Adjunction inequality for real algebraic curves

Abstract

The zero set of a real polynomial in two variable is a curve in $\mathbb R^2$. For a generic choice of its coefficients this is a non-singular curve, a collection of circles and lines properly embedded in $\mathbb R^2$. What topological arrangements of these circles and lines appear for the polynomials of a given degree? This question arised in the 19th century in the works of Harnack and Hilbert and was included by Hilbert into his 16th problem. Several partial results were obtained since then. However the complete answer is known only for polynomials of degree 5 or less. The paper presents a new partial result toward the solution of the 16th Hilbert problem. The proof makes use of the proof by Kronheimer and Mrowka of the Thom conjecture in $\mathbb C P^2$.