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{{short description|Geometric relation between the roots of a polynomial and those of its derivative}}

In [[complex analysis]], a branch of mathematics, the '''Gauss–Lucas theorem''' gives a [[geometry|geometric]] relation between the [[root of a function|root]]s of a [[polynomial]] {{mvar|P}} and the roots of its [[derivative]] {{mvar|P'}}. 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 {{mvar|P'}} all lie within the [[convex hull]] of the roots of {{mvar|P}}, that is the smallest [[convex polygon]] containing the roots of {{mvar|P}}. When {{mvar|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]].

[[File:Gauss Lucas Theorem animation.webm|thumb|Illustration of Gauss–Lucas theorem, displaying the evolution of the roots of the derivatives of a polynomial.]]

==Formal statement==
If {{mvar|P}} is a (nonconstant) polynomial with complex coefficients, all [[root of a function|zeros]] of {{mvar|P'}} belong to the convex hull of the set of zeros of&nbsp;{{mvar|P}}.{{sfn|Marden|1966|loc=Theorem (6,1)}}

==Special cases ==
It is easy to see that if <math>P(x) = ax^2+bx+c</math> is a [[second degree polynomial]], the zero of <math>P'(x) = 2ax+b</math> is the [[average]] of the roots of {{mvar|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.

For a third degree complex polynomial {{mvar|P}} ([[cubic function]]) with three distinct zeros, [[Marden's theorem]] states that the zeros of {{mvar|P'}} are the foci of the [[Steiner inellipse]] which is the unique ellipse tangent to the midpoints of the triangle formed by the zeros of {{mvar|P}}.

For a fourth degree complex polynomial {{mvar|P}} ([[quartic function]]) with four distinct zeros forming a concave [[quadrilateral]], one of the zeros of {{mvar|P}} lies within the convex hull of the other three; all three zeros of {{mvar|P'}} lie in two of the three triangles formed by the interior zero of {{mvar|P}} and two others zeros of {{mvar|P}}.<ref>{{cite journal |first=A. |last=Rüdinger |year=2014 |title= Strengthening the Gauss–Lucas theorem for polynomials with Zeros in the interior of the convex hull |journal=[[Preprint]] |arxiv= 1405.0689 |bibcode=2014arXiv1405.0689R }}</ref>

In addition, if a polynomial of degree {{mvar|n}} of [[real number|real coefficients]] has {{mvar|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.

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>

== Proof ==
{{Math proof|title=Proof|proof=  
By the [[fundamental theorem of algebra]],  <math>P</math> is a product of linear factors as

:<math> P(z)= \alpha \prod_{i=1}^n (z-a_i) </math>

where the [[complex number]]s <math>a_1, a_2, \ldots, a_n</math> are the – not necessarily distinct – zeros of the polynomial {{mvar|P}}, the complex number {{mvar|α}} is the leading coefficient of {{mvar|P}} and {{mvar|n}} is the degree of {{mvar|P}}. 

For any root <math>z</math> of <math>P'</math>, if it is also a root of <math>P</math>, then the theorem is trivially true. Otherwise, we have for the [[logarithmic derivative]]

:<math>0 =  \frac{P^\prime(z)}{P(z)} = \sum_{i=1}^n \frac{1}{z-a_i} = \sum_{i=1}^n \frac{\overline{z}-\overline{a_i}} {|z-a_i|^2}. </math>

Hence

:<math>
\sum_{i=1}^n \frac{\overline{z} } {|z-a_i|^2} = \sum_{i=1}^n \frac{\overline{a_i} } {|z-a_i|^2} </math>.

Taking their conjugates, and dividing, we obtain <math>z</math> as a convex sum of the roots of <math>P</math>:
:<math>z = \sum_{i=1}^n \frac{|z-a_i|^{-2}}{ \sum_{j=1}^n |z-a_j|^{-2}} a_i</math>
}}

== See also ==
{{Div col|colwidth=25em}}
* [[Marden's theorem]]
* [[Bôcher's theorem]]
* [[Sendov's conjecture]]
* [[Routh–Hurwitz theorem]]
* [[Hurwitz's theorem (complex analysis)]]
* [[Descartes' rule of signs]]
* [[Rouché's theorem]]
* [[Properties of polynomial roots]]
* [[Cauchy interlacing theorem]]
{{div col end}}

== Notes ==
{{reflist}}

== References ==
* {{cite journal |last1=Lucas |first1=Félix |date=1874 |title=Propriétés géométriques des fractionnes rationnelles |journal=C. R. Acad. Sci. Paris |volume=77 |pages=431–433 }}
* {{cite journal |last1=Lucas |first1=Félix |date=1879 |title=Sur une application de la Mécanique rationnelle à la théorie des équations|journal=[[C. R. Hebd. Séances Acad. Sci.]]|volume=LXXXIX|pages=224–226|url=https://gallica.bnf.fr/ark:/12148/bpt6k3046j/f226}}.
* {{cite book | last1=Marden | first1=Morris | title=Geometry of Polynomials | series=Mathematical Surveys and Monographs | volume=3 | publisher=American Mathematical Society, Providence, RI | edition=2nd | date=1966}}
* Craig Smorynski: ''MVT: A Most Valuable Theorem''. Springer, 2017, ISBN 978-3-319-52956-1, pp. 411–414

== External links ==
{{commonscat|Gauss–Lucas theorem}}
* {{springer|title=Gauss-Lucas theorem|id=p/g110080|mode=cs1}}
* [http://demonstrations.wolfram.com/LucasGaussTheorem/ Lucas–Gauss Theorem] by Bruce Torrence, the [[Wolfram Demonstrations Project]].
* [https://www.geogebra.org/m/nqn85uts Gauss-Lucas theorem] - interactive illustration
{{DEFAULTSORT:Gauss-Lucas Theorem}}
[[Category:Convex analysis]]
[[Category:Articles containing proofs]]
[[Category:Theorems in complex analysis]]
[[Category:Theorems about polynomials]]