A real algebraic, plane, projective curve
$A$
of degree
$m$
is given by a homogeneous polynomial of degree
$m$
in three variables, with real coefficients, defined up to multiplication by a non-zero scalar. If
$F$
is such a polynomial defining
$A$
, we denote by
${\bb C} A$
and
${\bb R} A$
, respectively, the sets of solutions of the equation
$F = 0$
in
${\bb C} P^2$
, respectively
${\bb R} P^2$
. We suppose that the curve
$A$
is non-singular, that is,
$F$
has no critical points in
${\bb C}^3\backslash 0$
. Then
${\bb C} A$
is a Riemannian surface of genus
$g = (m-1)(m-2)/2$
, and
${\bb R} A$
is a collection of
$L \le g+1$
circles embedded in
${\bb R} P^2$
. If
$L = g+1$
, we say that
$A$
is an
$M$
-curve. A circle embedded in
${\bb R} P^2$
is called oval, or pseudo-line, depending on whether it realizes the class 0 or 1 of
$H_1({\bb R} P^2)$
. If
$m$
is even, the
$L$
components of
${\bb R} A$
are ovals; if
$m$
is odd,
${\bb R} A$
contains exactly one pseudo-line, which will be denoted by
${\cal J}$
. Note that an oval separates
${\bb R} P^2$
into two pieces, a Möbius band and a disc. The latter is called the interior of the oval. An oval of
${\bb R} A$
is said to be empty if its interior contains no other oval of
${\bb R} A$
. Two ovals form an injective pair if one of them lies in the interior of the other one.