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\begin{document}

\title{Singularities of fractions and  behaviour of polynomials
at infinity}
\author{V.I.Arnold\thanks {Partially supported by Russian Basic Research
Foundation, grant 96-01-01104.}}
\date{}
\maketitle

\def\R{R}
\def\RRR{R^\times}
\def\RR{R^+}

\def\cm{{|\!\!\!\Rightarrow}}

\newcommand{\eqt}{\sim\nolinebreak\hspace{-0.9em} _{_\times}\hspace{0.25em}}

\newcommand{\ru}{\begin{picture}(10,10)\put(0,0){\line(1,1){10}}
                 \end{picture}}
\newcommand{\rd}{\begin{picture}(10,10)\put(0,10){\line(1,-1){10}}
                 \end{picture}}

\def\C{{\mathbb C}}
\def\F{{\cal F}}
\def\l{\lambda}
\def\d{\partial}

\def\la{\!\!\leftarrow\!\!}
\def\da{\!\!\downarrow\!\!}
\def\ua{\!\!\uparrow\!\!}
\def\ra{\!\!\rightarrow\!\!}

\section{Classification of singularities of fractions}

In this paper we mean by {\it fraction} the ratio (ordered pair) of
two  germs of (holomorphic or smooth) functions in $n$ variables at
a point  (which usually will be refered as the origin).
 For the sake of simplicity mostly the complex case is considered below.
The results  for the real case are similar, only certain formulas
contain non-equivalent real normal forms (which differ  usually
by $\pm$ signes at certain monomials).

The singularities of fractions will be classified up to the following
equivalence relations.

\medskip

\noindent {\bf Definition 1.} Two fractions are called {\it coinciding},
if their numerators and denominators are proportional (by {\it non-zero}
function).

\medskip

\noindent {\bf Example 1.} On $\C^2$ with coordinates $x$ and $y$ the
fraction $x^2/xy$ does not coincide neither with $x/y$, nor with $x^3/x^2y$,
but it coincides with $x^2e^x/xye^x$.

\medskip

\noindent {\bf Definition 2.} Two fractions are called {\it $\R$-equivalent},
if one of them becomes coinciding with the other when transfered by  a germ of
biholomorphism of the space of variables.

\medskip

\noindent {\bf Example 2.} The fraction $1/xy$ is $\R$-equivalent to the
fraction $1/(x^2+y^2)$.

\medskip

\noindent {\bf Definition 3.} Two fractions  are called $\RR$-{\it
equivalent }if one of them is $\R$-equivalent to the sum of the other one
with a germ of a holomorphic function.

\medskip

\noindent {\bf Example 3.} The fraction  $y^2/x$ is $\RR$-equivalent,
(though not $\R$-equivalent) to the fraction $(y^2+x)/x$.

\medskip

\noindent {\bf Remark.} The {\it sum} of the fraction $P/Q$ and the
holomorphic function $A$ is by definition the fraction $(P+AQ)/Q$.

\medskip

The main result of the paper is the begining of the classification
of fraction germs up to $\RR$-equivalence. First singularities
occur to be simple (to have no continious moduli). They are classified
by simple Lie algebras
$A_k, B_k, C_k, D_k, E_6, E_7$, $E_8, F_4$ (no $G_2$!) -- the same list
as in the classification of boundary singularities (comp. \cite{1}).

Germs of generic fractional functions are $\RR$-simple at any point.
Codimension in the space of all fractions on $\C^n$ of the subspace
of fractions, which have somewhere a non $\RR$-simple
germ, is equal to $4$ (for $n\geq 2$).

 Thus, non $\RR$-simple fraction germs appear first (in  an unremovable
way) in families of fractions on $n$-dimensional manifold, depending on four
parameters (provided that $n>1$).

\medskip

\noindent {\bf Definition.} The class $Y$ of fraction germs  {\it is adjacent
to} the class $X$, if the germs from the class $X$ belong to any arbitrary small
deformations of germ $Y$.  Here by small deformation of fraction
 $p(x)/q(x)$ we mean the fraction of the family
 $P(x,\l)/Q(x,\l)$, $P(\cdot,0)=p,Q(\cdot,0)=q$ with small $\l$.

\medskip

The table of adjacencies of $\RR$-simple fraction singularities is given in
Fig.~1.

$$
\begin{array}{ccccccccccccccccccccccc}
\!&\!\!&\!\!&\!\!&\!\!&\!\!&\!\!&\!\!&\!\!&\!\!&\!\!&\!\!&\!\!&\!\!&\! (P_8)
\!&\! \!&\! (X_9) \!&\!  \!&\! (J_{10}) \!&\!\!&\!\!&\!\!&\! \\
\!&\!\!&\!\!&\!\!&\!\!&\!\!&\!\!&\!\!&\!\!&\!\!&\!\!&\!\!&\!\!&\!\!&\! \da
   \!&\!     \!&\!  \da  \!&\!     \!&\!   \da    \!&\!\!&\!\!&\!\!&\! \\
\!&\!\!&\! A_0^* \!&\! \la \!&\! A_0
\!&\!\!&\!\!&\!\!&\!\!&\!\!&\!\!&\!\!&\!\!&\!\!&\! E_6
   \!&\! \la \!&\!   E_7 \!&\! \la \!&\! E_8      \!&\!\!&\!\!&\!\!&\! \\
\!&\!\!&\!\!&\!\!&\!\!&\!\!&\!\!&\!\!&\!\!&\!\!&\!\!&\!\!&\!\!&\!\!&\! \da
   \!&\!     \!&\!  \da  \!&\!     \!&\!   \da    \!&\!\!&\!\!&\!\!&\! \\
\!&\!\!&\!\!&\!\!&\!\!&\!\!&\!\!&\!\!&\!\!&\!\!&\!\!&\!\!&\!
D_4 \!&\! \la \!&\! D_5 \!&\! \la \!&\! D_6 \!&\! \la \!&\! D_7 \!&\! \la
\!&\! D_8 \!&\! \la \!&\! \cdots \\
\!&\!\!&\!\!&\!\!&\!\!&\!\!&\!\!&\!\!&\!\!&\!\!&\!\!&\!\!&\! \da \!&\! \!&\!
\da \!&\! \!&\! \da \!&\! \!&\! \da \!&\! \!&\! \da \!&\! \!&\! \\
\!&\!\!&\!\!&\!\!&\!\!&\!\!&\!\!&\!\!&\! A_1 \!&\! \la \!&\! A_2 \!&\!
\la \!&\! A_3 \!&\! \la \!&\! A_4 \!&\! \la \!&\! A_5 \!&\! \la \!&\! A_6
\!&\! \la \!&\! A_7 \!&\! \la \!&\! \cdots \\
\!&\!\!&\!\!&\!\!&\!\!&\!\!&\!\!&\!\!&\! \ua \!&\! \!&\! \ua \!&\! \!&\!
\ua \!&\! \!&\! \ua \!&\! \!&\! \ua \!&\! \!&\! \ua \!&\! \!&\! \ua \!&\! \!&\! \\
\!&\!\!&\!\!&\!\!&\!\!&\!\!&\! A_1^* \!&\! \la \!&\! B_2 \!&\! \la \!&\!
C_3 \!&\! \la \!&\! C_4 \!&\! \la \!&\! C_5 \!&\! \la \!&\! C_6
\!&\! \la \!&\! C_7 \!&\! \la \!&\! C_8 \!&\! \la \!&\! \cdots \\
\!&\!\!&\!\!&\!\!&\!\!&\!\!&\! \ua   \!&\!     \!&\! \ua \!&\!     \!&\!
\ua \!&\! (C_{2,2}) \!&\! \!&\! (P_8^\#)
 \!&\!\!&\!\!&\!\!&\!\!&\!\!&\!\!&\!\!&\!\!&\!\\
\!&\!\!&\!\!&\!\!&\!\!&\!\!&\! A_2^* \!&\! \la \!&\!
B_3 \!&\! \la \!&\! F_4 \!&\! \la \!&\!
(X_9^\#) \!&\!\!&\!\!&\!\!&\!\!&\!\!&\!\!&\!\!&\!\!&\!\!&\! \\
\!&\!\!&\!\!&\!\!&\!\!&\!\!&\! \ua   \!&\!     \!&\! \ua \!&\!     \!&\!
\ua \!&\!\!&\!\!&\!\!&\!\!&\!\!&\!\!&\!\!&\!\!&\!\!&\!\!&\!\!&\!\\
\!&\!\!&\!\!&\!\!&\! D_4^* \!&\! \ra \!&\! A_3^* \!&\! \la \!&\!
B_4 \!&\! \la \!&\! (J_{10}^\#)
 \!&\!\!&\!\!&\!\!&\!\!&\!\!&\!\!&\!\!&\!\!&\!\!&\!\!&\!\!&\! \\
\!&\!\!&\!\!&\!\!&\!  \ua  \!&\!     \!&\! \ua   \!&\!     \!&\!
\ua \!&\!\!&\!\!&\!\!&\!\!&\!\!&\!\!&\!\!&\!\!&\!\!&\!\!&\!\!&\!\!&\!\!&\!\\
(P_8^*) \!&\! \ra \!&\! E_6^* \!&\! \ra \!&\! D_5^* \!&\!
\ra \!&\! A_4^* \!&\! \la \!&\! B_5
\!&\!\!&\!\!&\!\!&\!\!&\!\!&\!\!&\!\!&\!\!&\!\!&\!\!&\!\!&\!\!&\! \\
\!&\! \!&\! \ua \!&\! \!&\! \ua \!&\! \!&\! \ua \!&\! \!&\! \ua
\!&\!\!&\!\!&\!\!&\!\!&\!\!&\!\!&\!\!&\!\!&\!\!&\!\!&\!\!&\!\!&\!\!&\!\\
(X_9^*) \!&\! \ra \!&\! E_7^* \!&\! \ra \!&\! D_6^* \!&\!
\ra \!&\! A_5^* \!&\! \la \!&\! B_6
\!&\!\!&\!\!&\!\!&\!\!&\!\!&\!\!&\!\!&\!\!&\!\!&\!\!&\!\!&\!\!&\! \\
\!&\! \!&\! \ua \!&\! \!&\! \ua \!&\! \!&\! \ua \!&\! \!&\! \ua
 \!&\!\!&\!\!&\!\!&\!\!&\!\!&\!\!&\!\!&\!\!&\!\!&\!\!&\!\!&\!\!&\!\!&\!\\
(J_{10}^*) \!&\! \ra \!&\! E_8^* \!&\! \ra \!&\! D_7^* \!&\! \ra \!&\!
A_6^* \!&\! \la \!&\! B_7
\!&\!\!&\!\!&\!\!&\!\!&\!\!&\!\!&\!\!&\!\!&\!\!&\!\!&\!\!&\!\!&\! \\
\!&\!\!&\!\!&\!\!&\! \ua \!&\! \!&\! \ua \!&\! \!&\! \ua
 \!&\!\!&\!\!&\!\!&\!\!&\!\!&\!\!&\!\!&\!\!&\!\!&\!\!&\!\!&\!\!&\!\!&\!\\
\!&\!\!&\!\!&\!\!&\! D_8^* \!&\! \ra \!&\! A_7^* \!&\!
\la \!&\! B_8
 \!&\!\!&\!\!&\!\!&\!\!&\!\!&\!\!&\!\!&\!\!&\!\!&\!\!&\!\!&\!\!&\! \\
\!&\!\!&\!\!&\!\!&\! \ua \!&\! \!&\! \ua \!&\! \!&\! \ua
 \!&\!\!&\!\!&\!\!&\!\!&\!\!&\!\!&\!\!&\!\!&\!\!&\!\!&\!\!&\!\!&\!\!&\!\\
\!&\!\!&\!\!&\!\!&\! \vdots \!&\!\!&\! \vdots \!&\! \!&\! \vdots
\!&\!\!&\!\!&\!\!&\!\!&\!\!&\!\!&\!\!&\!\!&\!\!&\!\!&\!\!&\!\!&\!
\end{array}
$$
$$
\begin{array}{c}
\begin{array}{cc}
A_k=\frac{y^{k+1}+z^2}{x}, & A_k^*=\frac{1}{x^{k+1}+y^2+z^2}; \\
D_k=\frac{y^2z + z^{k-1}}{x} & D_k^*=\frac{1}{x^2y+y^{k-1}+z^2}; \\
E_6=\frac{y^3+z^4}{x}  & E_6^*=\frac{1}{x^3+y^4+z^2}; \\
E_7=\frac{y^3+yz^3}{x} & E_7^*=\frac{1}{x^3+xy^3+z^2}; \\
E_8=\frac{y^3+z^5}{x} & E_8^*=\frac{1}{x^3+y^5+z^2} ;\\
B_k=\frac{x}{x^k+y^2+z^2} & C_k=\frac{x}{xy+y^k+z^2} ;
\end{array} \\
F_4=\frac{x}{x^2+y^3+z^2}.
\end{array}
$$
\begin{center}Fig. 1. Classification of $R^+$-simple fraction singularities
(formulas are written for the case of three variables).
\end{center}

