Gauss–Lucas theorem

In complex analysis, a branch of mathematics, the Gauss–Lucas theorem gives a geometric relation between the roots of a polynomial Template:Mvar and the roots of its derivative Template:Mvar. The set of roots of a real or complex polynomial is a set of points in the complex plane. The theorem states that the roots of Template:Mvar all lie within the convex hull of the roots of Template:Mvar, that is the smallest convex polygon containing the roots of Template:Mvar. When Template:Mvar has a single root then this convex hull is a single point and when the roots lie on a line then the convex hull is a 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

Formal statement

If Template:Mvar is a (nonconstant) polynomial with complex coefficients, all zeros of Template:Mvar belong to the convex hull of the set of zeros of Template:Mvar.Template:Sfn

Special cases

It is easy to see that if $P(x)=a{x}^2+bx+c$ is a second degree polynomial, the zero of $P^{\prime }(x)=2ax+b$ is the average of the roots of Template:Mvar. 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 Template:Mvar (cubic function) with three distinct zeros, Marden's theorem states that the zeros of Template:Mvar are the foci of the Steiner inellipse which is the unique ellipse tangent to the midpoints of the triangle formed by the zeros of Template:Mvar.

For a fourth degree complex polynomial Template:Mvar (quartic function) with four distinct zeros forming a concave quadrilateral, one of the zeros of Template:Mvar lies within the convex hull of the other three; all three zeros of Template:Mvar lie in two of the three triangles formed by the interior zero of Template:Mvar and two others zeros of Template:Mvar.

In addition, if a polynomial of degree Template:Mvar of real coefficients has Template:Mvar distinct real zeros $x_1<x_2<\cdots <x_n,$ we see, using Rolle's theorem, that the zeros of the derivative polynomial are in the interval $[x_1,x_n]$ which is the convex hull of the set of roots.

The convex hull of the roots of the polynomial

$p_n{x}^n+p_{n-1}{x}^{n-1}+\cdots +p_0$

particularly includes the point

$-\frac{p_{n-1}}{n\cdot p_n}.$

See also

• 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

References

• Template:Cite journal
• Template:Cite journal.
• Template:Cite book
• Craig Smorynski: MVT: A Most Valuable Theorem. Springer, 2017, ISBN 978-3-319-52956-1, pp. 411–414

External links

• Template:Springer
• Lucas–Gauss Theorem by Bruce Torrence, the Wolfram Demonstrations Project.
• Gauss-Lucas theorem - interactive illustration