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                          % References



\def\refKhV		{1}
\def\refKorchagin       {2} 




\rightheadtext{ Congruence modulo 8 for curves of degree 9 }
\topmatter
%-----

\title            Congruence modulo 8 for real 
                  algebraic curves of degree 9
\endtitle
%-----
\author           O.Ya.~Viro, S.Yu.~Orevkov
\endauthor
%-----
\address	St.Petersburg Branch of Steklov Math. Inst.
		and Upsala Univ.
\endaddress
%-----
\address	Steklov Math. Inst. (Moscow) and
		Univ. Paul Sabatier (Toulouse)
\endaddress
%-----
\endtopmatter

\document

\subhead    1. Introduction and statement of the result
\endsubhead
%
Let $A$ be a non-singular real algebraic curve of degree $m$ in 
$\RP^2$. Its connected components are embedded circles.
Those of them whose complement in $\RP^2$ is not connected are called 
{\it ovals}. 
One says that an oval $u$ lies {\it inside} an oval $v$ if $u$ is
contained is the orientable component of the complement of $v$.
%
%Let us call the {\it interior} of an oval
%the oriented component of its complement and the {\it exterior}
%of an oval --- the non-oriented one.
%
A union of $d$ ovals $v_1,\dots,v_d$ such that
$v_i$ is inside $v_{i+1}$, $1\le i<d$,
is called a {\it nest of the depth} $d$.
An oval is called {\it exterior} if it does not lye inside any other oval;
an oval is called {\it empty} if there is no other ovals inside it.
An oval is called {\it even} if it is contained inside an even number 
of other ovals, and {\it odd} otherwise.
Denote by $p$ and $n$ the number of even and odd ovals respectively.
One says that $A$ is an {\it $M$-curve} if it has the maximal possible
number of connected components which equals $M(m)=(m-1)(m-2)/2+1$.
If $A$ has $M(m)-i$ connected components then 
it is called an {\it $(M-i)$-curve} 
Let $\C A$ be the complexification of $A$. If 
$\C A\setminus A$ is not connected, $A$ is a curve {\it of type I};
if $\C A\setminus A$ is connected then 
$A$ is a curve {\it of type II}.

For curves of an even degree $m=2k$, in some cases, the difference
$p-n$ satisfies congruences. For example,

\noindent
{\it Gudkov-Rohlin congruence} $p-n\equiv k^2\mod 8$
for $M$-curves,

\noindent
{\it Gudkov-Krahnov-Kharlamov congruence }
$p-n\equiv k^2\pm 1\mod 8$ for $(M-1)$-curves,

\noindent
{\it Kharlamov-Marin congruence }
$p-n\not\equiv k^2+4\mod 8$ for $M$-curves of type II, and 

\noindent
{\it Arnold congruence } $p-n\equiv k^2\mod 4$
for curves of type I.

\noindent
These statements do not extend to curves of {\it odd} degrees.
So, for an $M$-curve of any odd degree $2k+1$ with $k\ge3$, 
the residue $p-n\mod8$ may take any values congruent to $k\mod2$. 
As far as we know, the following theorem is the first result of this kind.

\proclaim{ Theorem 1 } Let $A$ be a curve of degree
$m=2k+1=4d+1$ which has 
$4$ paiwise distinct nests of the depth $d$. 
Then
$$
\xalignat4
&\text{if $A$ is an $M$-curve then}
					&&p-n\equiv -k	&&\mod 8; 
							&&\qquad(1)
									\\
&\text{if $A$ is an $(M-1)$-curve then}
					&&p-n\equiv -k\pm1	&&\mod 8;			
									\\ 
&\text{if $A$ is an $(M-2)$-curve of type II then}
					&&p-n\not\equiv -k+4&&\mod 8;			
									\\
&\text{if $A$ is a curve of type I then}
					&&p-n\equiv -k	&&\mod 4;			
\endxalignat
$$ 
\endproclaim

It is clear that (1) for $d=2$ is equivalent to the fact that 
the number of exterior empty ovals of an $M$-curve of degree 9 with
4 nests is divisible by 4. This was conjectured by 
Korchagin \cite{\refKorchagin}.
Theorem 1 is obtained below (see Sect.~4) as a consequence of
Kharlamov-Viro congruence \cite{\refKhV} which generalizes the
classical congruences to the case of singular curves of even degrees.

