Franklin's Notes


Lucas' Theorem

The following is known as Lucas' Theorem:

Lucas' Theorem. If all zeroes of a polynomial $P(z)$ lie in the same half-plane, then all zeros of its derivative $P'(z)$ lie in the same half-plane.

To prove this theorem, we can start with a slightly weaker statements that is actually sufficient to prove the full theorem.

Proposition 1. If all zeroes of a polynomial $P(z)$ lie in the closed upper half-plane, then all zeroes of $P'(z)$ also lie in the closed upper half-plane.

Proof. Consider the rational function where $\alpha_i$ are the roots of $P$, repeated according to multiplicity. If they are all in the closed upper half-plane, and $z$ is some point in the open lower half-plane (complement of the closed upper half-plane), then $z-\alpha_i$ has a negative imaginary part, meaning that its reciprocal $(z-\alpha_i)^{-1}$ has a positive imaginary part, for each root $\alpha_i$. This would imply that $P'(z)/P(z)$ has a positive imaginary part for all $z$ in the open lower half-plane, meaning that $P'(z)/P(z)\neq 0$ for these values of $z$. Hence all zeroes of $P'(z)/P(z)$, and hence all zeroes of $P'(z)$, must also lie in the closed upper half-plane. $\blacksquare$

Having proven this, it is not too difficult to extend the result to an arbitrary half-plane.

Proposition 2. If all zeroes of a polynomial $P(z)$ lie in the same closed half-plane, then all zeroes of $P'(z)$ also lie in that closed half-plane.

Proof. Notice that any closed half-plane of $\mathbb C$ can be expressed by the inequality for some $a,b\in\mathbb C$. If all zeros of the polynomial $P(z)$ lie in this closed half-plane, then all zeros of the polynomial $P(az+b)$ lie in the closed upper half-plane. This means that all zeroes of $aP'(az+b)$ lie in the closed upper half-plane, and therefore all zeroes of $aP'(z)$ or $P'(z)$ lie in the closed half-plane given by as desired. $\blacksquare$

This can now be generalized further to a somewhat surprising result:

Proposition 3. For any polynomial $P$, the roots of $P'(z)$ lie in the convex hull of the roots of $P(z)$.

This result can be proven easily by noticing that the convex hull of the roots of $P$ can be expressed as a finie intersection of closed half-planes containing all roots of $P$.

complex-analysis

polynomials

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