A Hyper-Catalan Series Solution to Polynomial Equations, and the Geode

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

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.

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

I thought it was Hyper-Catan at first. this is cool, i guess, too.