Mathematicians Find Long-Sought Building Blocks for Special Polynomials

Problems in mathematics often have a simple “yes or no” structure: Is this statement true or false? But the most enduring and interesting problems propagate through generations, the products of decades of work, like the medieval cathedrals that took centuries to build. The answers to these questions open new doors and provide novel structures on which to continue building.

In the year 1900, the mathematician David Hilbert announced a list of 23 significant unsolved problems that he hoped would endure and inspire. Over a century later, many of his questions continue to push the cutting edge of mathematics research because they are intentionally vague.

“Hilbert had a kind of genius when he formulated his problems, which is that the questions were a bit open-ended,” said Henri Darmon of McGill University. “These really hard open questions are great for mathematics, because they sort of guide us.”

Shortly before Hilbert announced his list of problems, mathematicians discovered the building blocks for a specific collection of numbers associated with the rational numbers, those which can be expressed as a ratio of whole numbers. This discovery was the basis for the 12th problem on the list, which asks for the building blocks associated with number systems beyond the rational numbers.

After more than 50 years of collaborative effort, a recent preprint finally describes the building blocks Hilbert wanted for a broad family of number systems. But the answer relies on some very modern ideas.

“It’s something we’ve been looking for a long time, and they’ve really made a big breakthrough,” said Benedict Gross, an emeritus professor at the University of California, San Diego and Harvard University (and a former member of Quanta’s advisory board). “It’s completely different than what Hilbert had in mind. But that’s the way math is. You can never say how a problem is going to be solved.”

Digging for Roots

The edifice of Hilbert’s 12th problem is built upon the foundation of number theory, a branch of mathematics that studies the basic arithmetic properties of numbers, including solutions to polynomial expressions. These are strings of terms with coefficients attached to a variable raised to different powers, like x³ + 2x − 3. In particular, mathematicians often study the roots of these expressions, the values of x that make the polynomial equal zero.

Number theorists often classify polynomials by the type of coefficients they have. The ones with rational numbers as coefficients are common targets of study, because they’re relatively simple.

“We start with the rational numbers,” said Samit Dasgupta, a mathematician at Duke University and one of the authors of the recent work, along with Mahesh Kakde of the Indian Institute of Science. “This is sort of the fundamental system in number theory.”

Sometimes the roots of polynomials with rational coefficients are themselves rational numbers, but that’s not always the case. That means mathematicians who want to find the roots of all polynomials with rational coefficients need to look in an expanded number system: the complex numbers, which includes all rational and real numbers, plus the imaginary number i, the square root of −1.

When we plot the roots of a polynomial on the complex plane, with real numbers along the x-axis and purely imaginary ones along the y-axis, certain symmetries can emerge. These symmetries can be applied to rearrange the points, permuting their locations. If you can apply the symmetries in any order and get the same result, we say the polynomial is abelian. But if the order you apply the symmetries in changes the outcome, the polynomial is non-abelian. Number theorists are most interested in abelian polynomials, again for their simplicity, but they can be difficult to distinguish. For example, x² − 2 is abelian, but x³ − 2 is not.

“To get to the non-abelian stuff, you don’t have to move very far,” said Ellen Eischen of the University of Oregon.

Besides those symmetries, abelian polynomials also have another distinguishing feature, which involves trying to describe the roots of polynomials in simple and exact terms. For example, it’s easy to describe the roots of the polynomial x² − 3 exactly: They’re just the positive and negative square roots of 3. But it can be difficult to state the roots of more complicated polynomials with larger exponents.

Of course, there are workarounds. “You can solve numerically to approximate [the root of a polynomial],” said Eischen. “But if you want to write it down in an explicit way — which is what a lot of people would say feels more satisfying — we can only do that in limited ways.”

Abelian polynomials with rational coefficients, however, are special: It’s always possible to calculate their roots precisely from a fixed collection of building blocks. This discovery proved so powerful, it inspired Hilbert to pose his 12th problem, and it’s all thanks to a collection of numbers known as the roots of unity.

The Roots of Unity

The roots of unity are a seemingly simple concept with extraordinary power. Numerically, they’re the solutions to polynomials where a variable raised to a power is set equal to 1, such as x⁵ = 1 or x⁸ = 1. These solutions are complex numbers, and they are referred to by the number in the exponent. For example, the “fifth roots of unity” are the five solutions of x⁵ = 1.

But the roots of unity can be also described geometrically, without using equations. If you plot them on the complex plane, the points all lie on a circle of radius 1. If you think of the circle as a clock, you’ll always have a root of unity at 3 o’clock, where x = 1, since 1 to any power is still 1. The remaining roots of unity are equally spaced around the circle.

In the 1800s, prior to Hilbert’s list of problems, mathematicians discovered that the roots of unity could serve as “building blocks” for the particular collection of numbers that they wanted to study: roots of abelian polynomials with rational coefficients. If you take simple combinations of the roots of unity — adding, subtracting and multiplying them by rational numbers — you can describe all of these desired roots. For example, the square root of 5 is a root of the abelian polynomial x² − 5, and it can be expressed as the sum of various fifth roots of unity. Similarly, a root of x² − 2, the square root of 2, is formed using the eighth roots of unity. This is similar to the way the prime numbers are building blocks for the whole numbers.

