isotropic 2-surface to an integral
 surface approximating the initial integral curve at the same accuracy.
 Indeed, the difference between the values of $z$ along two intersecting
 integral curves whose projections are $b$-th order tangent in the
 symplectic space is $b+1$-th order tangent to $0$.

 \section{Diracians and contactians}

The classification of simple nonintegral curves in contact spaces is
 based on some algebraic preliminaries which I shall give
 in a slightly more general situation, than is needed for the proof of
 Theorem 2.

 Let $I$ be an ideal in the algebra of smooth (holomorphic, formal,...)
 functions on a symplectic (or Poisson) manifold.

 \medskip
 {\bf Definition.} The {\it Diracian} $DI$ of the ideal $I$ consists of those
 elements of $I,$ whose Poisson brackets with all the functions of $I$
 belong to $I$.

 \medskip

 {\bf Remark.} It seems, that Dirac has not considered these ideals.
 However his classification of constrains is a predecessor of the local
 symplectic algebra.

 \medskip
 {\bf Proposition 1.} {\it The Diracian of an ideal is an ideal.}

 \medskip
 {\bf Proof.} Let $ H\in DI,\,\, B\in I. $ Then for any $A\,$ one has
  $\{AH,B\} =
 A\{H,B\} + H\{A,B\} \in I,$ since $\{H,B\}\in I$ and $ H\in I.$
 Therefore, $AH\in DI.$

 \medskip
 {\bf Proposition 2.} {\it The Diracian of an ideal contains the square of
 this ideal.}

 \medskip
 {\bf Proof.} Let $ A,B,C \in I.$ Then $\{AB,C\} = A\{B,C\} + B\{A,C\}\in I.$
 Therefore $AB\in DI$ and $I^2 \subset DI.$

 \medskip

 {\bf Proposition 3.}{\it The Diracian of an ideal is a Lie algebra
 (with respect to the Poisson bracket).}

 \medskip
 {\bf Proof.} Let $ A,B \in DI,\quad C\in I.$
 According to the Jacoby identity, $\{\{A,B\},C\}=\{A,\{B,C\}\}+
 \{B,\{C,A\}\}.$
 Since $B\in DI,$ we get $\{B,C\}\in I.$ Since $A\in DI,$
 we conclude that $\{A,\{B,C\}\}\in I.$  Similarly,
 $\{B,\{C,A\}\}\in I.$ Therefore   $\{\{A,B\},C\}\in I,$
 whence $\{A,B\}\in DI.$

 \medskip

 {\bf Proposition 4.} {\it Let the ideal $I$ be generated by the generators
 $F_i.$ Then $H\in DI$ if and only if $\{H,F_i\}\in I.$}

 \medskip
 {\bf Proof.} $\{H,A F_i\} = \{H, A\}F_i + \{H,F_i\}A.\quad$ If
 $\{H,F_i\}\in I,$ we get $\{H,AF_i\}\in I,$ since $F_i\in I.$

 \medskip

 {\bf Proposition 5.} {\it Let the ideal $I$ be generated by the
 generators $F_i.$ Then the function $ F =\sum A_iF_i$ belongs
 to the Diracian of $I$ if and only if $\, \sum A_i\{F_i,F_j\}\in I $
 for every $j.$}

 \medskip
 {\bf Proof.} According to Proposition 4, we should verify the conditions
 $\{F,F_j\}\in I.$ But we have
 $$
 \{F,F_j\} = \sum A_i\{F_i,F_j\} +\sum F_i\{A_i,F_j\}
 $$
 and the last
 term belongs to $I.$

 In the three examples that follow we consider the ideals in the $\C$-algebra
 of the germs of holomorphic functions of 4 coordinates
 $x=q_1,\, y=q_2,\, u=p_1,\, v=p_2\,$
 at the origin of the space $\C^4$ equipped with the symplectic srtucture
 $dp_1\wedge dq_1 +dp_2\wedge dq_2.$ The word "generated" means "generated
 as a module over this algebra."

