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
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:
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
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...
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
A UNSW press release: https://www.unsw.edu.au/newsroom/news/2025/05/mathematician-...
> 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
I thought it was Hyper-Catan at first. this is cool, i guess, too.