Gauss–Lucas theorem

Formal statement

If P is a (nonconstant) polynomial with complex coefficients, all zeros of P' belong to the convex hull of the set of zeros of P.

Special cases

It is easy to see that if P(x) = ax^2+bx+c is a second degree polynomial, the zero of P'(x) = 2ax+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.

For a third degree complex polynomial P (cubic function) with three distinct zeros, Marden's theorem states that the zeros of 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 P.

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

In addition, if a polynomial of degree n of real coefficients has n distinct real zeros x_1 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}.

Proof

By the fundamental theorem of algebra, P is a product of linear factors as

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

where the complex numbers a_1, a_2, \ldots, a_n are the – not necessarily distinct – zeros of the polynomial P, the complex number α is the leading coefficient of P and n is the degree of P.

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

: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}.

Hence

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

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

Notes

References

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

References

1. Rüdinger, A.. (2014). "Strengthening the Gauss–Lucas theorem for polynomials with Zeros in the interior of the convex hull". [[Preprint]].