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.