 \medskip

 {\bf Example 1.} Consider the ideal $I$ of the functions, vanishing on
 the semicubical curve $A_{2,0}$ defined by the equations
 $ u^2 =x^3,\, y=v=0.$
 It is generated by the three functions $ ( H=u^2-x^3,\, y, \, v).$

 \medskip

 {\bf Proposition 6.}{\it The Diracian $DI$ is generated by
 $ ( H,\, v^2,\, uv,\, y^2).$}

 \medskip
 {\bf Proof.} Let $ F =AH +Bv +Cy \in DI.$
 Since $\{H,v\} =\{H,y\} =0,\quad \{v,y\}=1,$
 the criterium of Proposition 5 takes the form
 $B \in I,\, C\in I,$ whence $ F = AH + Kv^2 +Lvy +My^2,$
 as required.

 \medskip
{\bf Example 2.} Let I be the ideal of functions vanishing on the
semicubical parabola $ x^3 =y^2,\ u=v= 0.$ It is generated by  three
functions $ H = y^2 - x^3,\, u,\, v).$

\medskip

{\bf Proposition 7.}{\it The Diracian $DI$ is generated by $I^2$ and two
more functions
$$
H_e = 2xu +3yv,\qquad H_h = 2y u +3x^2 v
$$
(and hence by $ (H^2,\, H_e,\, H_h,\, u^2,\, uv,\, v^2)$).
}

\medskip
{\bf Proof.} Since $I^2$ belongs to $DI,$ we may replace all the functions by linear
(inhomogeneous) functions in $u$ and $v.$ A linear homogeneous function
$ B(x,y)u + C(x,y)v$ may be interpreted as a vector field
$\ksi = B\frac{\partial}{\partial x} + C\frac{\partial}{\partial y}$
on the plane $(x,y)$ (considering $u$ and $v$ as the coordinates of the
cotangent vector $udx +vdy$).

Applying the Proposition 5 to $F = AH +Bu +Cv,$ we first find the Poisson
brackets
$$
\{H,u\} = 3x^2,\quad \{H,v\} = -2y,\quad \{u,v\}=0.
$$
Then we rewrite the criterium of Proposition 5
$$
-3x^2 B +2y C \in I,\qquad 3x^2A \in I,\qquad -2yA \in I,
$$
where the coefficients $A, B, C$ depend only on $x$ and $y$
and $I$ denote the ideal, generated by $H$ in the algebra of functions of
$x$ and $y$.

The last conditions imply $ A\in I,\, AH\in I^2.$
The first condition has the form $\ksi H\in I.$
It means that the vector field $\ksi$ is tangent to the curve $H=0.$

Two tangent fields are the Euler quasihomogeneous field $\ksi_e =
2x\frac{\partial}{\partial x} +3y\frac{\partial}{\partial y}$
and the Hamilton field
$\ksi_h =
2y\frac{\partial}{\partial x} +3x^2\frac{\partial}{\partial y}.$
All other solutions are their combinations
$ \ksi = \alpha(x,y)\ksi_e + \beta(x,y)\ksi_h.$

Indeed, $\ksi_e H = 6H,$ hence, choosing $\alpha,$ we may obtain
any value
of the coefficient $G$ in $\alpha \ksi_e H = GH \in I.$ To find
all the solutions $\ksi$ of the equation $\ksi H = GH,$
it remains to solve in $B_0$ and $C_0$ the linear homogeneous
equation $\ksi_0 H = 0,$ which has the form
$ -3x^2B_0 +2y C_0 = 0.$ We obtain $ B_0 = 2y\beta,\,\, C_0 =3x^2\beta.$
Therefore $\ksi_0 = \beta \ksi_h,\,\, \ksi = \alpha\ksi_e +\beta\ksi_h.$

\medskip
{\bf Example 3.} Let $I$ be the ideal of functions, vanishing on the semicubically
singular curve $ x^3=y^2,\quad u=xy,\quad v=0.$ It is generated by three
functions $(H=y^2 - x^3,\, L = u -xy,\, v).$

\medskip

{\bf Proposition 8.} {\it The Diracian $DI$ is generated by $I^2$ and by two
more functions
$$
H_e = 2x^2u +3xyv+a,\qquad H_h =2yu +3x^2v+b,
$$
where $a=y^3-3x^3y,\quad b=-x^4-x^2y.$}

\medskip
I shall only use that $ H_e\, , \ H_h \in DI.$
This fact follows from the vanishing of the Poisson brackets
of these functions with $H,L$ and $v$ (according to Proposition 4).

