Hilbert 16th problem (概説論文)

D.A. Gudkov,
The topology of real projective algebraic varieties,
Usp. Mat. Nauk, {\bf 29--4} (1974), 3--79

Russ. Math. Surveys, {\bf 29--4} (1974), 1--79.

[Risler80]

Seminaire sur la geometrie algebrique reelle,
Publ. math\' ematiques de l'Universit\' e Paris VII, No.9.

J.-J. Risler,
Sur le 16eme probleme de Hilbert: Un resume et quelques questions,
in [Risler80], 11--25.

V.A. Rokhlin,
Complex topological characteristics of real algebraic curves,
Usp. Math. Nauk, {\bf 33--5} (1978), 77--89.

Russ. Math. Surveys, {\bf 33--5} (1978), 85--98.

G. Wilson,
Hilbert's sixteenth problem,
Topology, {\bf 17} (1978), 53--73.

Guillou, Marin,
A la recherche de la topologie perdue.

Progr. Math. 62, Birkh\"auser Boston, 1986 (book).

O.Ya. Viro,
Progress in the topology of real algebraic varieties over the last six years,
Uspekhi Mat. Nauk {\bf 41--3} (1986), 45--67.
Russian Math. Surveys {\bf 41--3} (1986), 55--82.

J. Bochnak, M. Coste, M-F. Roy,
G\' eom\' etrie alg\' ebrique r\' eelle,
Ergeb. Math. Grenzgeb. (3), {\bf 12}, Springer-Verlag, 1987 (book).

　→英語版あり．

R. Silhol,
Real Algebraic Surfaces,
Lecture Notes in Math., {\bf 1392}, Springer-Verlag, 1989 (book).

V.M. Kharlamov, A. Korchagin, G. Polotovski\u\i, O. Viro, (ed.)
Topology of Real Algebraic Varieties and Related Topics.
Dedicated to the memory of Dmitri\u\i Andreevich Gudkov,
Amer. Math. Soc. Transl., (2), {\bf 173}, Adv. in Math. Sci., {\bf 29},
Amer. Math. Soc., Providence, RI, 1996.

A. Degtyarev and V. Kharlamov,

Topological properties of real algebraic varieties: Rokhlin's way,

Russian Math. Surveys 55, no. 4 (2000), 735--814.

Viro の論文

O.Ya. Viro,
Generalized Petrovskii and Arnol'd inequalities and curves with singularities,
Usp. Mat. Nauk, {\bf 33--3} (1978), 145--146.

O.Ya. Viro,
Construction of $M$-surfaces,
Funkts. Anal. Prilozhen., {\bf 13--3} (1979), 71--72. English transl.,
Funct. Anal. Appl., {\bf 13} (1979), 212--213.

O.Ya. Viro,
Construction of multicomponent real algebraic surfaces,
Dokl. Akad. Nauk SSSR,{\bf 248--2} (1979), 279--282.
English transl., Soviet Math. Dokl.,
{\bf 20--5} (1979), 991--995.

O.Ya. Viro,
Curves of degree $7$, curves of degree $8$ and the Ragsdale conjecture,
Dokl. Akad. Nauk SSSR 254--6 (1980), 1306--1310. English transl.,
Soviet. Math. Dokl., {\bf 22--2} (1980), 566--570.

O.Ya. Viro,
Gluing of plane algebraic curves and constructions of curves of degree 6 and 7,
Lecture Notes in Math., {\bf 1060} (1984), 187--200.

O.Ya. Viro,
Progress in the topology of real algebraic varieties over the last six years,
Uspekhi Mat. Nauk {\bf 41--3} (1986), 45--67.
English transl., Russian Math. Surveys {\bf 41--3} (1986), 55--82.

O.Ya. Viro, (ed.)
Topology and Geometry -- Rohlin Seminar,
Lecture Notes in Math., {\bf 1346}, Springer-Verlag, 1988.

O.Ya. Viro,
Real algebraic plane curves: Constructions with controlled topology,
Algebra and analysis, {\bf 1--5} (1989);
English transl.
Leningrad Math. J., {\bf 1--5} (1990), 1059--1134.

O.Ya. Viro,
Complex orientations of real algebraic surfaces,
in Topology of manifolds and varieties,
Adv. Soviet Math., {\bf 18},
Amer. Math. Soc., Providence, (1994), pp. 261--284.

O.Ya. Viro,
Patchworking real algebraic varieties,
Preprint, Uppsala Univ., 1994:42

O.Ya. Viro,
Generic immersions of the circle to surfaces and the complex topology of real algebraic curves,
in Topology of real algebraic varieties and related topics,
Amer. Math. Soc. Transl. Ser. 2, {\bf 173},
Amer. Math. Soc., Providence, (1996), pp. 231--252.

