Finite Interpretations of a Hyper-Catalan Series Solution to Polynomial Equations and Visualizations

TL;DR

This paper explores finite interpretations of hyper-Catalan series solutions to polynomial equations, revealing that infinite series results can be viewed as finite identities at each level, with visualizations created using Python.

Contribution

It introduces a finite perspective on hyper-Catalan series solutions, connecting polygon subdivision combinatorics with polynomial algebra, and provides visual tools for understanding these relationships.

Findings

Hyper-Catalan series can be truncated to finite identities.

A correspondence between polygon subdivision operations and polynomial algebra is established.

Visualizations demonstrate the finite interpretations and algebraic relationships.

Abstract

The solution to the general univariate polynomial equation has been sought for centuries. It is well known there is no general solution in radicals for degrees five and above. The hyper-Catalan numbers $C[m_2,m_3,m_4,\ldots]$ count the ways to subdivide a planar polygon into exactly $m_2$ triangles, $m_3$ quadrilaterals, $m_4$ pentagons, etc. Wildberger and Rubine (2025) show the generating series $\mathbf{S}$ of the hyper-Catalan numbers is a formal series zero of the general geometric polynomial (meaning, general except for a constant of $1$ and a linear coefficient of $-1$). Using a variant of the series solution to the geometric polynomial that has the number of vertices, edges, and faces explicitly shown, We prove their infinite series result may be viewed as a finite identity at each level, where a level is a truncation of $\mathbf{S}$ to a given maximum number of vertices, edges, or faces (bounded by degree). We illustrate this result, as well as the general correspondence between operations on sets of subdivided polygons and the algebra of polynomials, with figures and animations generated using Python.