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| {{Redirect|M-curve|a curve describing the demographics of working women|Career woman#Limited work options}}
| | Hello and welcome. My title is Figures Wunder. My day occupation is a meter reader. One of the extremely best things in the world for me is to do aerobics and now I'm attempting to earn money with it. For many years he's been residing in North Dakota and his family enjoys it.<br><br>my webpage :: [http://apcbook.com.ng/index.php?do=/profile-446/info/ http://apcbook.com.ng/index.php?do=/profile-446/info] |
| [[File:ECClines-3.svg|thumb|300px|The [[elliptic curve]] (smooth degree 3) on the left is an M-curve, as it has the maximum (2) components, while the curve on the right has only 1 component.]]
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| In [[real number|real]] [[algebraic geometry]], '''Harnack's curve theorem''', named after [[Carl Gustav Axel Harnack|Axel Harnack]], describes the possible numbers of [[Connected space|connected component]]s that an algebraic curve can have, in terms of the degree of the curve. For any [[algebraic curve]] of degree ''m'' in the real [[projective plane]], the number of components ''c'' is bounded by
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| :<math>\frac{1-(-1)^m}{2} \le c \le \frac{(m-1)(m-2)}{2}+1.\ </math>
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| The maximum number is one more than the maximum [[geometric genus|genus]] of a curve of degree ''m,'' attained when the curve is nonsingular. Moreover, any number of components in this range of possible values can be attained.
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| [[File:Trott bitangents.png|thumb|The [[Trott curve]], shown here with 7 of its bitangents, is a quartic (degree 4) M-curve, attaining the maximum (4) components for a curve of that degree.]]
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| A curve which attains the maximum number of real components is called an
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| ''M-curve'' (from "maximum") – for example, an [[elliptic curve]] with two components, such as <math>y^2=x^3-x,</math> or the [[Trott curve]], a quartic with four components, are examples of M-curves.
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| This theorem formed the background to [[Hilbert's sixteenth problem]].
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| == References ==
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| *D. A. Gudkov, ''The topology of real projective algebraic varieties'', Uspekhi Mat. Nauk 29 (1974), 3–79 (Russian), English transl., Russian Math. Surveys 29:4 (1974), 1–79
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| *[[Carl Gustav Axel Harnack|C. G. A. Harnack]], ''Über Vieltheiligkeit der ebenen algebraischen Curven'', Math. Ann. '''10''' (1876), 189–199
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| *G. Wilson, ''Hilbert's sixteenth problem'', Topology '''17''' (1978), 53–74
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| [[Category:Real algebraic geometry]]
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| [[Category:Theorems in algebraic geometry]]
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Hello and welcome. My title is Figures Wunder. My day occupation is a meter reader. One of the extremely best things in the world for me is to do aerobics and now I'm attempting to earn money with it. For many years he's been residing in North Dakota and his family enjoys it.
my webpage :: http://apcbook.com.ng/index.php?do=/profile-446/info