\medskip

The symmetry of this table is closely related to the Lagrange duality,
found by I.G.Scher\-back  \cite{2} in the theory of boundary singularities.
Hovewer I have no the apriory proof of this symmetry and I was astonished
to discover it .

\medskip

\noindent {\bf Remark.} Besides the $\RR$-equivalence of fractions
(gauge corresponding to the additive
subgroup of the projective transformations of function value axis),
one can define as well the $\RRR$-equivalence (which  gauge coressponds
to the subgroup of affine transformations  $z\mapsto az+b$) and  $\R^P$-
eqiuvalence (corresponding to the whole projective group  $z\mapsto
(az+b)/(cz+d)$).

\medskip

Such an extension of underlying group might increase the number of
simple singularity classes, but this does not happen if the
dimension $n$ of variable space is greater than  $2$ .
For $n=2$ there appear additional $\RRR$-simple, but not $\RR$-simple,
singularities, whose  adjacencies are shown  in  Fig.~2 below.
They are to be compared with the lists of Giusti  \cite{3} and
Goryunov \cite{4}. Unfortunately I was unable to deduce
the table 2 from their lists.

Presumably, our results clear up  the Giusti list which seems to have
no good  structure. It consists of parts of different nature
($\RR$ and $\RRR$-simple), fluxing together due to the
equivalence relation (biholomorphic equivalence of complete intersections).
Our definitions correspond rather to the classification of flags of
submanifolds (see \cite{8}), which has a better theory of
basises of vanishing cycles and
Dynkin diagrams, than the theory of complete intersections.

\medskip

\noindent {\bf Theorem 1.} {\it
The $\RR$-simple fraction germs in $\C^n$ are listed (up to
the $\RR$-equivalence) in the following table:
$$
\begin{array}{ll}
A_k=(y^{k+1}+Q_2)/x, & A_k^*=1/(x^{k+1}+Q_1); \\
D_k=(y^2z+z^{k-1}+Q_3)/x, & D_k^*=1/(x^2y+y^{k-1}+Q_2); \\
E_6=(y^3+z^4+Q_3)/x, & E_6^*=1/(x^3+y^4+Q_2); \\
E_7=(y^3+yz^3+Q_3)/x, & E_7^*=1/(x^3+xy^3+Q_2); \\
E_8=(y^3+z^5+Q_3)/x, & E_8^*=1/(x^3+y^5+Q_2); \\
B_k=x/(x^k+Q_1), & C_k=x/(xy+y^k+Q_2); \\
F_4=x/(x^2+y^3+Q_2). &
\end{array}
$$
Here $k\geq 0$ for $A_k, A_k^*$; $k\geq 2$ for $B_k$; $k\geq 3$ for
$C_k$; and $k\geq 4$ for $D_k$. The letter $Q$ denotes the quadratic
form $Q_r=z_{r+1}^2+\cdots+z_n^2$, thus the coordinates on $\C^n$ are
denoted by:
$$
\begin{array}{ll}
(x;z_2,\ldots,z_n)\, (n\geq 1) & \mbox{for } A_k^* \mbox{ and } B_k; \\
(x,y;z_3,\ldots,z_n)\, (n\geq 2) & \mbox{for } A_k,D_k^*,E_k^*,C_k,F_4; \\
(x,y,z;z_4,\ldots,z_n)\, (n\geq 3) & \mbox{for } D_k,E_k.
\end{array}
$$
}

\medskip

\noindent {\bf Remark.} The singularity $A_0^*$ is $\R$-equivalent to $1/x.$
For a  generic fractional function the singular points of
this class form a hypersurface (of dimension
$n-1$ for functions in $n$ variables).

The singularity $A_0$ is $\R$-equvalent to $y/x$. For a generic fractional
function the singular points of this class form a submanifold
of codimension  $2$ (of dimension $n-2$ for functions in $n$ variables).

At the remaining points a generic fractional function is holomorphic
and therefore is $\RR$-equivalent to zero. The $\R$-classification of
these (holomorphic) germs is well known (see  \cite{5}).

The singularities from theorem 1 appear unremovably only in the families of
functions.  The simple singularity  denoted by $\cdot_k$ in the list,
can be found in generic family with $p\geq k$ parameters when the parameters
belong to a certain submanifold of dimension $p-k$ .
For these special values of parameters the singularity occurs at
isolated points of the source space of the function.

\medskip

\noindent {\bf Example.} Consider the singularity $A_1$. It occurs if
the restriction of the numerator to the smooth hypersurface of zeros of
denominator has Morse critical point.  Such points  are inherent  already
in generic fractions. But they belong to the class $A_1$ only if the
critical value is equal to zero.

This condition defines  certain submanifold of
codimension  $k=1$ in $p$-dimensonal parameter space. In this sense
the codimension of the class $A_1$ is equal to $1$.

\medskip

In Fig.~1. the classes of equal codimension are connected by  lines.
In brackets there are given the non $\RR$-simple "fencing" classes.
For function in two variables the class $C_{2,2}$ has the minimal
codimension (equal to $4$) among them. This class is formed by fractions
whose nominator and denominators have a common critical point with  zero
common critical value.

For greater number of variables $n>2$ the codimension of this class
grows (it equals $n+2$), but there appear other classes ($F^\#$) of
non $\RR$-simple (even non $\RRR$-simple) singularities with codimension
5 (they are described in theorem 2  below).

\medskip

\noindent {\bf Corollary.} {\it
Generic families of meromorphic functions in $n\geq 2$ variables
depending on $p\leq 3$ parameters contain only such functions whose
germ at any point of non-holomorphy is  $\RR$-simple and $\RR$-equivalent to
one of the following  simple germs

$$
0; A_0^*;A_0;A_1^*,A_1;A_2^*,B_2,A_2;A_3^*,B_3,C_3,A_3
$$
of theorem 1.
}

\medskip

\noindent {\bf Remark.} Proving theorem  1 (see $\S\S$ 3--4 below)
we actually will show more:

\medskip

\noindent {\bf Theorem 2.} {\it
Any germ of fraction on $\C^n$ either is $\RR$-simple (and therefore is
$\RR$-equavelent to one of the germs from theorem~1), or belongs to
a class of infinite codimension ($A_\infty$, $B_\infty$, $C_\infty$, 
$D_\infty$, $A_\infty^*$, $D_\infty^*$), or is adjacent to one of the 
following "fencing" classes of fractions:
$$
\begin{array}{ll}
(P_8)=(y^3+z^3+u^3+ayzu+Q_4)/x, & (P_8^*)=1/(x^3+y^3+z^3+axyz+Q_3); \\

(X_9)=(y^4+ay^2z^2+z^4+Q_3)/x, & (X_9^*)=1/(x^4+ax^2y^2+y^4+Q_2); \\

(J_{10})=(y^3+ayz^4+z^6+Q_3)/x, & (J_{10}^*)=1/(x^3+axy^4+y^6+Q_2); \\

(P_8^\#)=x/(xy+y^3+ay^2z+z^3+Q_3), & \\

(X_9^\#)=x/(x^2+axy^2+y^4+Q_2), & \\

(J_{10}^\#)=x/(x^3+ax^3y+y^3+Q_2), & \\

(C_{2,2})=(a_2y_2^2+\cdots+a_ny_n^2)/Q_0. &
\end{array}
$$
Here $a,a_i$ are parameters, $Q_r=y_{r+1}^2+\cdots+y_n^2$. Any germ from
classes $(P_8^\#)$ and $(X_9^\#)$ is adjacent to the classes
$B_4,F_4,C_4$, while the germs from the  class $(J_{10}^\#)$ are adjacent
only to the singularities $B_4$ and $F_4$.

