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 a M-curve of even degree 2 d {\displaystyle 2d} obeys the congruence
p − n ≡ d 2 ( mod 8 ) , {\displaystyle p-n\equiv d^{2}\,(\!{\bmod {8}}),}
where p {\displaystyle p} is the number of positive ovals and n {\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 k − 1 {\displaystyle k-1} , where k {\displaystyle k} 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
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