From my brief reading of it, it seems like the interesting bit here is the development of the Hypercatalan numbers as the coefficients of the infinite sums of roots of polynomials. Some partial results for special cases of the Catalan numbers and roots had been found in the past, but the full understanding of the structure they call the Geode enabled generalization of the previous findings.
> A UNSW mathematician has discovered a new method to tackle algebra’s oldest challenge – solving higher polynomial equations.
See also author's blog https://njwildberger.com/ and youtube channel Insights into Mathematics https://www.youtube.com/channel/UCXl0Zbk8_rvjyLwAR-Xh9pQ
Also the same author has a blog, with a post about this paper: https://njwildberger.com/ - blog also notes he's about to retire, too! Wow
you can find a walkthrough of his article here
Paper with Dean Rubine on Solving Polynomial Equations and the Geode (I) | N J Wildberger https://youtu.be/oIHd3zDDDCE?si=kdJnuRB1GYi_mg-5
Subdigons and Solving Polynomial Equations and the Geode (with Dean Rubine) II | N J Wildberger https://youtu.be/Jl0TWMN85G0?si=1IzlrHBBT009lLKq
The History behind Hyper-Catalan Series Solutions to Polynomial Equations -- with Dean Rubine https://youtu.be/nvH09WvvERY?si=Df4F9XeD5Yhgw0Xh
I had trouble finding a pdf link to the paper to download a regular pdf instead of using their in-browser viewer. It is here:
https://www.tandfonline.com/doi/pdf/10.1080/00029890.2025.24...
additionally he has another channel "Wild Egg Maths" where he delves into more advanced topics.
CGMthrowaway•9mo ago
On the Catalan number wikipedia page, scroll down to "A convex polygon with n + 2 sides..." to see the polygon dissection: https://en.wikipedia.org/wiki/Catalan_number