Classes of fractions which are  $\RR$-equivalent to those listed above
have the following codimension and $\RR$-modality:

{\rm
\begin{tabular}{l|ccc|ccc|ccc|c}
Class & $P_8$ & $X_9$ & $J_{10}$ & $P_8^*$ & $X_9^*$ & $J_{10}^*$ &
$P_8^\#$ & $X_9^\#$ & $J_{10}^\#$ & $C_{2,2}$  \\
\hline
codim & $7$ & $8$ & $9$ & $7$ & $8$ & $9$ & $5$ & $5$ & $5$ & $n+2$  \\
m & $1$ & $1$ & $1$ & $1$ & $1$ & $1$ & $1$ & $1$ & $1$ & $n-1$  \\
\hline
\end{tabular}
}
}

\medskip

\noindent {\bf Example.} Generic four-parametric families of fractions on
 $\C^2$ contain only germs which  are $\RR$-equivalent either to
$ay^2/(x^2+y^2)$ (for certain isolated values of parameters) or to the
$R^+$-simple fractions of the types $D_4, A_{k\leq 4}, C_{k\leq 4},
 F_4, B_{k\leq 4},$ $A^*_{k\leq
4}, D_4^*$.
\medskip

\noindent {\bf Remark.} The notations of the " fencing " boundary singularities
with sign $\#$ are chosen in such  way that  the double covering of
the bounary singularity of denominator (that is the substitution
$x=X^2$ in it) defines  a function from the corresponding class of
the fencing unimodal singularities of holomorphic functions $P_8,X_9,J_{10}$ in
variables $X,y,z,y_i$.


\section{Additional series of $\RRR$-simple fractions}

\noindent {\bf Definition.} Two fractions are called $\RRR$-{\it
equivalent}, if one of them is $\RR$-equivalent to the product of the
other with non-zero (at  the given point)  holomorphic function.

\medskip

In other words, the fraction $P/Q$ is $\RRR$-equivalent at $0$ to
 any fraction of the form
$(AP+BQ)/Q$,  provided that $A$ is non-zero at the origin.

\medskip

\noindent {\bf Example 1.} Let $P$ and $Q$  be generic quadratic forms in
 two variables $x$ and $y$. Then the fraction $P/Q$ is
$\RRR$-equivalent to the fraction $(x^2+y^2)/(xy)$.

\medskip

\noindent {\bf Example 2.} Let $P$ and $Q$ be generic quadratic forms in
$n>2$ variables. Then the fraction $P/Q$ is not $\RRR$-simple.

This fact follows immediately from the  eigenvalue theory of
pencils of quadratic forms.
One may suppose $Q=x_1^2+\cdots+x_n^2$,
$P=\l_1x_1^2+\cdots+\l_nx_n^2$. Eigenvalues of the form $aP+bQ$  with
respect to $Q$ are equal to  $a\l_i+b$. For $a=A(0), b=B(0)$ it implies,
that the classes of $\RRR$-equivalent fractions $P/Q$ correspond
 to submanifolds of
codimension 2 in the $n$-space of fractions having the
special form described above. If $n>2$ then all our fractions
are not simple (even with respect to liberal  $\RRR$-equivalence):
any neighbourhood of
every fraction contains a continuum of pairwise non-equivalent germs.

\medskip

The difference between  $n\leq 2$ and $n\geq 3$ cases discussed  above,
constitutes the main reason why the new  $\RRR$-simple series appear
for $n=2$ (and does not for $n>2$).

The adjacencies of these singularities are shown in Fig.~2.


$$
\begin{array}{ccccccccccc}
(A_1)  & \leftarrow & (A_2) & \leftarrow & (A_3) & \leftarrow & (A_4) &
\leftarrow & (A_5) & \leftarrow & \cdots \\
\uparrow & \ru & \uparrow & \ru & \uparrow & \ru & \uparrow & \ru &
\uparrow & \ru & \\

(B_2)  & \leftarrow & (C_3) & \leftarrow & (C_4) & \leftarrow & (C_5) &
\leftarrow & (C_6) & \leftarrow & \cdots \\
\uparrow & \ru & \uparrow & \ru & \uparrow & \ru & \uparrow & \ru &
\uparrow & \ru & \\

(B_3)  &    & C_{2,2} & \leftarrow & C_{3,2} & \leftarrow & C_{4,2} &
\leftarrow & C_{5,2} & \leftarrow & \cdots \\
\uparrow & \ru & \uparrow & \ru & \uparrow & \ru & \uparrow & \ru &
\uparrow & \ru & \\

(F_4) & \leftarrow & F_5 &    & C_{3,3} & \leftarrow & C_{4,3} &
\leftarrow & C_{5,3} & \leftarrow & \cdots \\
       &      & \uparrow & \ru & \uparrow & \ru & \uparrow & \ru &
\uparrow & \ru & \\

       &            & F_6 & \leftarrow & F_7  &   & C_{4,4} &
\leftarrow & C_{5,4} & \leftarrow & \cdots \\
       &    &    &    & \uparrow & \ru & \uparrow & \ru & \uparrow
       &    & \\

       &     &     &      & F_8  & \leftarrow & F_9  &
     & \vdots &  &  \\
     &   &   &  &   & & \uparrow &  &
 &  & \\

       &  &  & &  & & \vdots &  &  &  &
\end{array}
$$
$$
 F_{p+q+1}\to C_{p,q}\to (A_{p+q-1})
$$
\centerline{Fig. 2. Adjacencies of $\RRR$-simple , but not $\RR$-simple
germs,}
\centerline{to $\RR$-simple ones and among themselves (here $n=2$).}


\medskip

\noindent {\bf Theorem 3.} {\it
The $\RRR$-simple fraction germs in $\C^n$ are exhausted (up to
$\RRR$-equivalence) by $\RR$-simple germs and besides that for $n=2$
by the germs from the following list:
$$
C_{p,q}=\frac{x^p+y^q}{xy},\;\; F_{2k+1}=\frac{y^k}{x^2+y^3}, \;\;
F_{2k+2}=\frac{xy^{k-1}}{x^2+y^3},\;\;(p\geq q\geq 2, k\geq 2).
$$
 In the space of fractions with zero values of
numerator and denominator at the origin the codimension of the
corresponding class is equal respectively to $p+q,2k+1,2k+2$.
}

\medskip

\noindent {\bf Remark.} One may identify $C_{p,1}=C_{p+1},
C_{1,1}=B_2, F_3=B_3$ (all these germs as well as $F_4$ are already
$\RR$-simple).
In Fig.~2 the  $\RR$-simple germs are given in brackets.

\medskip

\noindent {\bf Corollary.} {\it
Generic  four-parameteric families of meromorphic functions in two
variables have only $\RRR$-simple germs at any point (of non-holomorphy).

Generic five-parametric families may contain non $\RRR$-simple germs, which
can not be removed by small perturbations.
}

\section{ Classificator of singularities of fractions }

The following 16 theorems permit to find the location of any fraction
$P/Q$ (where $P$ and $Q$ are  either holomorphic at the origin function on
$\C^n,n>2,$ or power series) in our classification .

Below the sign ``$\approx$" denotes $\R$-equivalence, the sign
``$\sim$"  means $\RR$-equivalence. The sign ``$\cm \bf {n}$"
is the abbreviation of ``see theorem  $\bf n$". Sign
``$\Rightarrow$" means  ``implies". And the sign ``$j^kf$" denotes
the $k$-jet (Taylor polynomial of degree $k$) of function  $f$ at the
 origin. By $x,y,z,u$ we denote coordinates on $\C^n$.

We will use the notations of singularity classes  ($A_k,\ldots,J_{10}^\#$),
as introduced in $\S$ 1.

\medskip

{\it
\begin{itemize}
\item[\bf 1.] If $j^0Q\ne 0$, then $P/Q\sim 0$.

\item[\bf 2.] If $j^0Q=0$, then

     \begin{itemize}
        \item[--] either $j^1Q\ne 0$, $\cm\bf 3$;
        \item[--] or $j^1Q=0$, $\cm\bf 9$.
     \end{itemize}

\item[\bf 3.] If $j^0Q=0,j^1Q\ne 0$, then $P/Q\sim
     \F(y_2,\ldots,y_n)/x, \cm \bf 4$.

\item[\bf 4.] If $P/Q\sim \F(y)/x$, then

     \begin{itemize}
         \item[--] either $j^0\F\ne 0$, $\cm\bf 5$;
         \item[--] or $j^0\F=0$, $\cm\bf 6$.
     \end{itemize}

\item[\bf 5.] If $P/Q\sim \F(y)/x,\,j^0\F\ne 0$,
then $P/Q\approx 1/x\in A_0^*$.