\medskip
{\bf Remark.} The contact Hamilton functions are rather sections of the
line bundle, defined by the contact structure, than functions.
Therefore the contact Diracians are rather submodules than ideals.

In the standard contact space
 one may identify contact Hamilton functions with ordinary
functions (see \cite{7}). {\it The contactian} $CI$ of an ideal
 of such functions
consists of those functions belonging to $I,$ whose contact flows preserve $I$.
This ideal, intermediate betwen $I$ and $I^2,$ is a Lie subalgebra with respect
to the contact Poisson brackets $\{.,\}_C$ (see \cite{7}).

The condition $K\in CI$ for a function $K$  of an ideal generated by $ A_i,$
may be written in the form $\{K,A_i\}_C \in I.$


\section{Semicubical nonintegrable curves}

The tangent 2-plane of a semicubical singularity of a curve in a
contact space is generically transversal to the contact hyperplane at the singular point.
Otherwise it belongs to the contact hyperplane.

\medskip
{\bf Case 1.} Let it be transversal. Then the smooth 2-surface,
containing the curve, can be transformed into coordinate $(z,q_1)$ --plane
by a contactomorphism
(according to the Darboux-Givental theorem, see \cite{4}).

The structure, induced by the contact structure on this plane is the fibration $dz=0.$
The fibered diffeomorphisms of the plane are extendable to contactomorphisms.

The normal forms of the semicubical
 singularities of curves in a fibered plane have been
obtained already in \cite{8,9}
(using the "invariants convolution" operation for
the Coxeter group $A_2$).

These normal forms are $q^2=z^3$ (the generic case -- the fiber direction
$dz =0$ is not tangent to the curve at the origin) and $z^3=q^2$ (the
degenerate case -- the fiber is tangent to the curve).

We thus get the cases $a^0$ and $b^1$ of Theorem 2.

\medskip
{\bf Remark.} These two normal forms are also related to
the Coxeter groups $D_5$ and $E_7$ corresponding to the
functions $z(q^2 - z^3)$ and $z(q^3 - z^2).$

\bigskip

{\bf Case 2.} The tangent 2-plane of the curve at the singular point
belongs to the contact hyperplane.

Project the curve to the symplectic space along the $z$--direction.
Reduce the resulting $A_2$--curve to one of the three normal forms
$(c,\quad d,\quad e)$ of Theorem 2 by a symplectomorphism (as in section 4).
Lifting it to a contactomorphism, we get $z=h(s),$ where
$h(s) = c s^4 + \dots$ (since the tangent 2-space belongs to the
contact hyperplane).

Suppose that $c\neq 0$ (as it is generically). Rescaling the coordinates,
we can construct a contactomorphism and a change of the coordinate $s$
such that $c$ will be equal to $1.$

\medskip

{\bf Theorem 3\ (c,e,f)\,}
{\it Every curve, defined by the above projection $\, c,\, e $ or $f$
and by $ h =s^4 + \dots,$ is contactomorphic to the curve with
 the same projection to the symplectic space, for which $ h = s^4.$}

\medskip

These three theorems are proved in sections 7, 8 and 9.

\medskip
 Suppose now that $h(s) = o(s^4)$ (cases $d,h,j$ of Theorem 2).

