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 a M-curve of even degree $2d$ obeys the congruence

: $p\; -\; n\; \backslash equiv\; d^2\backslash ,\; (\backslash !\backslash bmod\; 8),$

where $p$ is the number of positive ovals and $n$ 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 $k-1$, where $k$ is the number of maximal components of the curve.{{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}})

The theorem was proved by the combined works of Vladimir Arnold and Vladimir Rokhlin.{{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}}{{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| doi-access = free

}}{{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

}}