Gudkov's conjecture

In real algebraic geometry, Gudkov's conjecture, also called Gudkov’s congruence, (named after Dmitry Gudkov) was a conjecture, and is now a theorem, which states that an M-curve of even degree $$2d$$ obeys the congruence
 * $$ p - n \equiv d^2\, (\!\bmod 8),$$

where $$p$$ is the number of positive ovals and $$n$$ the number of negative ovals of the M-curve. (Here, the term M-curve stands for "maximal curve"; it means a smooth algebraic curve over the reals whose genus is $$k-1$$, where $$k$$ is the number of maximal components of the curve. )

The theorem was proved by the combined works of Vladimir Arnold and Vladimir Rokhlin.