*) https://www.tandfonline.com/doi/full/10.1080/00029890.2025.2460966 The American Mathematical Monthly A Hyper-Catalan Series Solution to Polynomial Equations, and the Geode N. J. Wildberger &Dean Rubine Received 27 Dec 2023, Accepted 07 Jun 2024, Published online: 08 Apr 2025
Abstract The Catalan numbers 𝐶𝑚 count the number of subdivisions of a polygon into m triangles, and it is well known that their generating series is a solution to a particular quadratic equation. Analogously, the hyper-Catalan numbers 𝐶𝐦 count the number of subdivisions of a polygon into a given number of triangles, quadrilaterals, pentagons, etc. (its type 𝐦), and we show that their generating series solves a polynomial equation of a particular geometric form. This solution is straightforwardly extended to solve the general univariate polynomial equation. A layering of this series by numbers of faces yields a remarkable factorization that reveals the Geode, a mysterious array that appears to underlie Catalan numerics. (引用終り) 以上 []
