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Gudkov's conjecture

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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

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
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