\item[\bf 6.] If $P/Q\sim \F(y)/x, \,j^0\F=0$, then

     \begin{itemize}
         \item[--] either $j^1\F\ne 0$, $\cm\bf 7$;
         \item[--] or $j^1\F=0$, $\cm\bf 8$.
     \end{itemize}

\item[\bf 7.] If $P/Q\sim \F(y)/x, \, j^0\F=0,\,
j^1\F\ne 0$, then $P/Q\approx y/x\in A_0$.

\item[\bf 8.] If $P/Q\sim \F(y)/x,\, j^1\F=0$, then

     \begin{itemize}
          \item[--] either $P/Q \in A_k\,(k\geq 1), \; D_k\,(k\geq 4),\;
                     E_6,\; E_7,\; E_8$;
          \item[--] or $P/Q$ is adjoint to $(P_8), (X_9)$ or
                           $(J_{10})$  and is not $\RR$-simple fraction;
          \item[--] or $P/Q$ belongs to the class of infinite codimension
           $A_\infty$ or to  $D_\infty$  (which in
holomorphic case contains the  reducible  fractions   and fractions
with non-isolated  singularity of numerator with zero critical value).
     \end{itemize}

\end{itemize}
}

\medskip

\noindent {\bf Remark.} The location of fraction $P/Q$ in $A-D-E$-
classification may be found applying the classificator from  \cite{5} to $\F$.

\medskip

{\it The classes of non $\RR$-simple fractions adjacent to $P_8,X_9$ and $J_{10}$
have codimensions respectively $7,8,9$ (provided that $n\geq 4,3,3$).}

That is the  germs from these classes  appear  unremovably  first in generic
families of fraction on $\C^n$, depending on $7,8$ and $9$ parameters
respectively.

The classes $D$ and $E$ are not void if $n\geq 3$.

Proceed with the classificator.

\medskip

{\it
\begin{itemize}

\item[\bf 9.] If $j^1Q=0$, then:

     \begin{itemize}
         \item[--] either $j^0P\ne 0$, $\cm\bf 10$;
         \item[--] or $j^1P\sim x$, $\cm\bf 11$;
         \item[--] or $j^1P=0$, $\cm\bf 16$.
     \end{itemize}

\item[\bf 10.] If $j^1Q=0,j^0P\ne 0$, then $P/Q=1/\F, j^1\F=0$.

{\rm
The classification of such fractions $P/Q$ (up to $\R$-equivalence)
reduces to  $\R$-classification of holomorphic function germs $\F$
having   a critical point with zero critical value  at the origin.
We get:
}

\begin{itemize}
\item[--] either $\F\approx a_k=x^{k+1}+y_2^2+\cdots + y_n^2\,(k\geq 1)$;
\item[--] or $\F\approx d_k=x^2y+y^{k-1}+y_3^2+\cdots + y_n^2\,(k\geq 4)$;
\item[--]
  \begin{tabular}{rcl}
   or $\F$ & $\approx$ & $e_6=x^3+y^4+y_3^2+\cdots + y_n^2$, or \\
         & $\approx$ & $e_7=x^3+xy^3+y_3^2+\cdots + y_n^2$, or \\
         & $\approx$ & $e_8=x^3+y^5+y_3^2+\cdots + y_n^2$;
  \end{tabular}
\item[--] or $\F$ is adjacent to one of the classes $(P_8), (X_9),
         (J_{10})$ of non $\R$-simple singularities of functions;
\item[--] or $\F$ belongs to one of the classes $A_\infty$ or $D_\infty$
of infinite codimension (in holomorphic case they consist of functions
having non-isolated critical point  with zero critical value).
\end{itemize}

{\rm
Respectively, we get:}

\begin{itemize}
\item[--] either $P/Q\approx 1/a_k\in A_k^*, k\geq 1$;
\item[--] or $P/Q\approx 1/d_k\in D_k^*, k\geq 4$;
\item[--] or $P/Q\approx 1/e_k\in E_k^*, k=6,7,8$;
\item[--] or the fraction $P/Q$ is adjacent to one of the classes
 $(P_8^*), (X_9^*),
(J_{10}^*)$ of  non  $\RR$-simple fractions;
\item[--] or $P/Q$ belongs to one of the  classes $A_\infty^*$
and $D_\infty^*$
of codimension $\infty$ (For holomorphic $P$ and $Q$, these classes
consist of fractions having non-isolated critical point of denominator
with zero critical value).
\end{itemize}
\end{itemize}

\medskip

{\rm
\noindent {\bf Remark.} The codimensions of classes $(P_8^*), (X_9^*),
(J_{10}^*)$ are equal to $7,8$ and $9$ respectively,  provided that
$n\geq 3,2,2$.
}

\medskip

\begin{itemize}
 \item[\bf 11.] If $j^1Q=0, j^1P=x$, than $P/Q\approx x/\F,
                        j^1\F=0$.

{\rm
The classification of these fractions up to $\R$-equivalence follows from
the theory of boundary singularities \cite{1}, applied to the function
$\F$. This theory implies that using diffeomorphisms preserving the " boundary"
$x=0$ (that is having the form $(x,y)\mapsto (xh(x,y),g(x,y))$)
one can reduce function $\F$ to one of the following forms:
}
\begin{itemize}
 \item[--] either $\F\approx b_k=x^k+y_2^2+\cdots + y_n^2 \, (k\geq 2)$;
 \item[--] or $\F\approx c_k=xy+y^k+y_3^2+\cdots + y_n^2 \, (k\geq 3)$;
 \item[--] or $\F\approx f_4=x^2+y^3+y_3^2\cdots + y_n^2$;
 \item[--] or $\F$ is adjacent to one of the classes $(P_8^\#), (X_9^\#),
(J_{10}^\#)$, which fence in simple boundary singularities (see. \cite{1});
 \item[--] or $\F$ belongs to one of the classes $B_\infty$ and
$C_\infty$ of infinite codimension.
\end{itemize}

{\rm
Thus we get respectively:
}

\begin{itemize}
\item[--] either $P/Q\approx Ex/b_k,k\geq 2, j^0E\ne 0, \cm\bf 12$;
\item[--] or $P/Q\approx Ex/c_k,k\geq 3, j^0E\ne 0, \cm\bf 13$;
\item[--] or $P/Q\approx Ex/f_4, j^0E\ne 0, \cm\bf 14$;
\item[--] or $P/Q$ is adjacent to one of the classes
$P_8^\#,X_9^\#,J_{10}^\#$ of fractions $Ex/\F$, where $\F$- is a "fencing "
boundary singularity, $j^0E\ne 0,\cm\bf 15$;
\item[--] or $P/Q$ belongs to one of the classes $B_\infty$ and $C_\infty$
of infinite codimenion (including, for example , the reducible fractions).
\end{itemize}

\item[\bf 12.] If $j^0E\ne 0, k\geq 2$, then $Ex/b_k\approx
x/b_k\in B_k$.

\item[\bf 13.] If $j^0E\ne 0, k\geq 3$, then $Ex/c_k\approx
x/c_k\in C_k$.

\item[\bf 14.] If $j^0E\ne 0$, then $Ex/f_4\approx
x/f_4\in F_4$.

\item[\bf 15.] The fractions $Ex/\F$, where $\F$ is a
"fencing " boundary singularity (and hence all fractions adjacent to them)
are not $\RR$-simple.  For all regular values  of parameter $a$ they are
$\RRR$-unimodal and $\RR$-unimodal.

\medskip

{\rm
\noindent {\bf Remark.} The codimension of the class of such fractions
is equal to $5$
(when $n\geq 2$).
}

\medskip

\item[\bf 16.] If $j^1Q=j^1P=0, n\geq 2$, the the fraction
$P/Q$ is not $\RR$-simple.

\medskip

{\rm
\noindent {\bf Remark.} The codimension of the class of such fractions
 is equal to $n+2\geq 4$.
}

\medskip

\end{itemize}
}



The theorems {\bf 1}--{\bf 16} imply that
all fractions of theorem~1 (from $\S$ 1)  are $\RR$-simple and vice versa.
Really, as we have seen any other fraction is adjacent to some class
of non $\RR$-simple (having moduli with respect to the $\RR$-equivalence
relation) fractions. The simplicity of the fractions from the list
of the theorem~1 is implied by the semi-continuity arguments of the multiplicity
$k$, of ranks and other similar invariants. These agruments show that
deforming a singularity of our list one gets only finite number of
other singularities. All of them also belong to the list.

\medskip

\noindent {\bf Example.} Deforming the fraction with $j^kP\ne 0$  one can
not get the fraction with $j^kP=0$.

\medskip

In particular, this implies that the singularities $A^*$, $D^*$, $E^*$ are 
not adjacent to $A$, $B$, $C$, $D$, $E$, $F$, and hence  to $P$, $X$, $J$. 
Similarly, the singularities $A, D, E$ are not adjacent to $A^*$, $D^*$, 
$E^*$, $B$, $C$, $F$, and
hence to $P^*$, $X^*$, $J^*$.  The singularities $B$ are not adjacent to
$A_2$, and the singularities  $C$ to $A_2^*$. In a similar way one can check
that all singularities of our list are adjacent only to the previous classes
of this very list.

\medskip

\noindent {\bf Remark.} However not all adjacencies are shown in
Fig.~1: for example, $E_k$ is adjacent to $A_{k-1}$.

\medskip


\section{Proofs of the clasificator theorems}

Theorems {\bf 1}, {\bf 2}, {\bf 4},{\bf 5},
{\bf 6}, {\bf 7} and {\bf 9} are evident.
Theorem
{\bf 3} is implied by Hadamard Lemma. Theorems {\bf 8},
{\bf 10}, {\bf 11}, {\bf 15} are essentially
the clasification of ordinary and boundary simple and fencing
singularities
of holomorphic functions (see \cite{1},\cite{5}). The only
really new  statement (which we shall now prove) is that
from  theorems {\bf 8}, {\bf 10}
and {\bf 15} concerning $\RR$- (and even about $\RRR$-)
unimodality of fencing  singularities of fractions.
 These singularities have the form $1/\F, \F(y)/x$ and
$x/\F(x,y)$ in theorems {\bf 10}, {\bf 8} and {\bf 11}
respectively.

