Gudkov's conjecture
In real algebraic geometry, Gudkov's conjecture, also called Gudkov’s congruence, (named after D. A. Gudkov) was a conjecture, and is now a theorem, which states that "a M-curve* of even degree 2d obeys p – n ≡ d2 (mod 8)", where p is the number of positive ovals and n the number of negative ovals of the M-curve. It was proved by the combined works of Vladimir Arnold and Vladimir Rokhlin.[1][2][3]
See also
Notes
- Note: M-curve stands for "maximal curve", it means a smooth algebraic curve in the reals whose genus is n-1, where n is the number of maximum components of the curve.[4]
References
- ↑ Sharpe, R. W. (1975), "On the ovals of even-degree plane curves", The Michigan Mathematical Journal, 22 (3): 285–288 (1976), MR 0389919
- ↑ 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
- ↑ Degtyarev, A. I.; Kharlamov, V. M. (2000), "Topological properties of real algebraic varieties: du côté de chez Rokhlin" (PDF), Uspekhi Matematicheskikh Nauk, 55 (4(334)): 129–212, doi:10.1070/rm2000v055n04ABEH000315, MR 1786731
- ↑ Arnold, Vladimir I. (2013). Real Algebraic Geometry. Springer. p. 95. ISBN 978-3-642-36243-9.
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