“Algebra’s Oldest Challenge” Gets A Sparkly New Answer

Well, according to a new paper from Norman Wildberger, an honorary professor in the University of New South Wales School of Mathematics and Statistics, and Dean Rubine, a computer scientist and volunteer middle school math coach in New Hampshire, it doesn’t have to be. By taking a particular sequence of numbers, extending them into multiple dimensions, and wrangling them into a vast and mysterious array they call the “Geode”, they’ve made a new inroad into a question usually accepted as impossible to answer: how to find a general method for solving higher-order polynomials – equations involving a variable raised to powers larger than four, like x⁵ – 3x³ + x – 1 = 0. 

And ironically, for a question often known as “algebra’s oldest challenge”, it all started with a denial that would have made the ancients proud.

What’s the problem?

Solving a polynomial equation is, despite the jargon, not a new problem. In fact, it’s one of the oldest: “Four millennia ago, the Babylonians could solve the system of equations 𝑥+𝑦=𝑠, 𝑥𝑦=𝑝, equivalent to a quadratic equation,” point out Wildberger and Rubine in the introduction to the new paper. And some of the foundational myths of math revolve, technically, around polynomial equations: it was, after all, the proof of an irrational solution to x² = 2 that doomed Hippasus to his watery grave all those years ago.

But for a problem that’s been around so long, it’s taken a very long time to solve. True, the ancients could figure out a solution to a quadratic equation – that is, one that involves x² as the highest term – and today it’s the kind of question math teachers pose to 15-year-olds rather than college geniuses. But scaling up by just one degree – that is, involving cubes – took not just millennia of mathematical advances, but the creation of an entirely new number system to figure out.

Move up another step, to the quartic equations – those that can involve an x⁴ term – and things start getting kind of ridiculous. There is, as with quadratics, a handy-dandy formula you can use to find the solutions, but it’s so unwieldy that nobody uses it if they don’t have to. It’s actually easier, almost always, to wrangle the equation into an entirely different base system, solve it in that, and then reshape it back into the original format, rather than tackle it head-on.

And then we get to quintics.

So here’s the problem with quintics: some of them can’t be solved. No, not as in “they’re really hard”, or “it’s still an open problem” – it’s been proven since 1824 that you can find degree-five polynomials, or higher, which are impossible to solve. 

At least, they’re impossible to solve with radicals. But that raises a couple of questions, doesn’t it? First, what are radicals? And second…  well, what if we use a different method?

A radical proposition

As complicated as any degree-four or less polynomial is to write out, there is a way in which they’re quite simple objects. After all, it may be circuitous; it may be time-consuming; but ultimately, any solution can be found with the right recipe of adding, subtracting, multiplying and dividing, and taking powers and roots.

The resulting numbers – values like ½, or –³√3, or 1+√5i, or 18,000,306 – are known as algebraic numbers, because they can be found algebraically (go figure). But the bits that are within the root symbols – also known as radical symbols, for reasons which will become clear in about two seconds – are called radicals, and they’re the most obvious sign of what Wildberger believes is a serious problem in modern math.

“[I don’t] believe in irrational numbers,” he said in a statement Thursday. They’re functionally and logically imprecise, he argued, requiring “an infinite amount of work and a hard drive larger than the universe” to calculate.

It’s an opinion not shared by many of his peers – but it’s one which “reopens a previously closed book in mathematics history,” he said. By scrupulously avoiding radicals and irrational numbers, he cooked up an alternative approach – one based instead on a different, infinite kind of polynomial, called power series.

At first glance, this doesn’t make things easier – we’ve gone from a polynomial which, while complicated, has at most six elements to it, to a new one which is infinitely long. But history is at least somewhat on Wildberger’s side here: “A power series solution to a polynomial equation is not a new idea,” he and Rubine point out. “In 1844, Gotthold Eisenstein found such a solution to 𝑥⁵+𝑥−𝑡=0, the simplest polynomial known not to have a general zero expressible in radicals.”

In fact, there’s a famous and well-established pattern for how these things go – at least for quadratic equations. Given such a problem, you can construct a solution using a sequence known as the Catalan numbers – they turn up as the coefficients in an infinitely long equation which, if “solved” in turn, will give the exact solution of the original problem.

It’s a lot harder, from a human perspective, than simply using the quadratic formula – though if you’re a computer, this is probably a fairly close approximation (ha) of how you’d solve a quadratic equation in any case. But once you step up a few degrees, the technique’s strengths become more apparent.

“The Catalan numbers are understood to be intimately connected with the quadratic equation,” the pair said. “Our innovation lies in the idea that if we want to solve higher equations, we should look for higher analogues of the Catalan numbers.”

“We’ve found these extensions,” they announced, “and shown how, logically, they lead to a general solution to polynomial equations.”

The shape of things

How, then, do you extend the Catalan numbers into higher dimensions? The answer lies in yet another pairing of mathematical fields: geometry and combinatorics.

“The Catalan numbers were introduced by Euler in 1751 to count subdivisions into n triangles of a fixed planar convex (𝑛+2)-gon, for a natural number n,” explain Wildberger and Rubine. There’s only so many things you can change about that setup, and the answer isn’t necessarily what you’d expect: instead of a bunch of boring triangles, Wildberger split his polygons into triangles, quadrilaterals, pentagons, and so on. 

However many possibilities there are of these subdivisions Wildberger calls the hyper-Catalan number for that type. Organizing them by face type reveals yet another new discovery: the mysterious “Geode”, a “fundamentally new array of numbers,” he said, “which extends the classical Catalan numbers and seem[s] to underlie them.”

“We expect that the study of this new Geode array will raise many new questions,” Wildberger said, “and keep combinatorialists busy for years.”

Still, while tantalizing, the Geode isn’t the main goal. Remember, we’re trying to solve polynomials here – and luckily, the hyper-Catalan numbers seem to be a perfect fit. They can’t be used to find exact solutions – of course, neither can radicals, according to Wildberger – but what they can do is create an infinite sequence that approximates it pretty well, if you cut it off after enough terms.

While issues may be open around convergence, the pair’s tests of their methods seem to have hit the mark: “Even just using a small portion of the full subdigon polyseries, we can obtain impressive results,” the paper boasts. 

What’s next?

So, where does math go from here? As you may have already been aware, we’ve been able to approximate solutions to higher-order equations for a while now – so what’s the point of all this?

Well, while a new way to solve quintics and higher is certainly a bonus, the real gem here is probably going to be Wildberger’s Geode. As the paper notes, “it is now […] an object of considerable interest.” 

That said, the method itself has also opened a few directions for future research. The Catalan numbers are but one of many ways to construct infinite power series – what happens if you use another one? How useful is this method compared to existing techniques for numerical approximation, and how applicable is it for computing?

But most of all, for Wildberger, the point seems to be one of principle.

“Formal power series give algebraic and combinatorially explicit alternatives to functions which cannot actually be concretely evaluated (such as nth root functions),” the paper concludes. “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,” the pair wrote. “The combinatorial and computational orientation is full of power, and we ought to harness it more fully.”

The paper is published in The American Mathematical Monthly.