The proofs of unimodality of these fractions are bases on
 quasi-homogeneity of fencing  singularities of holomorphic functions
$\F$. It permits to reduce the $\RRR$-equivalence of fractions to
 the $\R$-equivalence of holomorphic functions $\F$.

\medskip

\noindent {\bf Lemma 1.} {\it
Let $\F$  be quasi-homogeneous holomorhic function of non-zero weight.
Then any fraction being $\RRR$-equivalent to the fraction  $1/\F$,
is  in fact $\R$- equivalent to it.
}

\medskip

\noindent {\bf Proof.} A quasi-homogeneous function $\F$ of non-zero weight
is $\R$-equivalent to the product of itself by any non-zero  function
(see \cite{5} or lemma~3 below). Hence,  when  $H(0)\ne 0$ , we obtain
$$
(H/\F)+G=(H+\F G)/\F\approx 1/\F.
$$
Thus, the class of $\RRR$-equivalence of the fraction $1/\F$ is determined
by the class of biholomorphic equivalence of the denominator (provided that
the latter is quasihomogeneous). The lemma is proved.

\medskip

\noindent {\bf Corollary 1.} {\it
The singularity clases of fractions
$(P_8^*),(X_9^*), (J_{10}^*)$ from theorem~{\bf 10} are at
least unimodal with respect to $\RRR$-equivalence  (and hence
with respect to  $\RR$-equivalence of fractions). For almost all
values of parameter $a$ they are exactly  $\RR$-unimodal (and hence
$\RRR$-unimodal).
}

\medskip

Really, the singularities $P_8,X_9$ and $J_{10}$ of corresponding
holomorhic functions $\F$ are quasihomogeneous  (with positive
weights) and $\R$-unimodal.

This completes the proof of theorem {\bf 10}.

\medskip

\noindent {\bf Lemma 2.} {\it
Let $\F_1$ and $\F_2$ be such a quasihomogeneous holomorphic
functions in $y\in\C^m$
with non zero weights  that the  jerms of (irreducible) fractions
$\F_i/x$ at the point $0\in\C^{m+1}$ are $\RRR$-equivalent. Then
$\F_1\approx\F_2$ (that is one of
 the holomorphic function germs $\F_1$ and $\F_2$
at the origin in $\C^m$  might be obtained from the other via a
biholomorphic diffeomorphism).
}

\medskip

\noindent {\bf Proof.}
Holomorphic type of zero hypersurface of  numerator
($\F_i=0$) in the smooth hupersurface of zeros of  denominator ($x=0$)
is the same for $\RRR$-equivalent fractions. Meanwhile, the holomorphic
type of zero hupersurface of quasihomogeneous function with non zero
weghts determines this function up to $\R$-equivalence. This proves
lemma~2.

\medskip

\noindent {\bf Corollary 2.} {\it
Fraction singularities of classes $(P_8), (X_9), (J_{10})$
from theorem~{\bf 8} are at least unimodal with respect to
$\RRR$-equivalence relation (and hence to $\RR$-equivalence). For
almost all values of parameter $a$ these fractions are exactly
$\RR$-unimodal (and  $\RRR$-unimodal).
}

\medskip

This statement follows from Lemma~2 since singularities $P_8,
X_9, J_{10}$ of corresponding holomorphic functions $\F$ are
quasihomogeneous (with positive weights) and  $\R$-unimodal.

This completes the proof of theorem~{\bf 8}.

\medskip

Theorem~{\bf 11} is an immediate consequence of
classification of boundary singularities given in~\cite{5}. Really,
the diffeomorphism, which reduces the boundary
singularity of holomorphic function  $\F(x,y)$ on the manifold with the
boundary $x=0$ to its normal form $f$, sends function $x$ to  the function
$xE(x,y)$, where  $E(0)\ne 0$. So the fraction $x/\F(x,y)$ takes the form
$Ex/f(x,y)$. This proves theorem~{\bf 11}.

The proofs of theorems~{\bf 12}--{\bf 15} are bases
on the following evident statement on quasihomogeneous functions.

\medskip

\noindent {\bf Lemma 3.} {\it
Let $P$ and $Q$ be quasihomogeneous
functions with different
quasihomogeneous
degrees, let  $E$ be a holomorphic function with non zero value at the
origin. Then the fraction germ $EP/Q$ at the origin is biholomorphic
equivalent
to the fraction germ $P/Q$: $EP/Q\approx P/Q$.
}

\medskip

\noindent {\bf Proof.}
Denote by $z_i$ the quasihomogeneous coordinates and by $\alpha_i$
their weights. Thus we have
$$
\begin{array}{c}
P(t^{\alpha_1}z_1,\ldots,t^{\alpha_n}z_n)\equiv t^\beta P(z_1,\ldots,z_n), \\
Q(t^{\alpha_1}z_1,\ldots,t^{\alpha_n}z_n)\equiv t^\gamma Q(z_1,\ldots,z_n).
\end{array}
$$
Put $t(z)\equiv (\ln E(z))/(\beta-\gamma)$. Then the diffeomorphisms
$(Z_i\equiv t^{\alpha_i}(z)z_i)$ is the required equivalence, as
$$
(P/Q)(Z(z))\equiv E(z)(P/Q)(z).
$$
Lemma 3 is proved.

Normal forms $b_k,c_k,f_4$ are quisihomogeneous polynomials
(whose degrees are greater than that of  $x$, provided that $k\geq 2$).
Therefore lemma~3 implies theorems~{\bf 12}--{\bf 14}
directly  (the $\R$-equivalence constructed in the proof of
lemma~3 preserving the boundary $x=0$).

\medskip

\noindent {\bf Corollary 3.} {\it
Fencing fraction singularities of classes $(P_8^\#),
(X_9^\#)$, $(J_{10}^\#)$  are
at most $\RR$-unimodal (and hence at most $\RRR$-unimodal)
provided that the values of parameter  $a$ are regular.
}

\medskip

In fact, by lemma~3 these fractions are $\R$-equivalent themselves to
normal forms of the type $x/f(x,y)$, where the quasihomogeneous
polynomial $f$ contains  only one parameter $a$.
All germs of their small deformations (values of $a$ being regular)
are either $\RR$-simple or belongs to the same class. The corollary
is proved.

\medskip

To complete the proof of theorem~{\bf 15} claiming the
unimodality of these fractions it rests to show that
the module $a$ of the normal form of fencing fraction  $x/f$
remains invariant not only with respect to $\R$-equivalences,
which preserves the boundary $x=0$, but also to $\RR$-equivalences
(in fact, even to $\RRR$-equivalences).

\medskip

\noindent {\bf Lemma 4.} {\it
Let $f(x,y)$ be a quasihomogeneous function holomorphic at the origin,
whose weight is greater than that of variable $x$. Then any fraction
which is $\RRR$-equivalent to the fraction  $x/f$ is ,in fact,
$\R$-equivalent to it.
}

\medskip

\noindent {\bf Proof.}
Consider a fraction $Ex/f+G$, which is $\RRR$-equivalent to $x/f$.
Here  $E(0)\ne 0$. By lemma~3 the first term is $\R$-equivalent to
$x/f$, hence the sum is $\R$-equivalent to $x/f+G'=(x+G'f)/f$.
Consider $x+G'f=X$ as a new variable. Then
$$
f(x,y)\equiv f(X,y)+[f(X-G'f(x,y),y)-f(X,y)].
$$
Now the expresion in square brackets might be written in the form
$K(X,y)f(x,y)$, where function $K$ is holomorphic at the origin.
The value of $K$ at the origin equals zero, since $\d f/\d X$  vanishes
at the origin (as the weight of $f$ is greater than that of $x$).
Therefore $f(x,y)\equiv H(X,y)f(X,y)$, where the function $H$ being equal to
$(1-K)^{-1}$ is holomorphic at the origin. Thus we get the
identity
$$
(x+G'f)/f(x,y)\equiv X/(H(X,y)f(X,y)).
$$
This proves  $\R$-equivalence of the sum $Ex/f+G$ and the fraction $x/(Hf)$.
The latter is $\R$-equivalent to the fraction  $x/f$ by lemma~3,
hence finally
$$
Ex/f+G\approx x/f.
$$
Lemma 4 is proved.

\medskip

\noindent {\bf Lemma 5.} {\it
Let $f_1,f_2$ be quasihomogeneous holomorphic functions whose weights differ
from
that of quasihomogeneous variable $x$. Than the
$\R$-equivalence if irreducible fractions
$x/f_1$ and $x/f_2$ implies boundary  $\R$-equivalence of
denominators (i.e. the equivalence of function germs  $f_1$ and
$f_2$ at the origin via a diffeomorphism preserving the
hypersurface $x=0$).
}

\medskip

\noindent {\bf Proof.}
Initial  $\R$-equivalence  maps $x$ to $Ex$, and
 $f_1$ to $Ef_2$, where the holomorphic function $E$ is non zero at the
origin.
Thus the diffeomorphism preserving the hyperplane $x=0$ and converting
$f_1$ into $Ef_2$ is already found. By lemma~3 $Ef_2\approx f_2$
via the diffeomorphism preserving hyperplane  $x=0$. Lemma~5 is proved.

\medskip

\noindent {\bf Corollary 4.} {\it
Let  be quasihomogeneous functions in
$x$ and $y$. Let weight  of $f_1$ and $f_2$ be greater that the weight of
variable $x$. Then $\RRR$-equivalence of irreducible fractions
$x/f_1$ and $x/f_2$ implies boundary  $\R$-equivalence of the denominators
(i.e. there exists a diffeomorphism preserving hyperplane $x=0$ and
converting $f_1$ into $f_2$).
}

\medskip

\noindent {\bf Proof.}
The fractions are $\R$-equivalent by lemma~4. By lemma~5 there exists the
boundary  $\R$-equivalence of the denominators.