 \medskip

 {\bf Theorem 4.}{\it No curve, defined in the contact 3-space with the
 contact structure $dz + pdq =0$ by the equations
 $\, q=s^2,\, p=s^3,\, z =cs^5\,$ is simple : $c$ is a modulus.}

 \medskip
 {\bf Proof.} The contact vector field, defined by the contact
  Hamilton function $K,$ has the form (see \cite{7}):
  $$
  \dot p = - K_q + p K_z , \qquad \dot q = K_p ,\qquad \dot z = K - p K_p.
  $$

The existence of the required equivalence would imply the solvability of
  the homological equation
  $$
  \left\{\begin{array}{ccc}
  2su(s) +K_p & = & 0,\\
  3s^2u(s) - K_q + p K_z & = & 0, \\
  5cs^4 u(s) + K - p K_p & = & s^5
  \end{array}
  \right.
  $$
  in the unkowns $u$ and $K$ (representing the deformations of
   the independent and dependent variables).

   Function $K$ should here be evaluated at the point $(q=s^2,\, p=s^3,\,
   z=cz^5).$
   One also should have  $u(0) = K(0) = K_p(0) =K_q(0) = 0\,$
   (the singular point remaining at the origin).

   These linear equations are contradictory. Indeed, they are
   quasihomogeneous (for the weights of variables
   ${\rm deg}\, s =1,\quad{\rm deg}\, q =2,\quad{\rm deg}\, p =3,\quad
    {\rm deg}\, z =5).$
  Equating terms of degrees 2, 3 and 5 in the first, second
  and third equations, we get for the coefficients of the Taylor series
  $u(s) = as + \dots,\!$ $ K = Apq + B z + \dots\, $ the relations
  $2a + A = 0,\quad 3a + (B - A) = 0,\quad 5ca + B c = 1, $
  which are contradictory.

  \medskip

{\bf  Corollary.}{\it There are no simple singularities among
the semicubical curves with $h(s) = o(s^4).$}

\medskip
{\bf Proof.}
Consider the class $(d)$ of curves, whose projection
to the symplectic plane is of type $c$ but for which
$h(s) = cs^5 +({\rm higher\, order\, terms}).$

There are no simple curves is this class, since the contradiction,
described above, remains when the higher order terms are taken
into account.

But all the semicubical singularities, for which $ h(s) =o(s^4)$,
are adjacent to the curves of class $(d)$ (since classes $e$ and $f$
are adjacent to $c$). Whence the corollary.

\medskip
{\bf Remark.} The result remains true in any dimension, since the
nonsimple curves of class $(d)$ remain nonsimple in any dimension
(it follows, for instance, from the Darboux-Givental theorem but
can also be deduced from the calculations above).

\section{Proof of Theorem 3c}

We need the following three statements on the representations
 of holomorphic functions
by combinations of other holomorphic functions. They remain true if
the given functions depend holomorphically on the parameters (of which the
representation coefficients will then depend holomorphically).

\medskip
{\bf 1$^o$.} {\it If $A(s^2,s^3) = 0,$ one has}
 $ A(x,y) = B(x,y)(y^2 - x^3).$

 \medskip
Indeed, one always can write
$A = a(x) + y b(x) + B(x,y)(y^2 -x^3),$ and if $A(s^2,s^3) = 0$
one should have $a\equiv b \equiv 0.$

\medskip
{\bf 2$^o.$} {\it Any function $s^2 f(s)$ is the value of a holomorphic
function $F(s^2,s^3).$}

\medskip
Indeed, if $ f(s) = u(s^2) + s v(s^2),$ it suffices to take
$ F = x u(s) + y v(x).$

\medskip

{\bf 3$^o.$} {\bf Lemma.} {\it If $ R(s^2, s^3) = s^a( c +s r(s)),
\quad c\neq 0,$
and $ M(s^2, s^3) = s^{a+2} m(s),$ then there exists a representation
$$
M(x,y) = B(x,y) R(x,y) + C(x,y) ( y^2 - x ^3).
$$
}

\smallskip
Indeed, $\, s^2 m(s) \left( c -sr(s)\right) = F(s^2, s^3),\quad
c^2 -s^2r^2(s) = G(s^2, s^3),\quad G(0)\neq 0,$ according to {\bf 2$^0$}.
For $ B =F/ G $ one gets $ (M -BR)(s^2,s^3) = 0 $ and the existence
of $C$ follows from {\bf 1$^0$}. Lemma {\bf 3$^o$} is proved.


