Two mathematicians have used a new geometric approach in order to address a very old problem in algebra. In school, we often learn how to multiply out and factor polynomial equations like (x² – 1) or (x² + 2x + 1). In real life, these equations get very messy, very fast. In fact, mathematicians typically only approximate the solutions for ones above a certain value, known as higher degree (or higher order) polynomials.
In this paper, however, the authors posit that they can use a metric from geometry called a Catalan number, or Catalan series, to find exact solutions to higher degree polynomials.
The Catalan numbers are a natural observed consequence of a bunch of different mathematical scenarios, and can be found by engaging in such efforts as distilling Pascal’s triangle of polynomial coefficients. They help graph theorists and computer scientists to plan data structures called trees by showing how many different tree arrangements can be made within certain parameters. In this case, they also quantify how many ways you can divide a polygon of any size into a particular number of triangles.
The leading mind behind this work, mathematician Norman “N.J.” Wildberger, is an honorary professor at the University of New South Wales in Australia—a term likely reflecting the fact that he retired in 2021 after teaching at the university since 1990. Wildberger also self identifies as a “heretic” of certain mathematical foundations, exemplified in part by his longtime belief that we should stop using infinity or infinite concepts in some parts of math.
That opposition to infinite or irrational numbers is key to this research. For many, many years now, people studying algebra have known that we simply “can’t” solve certain polynomials. They can’t be taken apart into a mathematical term that fits under a square root (a.k.a. radical) sign at all. But, in Wildberger’s view, focusing on this divide and dwelling inside the radical is a hindrance. We should “sidestep” it altogether.
To make this argument, Wildberger teamed up with Dean Rubine, a computer scientist who has worked for Bell Labs and Carnegie Mellon University. However, for decades now, Rubine has helped to lead the number-crunching at a secretive hedge fund that focuses on algorithms (more later on his role in this publication).
The paper has a teacherly quality, reading somewhat like a chapter from a good textbook. The authors lay out and define their terms, then build their arguments one by one into a complete picture. What results is the ‘hyper-Catalan’ array, which contains the classic Catalan numbers as well as an extension that includes other numbers that satisfy the conditions to solve polynomials. (Remember, the hyper-Catalan number series doesn’t need to line up with all the other uses of the Catalan numbers—rather, the Catalans are a basis from which to begin building a unique set that solves the polynomial problem.) This all wrapped up in an array called the Geode, which encompasses the entirety of the hyper-Catalan number series.
After stepping through the work leading up to and including the Geode array, Wildberger gets in one last jab:
[F]ormal power series give algebraic and combinatorially explicit alternatives to functions which cannot actually be concretely evaluated (such as nth root functions). Hence they ought to assume a more central position. This is a solid, logical way of removing many of the infinities which currently abound in our mathematical landscape.
Having been authored by an aging iconoclast and a longtime quantitative executive, this work may have more of an uphill climb to be broadly recognized. It’s also published in the peer reviewed American Mathematical Monthly—a broad interest journal associated with the Mathematical Association of America. The journal accepts advertisers, offers paid editing services, and offers an option where authors can pay to make their articles open access—often a few thousand dollars or more. (The latter is, unfortunately, the normalized model and cost of open access publishing.) In this case, this less-orthodox approach could be a result of the subject matter simply not being on most people’s radar anymore. But it also fits right into Wildberger’s lifelong quest to trim the mathematical fat and present clear, simple ideas for as many people as possible.
On the tech forum Hacker News (from startup incubator Y Combinator), Rubine explained in a post that he’d closely followed Wildberger’s work on this problem since 2021, when Wildberger declared he was going to solve this problem on his YouTube channel.
“[H]e was doing a series where he’d teach amateurs how to do math research,” Rubine said. “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.”
With that kind of determination, Wildberger may, after all, be an apt opponent for infinity itself. His democratic, open-door approach to mathematical thinking is really admirable. And in the paper, he and Rubine point out a number of questions that this theory opens up. We’ll see if others in the mathematics community pick up some of these questions. I, for one, hope so, because 41 more videos is a long time to wait for the next breakthrough.
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