\medskip

\noindent {\bf Corollary 5.} {\it
Fractions of the classes $(P_8^\#), (X_9^\#), (J_{10}^\#)$
from theorem~$2$ are at least unimodal with respect to
$\RRR$-equivalence relation (the more so to  $\RR$-equivalence).

}

\medskip

\noindent {\bf Proof.}
The $\RRR$-equivalence of fractions implies boundary
$\R$-equivalence of the denominators. Since parameter $a$ is module
of boundary $\R$-equivalence of denominators
(according to the classification of fencing boundary singularities,
see~\cite{1}), then it ought to be module for
$\RRR$- (as well as for $\RR$-) classification of fractions.

This corollary and corollary~3 imply that
{\it the fencing fraction singularities, mentioned in these statements are exactly
unimodal for non singular values of parameter $a$  with respect to both of
$\RRR$- and  $\RR$- equivalences.
}

This completes the proof of theorem~{\bf 15}.

Theorem~{\bf 16} is proved in $\S$ 2 (example 2).

Theorems {\bf 1}--{\bf 16} together imply both
theorem~1 $\S$~2 and theorem~2 (the statement of the latter
about adjacency and codimension is a consequence
of boundary singularity theory and corollary~4).

\section{Classification of $\RRR$-simple fractions germs}

All $\RR$-simple fraction germs are so more  $\RRR$-simple. The above
classificator and corollaries~1,~2,~5 imply  the following statement.
Every  non $\RR$-simple fraction germ:

\begin{itemize}
\item[--] either belongs to one of the classes of infinite codimension;
\item[--] or is adjacent to one of the fencing   classes, which
consists of $\RRR$-unimodal germs;

\item[--] or belongs to the class of non $\RR$-simple fraction germs
$P/Q$ from theorem~{\bf 16} (their numerator and denominator
have common critical point with zero critical values).
\end{itemize}


Singularities from the latter class are not $\RRR$-simple provided that
the fractions are defined on $\C^n,\,n>2$ (example~2, $\S$ 2).
Hence, the list of $\RRR$-simple fraction singularities consists of
two parts:
a) $\RR$-simple singularities; b) special fraction singularities
in two variables, which are $\RRR$-simple but not $\RR$-simple.

These special classes are denoted in $\S$ 2 by
$C_{p,q}$ and $\F_m$
(to be more strict, there are also infinitely degenerated
limit cases $C_{\infty,q}, C_{\infty,\infty}$ and $\F_\infty$, forming
the classes of infinite codimension). The location of each
fraction in $\RRR$-equivalence hierarchy  is determined by
the following classificator (here
$\eqt$ means $\RRR$-equivalence).

\medskip

\noindent {\bf Classificator of $\RRR$-simple fraction singularities
$P(x,y)/Q(x,y)$.}

\medskip

{\it
\begin{itemize}
\item[{\bf 1}.] If $j^1P=j^1Q=0$, then

\begin{itemize}
\item[--] either $j^2Q\approx xy$, $\cm {\bf 2}$;
\item[--] or $j^2Q\approx x^2$, $\cm {\bf 5}$;
\item[--] or $j^2Q=0$, $\cm {\bf 13}$.
\end{itemize}

\item[{\bf 2}.] If $j^2Q=xy$, then $P/Q\approx R/(xy)$,
$\cm{\bf 3}$.

\item[{\bf 3}.] If $Q=xy,j^pR(x,0)=ax^p,j^qR(0,y)=by^q$, then

\begin{itemize}
\item[--] either $ab\ne 0$, $\cm {\bf 4}$;
\item[--] or $R/Q$ belongs to a class of infinite codimension.
\end{itemize}

\item[{\bf 4}.] If $Q=xy,j^pR(x,0)=ax^p, j^qR(0,y)=by^q,ab\ne 0$,
then

$R/Q\eqt (x^p+y^q)/(xy) \Rightarrow C_{p,q}$.

\item[{\bf 5}.] If $j^2Q=x^2, j^1P=0$, then

\begin{itemize}
\item[--] either $j^3Q\approx x^2+y^3$, $\cm {\bf 6}$;
\item[--] or  $j^3Q\approx x^2$, $\cm {\bf 11}$.
\end{itemize}

\item[{\bf 6}.] If $j^3Q=x^2+y^3$, then $P/Q\approx R/(x^2+y^3)$,
$\cm{\bf 7}$.

\item[{\bf 7}.] If $Q=x^2+y^3$, then $R/Q\sim
(\alpha(y)+x\beta(y))/Q$, $\cm{\bf 8}$.

\item[{\bf 8}.] If $Q=x^2+y^3,R=\alpha(y)+x\beta(y),j^1R=0$, then

\begin{itemize}
\item[--] either for certain $k\geq 2\; j^k\alpha=ay^k,a\ne 0,
j^{k-2}\beta = 0$, $\cm {\bf 9}$;
\item[--] or for certain  $k\geq 2\; j^k\alpha=0,j^{k-1}\beta =
by^{k-1}, b\ne 0$, $\cm {\bf 10}$;
\item[--] or $R/Q$ belongs to a class of infinite codimension.
\end{itemize}

\item[{\bf 9}.] If $Q=x^2+y^3, R=\alpha+x\beta, j^k\alpha
=ay^k,k\geq 2,a\ne 0, j^{k-2}\beta =0$, then $R/Q\eqt y^k/(x^2+y^3),
\Rightarrow \F_{2k+1}$.

\item[{\bf 10}.] If $Q=x^2+y^3, R=\alpha+x\beta, j^k\alpha=0,
j^{k-1}\beta = by^{k-1},k\geq 2,b\ne 0$, then $R/Q\sim  xy^{k-1}/(x^2+y^3),
\Rightarrow \F_{2k+2}$.

\item[{\bf 11}.] If $j^3Q=x^2, j^1P=0$, then the fraction
$P/Q$ is ajacent to the fraction class
$x/(x^2+axy^2+y^4)$, $\cm {\bf 12}$.

\item[{\bf 12}.] Fractions from the class
$X_9^\#=x/(x^2+axy^2+y^4)$ are $\RRR$-unimodal for non singular
values of parameter  $a$.

\item[{\bf 13}.] If $j^1P=0,j^2Q=0$, then the fraction $P/Q$
is adjacent to the fraction class
$x/(x^3+ax^2y+y^3)$, $\cm{\bf 14}$.

\item[{\bf 14}.] The fractions from the class
$J_{10}^\#=x/(x^3+ax^2y+y^3)$ are $\RRR$-unimodal for
non singular values of parameter $a$ .

\end{itemize}
}

\medskip

\noindent {\bf Proofs of the theorems on the classification
of $\RRR$-simple fractions.}

\medskip

Theorems {\bf 1}, {\bf 3}, {\bf 8}, {\bf 11},
{\bf 13} are evident.

Theorems {\bf 2} and {\bf 5}, {\bf 6}
are actually the $\R$-classification of holomorphic functions  of classes
$A_1$ (Morse Lemma) and $A_2$.

Theorems {\bf 12} and {\bf 14} are the particular
cases of the theorem~{\bf 15} $\S$ 3.

The remaining statements (theorems  {\bf 4}, {\bf 7}, {\bf 9},
{\bf 10}) are proven below .

\medskip

\noindent {\bf Proof of theorem {\bf 4}.} By the hypothesis,
$R=ax^p+by^q+(xy)S(x,y)$ (here we may assume $a$ to depend on
$x$ and $b$  -- on $y$ only), $ab\ne 0$ at the origin. Therefore,
there exists a $\RR$-equivalence
$$
R/(xy)\sim (ax^p+by^q)/(xy)=g(x,y).
$$
Now transform the fraction $f=(x^p+y^q)/(xy)$ by neans of substitution
$x=AX, y=BY$.
We'll find such $A(X,Y)$ and $B(X,Y)$ , that $f(x,y)=g(X,Y)$.

If $N=pq-p-q\ne 0$, then $A=a^{(q-1)/N}b^{1/N}, B=a^{1/N}b^{p-1/N}$
is the desired  $\R$-equivalence normalising the germ $g\approx
f$. Despite this fact, all our singularities are even non $\RR$-simple
(due to unique exeptional case  $p=q=2$, when $N=0$). In this latter
case  we choose $X=\sqrt{a}x, Y=\sqrt{b}y$.  The required
$\RRR$-equivalence takes the form $g(x,y)= c(X^2+Y^2)/(XY), c=1/\sqrt{ab}$
(i.e. $g\eqt f$).

\medskip

\noindent {\bf Proof of theorem {\bf 7}.} Decompose  function $R$
into its even and odd parts with respect to  $x$
$$
R(x,y)=A(x^2,y)+xB(x^2,y).
$$
When  $z=x^2+y^3$ Hadamard lemma implies
$$
\begin{array}{rcl}
R(x,y) & = & A(z-y^3,y)+xB(z-y^3,y) = \\
       & = & A(-y^3,y)+xB(-y^3,y)+zC(x,y) = \\
       & = & \alpha(y)+x\beta(y)+QC,
\end{array}
$$
this proves theorem {\bf 7}.

\medskip

We preface the proofs of theorems {\bf 9} and {\bf 10}
with the following general remark.

Consider holomorphic (at the origin) vector fields tangent to level
hypersurfaces of function  $Q$. Such vector fields form  module $M$
over the ring  of functions holomorphic at the origin. Phase flows of
these fields preserve function  $Q$.

\medskip

\noindent {\bf Example.} In the case  of the
function $Q=x^2+y^3$ in two variables  we are interested  now in
module  $M$ is generated by  Hamiltonian field
$H=3y^2\d/\d x-2x \d/\d y$ and consists of the fields  $hH$, where
function $h$ are holomorphic at the origin.


