This is pretty neat. Every univariate polynomial has a series solution whose coefficients come from some intricate combinatorics whose meaning is still partly obscure.

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...

Earlier comments found at https://news.ycombinator.com/item?id=43869093

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.

Amazing paper. The depth, the creativity, and making it rigorous to connect these ways of counting shapes into solving polynomials, but still also a sense that it's not the final answer, that it's characterizing some fundamental patterns, but it's also a step on the way to more understanding. Really cool. One of the authors also has a YouTube channel: 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

N J Wildberger has been doing Math YouTube long before it was popular , in his channel "Insight into mathematics" you can find a walkthrough of the Article presented as he always does in a very pleasant and engaging manner.

additionally he has another channel "Wild Egg Maths" where he delves into more advanced topics.

Here is the layman's article of the paper linked: https://www.popsci.com/science/algebra-oldest-problem-solved...

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