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]

See also

• Hilbert's sixteenth problem
• Tropical geometry

References

1. Arnold, Vladimir I. (2013). Real Algebraic Geometry. Springer. p. 95. ISBN 978-3-642-36243-9.
2. Sharpe, Richard W. (1975), "On the ovals of even-degree plane curves", Michigan Mathematical Journal, 22 (3): 285–288 (1976), MR 0389919
3. Khesin, Boris; Tabachnikov, Serge (2012), "Tribute to Vladimir Arnold", Notices of the American Mathematical Society, 59 (3): 378–399, doi:10.1090/noti810, MR 2931629
4. Degtyarev, Alexander I.; Kharlamov, Viatcheslav M. (2000), "Topological properties of real algebraic varieties: du côté de chez Rokhlin" (PDF), Uspekhi Matematicheskikh Nauk, 55 (4(334)): 129–212, arXiv:math/0004134, Bibcode:2000RuMaS..55..735D, doi:10.1070/rm2000v055n04ABEH000315, MR 1786731