Lucas Enjoys MathJax Too

In our previous post, we explored the concept of a convex hull. This time I want to look at a beautiful implication of that geometric boundary, one that perfectly bridges pure mathematics, classical physics, and even modern artificial intelligence. In the world of LLMs, the convex hull represents the boundary of an AI's knowledge base. Simply step outside that geometric perimeter of training data, and the model begins to hallucinate.

But I won't let LLMs completely hijack this post, because boundaries matter in human history, too. In fact, a simple boundary of language such us sharing a last name can completely alter a person's legacy. This post is my small tribute to the French mathematician Félix Lucas. Because the late 19th century featured two prominent mathematicians named Lucas, Félix is routinely overshadowed by his contemporary, Édouard Lucas (the number theorist famous for inventing the Tower of Hanoi). Yet, Félix Lucas’s true legacy deserves its own spotlight: his pioneering work in potential theory used the physics of gravity and electricity to beautifully map the layout of complex roots.

To be entirely fair, the history of this theorem is a tale of two discoveries. Gauss was the first to stumble upon the property, locking it away in his private notebooks without publishing it. When I first installed the MathJax library here back in 2017, I let Gauss take the spotlight to show off just how beautiful mathematical typesetting could look on my site in a post titled "Gauss Enjoys MathJax". Now, over a decade later, I think it is the perfect time for a follow-up. The real history also has a matching second chapter. Thirty-nine years after Gauss's private discovery, Félix Lucas independently derived, formally stated, and published the proof in Paris's prestigious Comptes rendus de l'Académie des sciences. To credit both the private spark and the public proof, history permanently tied their names together. After all these years, it is finally time to let Lucas enjoy our MathJax library too. Here is the elegant rule that the Gauss-Lucas theorem establishes:

Let $P(z)$ be a polynomial of degree $n$ with roots $z_1, z_2, \dots, z_n$ (not necessarily distinct). We can write it as: $$P(z) = a_n (z - z_1)(z - z_2)\cdots(z - z_n)$$

Without loss of generality, imagine placing equal positive electric charges on the complex plane at the locations of these original roots. If we want to map out where these repulsive forces cancel each other out, we just have to solve the following equation of equilibrium for 2D case:

$$F_{net}(z) = \sum_{k=1}^{n} k \frac{ e \cdot (z - z_k)}{|z - z_k|^2} = 0$$

but that's the same as

$$\sum_{k=1}^{n} \frac{z - z_k}{|z - z_k|^2} = \sum_{k=1}^{n} \frac{1}{\bar{z} - \bar{z}_k} = 0$$

Or, if we take the complex conjugate of the entire expression, we can drop the bar notation and work directly with standard complex coordinates:$$\sum_{k=1}^{n} \frac{1}{z - z_k} = 0$$

which, in its turn, is nothing but the derivative of $\ln(P(z))$.

$$\frac{P'(z)}{P(z)} = \sum_{k=1}^n \frac{1}{z - z_k} = 0$$

The Theorem

Let $P(z)$ be a non-constant polynomial with complex coefficients. If $z_k \in Z$ is the set of all roots of $P(z)$, and $H$ is the convex hull of $Z$, then every root of the derivative $P'(z)$ also lies within $H$.

Proof

Now, let $w$ be a root of the derivative, meaning $P'(w) = 0$. If $w$ is also a root of $P(z)$ (a multiple root), it is automatically inside the convex hull $H$. The theorem holds trivially. But if $w$ is not a root of $P(z)$, then $P(w) \neq 0$. Since $P'(w) = 0$, we get

$$\sum_{k=1}^n \frac{1}{w - z_k} = 0$$

If we take the complex conjugate of the entire equation (like we did earlier coming from net force), we get:$$\sum_{k=1}^n \frac{w - z_k}{|w - z_k|^2} = 0$$

Let's separate $w$ from the roots $z_k$. To make the algebra cleaner, let's define a positive real weight for each root:

$$C_k(w) = \frac{1}{|w - z_k|^2} > 0$$

Plugging $C_k$ back into our equation gives:

$$\sum_{k=1}^n C_k(w - z_k) = 0$$

Distribute the summation:

$$w \sum_{k=1}^n C_k(w) = \sum_{k=1}^n C_k(w) z_k$$

Finally, solve for $w$:

$$w = \frac{\sum_{k=1}^n C_k(w) z_k}{\sum_{k=1}^n C_k(w)}$$

Now if we look closely at that final formula we notice that the weights $C_k$ are positive real numbers, this formula is exactly the definition of a weighted average (or a convex combination) of the original roots $z_k$ or in other words a point within the convex hull $H$ of $Z$. $\blacksquare$

An equilibrium point of pulling forces can never magically drift outside the perimeter of the objects pulling it, it is trapped in the convex hull. It takes a problem that seems like it should require brutal coordinate geometry and solves it using nothing more than a clever weighted average. It’s pure mathematical elegance.