“Gudkov's conjecture”的意思、由来-开放百科全书

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

obeys the congruence

where

is the number of positive ovals and

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

, where

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

References

1. ^{{cite book|first=Vladimir I.|last=Arnold|authorlink=Vladimir Arnold| title=Real Algebraic Geometry|url=https://books.google.com/books?id=xchAAAAAQBAJ&pg=PA95|year=2013|publisher=Springer|isbn=978-3-642-36243-9|page=95}}
2. ^{{citation | last = Sharpe | first = Richard W. | issue = 3 | journal = Michigan Mathematical Journal | mr = 0389919 | pages = 285–288 (1976) | title = On the ovals of even-degree plane curves | url = http://projecteuclid.org/euclid.mmj/1029001529 | volume = 22 | year = 1975}}
3. ^{{citation | last1 = Khesin | first1 = Boris | author1link= Boris Khesin | last2 = Tabachnikov | first2 = Serge | author2link = Sergei Tabachnikov | doi = 10.1090/noti810 | issue = 3 | journal = Notices of the American Mathematical Society | mr = 2931629 | pages = 378–399 | title = Tribute to Vladimir Arnold | volume = 59 | year = 2012}}
4. ^{{citation | last1 = Degtyarev | first1 = Alexander I. | last2 = Kharlamov | first2 = Viatcheslav M. | doi = 10.1070/rm2000v055n04ABEH000315 | issue = 4(334) | journal = Uspekhi Matematicheskikh Nauk | mr = 1786731 | pages = 129–212 | title = Topological properties of real algebraic varieties: du côté de chez Rokhlin | url = http://www.fen.bilkent.edu.tr/~degt/papers/rokh.pdf | volume = 55 | year = 2000| arxiv = math/0004134 | bibcode = 2000RuMaS..55..735D }}