The thesis presents Harnack’s inequality for smooth real projective plane
curves. The inequality gives a sharp upper bound for the number of connected components of the real locus of a curve in terms of the degree of the
curve, or equivalently in terms of the genus of its complexification. We first
introduce the basic notions of algebraic curves, Riemann surfaces, real structures, and homology over F2. The inequality is then proved on several levels.
First, we present the classical geometric proof, which uses an auxiliary curve,
a parity argument, and B´ezout’s theorem. Next, we discuss Klein’s proof,
where the real components of the curve appear as boundary components of
the quotient of the Riemann surface by complex conjugation. Finally, we
derive the inequality from the Smith-Thom inequality, which compares the
total Betti number of a space with involution and the total Betti number
of its fixed point set. The thesis shows that Harnack’s inequality is not
merely an isolated geometric estimate, but a consequence of a more general
topological comparison. For real projective plane curves, the Smith-Thom
inequality becomes, after realising the relevant Betti numbers, a comparison
between the number of components of the real locus and the genus of the
complexification. This comparison is precisely Harnack’s inequality.
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