The talk is devoted to real plane algebraic curves with finitely many real points.
We study the following question: what is the maximal possible number of real points
of such a curve provided that it has given (even) degree and given geometric genus?
This question is related to the first part of Hilbert’s 16-th problem (topology of real algebraic varieties)
and to Hilbert’s 17-th problem (more precisely, positivity of real polynomials vs. their representations as sums of squares).
We obtain a complete answer to the above question in the case where the degree is sufficiently large
with respect to the genus, and prove certain lower and upper bounds for the number in question in the general case.
This is a joint work with E. Brugallé, A. Degtyarev and F. Mangolte.