\medskip
Now consider the curve $\Gamma$  in $\C^3$ with the contact structure 
$dz + pdq = 0$
(note the sigh), defined by the equations $ H(p,q) =0,\ z =F(p,q)$
(we are mostly interested in the case $H = p^2 - q^3,\ F =q^2$
of the standard curve $(q = s^2,\, p = s^3,\, z =s^4)$).

The contact Hamiltonian $K$ defines the contact vector field (see \cite{7},
appendice 4, for the choice of the sign of $z$) :

\begin{equation}\label{1}
\frac{dp}{dt} = - K_q + pK_z  ,\quad \frac{dq}{dt} =  K_p,\quad \frac{dz}{dt} = K - p K_p.
\end{equation}

We choose $ K = A(p,q) + \left(z - F(p,q)\right)B(p,q).$ Denote by
$P(t),\, Q(t),\, Z(t)$ the motion, starting at $t = 0$ from a point
$(p,q,z)$ of $\Gamma,$ defined by the contact Hamiltonian $K.$

Define the initial velocity $ \dot H =
 \left.\frac{d}{dt}\right|_{t=0} 
H(P(t), Q(t)).$
We   find from (\ref{1})
\begin{equation}\label{2}
\dot H = \{A,H\} + B\left(\{H,F\} + p H_p\right),
\end{equation}
where $\{I,J\}  = I_p J_q - I_q J_p$ is the Poisson bracket of
functions $I$ and $J$.

Indeed, on $\Gamma$ we have
\begin{equation}\label{3}
K = A,\quad K_p = A _p - F_p B,\quad K_q = A_q - F_q B,\quad K_z =B,
\end{equation}
therefore
$ H_p\dot p + H_q \dot q = H_p (-A_q + F_qB +pB) + H_q(A_p - F_q B).$

\smallskip
Now suppose, that $\dot H = 0$ on $\Gamma.$ Write the vertical
component of the moving curve as $Z(t) =F\left( P(t),Q(t),t\right)$
and calculate the initial velocity of the change of the function $F$
$$
\frac{\partial F}{\partial t}
 = \left.\frac{\partial}{\partial t}\right|_{t=0}F(P,Q,t).
$$
We get from (\ref{1}) and (\ref{3}) the formula
\begin{equation}\label{4}
\frac{\partial F}{\partial t} = \{F,A\} + A - p A_p.
\end{equation}

Indeed,
$\dot z = F_p\dot p + F_q \dot q + \frac{ \partial F}{\partial t},\,$ hence
$$
\frac{ \partial F}{\partial t} = \dot z - F_p\dot p -F_q \dot q =
A-p A_p +pF_pB -F_p(-A_q +F_q B +pB) -F_q(A_p -F_pB),
$$
as required.

To kill the perturbation $m$ in the equations of the curve
$$
q=s^2,\quad p=q^3,\quad z =s^4 + m(s),\quad m =s^5n(s),
$$
using the homotopy method, we shall construct a (time-dependent)
contact Hamilton function $ K = A + (z-F)B,$ where at moment $t\,\,$
$F(s^2,s^3) = s^4 + t m(s)$ and where the coefficients $A$ and $B$
verify :


\noindent
{\bf (a)} the condition $\dot H = 0$ on $\Gamma _t$ of the preservation
of
the semicubical projection of the curve to the symplectic plane,

\noindent
{\bf (b)} the homological equation $ \frac{ \partial F}{\partial t} = m $
on $\Gamma _t\,,$

\noindent
{\bf (c)} the origin fixing conditions $ A(0) =A_p(0) = A_q(0) = B(0) = 0.$

