Mathematicians have made a huge leap forward on one of the field’s all-time favorite problems.
Curves—squiggly lines through space, such as the trajectory of a comet or the trend of the stock market—are some of the simplest objects in mathematics. But even though they have been studied for thousands of years, mathematicians still have some fundamental questions about them unanswered.
Number theorists have typically looked for special points on a curve with coordinates x–this Grids that are either whole numbers or fractions. These rare points are often interconnected in complex and meaningful ways. “We are mathematicians, and we care about structure,” says Barry Mazur, the Gerhard Gade University Professor at Harvard University. That structure can sometimes be useful; For example, rational points on so-called elliptic curves gave rise to an entire branch of cryptography.
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But there is a vast set of curves, composed of many infinite families, and each with its own structure of rational points. Number theorists have dreamed of finding a concrete mathematical rule that applies to every curve. But such a one-sided formula has been beyond their understanding for a long time.
This changed a few weeks ago. In a preprint paper posted Feb. 2, three Chinese mathematicians put first hard upper bound There can be any number of rational points on any curve. The mathematical results are limitless.
“This is a really surprising result that sets a new standard for what to expect,” says mathematician Hector Peston of the Pontifical Catholic University of Chile, who was not involved in the work.
Finite or infinite?
Curves are represented mathematically by simple equations called polynomials. They are essentially a handful of variables that have been multiplied and added together.
think about the equation x2 + this2 = 1. If x And this A coordinate plane has two axes, this equation represents a circle. Each point on the circle corresponds to a different solution of this equation. For example, talk x = 1 more this = 0, written as the coordinate pair (1, 0), is on the circle: If you put those values x And this In the equation, you get 1 = 1, which is a valid solution.
Some solutions, including (1, 0) and (3⁄5, 4⁄5), are “rational”, meaning both x And this are either whole numbers or ratios of whole numbers. Other solutions, such as (1⁄√2, 1⁄√2), are “irrational”. Plug in these values x And thisAnd you get a valid solution to the equation – the coordinates fall directly onto the circle. But you can never express them as whole numbers and their ratios.
Ancient Greek mathematicians were obsessed with finding rational points along curves. He wondered how many of these special points there were in a given curve. This is one of the simplest questions in mathematics, but it has puzzled mathematicians for millennia. “These problems are at the heart of number theory,” says Shenxuan Zhou, a mathematician at the Toulouse Mathematics Institute who co-authored the new result.
The circle – a special type of curve – has infinitely many rational points. The same is true for any other curve, where neither x and neither this is raised to a power greater than 2. These “degree 2” equations always either have no rational points or infinitely many. One degree higher, the number of rational points on a curve of degree 3 is sometimes infinite and sometimes finite.
But in 1922 Louis Mordel made a famous conjecture that indicated a rapid change in the situation for higher-degree equations. It states that when a curve has degree 4 or more, there will always be a finite number of rational points.
Sixty-one years later Gerd Faltings proved Mordell right; He was awarded the Fields Medal, the highest honor in mathematics. But Mordell’s conjecture, now called Falting’s theorem, says nothing about this. How many? There are points near these curves.
Since then, mathematicians have found a formula to answer this question. “That’s all we know Is A formula,” says Peston. “It’s out there somewhere, and it’s good, but we want it.”
A rule for every curve
This is where new evidence comes to light. Its authors present a formula that can be applied to any curve in the mathematical universe, regardless of its degree. It doesn’t tell exactly how many rational points there are in that curve, but it gives an upper bound on that number.
Previous formulas of this type either did not apply to all curves or depended on the specific equation used to define them. The new formula is something mathematicians have hoped for since Falting’s proof, a “uniform” statement that applies to all curves regardless of the coefficients in their equations. “This one statement gives us a broader understanding,” Mazur says.
It depends on only two things. The first is the degree of the polynomial that defines the curve – the higher the degree, the weaker the statement becomes. The second thing the formula relies on is something called a “Jacobian variety”, a special surface that can be constructed from any curve. Jacobian varieties are interesting in their own right, and the formula also provides a fascinating way to study them.
The new result is a first step toward knowing how many points the curves contain, not just whether they contain an infinite number of points. “There are more questions on the horizon,” says Pasten. “We can be more ambitious now.”
Curves are also the first step into the mathematical world of shapes created by equations. Polynomial Equations with Extra Variables x And this Can generate more complex objects, such as surfaces or their higher-dimensional analogs, called “manifolds”. Manifolds are central to modern mathematics as well as theoretical physics, where they are used to map space and time.
All these questions about rational points matter for those higher-dimensional objects as well. Peston and the mathematician Gerson Caro definitively put an upper bound on the number of rational points. Appeared in 2023 paperFor example. The new result gives Peston hope for further progress in this broader search.
This discovery is one of several new recent results about rational points on curves. Overall, this surge may signal a new chapter in this millennium-old saga.
“This is an exciting, fast-moving field,” says Mazur. “Something big is happening right now.”