%This paper appeared as an attempt to find a new general congruence
%whose a particular case would be the Korchagin's conjecture.
%However, this attempt failed because it happened (see the proof of
%Theorem 1) that (1) is a particular case of a previousely known
%congruence --- Viro-Kharlamov congruence \cite{\refKhV} for 
%singular curves.

\subhead 2. Brown - van der Blij invariant 
\endsubhead
%
By a {\it quadratic space} we mean a triple $(V,\circ,q)$
composed of a vector space $V$ over the field $\Z_2$,
a bilinear form 
$V\times V\to\Z_2$, $(x,y)\mapsto x\circ y$, and a function
$q:V\mapsto\Z_4$ which is quadratic with respect to $\circ$
in the sense that 
$q(x+y)=q(x)+q(y)+2x\circ y$.
A quadratic space is determined by its {\it Gram matrix} with respect
to a base $e_1,\dots,e_n$ of $V$, i.e. the matrix
$Q=(q_{ij})$ where $q_{ii}=q(e_i)$ and $q_{ij}=e_i\circ e_j$
for $i\ne j$ (the diagonal
entries are defined $\mod 4$, the others $\mod 2$; note that
$q(x)\equiv x\circ x \mod 2$).
It is easy to see that by elementary changes of the base,
one can put the Gram matrix to the block-diagonal form 
$\diag(d_1,\dots,d_t)\oplus Q_1\oplus\dots\oplus Q_s$ 
where each block $Q_i$ is either 
$\left(\smallmatrix0&1\\1&0\endsmallmatrix\right)$, or
$\left(\smallmatrix2&1\\1&2\endsmallmatrix\right)$. 
If all $d_i\ne2$, we say that the form 
$q$ is {\it informative} and in this case 
we define its {\it Brown - van der Blij invariant}
$B(q)=\sum B(d_i) + \sum B(Q_i) \mod 8$ where
$B(0)=0$, $B(1)=1$, $B(-1)=-1$,
$B\left(\smallmatrix0&1\\1&0\endsmallmatrix\right)=0$, and
$B\left(\smallmatrix2&1\\1&2\endsmallmatrix\right)=4$.

\subhead 3. Kharlamov-Viro congruence for nodal curves
\endsubhead
%
Let $A$ be a curve in $\RP^2$
of degree $2k$ defined by $f=0$ and let
each of its singular points be the point of transverse intersection
of two smooth real local branches.
$A$ is called an {\it $M$-curve} (a curve {\it of type I}) 
if the normalization of any its irreducible component is an 
$M$-curve (a curve of type I).
A curve which is not of type I, is of type II.
%$A$ is a curve  {\it of type II} if it is not of type I.
Let $x_1,\dots,x_s$ be the singular points and $\Gamma_A$
be the union of the connected components of $A$ passing through them.
Let $b=0$ if $\RP^2_+=\{f\ge0\}$
is contractable in $\RP^2$ and $b=(-1)^k$ otherwise.
%Let us denote by $L$ a simple closed loop in $\Gamma_A$,
%non-contractible in $\RP^2$ (if such a loop exists).
%
%Let $C_1,\dots,C_r$ be the components of the complement of $\Gamma_A$
%lying on the same side of $\Gamma_A$ as $\RP^2_+$.
%

Suppose that $\Gamma_A$ is connected.
Let us define a quadratic space $(V,\circ,q)$ as follows.
Let $C_1,\dots,C_r$ be the oriented components of 
$\RP^2\setminus\Gamma_A$ on which $f>0$ near $\Gamma_A$.
Let $(V_0,\circ,q_0)$ be the quadratic space with the orthogonal base
$e_1,\dots,e_s$ such that  
$q_0(e_1)=\dots=q_0(e_s)=-1$. 
%Let $V$ be the subspace of $V_0$ generated by 
Set $c_i=\sum_{j\in\alpha_i}e_j$ where
$\{x_j\}_{j\in\alpha_i}$ are the singular points through which
$\partial C_i$ passes only once.
In the cases when either $\Gamma_A$ is contractible in $\RP^2$ or,
as in Sect.~4, there is a {\it branch of $\Gamma_A$} (i.e. a smoothly
immersed circle) which is non-contractible in $\RP^2$, we
define $V\subset V_0$ as the subset generated by $c_1,\dots,c_r$
and we set $q=q_0|_V$.

