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In mathematics, a polynomial lemniscate or ''polynomial level curve'' is a plane algebraic curve of degree 2n, constructed from a polynomial ''p'' with complex coefficients of degree ''n''. For any such polynomial ''p'' and positive real number ''c'', we may define a set of complex numbers by This set of numbers may be equated to points in the real Cartesian plane, leading to an algebraic curve ''ƒ''(''x'', ''y'') = ''c''2 of degree 2''n'', which results from expanding out in terms of ''z'' = ''x'' + ''iy''. 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. == Erdős lemniscate == A conjecture of 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 monic, which Erdős conjectured was attained when ''p''(''z'') = z''n'' − 1. This is still not proved but Fryntov and Nazarov proved that ''p'' gives a local maximum.〔 〕 In the case when ''n'' = 2, the Erdős lemniscate is the Lemniscate of Bernoulli : 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 genus of (''n'' − 1)(''n'' − 2)/2. By inverting the Erdős lemniscate in the unit circle, one obtains a nonsingular curve of degree ''n''. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「polynomial lemniscate」の詳細全文を読む スポンサード リンク
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