The paper, by myself and Dean Rubine, is "A Hyper-Catalan Series Solution of Polynomial Equations, and the Geode" and it will appear in the May 2025 volume of the American Mathematical Monthly (see link below for online Open Access).
The basic idea is to resurrect the historical problem of solving polynomial equations, and offer a series solution that completely sidesteps the usual "solution by radicals" approach coming from Galois theory. Instead we develop a series solution, whose coefficients are "hyper-Catalan numbers" which form a multi-dimensional array which extend the Catalan numbers, and count arbitrary diagonal subdivisions of planar polygons instead of just triangular subdivisions.
In this video we give one of the basic solution formulas, to a special polynomial equation which we call the "geometric case" : in which the constant function is 1, and the linear coefficient is -1. We apply this to recover a formula of Eisenstein (1850) for a series solution to the Bring radical of a degree 5 polynomial. We also show how to use the general series to obtain approximate numerical solutions, illustrating the ideas with Wallis' famous cubic equation, which also allows us to explain a boot-strapping method.
The main argument is first presented in the familiar case of ordinary Catalan numbers (A000518 in the Online Encyclopedia of Integer Sequences) and quadratic equations. We talk about an algebra of multisets of triagons, which are planar polygons subdivided by diagonals into just triangular pieces. An Exercise from Concrete Mathematics, the lovely text by Graham, Knuth and Patashnik, provides a key idea (which they further attributed to George Polya).
In the next video we will move to the more elaborate situation where arbitrary subdivisions of a polygon support a rich algebraic / combinatorial structure.
This work is an outgrowth of our video series "Solving Polynomial Equations" available to Members of the Wild Egg Maths YouTube channel at https://www.youtube.com/playlist?list=PLzdiPTrEWyz4CVqTYS1fwInPOZxBxDaQf
The paper is available online at https://www.tandfonline.com/doi/full/10.1080/00029890.2025.2460966