\medskip

Consider the ideal $(Q,MP)$, generated by $Q$ and by all functions
from $MP$, where $P$ is some chosen function. Assume this ideal to
have finite codimension, so the factor by it is generated (as a linear
space) by a finite set of functions $e_1,\ldots,e_N$. In other words,
suppose that for any function (holomorphic at the origin)
there exists a decomposition

$$
f=\sum c_ie_i+gQ+hmP,
$$
where $c_i\in\C$ are constants, functions $g$ and $h$ are holomorphic
at the origin and  $m\in M$.

\medskip

\noindent {\bf Example 1.} $Q=x^2+y^3, P=ay^k, a\ne 0, k\geq 2$. In this
case $HP=-2akxy^{k-1}$. Therefore ideal $(Q,MP)=(x^2+y^3, xy^{k-1})$
has finite codimension. As the generators $e_i$ one can choose
the monomials $y^s,s<k+2$ and $xy^s, s<k-1$.

\medskip

\noindent {\bf Examples 2.} $Q=x^2+y^3, P=axy^{k-1}, a\ne 0, k\geq 2$.
In this case $HP=ay^{k-2}[3y^3-2(k-1)x^2]$. Hence ideal
$(Q,MP)=(x^2+y^3, y^{k+1}, x^2y^{k-2})$ has finite codimension.
As functions $e_i$ one can take monomials $y^s$ and $xy^s, s<k+1$.

\medskip

Suppose now that function $Q$ is quasihomogeneous  (with positive
weights of the arguments) and functions $P$ and $e_i$ are
quasihomogeneous as well. Note, that in examples~1~and~2 all
this assumptions are fulfilled.

Distinguish those  $e_i$ , whose quasihomogeneous degree is greater
than that of quasihomogeneous function $P$. Denote them by
$f_j$  and call them {\it upper generators}.

\medskip

\noindent {\bf Example 3.} In example 1 there is only one upper
generator:
$f=y^{k+1}$. In example~2  it is alone  ($f=xy^k$) as well.

\medskip

Let now $S$ be a quasihomogeneous function (holomorphic at the origin),
whose degree is greater than that of quasihomogeneous function $P$.

\medskip

\noindent {\bf Lemma.} {\it
Fraction $(P+S)/Q$ is $\RR$-equivalent to a fraction of the form $(P+T)/Q$,
where $T$ is a scalar linear combination of upper generators  $f_j$.
}

\medskip

\noindent {\bf Proof.}
We adapt the algorithm of  quasihomogeneous function  normalisation
(see \cite{5}) to our situation, when the variable changes preserve
all the values of $Q$ (they are phase flow mappings of fields from $M$).
As a result we reduce  $P+S$ to the form  $P+T+gQ$, preserving $Q$:
$$
(P+S)/Q\approx (P+T+gQ)/Q\sim (P+T)/Q.
$$
Lemma is proved.

\medskip

\noindent {\bf Proof of theorem {\bf 9}.} Apply lemma to
$Q=x^2+y^3,P=ay^k,S=R-P$. Quasihomogeneous order  of zero of
perturbation $S$ is greater than the degree of $P$ by the
assumption of the theorem. Acoording to example~3, we obtain
the following $\RR$-equivalence
$$
R/Q\sim Ey^k/Q,\; E=a+by,\; E(0)=a\ne 0.
$$
By definition  of $\RRR$-equivalence, $R/Q \eqt y^k/Q$. This proves
the theorem.

\medskip

\noindent {\bf Remark.} If $k\ne 3$ we get even
$\RR$-equivalence $R/Q\sim y^k/Q$ (by lemma~3~$\S$~4).

\medskip

\noindent {\bf Proof of theorem {\bf 10}.} Apply lemma to
$Q=x^2+y^3,P=axy^{k-1}, S=R-P$. quasihomogeneous order of zero of
perturbation $S$ is greater than the degree of $P$ by theorem assumption.
According to example~3, we get the $\RR$-equivalence
$$
R/Q\sim Exy^{k-1}/Q,\; E=a+by,\; E(0)=a\ne 0.
$$
Quasihomogeneous degrees of qusihomogenious functions $xy^{k-1}$
and $x^2+y^3$
are different (for arbitrary  $k$). By lemma~3~$\S$~4 we get the
$\RR$-equivalence $R/Q\sim xy^{k-1}/Q$. This proves the theorem.


\section{ Singularities of polynomials at infinity.}

Consider the family of  level hypersurfaces of a polynomial in  $m$
complex variables near  an infinitely distant point  of $m$-dimensional
projective space.  The formulas given below describes this family
of hypersurfaces (up to a biholomorphic equivalence) for generic
polynomials and for generic familes of polynomials, depending on a few
parameters, provided that the degrees of polynomials are large enough.
Simplest singularies happen to be simple (non simple singularities
arise in generic families only when the number of parameters are
at least $5$). All simple singularities (having no moduli for
holomorphic equivalence relation) are described by means of ordinary
(for boundary singularity theory) list of Weyl  groups
$A_k (k\geq 1), B_k (k\geq 2), C_k (k\geq3),
D_k (k\geq 4), E_k (k=6,7,8), F_4$. The role of boundary is
played  of course by infinite hyperplane.

Consider polynomial $f(x,y_2,\ldots,y_m)$ in $\C^m, m\geq 2$.
Near infinite distant point at the  $x$ axis  introduce new
affine coordinates  $Y=y/x,Z=1/x$ centered at this point
$([x:y:1]=[1:Y:Z])$. The explicit substitution of $1/Z$ instead of $x$ and
of $Y/Z$ instead of $y$ proves the following lemma.

\medskip

\noindent {\bf Lemma.} {\it
In this new variables  polynomials $f$ is represented by fraction
$f(x,y)=H(Y,Z)/Z^n$, where $H$ is a polynomial of degree $n$
(its coefficients are obtained from that of polynomial  $f$ by
a permutation),
and $Z=0$ is the equation of infinite hyperplane.
}

\medskip

We will study the fraction germes of this special form
$H/Z^n$ with the numerator $H$ being an arbitrary holomorphic function.
Such a fraction will be called {\it $\R$-simple}, if  its neigbourhood
{\it in the space of these fractions } is exhausted by a finite number
of $\R$-equivalence classes.

\medskip

\noindent {\bf Theorem.} {\it
Simple (having no moduli with respect to biholomorphic
$R$-equivalence) fraction singularities of the form $H/Z^n$
in $\C^m$  are exhausted (up to $\R$-equivalence) by the following
list:
$$
\begin{array}{l}
A_0=Y_m/Z^n;\; A_k=(Y^{k+1}+Z+Q_2)/Z^n,k\geq 1 ; \\
B_k=(Y^2+Z^k+Q_2)/Z^n, k\geq 2;\\
C_k=(Y^k+YZ+Q_2)/Z^n, k\geq 3; \\
D_k=(Y^2U+U^{k-1}+Z+Q_3)/Z^n, k\geq 4; \\
E_6=(Y^3+U^4+Z+Q_3)/Z^n ;\\
E_7=(Y^3+YU^3+Z+Q_3)/Z^n ;\\
E_8=(Y^3+U^5+Z+Q_3)/Z^n ;\\
F_4=(Y^3+Z^2+Q_2)/Z^n .
\end{array}
$$
These singularities are $\R$-simple proveded that $n$ is
sufficiently large  (namely, when
$n>1$ for $A_k, C_k, D_k, E_k$,   when $n>k$ for $B_k$
and when $n>2$ for $F_4$).

Codimension of simple singularity class  indexed by $k$ is equal to $k$.

All non $\R$-simple singularities form
the subset of codimension five (provided that $n$ is
sufficiently large).
}

\medskip

\noindent {\bf Remark}. A class of fraction germs $H/Z^n$  has
codimension  $c$, if the germs from this class  arise (at
isolated points) in generic fraction families of the form $H/Z^n$,
depending on $p\geq c$ parameters (when parameters belongs to
a certain $p-c$-dimensional submanifold in $p$-dimensional parameter space).

\medskip

\noindent {\bf Example.}  Class $A_1$ has codimension $1$, as
the corresponding function $H$  is to have a critical point  with
{\it zero critical value} on  manifold $Z=0$.

\medskip

The adjacency table of $\R$-simple and fencing  singularities
of the form $H/Z^n$ (with large enough $n$) is given in Fig.~3.
Fencing  singularities are  determined by formulas

$$
\begin{array}{l}
(P_8)=(Z+Y^3+U^3+V^3+aYUV+Q_4)/Z^n, \\
(X_9)=(Z+Y^4+aY^2U^2+U^4+Q_3)/Z^n,\\
(J_{10})=(Z+Y^3+aYU^4+U^6+Q_3)/Z^n;\\
(P_8^\#)=(ZY+Y^3+aY^2U+U^3+Q_3)/Z^n,\\
(X_9^\#)=(Z^2+aZY^2+Y^4+Q_2)/Z^n,\\
(J_{10}^\#)=(Z^3+aZ^2Y+Y^3+Q_2)/Z^n.
\end{array}
$$