\medskip

The conditions (b) and (a) can be written as equations
$$
\begin{array}{cr}
\{F,A\} + A - pA_p  =  m(s),\ \ \ \ \ \ \ \ & (b) \\
 \{A,H\} + B\left(\{H,F\} + p H_p\right) =  0 &\ \ \ \ (a)
 \end{array}
 $$
 along the semicubical parabola $ q =s^2,\quad p = s^3.$

 To solve the linear system (b),(a) in the unknowns $A$ and $B,$
 consider the quasihomogeneous grading ${\rm deg}\, q = 2,\quad
 {\rm deg}\, p =3,\quad {\rm deg}\, H = 6.$
 First we find the solutions, verifying (a) exactly but (b) only
 approximatly:
 $$
 \begin{array}{ccccr}
 \{F,A\} + A - pA_p & =& s ^n \  {\rm mod}\ s^{n+1},\quad & n\geq 5.
 &(b')
\end{array}
$$
We shall find a solution of this equation of quasihomogeneous
degree $n+1.$ Since the equation (b) is linear, we thus reduce the
equation (b) to a similar equation with a high order zero of the
right hand side $m,$ which we shall solve explicitly.


To find the solution of the approximating equation
$(b')$,
we replace $F$ by its lower order part $q^2$ ( which is also the unperturbed $F$).
The lowest order term of the left hand side is $ -2qA_p.$ To make it equal to
$s^n$ we choose $ A = p^2q^{a-1}/4$ if $n$ is odd and
$ A = pq^{b-1}/2$ if $n$ is even (${\rm deg}\ A = n+1 \geq 6$).
We thus get the required solution $A$ of equation $(b')$.

 Choose $B$ verifying  condition (a), which we can write
 in the form
 $$
 M = B R + C H,\quad M =\{H,A\},\quad R =\{H,F\} + p H_p.
 $$


 For $F=q^2$ we get $R = 4s^5 + O(s^6).$ The higher order terms in $F$
 do not change the principal part $4s^5$ of $R.$ We also have
 for the quasihomogeneous $A$
 $$
 {\rm deg}\,M\leq {\rm deg}\,H +{\rm deg}\,A -{\rm deg}\,p -{\rm deg}\,q =
 {\rm deg}\,A + 1 \geq 7.
 $$
 Equation (a) is thus solvable in $B$ according to Lemma $3^o$ above.

 Suppose now that the right hand side $m$ of equation (b) is divisible by
 $s^7.$ In this case we put $ A = f H$ and get for $f$ the equation
 $-f R = m$ along the semicubical parabola. This equation is solvable
 according to Lemma $3^o.$ Now $\{A,H\} = 0$ along the parabola, therefore
 equation (a) is satisfyed for $B=0.$

 We have thus solved the homological equation for any $m$ divisible by
 $s^5.$ The flow of the contact vector field that we have constructed
 sends the nonperturbed curve $ (z = s^4)$ onto the perturbed one
 $\left(z = s^4 + m(s)\right).$

 \section{Proof of Theorem 3e}

 Consider the curve $\Gamma \quad ( x =s^2,\, y =s^3,\, u=v=0)$
 in the space $\C^4$ equipped with the Darboux coordinates
 $q_1 =x,\,q_2=y,\,p_1 =u,\,p_2 = v.$
 We shall reduce to the  contact normal
  form the curve in the 5-space, equipped with
 the contact structure $ dz +pdq =0,$ whose projection to $\C^4$ is
 $\Gamma$  and for which $z = s^4 +m(s),\quad m =s^5 u(s).$

 We should construct the contact homotopy over $\Gamma$ of the curves
 $z = F(x,y,t) = x^2 +t M(x,y),\quad M(s^2,s^3) = m(s).$
 Our contact time-dependent Hamilton function $K$ will not
 depend on $z.$ To preserve the projection $\Gamma$ we shall choose $K$
 in the Diracian of the ideal $I$ of the curve $\Gamma.$

