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Mathematicians Find Long-Sought Building Blocks for Special Polynomials | Quanta Magazine
Hilbert’s 12th problem asked for novel analogues of the roots of unity, the building blocks for certain number systems. Now, over 100 years later, two mathematicians have produced them.
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Mathematicians Find Long-Sought Building Blocks for Special Polynomials Jaki King for Quanta Magazine
Hilbert’s 12th problem asked for novel analogues of the roots of unity, the building blocks for certain number systems. Now, over 100 years later, two mathematicians have produced them.
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.