O.Ya. Viro, V.I. Zvonilov,
An inequality for the number of nonempty ovals of a curve of odd degree,
Algebra i Analiz. {\bf 4} (1992) (Russian);
English transl.
St. Petersburg Math. J. {\bf 4} (1993), 539--548.

G. Wilson; Hilbert's sixteenth problem

G. Wilson,
Hilbert's sixteenth problem,
Topology, 17 (1978), 53--73.

とても重要な論文．

微分トポロジー的な方法で，Hilbert 16問題に関するロシア人たちの一連の結果を再証明している．

曲線で分岐するCP^2の２重被覆が単連結であることが，初等的に証明されている．

Rislerの論文ほか

T.M. Rassias,
On Hilbert's sixteenth problem,
C. R. Math. Rep. Acad. Sci. Canada, {\bf 1} (1979), 203--205.

J.-J. Risler,
S\' eminaire sur la g\' eom\' etrie alg\' ebrique r\' eelle,
Publ. math\' ematiques de l'Universit\' e Paris VII, No.9.

J.-J. Risler,
Sur le 16$^{\mbox{\rm {\` eme}}}$ probl\` eme de Hilbert:
Un r\' esum\' e et quelques questions,
in \cite{Risler80}, 11--25.

J.-J. Risler,
Un analogue local du th\' eor\` eme de Harnack,
Invent. Math., {\bf 89} (1987), 119--137.

J.-J. Risler,
Construction d'hypersurfaces r\' eelles (d'apr\` es Viro),
S\' eminaire Bourbaki, 45\` eme ann\' ee (1992--93), n$^\circ$ 763

English transl., Ast\' erisque, 216 (1993), 69--86.

Rokhlin の論文

V.A. Rokhlin,
{\it
Proof of Gudkov conjecture,
}
Funkts. Anal. Prilozhen., {\bf 6--2} (1972),
62--64. English transl., Funct. Anal. Appl.,
{\bf 6} (1972), 62--64.

\bibitem{Rokhlin72b}
V.A. Rokhlin,
{\it
Congruences modulo 16
in Hilbert's sixteenth problem I,
}
Funkts. Anal. Prilozhen., {\bf 6--4} (1972),
English transl., Funct. Anal. Appl.,
{\bf 6} (1972), 301--306.

\bibitem{Rokhlin73}
V.A. Rokhlin,
{\it
Congruences modulo 16
in Hilbert's sixteenth problem II,
}
Funkts. Anal. Prilozhen., {\bf 7--2} (1973),
91--92. English transl., Funct. Anal. Appl.,
{\bf 7} (1973), 163--164.

\bibitem{Rokhlin74}
V.A. Rokhlin,
{\it
Complex orientations of real algebraic curves, }
Funkts. Anal. Prilozhen., {\bf 8--4} (1974),
71--75. English transl., Funct. Anal. Appl.,
{\bf 8} (1974), 331--334.

\bibitem{Rokhlin78}
V.A. Rokhlin,
{\it
Complex topological characteristics of real algebraic
curves,
}
Usp. Math. Nauk, {\bf 33--5} (1978),
77--89. English transl., Russ. Math. Surveys,
{\bf 33--5} (1978), 85--98.

\bibitem{Rokhlin80}
V.A. Rokhlin,
{\it
New inequalities in the topology of real algebraic curves,
}
Funkts. Anal. Prilozhen., {\bf 14--1} (1980),
37--43. English transl., Funct. Anal. Appl.,
{\bf 14} (1980), 29--33.

曲線の構成(Polotovskiiなど)

D.A. Gudkov,
Construction of a curve of the sixth order of type $\dfrac{1}{5}5$,
Izv. Vyssh. Uchebn. Zaved. Mat., {\bf 3 (130)} (1973), 28--36.

G.M. Polotovskii,
A catalogue of M-decomposing curves of sixth order,
Dokl. Acad. Nauk SSSR, {\bf 236--3} (1977), 548--551.

English transl., Soviet Math. Dokl., {\bf 18--5}, (1977).

G.M. Polotovskii,
$(M-1)$ and $(M-2)$-ramified curves of sixth order,
in [LA], 130--148.

[LA] E.A. Leontovich-Andronova,
Methods of Qualitative Theory of Differential Equations,
Interuniv. Collect. Gor'ki., (1978).

D.A. Gudkov,
On the topology of algebraic curves on a hyperboloid,
Usp. Mat. Nauk, {\bf 34--6} (1979), 26--32.

English transl., Russ. Math. Surveys, {\bf 34--6} (1979), 27--35.

２次変換によってRP^2上の曲線をhyperboloid上に写すことにより曲線を構成

G.M. Polotovskii,
On the classification of non singular curves of degree 8,
Lecture Notes in Math., 1346 (1988), Springer, pp. 455--485.