 The elements of $I^2$ are not moving the curve.  Hence we try
 $$
 K = f H_e + g H_h,\quad H_e =2xu +3yv,\quad H_h =2yu + 3x^2v
 $$
(according to example 2 of section 5), where $f=f(x,y),\ g = g(x,y).$
This choice implies $\dot H =0$ on $\Gamma.$ The velocity of the
deformation of function $F,$ defining the moving curve, is calculated
as in equation (4) of section 7. Taking into account, that we have now
$ K =A,\, B=0,\, A=pA_p,\, H_e = H_h =0$ on $\Gamma,$
we get along $\Gamma$
$$
\frac{\partial F}{\partial t} = f\{F,H_e\} + g\{F,H_h\}.
$$

For the unperturbed curve $F_0 = x^2$ we find $\{F_0,H_e\} = -4x^2,
\ \{F_0,H_h\} = -4xy.$

Therefore for the perturbed curve we get along $\Gamma$ at any time
moment
$$
\{F,H_e\} = -4 s^4 + \dots,\qquad \{F,H_h\} = -4 s^5 + \dots ,
$$
where the dots mean higher order terms. Now it is easy to solve the
homological equation in $f,g$
$$
f\{F,H_e\} + g\{F,H_h\} = m(s),\quad m =s^5 n(s).
$$
We choose $g = -{n(0)}/{4} $ and get for $f$ the equation
$f(-4s^4 + \dots ) = s^6 k(s)$ along $\Gamma,$
solvable in $f(x,y)$ according to Lemma $3^o$ of section 7.

The contact flow defined by contact Hamiltonian that we have
constructed sends the nonperturbed curve onto the
perturbed one.


\section{ Proof of Theorem 3f}

Now $\Gamma$ is defined in $\C^4$ by the equations
$ x =s^2,\ y=s^3,\ u =s^5,\ v=0.$ Choose
$H = y^2 -x^3,\quad L =u -xy.\,$ As in section 8, use the generators of the
$D(I)/I^2$ provided by Example 3 of section 5 :
$$
K = f(x,y)H_e + g(x,y) H_h,
$$
$$
H_e =  2x^2u + 3xyv +a, \qquad H_h =  2yu +3x^2v +b,
$$
$$
a = y^3 -3x^3y,\qquad  b = -x^4 - xy^2.
$$

Then along $\Gamma$ we have
$$
H = L = H_e =H_h =\{H,H_e\} = \{H,H_h\} =\{L,H_e\} = \{L,H_h\} = 0.
$$
We also find $ K- p K_p = af +bg$ and along $\Gamma\quad$
$a= - 2s^9,\quad b = -2s^8.$

Denote the coordinate that we wish to normalise by the homotopy
method as $ z = F(x,y,t) = x^2 + t M(x,y),\quad M(s^2,s^3) = s^5 n(s).$

The homological equation $\{F,K\} + K - pK_p = M $ along $\Gamma$
takes the form
$$
\begin{array}{lr}
f(\{F,H_e\} +a) + g(\{F,H_h\} +b ) = m(s),\qquad m =s^5n(s) &\ \ \ \ \ \ \ \ (*)
\end{array}
$$
For the unperturbed curve $F_0 = x^2$ we get
$\{F_0,H_e\} = -4x^3,\quad \{F_0,H_h\} = -4xy.$
Therefore along $\Gamma$ we have
$$
\{F,H_e\} + a = -4 s^6{\rm mod}\,s^7,\qquad \{F,H_h\} + b = - 4 s^5{\rm mod}\,s^6.
$$
To solve the homological equation $(*)$ we first solve it ${\rm mod}\,s^6,$
choosing $f =0,\ \ g = -n(0) /4.$
Then we solve ${\rm mod}\,s^7$ the equation with the right hand side
$s^6k(s),$ choosing $ g =0, f = -k(0)/4.$

Finally, to solve the remaining equation with right hand side $s^7l(s)$
we choose $f =0$ and find $g(x,y)$ using  Lemma $3^o$ of section 7.

We thus obtain the required contact flow, sending the nonperturbed
curve onto the perturbed one.




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