“These Geniuses Solved the Impossible!”: Revolutionary Breakthrough in Algebra Stuns Mathematicians Worldwide, Defying Decades of Established Beliefs

For centuries, a seemingly simple question has puzzled mathematicians: Can all polynomial equations be solved, no matter their complexity? Since the 19th century, the consensus was clear: no. While quadratic, cubic, and quartic equations can be solved with explicit formulas, no universal formula exists for polynomials of degree higher than four. This has been a fundamental teaching in advanced algebra courses. However, recent developments have sparked intrigue and debate in the mathematical community. Two researchers have proposed a novel approach that may unravel this centuries-old enigma, challenging established doctrines and offering a new perspective on a problem once deemed unsolvable.

An Unlikely Alliance: A ‘Heretic’ Mathematician and an Algorithm Expert

Behind this groundbreaking discovery are Norman “NJ” Wildberger and Dean Rubine. Wildberger, a professor of mathematics at the University of New South Wales in Australia, is renowned for his critical stance on traditional mathematical foundations. He advocates for the abandonment of concepts like infinity and irrational numbers in certain branches of mathematics, earning him the moniker “the heretic mathematician.” His radical views have often sparked controversy in academic circles.

Joining him is Dean Rubine, an accomplished computer scientist with a background at Bell Labs and Carnegie Mellon, currently serving as a Chief Technology Officer at a hedge fund specializing in algorithms. Their collaboration began on YouTube, where Wildberger has been sharing educational videos since 2021, challenging the mathematical community to reconsider a “solvable problem”: finding a new method for resolving general polynomials. Intrigued by this challenge, Rubine followed Wildberger closely. Two years and 41 videos later, while Wildberger had not yet published a scientific paper, Rubine took the initiative to structure their work into a co-authored publication.

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The Secret Weapon: Catalan Numbers

To bypass traditional mathematical obstacles, the duo employs a well-known mathematical structure familiar to both geometers and computer scientists: Catalan numbers. These natural numbers appear in numerous combinatorial problems, such as the organization of binary trees, parenthesizing expressions, and polygon triangulations. Wildberger and Rubine postulate that these numbers can also serve as a geometric and combinatorial foundation for reconstructing solutions to complex polynomial equations.

Exploring this idea, they have developed a new mathematical structure called the hyper-Catalan tableau, an extension of Catalan numbers enriched to meet the conditions posed by certain polynomials. This set of structures forms what they call the “Geode,” a tool that allows for a novel mapping of solutions. This innovative approach could potentially reshape how mathematicians tackle polynomial equations.

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Reconciling Algebra and Geometry

While traditional mathematical methods strive to express polynomial roots in terms of radicals or transcendental functions, Wildberger and Rubine propose a more geometric approach based on the logic of arrangements and symmetries. They argue that the real obstacle lies not in the equation itself but in the methods used to solve it. By refusing to employ certain “non-constructive” concepts, such as nth roots or infinity, they restore focus to more concrete tools like formal series.

Formal series allow for the manipulation of symbolic expressions without the need to precisely evaluate each term. As they write in their article, “Formal series offer explicit algebraic and combinatorial alternatives to functions that cannot be concretely evaluated.” These series, they argue, should play a more central role in modern mathematics, offering a tangible alternative to abstract mathematical functions.

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Reactions and Perspectives

Their article, published in the American Mathematical Monthly, a peer-reviewed journal of the Mathematical Association of America, is both rigorous and educational. Each concept is carefully introduced, definitions are precise, and arguments are meticulously constructed. The tone is akin to that of a university textbook, making their work accessible to anyone with a solid foundation in mathematics.

However, how the scientific community will respond remains to be seen. The unconventional nature of their approach and Wildberger’s iconoclastic personality may work against them in certain academic circles. Yet, the ideas are there, and so is the potential. On the Hacker News forum, Rubine commented, “When Wildberger said he was going to solve the general polynomial, I thought it was a joke. But he was serious. Two years later, he had a method, and it was just a matter of writing it down.”

Towards a Subtle Revolution?

Their work does not claim to solve everything. It does not contradict established mathematical results, particularly those established by Galois theory, which prove that it is impossible to find a single radical formula for all polynomials of degree higher than four. However, where traditional methods falter, Wildberger and Rubine offer an alternative path. Their approach is constructive, rigorous, and potentially applicable to fields such as cryptography, symbolic analysis, or algorithmics.

With their “hyper-Catalan Geode,” they open a breach in a mathematical fortress once thought impenetrable. One thing is certain: their work is already raising many questions. And perhaps, over time, other researchers will seize upon it to deepen—or challenge—it. Could this be the beginning of a new era in mathematical theory?

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