Gudkovの論文

D.A. Gudkov,
Construction of a curve of the 6th order of type $\dfrac{1}{5}5$,
Izv. Vyssh. Uchebn. Zaved. Mat., {\bf 3 (130)} (1973), 28--36.

D.A. Gudkov, A.D. Krakhnov,
Periodicity of the Euler characteristic of real algebraic (M-1)-varieties,
Funkt. Anal. Prilozhen., {\bf 7--2} (1973), 15--19.

= Funct. Anal. Appl., {\bf 7} (1973), 98--102.

D.A. Gudkov,
The topology of real projective algebraic varieties,
Usp. Mat. Nauk, 29-4 (1974), 3--79

= Russ. Math. Surveys, {\bf 29--4} (1974), 1--79.

D.A. Gudkov, G.A. Utkin,
The topology of curves of the sixth order and surfaces of the fourth order,

Gor'kov. Gos. Univ. Uchen. Zap., {\bf 87} (1969), 3--213.

= Nine papers on Hilbert's 16th problem,
Amer. Math. Soc. Transl. (2) {\bf 112} (1978).

D.A. Gudkov,
On the topology of algebraic curves on a hyperboloid,
Usp. Mat. Nauk, {\bf 34--6} (1979), 26--32.

= Russ. Math. Surveys, {\bf 34--6} (1979), 27--35.

ここで，hyperboloidとは，RP^3の中のnonsingular quadric (2次曲面)で，トーラスに同相なもののこと．これは，RP^1 x RP^1 とみなせる．球面S^2に同相な場合は，ellipsoidと呼ばれる．ほかには，空集合の場合がある（同じ定義方程式の複素零点集合は空でないにも関わらず）

Gudkovのこの論文では，RP^2上の代数曲線を，RP^2からhyperboloidへの２次変換（双有理変換の一種）によってhyperboloid上に写すことにより，hyperboloid上の曲線を構成している．その際には，order がどのように変化するかについての公式が重要である．

D.A. Gudkov,
Generalization of a theorem of Brusotti for curves on a surface of second order,

Funct. Anal. Appl., {\bf 14} (1980), 15--18.

Petrovskii の論文

I.G. Petrovskii,
On the topology of real plane algebraic curves,
Ann. of Math., (2) {\bf 39} (1938), 189--209.

I.G. Petrovskii,
On the topological properties of algebraic lines and surfaces,
Vestnik Moskov. Gos. Univ., {\bf 11} (1949), 23--27.

O.A. Oleinik,
Some estimates for the Betti numbers of real algebraic surfaces,
Dokl. Akad. Nauk SSSR, {\bf 67} (1949), 425-426.

O.A. Oleinik, I.G. Petrovskii,
On the topology of real algebraic surfaces,
Dokl. Acad. Nauk SSSR, {\bf 67} (1949), 31--32.

O.A. Oleinik, I.G. Petrovskii,
On the topology of real algebraic surfaces,
Izv. Akad. Nauk SSSR Ser Mat., {\bf 13} (1949), 389--402. English transl.,
Amer. Math. Soc. Transl., {\bf 7} (1952), 399--417.

O.A. Oleinik,
On the topology of real algebraic space curves,
Dokl. Akad. Nauk SSSR, {\bf 70} (1950), 13-14.

O.A. Oleinik,
On algebraic curves on an algebraic surface,
Usp. Mat. Nauk, {\bf 6--4} (1951), 209--210.

O.A. Oleinik,
Estimates of the Betti numbers of real algebraic hypersurfaces,
Mat. Sb., {\bf 28} (1951), 635--640.

O.A. Oleinik,
On the topology of real algebraic curves on an algebraic surface,
Mat. Sb., {\bf 29} (1951), 133-156.

O.A. Oleinik, I.G. Petrovskii,
On the topology of real algebraic surfaces,
Usp. Mat. Nauk, {\bf 6--4} (1951), 208--209.

Thom の論文

R. Thom,
Sur l'homologie des varietes algebriques reelles,
in

Differential and Combinatorial Topology.
(A symposium in honor of M. Morse),
Princeton University Press, (1965), 255--265.

Hilbert 16th problem (19C頃の論文)

A. Harnack,
Uber die Vieltheiligkeit der ebenen algebraischen Curven,
Math. Ann. 10 (1875), 189--199.

F. Klein,
Eine neue Relation zwischen den Singularitaten einer algebraischen Kurve,
Math. Ann. 10 (1876), 199--210.

D. Hilbert,
Uber die reellen Zuge algebraischer Curven,
Math. Ann. 38 (1891), 115--138.

V. Ragsdale,
On the arrangement of the real branches of plane algebraic curves,
Amer. J. Math. 28 (1906), 377--404.

K. Rohn,
Die Maximalzahl und Anordnung der Ovale bei der ebenen Kurve 6
Ordnung und bei der Flache 4 Ordnung,
Math. Ann. 73 (1913), 177--229.