CUBICS AS TOOLS TO STUDY THE TOPOLOGY OF $M$-CURVES OF DEGREE 9 IN ${\bb R} P^2$

Abstract

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.