So the roots of unity are the exact building blocks you need to perfectly describe the roots of abelian polynomials with rational coefficients. On the flip side, any combination of the roots of unity will result in a number that is the root of some abelian polynomial with rational coefficients. The two are inextricably linked.

What Hilbert wanted when he posed his 12th problem was for mathematicians to find the building blocks of the roots of abelian polynomials with coefficients from number systems beyond rational numbers. In other words, what is the analogue of the roots of unity for other number systems?

Go Beyond

It’s an ambitious question, but that’s why it landed on Hilbert’s list in the first place. He suspected it could be answered because, as he was writing it, he had an idea about how to describe the building blocks for one other type of number system, known as imaginary quadratic fields. (Roughly, this system includes only the rational numbers and the square root of a negative number.) His guess was proved correct several decades later.

“There are two cases — the case of [the rationals] and the case of the imaginary quadratic fields — that guided Hilbert in formulating his question,” said Alice Pozzi of Imperial College London.

Hilbert expected the building blocks of other number systems to be described in similar terms to the two cases he already knew about. This meant using complex analysis, the branch of mathematics that studies functions using complex numbers.

But in the 1970s — decades after Hilbert laid the foundations for his 12th problem — the mathematician Harold Stark conjectured that L-functions might help crack it instead. These are a type of function that adds together infinitely many numbers. The Riemann zeta function, which is the subject of another problem on Hilbert’s list, is a famous example:

ζ(s) = 1 + $latex \frac{1}{2^s}$ + $latex \frac{1}{3^s}$ + $latex \frac{1}{4^s}$ + $latex \frac{1}{5^s}$+ $latex \frac{1}{6^s}$ + $latex \frac{1}{7^s}$ + ….

For centuries, mathematicians have known that L-functions produce mysterious and intriguing results. They show how infinite sequences of simple fractions can be used to build numbers related to pi and other important constants.

Building on this intuition, Stark was able to come up with an analogue of the roots of unity for other number systems using L-functions. But while mathematicians believe that Stark’s conjecture is true, having tested it extensively using computer analysis, they have not had any success proving it.

“The Stark conjecture is really difficult as far as we know,” said Darmon. “There’s been virtually no progress at all, and it’s been 50 years.”

Ultimately, what Stark did was provide a recipe that claims to find, using L-functions, the building blocks for the roots of abelian polynomials with coefficients from other number systems. It’s just that no one knows how to show that the recipe works.

Worse yet, Stark’s recipe only gives half of the information you need to actually describe the building blocks. This is equivalent to having only the longitude of a location — you also need the latitude to find a specific spot.

In the 1980s, Gross continued the work by publishing a modified version of Stark’s recipe, this time using newer ingredients. Stark, like Hilbert, had thought in terms of complex numbers, but Gross instead used the p-adic numbers. These are alternatives to the standard numbers that use different methods to decide when two numbers are close together.

“You can build up the whole theory of calculus from scratch again, where you use this new notion of what it means for things to be close,” said Dasgupta.

Many concepts in mathematics can be rewritten using p-adic numbers, and that includes L-functions. In fact, within modern number theory, p-adic L-functions are viewed as a natural companion to complex L-functions.

“They comprise a very coherent family,” said Barry Mazur of Harvard. “They work together.”

Even so, at first Gross’ translation from complex numbers to p-adic numbers didn’t seem to bring mathematicians any closer to proving Stark’s conjecture.

“I was just totally intimidated by it, because I thought, ‘Well, this is as hard as the original conjecture, but it’s just in a different form,’” said Gross. But in the following decades, Gross’ p-adic conjecture began to look more amenable to solving than its complex counterpart, as number theorists developed the theory of p-adic numbers.

“There’s a whole world now of p-adic analysis that’s very rich and has led to a lot of interesting results,” said Darmon. Many important problems in mathematics turned out to be easier to solve using p-adic numbers rather than complex numbers — Hilbert’s 12th problem included.

Chipping Away

In March of this year, Dasgupta and Kakde posted a paper that used p-adic L-functions to answer, for the first time, Hilbert’s question for a separate large collection of number systems. Known as totally real fields, these systems are an extension of the rational numbers that also incorporates one root of a given polynomial. (For example, incorporating the square root of 2 expands the rational numbers to include numbers like $latex \frac{\sqrt{2}}{3}$, 1 + $latex \sqrt{2}$, and 2 – 3$latex \sqrt{2}$.)

Dasgupta first came up with the final formula they’d need — a refinement of Gross’ conjecture — in his 2004 doctoral thesis.

“Half my life I’ve been working on this,” said Dasgupta. “So it’s very satisfying to have finally completed the proof.”

The first step in the process, which took place over the past decade, was to finally prove Gross’ conjecture in a series of two papers, using the latest developments in p-adic number theory. But this gave only half the information, because Gross’ conjecture — like Stark’s —gives only one of the two numbers needed to precisely describe the building blocks.

For the last three years, Dasgupta and Kakde have worked to prove a version of Gross’ conjecture that provides both numbers, even when it still seemed impossible.

“Maybe it’s just that both of us are very optimistic,” said Kakde. “At times it did seem that these obstacles were quite serious, and at times they were quite serious, but fortunately, progress kept happening.”

Last year they had a breakthrough. They were able to prove that the precise building blocks associated to totally real fields exist. In other words, they knew that the desired objects were somewhere out there, and this insight led them in the right direction. It gave them the key equations for proving the exact formula that fully describes the building blocks.