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| {{Redirect|M-curve|a curve describing the demographics of working women|Career woman#Limited work options}}
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| [[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.]] | |
| 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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