Monodromy of the Gauss–Lucas Theorem
This applet is inspired by a paper Nick Salter (2025) on the monodromy of stratified braid groups. In it, among other things, Nick investigates an interesting and natural question arising from a classical theorem in complex analysis:
Let \(p(z) \in \mathbb{C}[z]\) be a non-constant polynomial. Then the roots of the derivative \(p'(z)\), i.e., the critical points of \(p(z)\), must always lie inside the convex polygon formed by the polynomial’s roots.
If we continuously change the coefficients of \(p(z)\), the roots and critical points will smoothly vary around the complex plane. If we move the coefficients about in a closed loop—bringing the polynomial exactly back to where it started—the configuration of roots and critical points will be identical, since we’re considering the same polynomial that we started with! However, the individual roots may have swapped places. This permutation of the roots after completing a loop in the parameter space is called monodromy. Topologists refer to the paths these points trace out over time as braids.
Because of the Gauss–Lucas theorem, we know the critical points are trapped—they can never escape the convex hull of the roots at any point during this process. Nick asks: Can we force the roots to carry out any hypothetical trajectory, provided we never break the Gauss–Lucas boundary? Indeed, he proved there are “legal” braids which respect the convex hull constraint but cannot be realized by any actual continuous path of polynomials.
Monodromy Applet
This applet shows two pictures of the complex plane. On the left, we have the Coefficient Space (\(\mathbb{C}_{\text{coeff}}\)), which reflects the actual coefficients of our polynomial \[ p(z) = z^5 + a_2 z^3 + a_3 z^2 + a_4 z + a_5. \] On the right, we draw the roots (the \(z\) such that \(p(z) = 0\)), the critical points (when \(p'(z)=0\)), and so on, together with the associated convex hulls.
How to Interact
- Drag Coefficients: Click and drag any coefficient dot (\(a_2, a_3, a_4\), or \(a_5\)) in the left panel to modify \(p(z)\).
- Clicking a coefficient reveals the branch points in the parameter space where roots (and critical points) collide.
- Compute Monodromy: While holding a coefficient, we can drag it through a wide loop so that it goes around one of the branch points, then return it to its starting position. At the same time, in the Root Space (\(\mathbb{C}_{\text{root}}\)) panel, we will see that some of the roots and critical points have traded places!
- Toggle Layers: Click the items in the Root Space legend to show or hide the convex hulls for the various derivative layers.
- Remember, Gauss–Lucas tells us that each convex hull will be nested in the one before it!
Note that, in this applet, we consider a depressed quintic. This doesn’t reflect anything distressing about the polynomial’s emotional state—rather, it just means that the coefficient on the second-highest power of \(z\) is \(0\). We can always depress a polynomial (of degree \(n\)) using a simple change of variables like \[ z \mapsto z + \frac{a_1}{n}. \] The \(n=2\) version, which we learn in school when exploring quadratic relationships, is called completing the square.
For our purposes, depressing the quintic makes some of the math easier, and also forces the pictures we draw in the Root Space to always be centered at zero. Moreover, note that the fourth derivative is \(p^{(4)}(z) = 120 z\), so the origin can be thought of as a final point in our hierarchy of nested critical points!