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| [[Image:Cyc7.png|thumb| <math>|z^6+z^5+z^4+z^3+ </math>
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| <math> z^2+z+1|=1</math>]]
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| In mathematics, a '''polynomial lemniscate''' or ''polynomial level curve'' is a [[algebraic curve|plane algebraic curve]] of degree 2n, constructed from a polynomial ''p'' with complex coefficients of degree ''n''.
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| For any such polynomial ''p'' and positive real number ''c'', we may define a set of complex numbers by <math>|p(z)| = c.</math> This set of numbers may be equated to points in the real Cartesian plane, leading to an algebraic curve ''ƒ''(''x'', ''y'') = ''c''<sup>2</sup> of degree 2''n'', which results from expanding out <math>p(z) \bar p(\bar z)</math> in terms of ''z'' = ''x'' + ''iy''.
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| When ''p'' is a polynomial of degree 1 then the resulting curve is simply a circle whose center is the zero of ''p''. When ''p'' is a polynomial of degree 2 then the curve is a [[Cassini oval]].
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| == Erdős lemniscate ==
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| [[Image:Erdos5.png|thumb|Erdős lemniscate of degree ten and genus six]]
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| A conjecture of [[Paul Erdős|Erdős]] which has attracted considerable interest concerns the maximum length of a polynomial lemniscate ''ƒ''(''x'', ''y'') = 1 of degree 2''n'' when ''p'' is [[Polynomial#Classifications|monic]], which Erdős conjectured was attained when ''p''(''z'') = z<sup>''n''</sup> − 1.
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| This is still not proved but Fryntov and [[Fedor Nazarov|Nazarov]] proved that ''p'' gives a
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| local maximum.<ref>
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| {{cite journal|
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| first1=A|
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| last1=Fryntov|
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| first2=F|
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| last2=Nazarov|
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| title=New estimates for the length of the Erdos-Herzog-Piranian lemniscate|
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| year=2008|
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| journal=Linear and Complex Analysis|
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| volume=226|
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| pages=49–60|
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| url=http://arxiv.org/abs/0808.0717}}</ref> In the case when ''n'' = 2, the Erdős lemniscate is the [[Lemniscate of Bernoulli]]
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| :<math>(x^2+y^2)^2=2(x^2-y^2)\,</math>
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| and it has been proven that this is indeed the maximal length in degree four. The Erdős lemniscate has three ordinary ''n''-fold points, one of which is at the origin, and a [[geometric genus|genus]] of (''n'' − 1)(''n'' − 2)/2. By [[inversive geometry|inverting]] the Erdős lemniscate in the unit circle, one obtains a nonsingular curve of degree ''n''.
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| == Generic polynomial lemniscate ==
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| In general, a polynomial lemniscate will not touch at the origin, and will have only two ordinary ''n''-fold singularities, and hence a genus of (''n'' − 1)<sup>2</sup>. As a real curve, it can have a number of disconnected components. Hence, it will not look like a lemniscate, making the name something of a misnomer.
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| [[Image:Mandelcurve2.png|thumb|Mandelbrot curve M<sub>2</sub> of degree eight and genus nine]]
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| An interesting example of such polynomial lemniscates are the Mandelbrot curves.
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| If we set ''p''<sub>0</sup> = ''z'', and ''p''<sub>''n''</sub> = ''p''<sub>''n''−1</sub><sup>2</sup> + ''z'', then the corresponding polynomial lemniscates M<sub>n</sub> defined by |''p''<sub>''n''</sub>(''z'')| = 1 converge to the boundary of the [[Mandelbrot set]].
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| The Mandelbrot curves are of degree 2<sup>n+1</sup>.<ref>{{citation|title=High-Dimensional Chaotic and Attractor Systems: A Comprehensive Introduction|publisher=Springer|year=2007|isbn=9781402054563|page=492|url=http://books.google.com/books?id=mbtCAAAAQBAJ&pg=PA492|first1=Vladimir G.|last1=Ivancevic|first2=Tijana T.|last2=Ivancevic}}.</ref>
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| == Notes ==
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| {{Reflist}}
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| ==References==
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| *[[Alexandre Eremenko]] and [[Walter Hayman]], ''On the length of lemniscates'', Michigan Math. J., (1999), '''46''', no. 2, 409–415 [http://projecteuclid.org/Dienst/UI/1.0/Summarize/euclid.mmj/1030132418]
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| *O. S. Kusnetzova and V. G. Tkachev, ''Length functions of lemniscates'', Manuscripta Math., (2003), '''112''', 519–538 [http://arxiv.org/abs/math.CV/0306327]
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| *[http://www.mathcurve.com/courbes2d/cassinienne/cassinienne.shtml "Cassinian curve" at Encyclopédie des Formes Mathématiques Remarquables]
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| {{DEFAULTSORT:Polynomial Lemniscate}}
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| [[Category:Curves]]
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| [[Category:Algebraic curves]]
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