Number Theory meets Differential Equations: Using Homotopy Continuation to Compute Shabat Polynomials

A Belyĭ pair \(\bigl( S,\, z(w) \bigr)\) is a compact, connected Riemann surface \(S\) together with a rational function \(z \colon S \to \mathbb{P}^1(\mathbb{C})\) which has at most three critical values \(\bigl\{ 0,\, 1,\, \infty \bigr\}\). The preimage \(z^{-1}\bigl( [0,1] \bigr) \subseteq S \hookrightarrow \mathbb{R}^3\) is a bipartite graph called a Dessin d’Enfant. When \(S = \mathbb{P}^1(\mathbb{C}) \simeq S^2(\mathbb{R})\) is the Riemann sphere, a tree—namely, a planar graph with one face—can be realized as the Dessin d’Enfant of a polynomial. Named in honor of Georgii Borisovich Shabat, a Belyĭ map \(z \colon S \to \mathbb{P}^1(\mathbb{C})\) which is also a polynomial is called a Shabat polynomial.

In this project, we seek to compute more examples of Shabat polynomials. We introduce a trick which reduces the computation of such polynomials to finding solutions to a system of polynomial equations. We explore computing these solutions via two methods: (1) Newton’s method, to find approximate values, and (2) the homotopy continuation method, to deform known solutions into desired yet unknown ones. Simply put, in this talk we give an introduction to using differential equations to answer questions in number theory.

This project was completed through the Research Experience for Undergraduate Faculty (REUF) at AIM/ICERM in Summers 2025 and 2026 (DMS‑2015462).

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