Solving Polynomial Equations and the Geode - Part I

Explore a mathematical lecture discussing a paper by N J Wildberger and Dean Rubine titled "A Hyper-Catalan Series Solution of Polynomial Equations, and the Geode," scheduled for publication in the May 2025 volume of the American Mathematical Monthly. Discover a novel approach to solving polynomial equations that bypasses traditional Galois theory and "solution by radicals" methods, instead developing a series solution with coefficients called "hyper-Catalan numbers." Learn about these multi-dimensional arrays that extend Catalan numbers by counting arbitrary diagonal subdivisions of planar polygons. The lecture presents a solution formula for the "geometric case" of polynomial equations, applies it to recover Eisenstein's 1850 formula for degree 5 polynomials, and demonstrates numerical approximation techniques using Wallis' cubic equation. The presentation begins with familiar ordinary Catalan numbers and quadratic equations, introducing an algebra of multisets of triagons (planar polygons subdivided into triangular pieces), and references a key idea from Graham, Knuth, and Patashnik's "Concrete Mathematics."