$$
\begin{array}{ccccccccccccccc}
  &   &   &   &   &   &   & \rd  &   &   &   &   &   &   &   \\
 & & & & & & & & (P_8) & \leftarrow & (X_9) & \leftarrow & (J_{10}) & & \\
  & & & & & & & \rd & \downarrow & \rd & \downarrow & \rd &  \downarrow
& \rd & \\
 &  &  &  &  &  &  &  & E_6 & \leftarrow & E_7 & \leftarrow & E_8 &  &  \\
 & & & & & \rd & & \rd & \downarrow & \rd & \downarrow & \rd &
\downarrow & \rd &  \\
  & & & & & & D_4 & \leftarrow & D_5 & \leftarrow & D_6 & \leftarrow &
D_7 & \leftarrow & \cdots \\
 & \rd & & \rd & & \rd & \downarrow & \rd & \downarrow & \rd &
\downarrow & \rd &  & \rd & \\
A_0 & \leftarrow & A_1 & \leftarrow & A_2 & \leftarrow & A_3 &
\leftarrow & A_4 & \leftarrow & A_5 & \leftarrow & A_6 & \leftarrow &
\cdots \\
 & \ru & \uparrow & \ru & \uparrow & \ru & \uparrow & \ru & \uparrow &
\ru & \uparrow & \ru & \uparrow & \ru &  \\
 &  & B_2 & \leftarrow & C_3 & \leftarrow  & C_4 & \leftarrow  & C_5 &
\leftarrow  & C_6 & \leftarrow  & C_7 & \leftarrow  & \cdots \\
 & \ru & \uparrow & \ru & \uparrow & \ru & \uparrow & \ru &  &  &  &  &  &
&  \\
 &  & B_3 & \leftarrow  & F_4 & \leftarrow  & (P_8^\#) &  &  & & & & & & \\
 & \ru & \uparrow & \ru & \uparrow &  & | & & & & & & & & \\
 & & B_4 & \leftarrow  & (J_{10}^\#) & - & (X_9^\#) & & & & & & & & \\
 & \ru & \uparrow &\ru & & & & & & & & & & &  \\
 & & B_5 & & & & & & & & & & & & \\
 & \ru & \uparrow & & & & & & & & & & & & \\
 & & \cdots & & & & & & & & & & & &
\end{array}
$$
$$
A_k=\frac{Y^{k+1}+Z+U^2}{Z^n}; \; B_k=\frac{Y^2+Z^k+U^2}{Z^n}; \;
C_k=\frac{Y^k+YZ+U^2}{Z^n};
$$
$$
D_k=\frac{Y^2U+U^{k-1}+Z}{Z^n}; \;E_6=\frac{Y^3+U^4+Z}{Z^n}; \;
E_7=\frac{Y^3+YU^3+Z}{Z^n};
$$
$$
E_8=\frac{Y^3+U^5+Z}{Z^n}; \; F_4=\frac{Y^3+Z^2+U^2}{Z^n}.
$$
\centerline{Fig. 3. Adjacencies of $\R$-simple fraction singularities
of the form $H/Z^n$.}
\medskip

Here by  $Q_r$ we denote the quadratic form $Y_{r+1}^2+\cdots+Y_m^2$.
Thus, the coordinates in $\C^m$ are denoted by
$(Z,Y,Y_3,\ldots,Y_m)$ for $A_k, B_k, C_k, F_4, (X_9^\#),
(J_{10}^\#)$,  by
$(Z,Y,U,Y_4,\ldots,Y_m)$ for $D_k,E_k,(X_9),(J_{10})$, $(P_8^\#)$ and by
$(Y,Z,U,V,Y_5,\ldots,Y_m)$ for $(P_8)$.

Fencing  singularities are even not $\RRR$-simple (they have
moduli with respect to  $\RRR$-eqivalence of fractions of the form
$H/Z^n$). Besides these fencing
not  $\RRR$-simple singularities and those adjacent to them
there exist also  classes
of non $\R$-simple singularities $H/Z^n$
$A_\infty, B_\infty, C_\infty, D_\infty$ of  infinite codimension.
Singularities with
non suffuciently large degrees $n$ of denominators ($n\leq k$ for $B_k$,
$n=1$ for $A_k, C_k, D_k, E_k$, $n=1$ and $2$ for $F_4$) are
not  $\R$-simple as well.

In the fig.~3 the singularities of equal codimensions are  connected by
lines. The formulas in the figure correspond to the case $m=3$.

\medskip

\noindent {\bf Example.} General four-parameter families
of fractions $H/Z^n$ contain only simple singularities:
$D_4, A_4, C_4, F_4, B_4$ -- at isolated points, $A_3,
C_3, B_3$ -- on certain curves, $A_2$,$B_2$ --on some
surfaces, $A_1$ --  on three dimensional submanifolds
and $A_0$ at the remaining points of four-dimensional parameter space.
Here we suppose, that $n>4$, otherwise  class $B_4$ in not $\R$-simple.

\medskip

\noindent {\bf Proof of the theorem.} Boundary singularity classification
implies that either function $H$ might be reduced to the corresponding
normal form of the numerator from the theorem  by a diffeomorphism, which
preserve hyperplane $Z=0$, or this function is adjacent to one of
the six fencing  unimodal singularity classes,  or belongs to one
of the following classes of infinite codimension $A_\infty, \ldots, D_\infty$.

In the first  case the fraction might be reduced (by biholomorphic
transformation) to the form
$$
H/Z^n\approx EH_0/Z^n,
$$
where $E(0)\ne 0$. Normal forms  $H_0$ for all these classes are
quasihomogeneous. If $n$ is sufficiently large (precise limits are
described in the theorem), then the quasihomogeneous degree of the
numerator is less than that of the denominator. According
to lemma~3~$\S$~4 coefficient  $E$ might be
exterminated by an appropriate $\R$-equivalence. So in this case
$$
H/Z^n\approx H_0/Z^n.
$$

Any fencing  singularity is non $\RRR$-simple as the singularity of
fraction $H/Z^n$. This might be proved  by lemma~2~$\S$~4
arguments (as the fencing singularities are quasihomogeneous).
The equality of codimension of classes $A_\infty, \ldots, D_\infty$
to infinity  is evident. The theorem is proved.

\medskip

\noindent {\bf Corollary.} {\it
Generic family of fractions $H/Z^n$, containing  simple
singularity $H_0/Z^n$ for zero parameter value , can be reduced
for the nearby parameter values to the form  $\F/Z^n$, where $\F$
is a versal deformation of boundary singularity of
$H$ (using a holomorphic change of variables holomorphically
depending on parameters and a holomorphic change of parameters).
}

\medskip

\noindent {\bf Example.} One-parameter  generic family  of fractions
$H/Z^n$ in $\C^2$ might be reduced to the form

\begin{itemize}
\item[(1)] $A_1=(Y^2+Z+\l)/Z^n$,
\end{itemize}
where $\l$ is a parameter.

Two-parameter generic family of fractions
have local normal forms

\begin{itemize}
\item[(2)] $A_2=(Y^3+\l_1Y+\l_2+Z)/Z^n$,
\item[(3)] $B_2=(Y^2+Z^2+\l_1Z+\l_2)/Z^n$.
\end{itemize}

Finally three-parameter generic families  have normal forms:


\begin{itemize}
\item[(4)] $A_3=(Y^4+\l_1Y^2+\l_2Y+\l_3+Z)/Z^n$,
\item[(5)] $B_3=(Y^2+Z^3+\l_1Z^2+\l_2Z+\l_3)/Z^n$,
\item[(6)] $C_3=(Y^3+\l_1Y^2+\l_2Y+\l_3+YZ)/Z^n$.
\end{itemize}

Appliing these formulae to generic
families of polynomials in two variables at infinity, we obtain

\medskip

\noindent {\bf Corollary.} {\it
If the  polynomial  degree  $n$ is sufficiently large
($n>p$, where $p$ is the number of family  parameters),
then (provided that $p<5$)
a family of polynomials , which depends generically on
$p$ parameters, has local normal form  $f=\F/Z^n$, where
$\F$ is a versal deormation of the corresponding  simple
boundary singularity $(A_k, B_k, C_k, F_4)$.
}

\medskip

\noindent {\bf Example.} Level lines of generic polynomial
$(p=0)$ intersect  transversally the infinity line ($A_0$ case).
Formula $A_0=Y/Z^n$ implies that the level lines
$(Y=cZ^n)$ of generic polynomial  $f$ of degree $n>1$ are tangent to
each other at generic point at infinity-line (they have  there a
common point of multiplicity  $n$).

\medskip

In generic one parameter family of polynomials of degree $n\geq 2$
the level lines might have  tangency
with the infinity  line for isolated parameter values
(formula (1) for  $\l=0$  determines
the level line equation of polynomial  $f$: $Z=-Y^2+cZ^n$).
The latter implies that all level lines $f=c$ are quadratically tangent
to infinity line. Meanwhile  these lines become more tangent to each other
(than for regular point). At the infinity they intersect at a
common point with multiplicity  $2n$.

Formula (1) shows how the bifurcation (i.e. the transition from
transversal intersection with the infinity to quadratic tangency
($Z=-Y^2-\l+cZ^n$) looks like. Naturally,
the conjunction of two intersection points   (distance between them
being of the order $\sqrt{\l}$ when $\l\ne 0$) causes for $\l=0$ the
augmentation of their intersection multiplicity : Instead of two
intersection points of multiplicity $n$ there arise one of multiplicity
$2n$.

Similarly, for $n\geq 3$ generic bifurcations in two
parameter families of polynomials of degree $n$ are described by
formulae (2), (3). When $n\geq 4$ generic three parameter
bifurcations are determined by formulae (4), (5), (6).


Our formulae imply various topological and analitic
results on singularities and bifurcations at infinity of  level surfaces
of families of  polynomials in  arbitrary number of variables:
theory of vanishing cycles, corresponding monodromy group theory
(generalised braids group representations), asymptotic behaviour of
various integrals, related to polynomials, corresponding period mappings,
Poisson and Frobenius structures etc.

 Even simplest examples (of small dimension and codimension)
in this field provides  non trivial results.
I would like to emphasize the possibility to apply these ideas
to the study of non subregular orbits of coadjoint representation
of simple complex Lie algebra and of complicated hierarhy of conditions
at infinity, appearing when "Milnor fibrations" of these non isolated
singularities are investigated (cr.~\cite{6},~\cite{7}).


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Steklov Mathematical Institute  of Russian Academy of Sciences
and Universite Paris-Dauphine.

\end{document}
