Real plane algebraic curves with finitely many points
Ponente(s): Erwan Brugallé Bouttier
Given a real polynomial of degree $d$ in two variable $P(x,y)$, the
equation $P(x,y)=0$ has generically either no or infinitely many real
solutions. When $d$ is even, it may nevertheless happen that such
equation has a positive, yet finite, number of real solutions. In that
case, it is natural to wonder what is the maximal possible number of
real solutions in terms of $d$. Surprisingly, this simple problem turns
out to be quite difficult and remains unsolved for $d\ge 10$. In this
talk I will report some recent developments concerning this problem.
This is a joint work with Alex Degtyarev, Ilia Itenberg, and Frédéric
Mangolte.