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Restrictions on arrangements of ovals of projective algebraic curves of odd degree

Author(s): Anatoly B. Korchagin
Journal: Proc. Amer. Math. Soc. 129 (2001), 363-370.
MSC (2000): Primary 14P25; Secondary 14H50, 14P05
Posted: August 30, 2000
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Abstract: This paper investigates the first part of Hilbert's 16th problem which asks about topology of the real projective algebraic curves. Using the Rokhlin-Viro-Fiedler method of complex orientation, we obtain new restrictions on the arrangements of ovals of projective algebraic curves of odd degree $d = 4k + 1$, $k \geq 2$, with nests of depth $k$.

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A. Harnack, Uber die Vieltheiligkeit der ebenen algebraischen Kurven, Math. Ann. 10(1876), 189-199.

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V. A. Rokhlin, Complex topological characteristics of real algebraic curves, Uspekhi Mat. Nauk, 33(1978), no. 5, 77-89; English transl. in Russian Math. Survey, 33(1979). MR 81m:14024

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O. Ya. Viro, Achievements in the topology of real algebraic varieties over the last six years, Russian Math. Surveys, 41(1986), 52-82. MR 87m:14023

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O. Ya. Viro, Real plane algebraic curves: construction with controlled topology, Leningrad Math. J., 1(1990), no. 5, 1059-1134. MR 91b:14078

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Additional Information:

Anatoly B. Korchagin
Affiliation: Department of Mathematics and Statistics, Texas Tech University, Lubbock, Texas 79409-1042
Email: korchag@math.ttu.edu

DOI: 10.1090/S0002-9939-00-05568-4
PII: S 0002-9939(00)05568-4
Keywords: Complex orientations of algebraic curves, positive and negative ovals, positive and negative injective pairs, chains of ovals
Received by editor(s): April 14, 1998
Received by editor(s) in revised form: May 4, 1999
Posted: August 30, 2000
Communicated by: Leslie Saper
Copyright of article: Copyright 2000, American Mathematical Society

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