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 [math]\displaystyle{ 2d }[/math] obeys the congruence

[math]\displaystyle{ p - n \equiv d^2\, (\!\bmod 8), }[/math]

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

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

See also

• Hilbert's sixteenth problem
• Tropical geometry