In the case when $\Gamma_A$ is not contractible in $\RP^2$
but all its branches are,
let us choose a simple closed curve in $\Gamma_A$ which is not contractible
in $\RP^2$.
Let $(V'_0,\circ,q'_0)$ be the quadratic space with the base
$(e_0,\dots,e_s)$ which contains $V_0$ as a quadratic subspace
($q'_0|_{V_0}=q_0$) and let $q'_0(e_0)=(-1)^k$, 
$e_0\circ e_j = 0$ iff $L\sim 0$ in $H_1(RP^2_+, RP^2_+\setminus x_j)$.
Let $V\subset V'_0$ be the subspace generated by
$c_1,\dots,c_r$, and $e_0+\sum_{j\in\alpha_0}e_j$ where 
$\alpha_0 = \{ j \,|\, L\not\sim 0$ in $H_1(RP^2_-, RP^2_-\setminus x_j)\}$, 
and let $q=q'_0|_V$.

If $\Gamma_A$ is not connected, we define $(V,\circ,q)$ as the
direct sum of quadratic spaces associated as above to each connected
component of $\Gamma_A$.

\proclaim{ Theorem 2}
Suppose that each branch of $A$ 
%(i.e. a smoothly immersed circle) 
which is contractible in $\RP^2$ cuts other branches at
$n\equiv0\mod4$ singular points 
and each branch which is not contractible in $\RP^2$, 
at $n\equiv(-1)^{k+1}\mod4$ singular points. 
%(in the both cases we count only those points
%through which the given branch passes only once).
If $A$ is an $M$-curve then
$\chi(\RP^2_+)\equiv k^2 + B(q) + b\mod8$ and also the corresponding
analogues of Gudkov-Krahnov-Kharlamov, Kharlamov-Marin, and Arnold 
congruences take place.
\endproclaim

%\midinsert
%\epsfxsize 100mm
%\centerline{\epsfbox{m8d9-12.eps}}
%\botcaption{ \hbox to 17mm{} (Fig.~1 
%             \hbox to 42mm{} (Fig.~2  }
%\endcaption
%\endinsert

Theorem 2 is a corollary of Theorem (3.B) on curves with arbitrary
singularities from the paper by Kharlamov and Viro \cite{\refKhV}.
Theorem 2 is formulated here because there are mistakes in \cite{\refKhV}
in the discussion of the corresponding particular case (4.I), (4.J) 
of Theorem (3.B). 

\subhead 4 Proof of Theorem 1
\endsubhead
Let us choose any three pairwise distinct nests of the depth $d$ 
and a point inside the innermost oval of each of them.
Theorem 1 follows from Theorem 2
applied to the union of $A$ and the three straight lines passing
through the three chosen points.
Indeed, the union of the three chosen lines and the non-contractible
branch of $A$ divides $\RP^2$ into 4 triangles and
3 quadrangles (curvilinear). 
All ovals not belonging to the three chosen nests lye in
the quadrangles (otherwise would exist a conic having too many
intersections with $A$). Therefore, after the suitable choice of the sign,
one has $\chi(\RP^2_+)=\chi(\bigcup\overline C_j)+p'-n'$ where
$p'$ and $n'$ are the numbers of even and odd ovals, not belonging
to the three chosen nests. 
$B(q)$ can be computed according to Sect.~2.


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\Refs

\ref\no \refKhV
\by     V.M.~Kharlamov, O.Ya.~Viro
\paper	Extension of the Gudkov-Rohlin congruence
\jour   Lect. Notes. Math. \vol 1346 \yr 1988 
\pages  357--406
\endref

\ref\no \refKorchagin
\by     A.B.~Korchagin
\paper  Construction of new M-curves of 9th degree
\jour   Lect. Notes. Math. \vol 1524 \yr 1991 
\pages  407--426
\endref


\endRefs

\enddocument