A mathematician has developed an algebraic solution to an equation that was long thought to be unsolvable.
A groundbreaking discovery from a UNSW Sydney mathematician may finally offer a solution to one of algebra’s toughest problems: how to solve high-degree polynomial equations.
Polynomials are equations where a variable (like x) is raised to various powers. A simple example is: 1 + 4x – 3x² = 0.
These equations aren’t just for math textbooks, they’re essential in everything from predicting planetary motion to coding software.
But when it comes to “higher-order” polynomials, where x is raised to the fifth power or beyond, mathematicians have struggled for centuries to find a universal solving method. That could be about to change, thanks to this exciting new breakthrough.
Now, UNSW Honorary Professor Norman Wildberger has revealed a new approach using novel number sequences, outlined in a recent publication with computer scientist Dr. Dean Rubine.
“Our solution reopens a previously closed book in mathematics history,” Prof. Wildberger says.
The polynomial problem
Solutions to degree-two polynomials have been around since 1800 BC, thanks to the Babylonians’ ‘method of completing the square’, which evolved into the quadratic formula familiar to many high school math students. This approach, using roots of numbers called ‘radicals’, was later extended to solve three- and four-degree polynomials in the 16th century.
Then, in 1832, French mathematician Évariste Galois showed how the mathematical symmetry behind the methods used to resolve lower-order polynomials became impossible for degree five and higher polynomials. Therefore, he figured, no general formula could solve them.
Approximate solutions for higher-degree polynomials have since been developed and are widely used in applications but, Prof. Wildberger says, these don’t belong to pure algebra.
Radical rejection behind new method
The issue, he says, lies in the classical formula’s use of third or fourth roots, which are radicals.
The radicals generally represent irrational numbers, which are decimals that extend to infinity without repeating and can’t be written as simple fractions. For instance, the answer to the cubed root of seven, 3√7 = 1.9129118… extends forever.
Prof. Wildberger says this means that the real answer can never be completely calculated because “you would need an infinite amount of work and a hard drive larger than the universe.”
So, when we assume 3√7 ‘exists’ in a formula, we’re assuming that this infinite, never-ending decimal is somehow a complete object.
This is why, Prof. Wildberger says, he “doesn’t believe in irrational numbers.”
Irrational numbers, he says, rely on an imprecise concept of infinity and lead to logical problems in mathematics.
Prof. Wildberger’s rejection of radicals inspired his best-known contributions to mathematics, rational trigonometry, and universal hyperbolic geometry. Both approaches rely on mathematical functions like squaring, adding, or multiplying, rather than irrational numbers, radicals, or functions like sine and cosine.
His new method to solve polynomials also avoids radicals and irrational numbers, relying instead on special extensions of polynomials called ‘power series’, which can have an infinite number of terms with the powers of x.
By truncating the power series, Prof. Wildberger says, they were able to extract approximate numerical answers to check that the method worked.
“One of the equations we tested was a famous cubic equation used by Wallis in the 17th century to demonstrate Newton’s method. Our solution worked beautifully,” he said.
New geometry for a general solution
However, Prof. Wildberger says the proof for the method is, ultimately, based on mathematical logic.
His method uses novel sequences of numbers that represent complex geometric relationships. These sequences belong to combinatorics, a branch of mathematics that deals with number patterns in sets of elements.
The most famous combinatorics sequence, called the Catalan numbers, describes the number of ways you can dissect a polygon, which is any shape with three or more sides, into triangles.
The numbers have important practical applications, including in computer algorithms, data structure designs, and game theory. They even appear in biology, where they’re used to help count the possible folding patterns of RNA molecules. And they can be calculated using a simple two-degree polynomial.
“The Catalan numbers are understood to be intimately connected with the quadratic equation. Our innovation lies in the idea that if we want to solve higher equations, we should look for higher analogues of the Catalan numbers.”
Prof. Wildberger’s work extends these Catalan numbers from a one-dimensional to multi-dimensional array based on the number of ways a polygon can be divided using non-intersecting lines.
“We’ve found these extensions, and shown how, logically, they lead to a general solution to polynomial equations.
“This is a dramatic revision of a basic chapter in algebra.”
Even quintics – a degree five polynomial – now have solutions, he says.
Aside from theoretical interest, he says, the method holds practical promise for creating computer programs that can solve equations using the algebraic series rather than radicals.
“This is a core computation for much of applied mathematics, so this is an opportunity for improving algorithms across a wide range of areas.”
Geode’s unexplored facets
Prof Wildberger says the novel array of numbers, which he and Dr. Rubine called the “Geode”, also holds vast potential for further research.
“We introduce this fundamentally new array of numbers, the Geode, which extends the classical Catalan numbers and seem to underlie them.
