Photo courtesy of Dean Rubine.
General polynomial equations are deemed to be humanity’s oldest algebra problem and Oyster River Middle School (ORMS) MATHCOUNTS coach Dr. Dean Rubine and retired University of New South Wales mathematician Dr. Norman Wildberger deciphered the enigma.
A Hyper-Catalan Series Solution to Polynomial Equations, and the Geode, was published in April 2025 in the American Mathematical Monthly.
Rubine’s interest in investigating the algebraic problem dates back to sixth grade, when he was first taught about polynomials and the subject kept coming up in class. As it turns out, many math problems require solutions to polynomial equations.
“The Fundamental Theorem of Algebra says every non-constant polynomial has a complex root; the natural question is how to find it,” says Rubine. “These days, with computer algebra systems, it's easy to get numerical approximations to solutions. But what we'd consider an ‘exact’ solution in math, an expression with square roots and cube roots and the like, is generally not possible to write down for polynomials degree five and higher. I used to wonder what exact solutions for those would look like.”
The collaboration with Wildberger was years in the making. Rubine had been following the mathematician’s work since he first watched him on YouTube in 2016. In February 2021, Wildberger launched a new series on his YouTube channel where he taught amateur mathematicians how to do research.
At the time, Wildberger said the first problem the group was going to solve was the general polynomial equation, which Rubine thought was banter because it was impossible past degree four. But Wildberger was serious, and 41 YouTube videos later, he had solved the enigma.
After two years went by with no paper from Wildberger, Rubine started writing a draft in June 2023. He sent the draft to Wildberger two weeks later along with an invitation to finalize the work together, which Wildberger agreed to.
“Although we both have our fingerprints on all the parts, my goal was to lighten up Professor Wildberger’s load by writing the paper and acting as a junior researcher to get the result published,” said Rubine. “I wrote up and typeset the math, produced figures which required writing computer programs, produced examples, assembled the bibliography, researched and wrote up the history, and was the corresponding author. I also paid the publishing fees so that the article could be available to everyone.”