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| In [[complex analysis]], a branch of mathematics, the '''Gauss–Lucas theorem''' gives a [[geometry|geometrical]] relation between the [[root of a function|root]]s of a [[polynomial]] ''P'' and the roots of its [[derivative]] ''P<nowiki>'</nowiki>''. The set of roots of a real or complex polynomial is a set of [[point (geometry)|points]] in the [[complex plane]]. The theorem states that the roots of ''P<nowiki>'</nowiki>'' all lie within the [[convex hull]] of the roots of ''P'', that is the smallest [[convex polygon]] containing the roots of ''P''. When ''P'' has a single root then this convex hull is a single point and when the roots lie on a [[line (geometry)|line]] then the convex hull is a [[line segment|segment]] of this line. The Gauss–Lucas theorem, named after [[Carl Friedrich Gauss]] and [[Félix Lucas]] is similar in spirit to [[Rolle's theorem]].
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| ==Formal statement==
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| If ''P'' is a (nonconstant) polynomial with complex coefficients, all [[root of a function|zeros]] of ''P<nowiki>'</nowiki>'' belong to the convex hull of the set of zeros of ''P''.<ref>Marden (1966), Theorem (6,1).</ref>
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| ==Special cases ==
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| It is easy to see that if ''P''(x) = ''ax''<sup>2</sup> + ''bx'' + ''c '' is a [[second degree polynomial]],
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| the zero of ''P<nowiki>'</nowiki>''(''x'') = 2''ax'' + ''b'' is the [[average]] of the roots of ''P''. In that case, the convex hull is the line segment with the two roots as endpoints and it is clear that the average of the roots is the middle point of the segment.
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| In addition, if a polynomial of degree ''n'' of [[real number|real coefficients]] has ''n'' distinct real zeros <math>x_1<x_2<\cdots <x_n\,</math>, we see, using [[Rolle's theorem]], that the zeros of the derivative polynomial are in the interval <math>[x_1,x_n]\,</math> which is the convex hull of the set of roots.
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| The convex hull of the roots of the polynomial <math> p_n x^n+p_{n-1}x^{n-1}+\cdots p_0 </math> particularly includes the point <math>-\frac{p_{n-1}}{n\cdot p_n}</math>.
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| == Proof ==
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| Over the complex numbers, ''P'' is a product of prime factors
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| :<math> P(z)= \alpha \prod_{i=1}^n (z-a_i) </math> | |
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| where the complex numbers <math>a_1, a_2, \ldots, a_n</math> are the – not necessary distinct – zeros of the polynomial P, the complex number <math>\alpha</math> is the leading coefficient of P and n is the degree of P. Let z be any complex number for which <math>P(z) \neq 0</math>. Then we have for the [[logarithmic derivative]]
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| :<math> \frac{P^\prime(z)}{P(z)}= \sum_{i=1}^n \frac{1}{z-a_i}. </math> | |
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| In particular, if z is a zero of <math>P^\prime</math> and still <math>P(z) \neq 0</math>, then
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| :<math> \sum_{i=1}^n \frac{1}{z-a_i}=0\ </math>
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| or
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| :<math>\ \sum_{i=1}^n \frac{\overline{z}-\overline{a_i} } {\vert z-a_i\vert^2}=0. </math>
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| This may also be written as
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|
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| :<math> \left(\sum_{i=1}^n \frac{1}{\vert z-a_i\vert^2}\right)\overline{z}=
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| \left(\sum_{i=1}^n\frac{1}{\vert z-a_i\vert^2}\overline{a_i}\right). </math>
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| Taking their conjugates, we see that <math>z</math> is a weighted sum with positive coefficients that sum to one, or the [[Barycentric coordinates (astronomy)|barycenter on affine coordinates]], of the complex numbers <math>a_i</math> (with different mass assigned on each root whose weights collectively sum to 1).
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| If <math>P(z)=P^\prime(z)=0</math>, then <math>z=1\cdot z+0\cdot a_i</math>, and is still a [[convex combination]] of the roots of <math>P</math>.
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| == See also ==
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| * [[Marden's theorem]]
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| * [[Bôcher's theorem]]
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| * [[Sendov's conjecture]]
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| * [[Rational root theorem]]
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| * [[Routh–Hurwitz theorem]]
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| * [[Hurwitz's theorem (complex analysis)]]
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| * [[Descartes' rule of signs]]
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| * [[Rouché's theorem]]
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| * [[Sturm's theorem]]
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| * [[Properties of polynomial roots]]
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| * [[Gauss's lemma (polynomial)]]
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| * [[Polynomial function theorems for zeros]]
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| * [[Content (algebra)]]
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| == Notes ==
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| {{reflist}}
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| == References ==
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| * Morris Marden, ''Geometry of Polynomials'', AMS, 1966.
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| == External links ==
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| * [http://demonstrations.wolfram.com/LucasGaussTheorem/ Lucas–Gauss Theorem] by Bruce Torrence, the [[Wolfram Demonstrations Project]].
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| {{DEFAULTSORT:Gauss-Lucas Theorem}}
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| [[Category:Convex analysis]]
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| [[Category:Articles containing proofs]]
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| [[Category:Theorems in complex analysis]]
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