Arf Invariants of Real Algebraic Curves

. Let CA be the complex curve in CP(2) given by the same polynomial as RA. Thus RA = CA∩RP(2). RA is a M-curve precisely when CA \ RA consists of two punctured spheres which are interchanged by complex conjugation. Arbitrarily choose one of these components, say CA. The complex structure on CA induces an orientation on CA, and thus on each immersed circle of RA. Of course if we choose the other component, we would get the opposite orientation on each immersed circle of RA. An orientation on each of the components up to reversing all the orientations simultaneously is called a semi-orientation. Thus each M-curve has receives a semi-orientation, called the complex orientation [R]. An oval is a 2-sided simple closed curve in RP(2), the real projective plane. The inside of an oval is the component of its complement which is a disk. The outside of an oval is the component of its complement which is a Mobius band. Suppose C is the disjoint collection of oriented ovals. We call such a collection a simple curve. An oval of C is called even if it lies inside and even number of other ovals of C. An oval of C is called odd if it lies inside an odd number of other ovals of C. Let p(C) denote the number of even ovals in C, and n(C) denote the number of odd ovals in C. If one of ovals of C lies inside another oval of C or vise versa, we say they are linked. We say C is odd if each oval is linked with an odd number of other ovals. It follows that an odd curve must have an even number of components. We say C is even if each oval is linked with an even number of other ovals and the total number of ovals is odd. Let Π(C) denote the number of pairs of linked ovals for which the orientations on the curves extend to an orientation of the intervening annulus. Let Π−(C) denote the number of pairs of linked ovals for which the orientations on the curves do not extend to an orientation of the intervening annulus. We will usually write simply n, p, Π± without C, except in instances where it might be unclear to which curve these numerical characteristics refer.