“We expect that the study of this new Geode array will raise many new questions and keep combinatorialists busy for years.
“Really, there are so many other possibilities. This is only the start.”
Reference: “A Hyper-Catalan Series Solution to Polynomial Equations, and the Geode” by N. J. Wildberger and Dean Rubine, 8 April 2025, The American Mathematical Monthly.
DOI: 10.1080/00029890.2025.2460966
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10 Comments
A few things to note: No established mathematics has been overturned; it is still true that there are polynomial equations that cannot be solved in terms of radicals. Wildberger’s approach gives approximations to the roots (which are irrational numbers, whether he “believes” in them or not), just like several existing methods. No other mathematicians are quoted giving their views on this work. The work was not published in a research journal, but in an expository journal intended for educators.
“No other mathematicians are quoted giving their views on this work. The work was not published in a research journal, but in an expository journal intended for educators.”
Same was true of The Calculus.
More importantly than this/that is the reason Newton chose to avoid research journals, worth understanding.
research journals appear to be (and indeed behave like) societies… perhaps he was seeking to avoid that proxy for rigorous review and instead let the otherwise sufficiently intelligent educators make up their own minds.
You would be surprised to know that the history of science demonstrates that non-linear innovation by necessity departs from well understood concepts and is most often rejected. Indeed, the history of Maths follows the same trajectory. Not surprisingly, the same is true of almost all fields. Why? Start looking for answers in those journal societies, I recommend.
No, they are NOT ‘essential for coding software’.
I have been writing computer programs for 45 years and I have never used polynomials in any of my software programs.
Yes you have. You just didn’t know it.
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Confusingly written article and/or underlying presentation.
1. Approximations don’t belong in algebra.
Then he goes on to provide an approximate solution.
2. He doesn’t believe in ‘radicals’ due to infinity.
Then he goes on to provide an infinite series representation.
3. Polygon divided by non-intersecting lines ? You mean planes?
Which is it?
I agree, I wanted some deeper treatment of the mathamatics.
Co-author here. The above are all are more or less reasonable prima facie objections I can address. Let me urge folks to ignore these hyped articles and read the paper, which is not overly difficult as pure math goes. It’s doi 10.1080 slash 00029890.2025.2460966
Yes, Joe Public, Galois Theory still is as valid as ever; we don’t claim otherwise. As for only giving an approximation to the roots, yes and no. Yes in the sense that for most polynomials that’s all there is: there are only approximate rational complex roots accessible; we can work a bit harder and improve our approximation. No in the following sense: The hard thing to get one’s head around is that a zero of the general polynomial isn’t a complex number. The exact zero of the general polynomial is a power series (with some negative powers). It’s a more combinatorial perspective; issues of convergence are secondary. That’s why we call them generating series, not generating functions. The solution sort of evaporates when we start putting in numbers for the variables, and convergence is definitely not assured.
Our solution appears infinite, but it’s best understood as an ongoing family of finite solutions, one at each level or degree. For example, we define SV=S[vt_2,v^2t_3,v^3t_4,…] and h(x)=1 – x +t_2 v x^2 + t_3 v^2 x^3 + … then we can show h(SV mod v^(d+1)) = 0 mod v^{d+1} which is a totally finite computation. We cut that from the paper because we were over the 20 page limit, and it’s a typical combinatorial interpretation of a formal power series; we’ll cover it in a future publication.
The American Mathematical Monthly is the most widely read mathematics journal in the world, and one of the most prestigious, with expository articles often with substantial research content. Calling it ‘intended for educators’ is a bit derogatory and not really true.
Razzi, I’ve been a programmer for 50 years, starting on Unix in 1975. I actually have had occasion to solve polynomials in programs, but, yes, at best this work might find its way into numerical or symbolic solvers. I generally bite my tongue when NJ Wildberger opines on computer science; after all he’s kind enough to let me pretend to be a mathematician.
Amateur, I’ve addressed above how this is both a method of approximation and an exact solution. My view is the ongoing sum is both simpler and more informative than hiding the infinite processes away in nested radicals. As I explained above, the infinite series result can be interpreted as a family of finite results.
I don’t know where you got planes from; we’re just looking at combinatorial objects on the page. In the paper a triagon is a roofed polygon subdivided into triangles by non-crossing diagonals. It’s the usual Calatan number story, and we show how to use the multiset of triagons to solve a quadratic equation. Then we generalize into subdigons; a subdigon is a roofed polygon subdivided by non-crossing diagonals into triangles, quadrilaterals, pentagons, etc. We show how the associated generating series solves the general geometric polynomial.
I’m happy to further answer questions, so feel free to ask.
I have the impression this leads to solutions writeable in closed-form where at least some of the terms can be infinite series, with geometry and symmetry as the basis. That should be interesting insofar as it produces general solutions.
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