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 ${\displaystyle 2d}$ obeys the congruence

${\displaystyle p-n\equiv d^{2}\,(\!{\bmod {8}}),}$

where ${\displaystyle p}$ is the number of positive ovals and ${\displaystyle n}$ 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 ${\displaystyle k-1}$, where ${\displaystyle k}$ is the number of maximal components of the curve.^[1])

The theorem was proved by the combined works of Vladimir Arnold and Vladimir Rokhlin.^[2][3][4]