New Proof Illuminates the Hidden Structure of Common Equations

In a recent paper, Manjul Bhargava of Princeton University has settled an 85-year-old conjecture about one of math’s most ancient obsessions: the solutions to polynomial equations such as x² – 3x + 2=0. “It’s a great problem, famous old question,” said Andrew Granville, a professor at the University of Montreal. “[Bhargava] had an interesting, somewhat different approach, which was very creative.”

To understand polynomials, mathematicians study their roots, the values of x that make the polynomial equal zero. If you plug the number 1 or 2 into x² – 3x + 2, you’ll get zero, making 1 and 2 the roots of that polynomial.

The equation x² – 5=0 is a bit trickier. The polynomial can’t be solved using a rational number — a fraction made by dividing two integers. So mathematicians define a new number that solves the equation and call it $latexsqrt{5}$. But all we know about $latexsqrt{5}$ is that its square is 5. Once you have $latexsqrt{5}$, you can easily multiply it by –1 to get a second root: –$latexsqrt{5}$.

These two equations differ in another critical way. The roots of x² – 5=0 help solve lots of other equations in our mathematical system, like x² – 20=0. (Note that here, our mathematical system is limited to polynomials and rational numbers.) But if we start using them this way, we’ll find that $latex sqrt{5}$ and –$latexsqrt{5}$ are completely interchangeable. Both $latex 2sqrt{5}$  and –$latex2sqrt{5}$ work equally well as solutions of x² – 20=0 — and, more generally, in any context. Anywhere $latex sqrt{5}$ is helpful, so is –$latexsqrt{5}$.

This situation is by far the most common scenario. It’s rare to have rational equations with roots that are not interchangeable, as in our first example, where if you start using 2 in your mathematical system in place of 1, nonsense will ensue. “If you switch 1 and 2, then all of arithmetic just kind of dies,” said Frank Thorne, a collaborator of Bhargava’s and a professor at the University of South Carolina.

The van der Waerden conjecture, formulated in 1936 by the Dutch mathematician Bartel Leendert van der Waerden, attempted to quantify how many polynomials have noninterchangeable roots. Progress for decades was steady but slow. But recently that progress accelerated, blown ahead on the broader trade winds advancing number theory as a whole. “Especially more concrete, classical questions in number theory, they really made a comeback in the past 20 years,” said Rainer Dietmann, a professor at Royal Holloway University of London.

For that, some credit Bhargava, who in 2014 received the Fields Medal, widely considered to be math’s highest honor. “Bhargava kicked down a bunch of doors and invited people to go exploring,” said Thorne.

In the summer of 2021, a torrent of new work on the van der Waerden conjecture appeared. Dietmann and his collaborator Sam Chow of the University of Warwick, having already solved a few key cases, posted a new paper making significant headway on June 28. The following week, another team of six shared their own preprint.

In the midst of all this, Bhargava gave an online talk on July 1. In the talk, Bhargava presented a proof of a slightly modified van der Waerden conjecture. “He’d gotten within a hair’s breadth,” said Thorne.

Just two weeks later, in an online presentation during the Mathematical Congress of the Americas, Bhargava shared new work proving van der Waerden’s conjecture in its entirety. He posted his paper online in November.