A Hyper-Catalan Series Solution to Polynomial Equations, and the Geode | Hacker News Reader

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

tandfonline.com

26 pointsCGMthrowaway10 comments

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

He retired from his teaching position but he still research math and present his findings on his channel.

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

Coauthor here-- thanks for the kind words. NJW retired right before he started this project, in Feb 2021. On his WildEgg channel he said he was doing a series where he'd teach amateurs how to do math research, behind the members paywall. For the first problem, he said he'd solve the general polynomial. I thought it was a joke, because everybody 'knows' that we can't go beyond degree four. But no, 41 videos later, he had done it. Two years after that he still hadn't written it up, so I wrote a draft and sent it to him, which evolved into this paper.

You're welcome. And that's a great story! For me one thing it highlights is how you need to be just as good at marketing your work as you do at doing the work - or, in NJW's case have a good friend who's good at it! :)

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

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...

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

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.

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

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