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
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.
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
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
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