October 9, 2024
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A century question continues to be answered in basic mathematics
Mathematicians have made a lot of progress over a century ago on a question called Mordell’s conjecture.
A boy working on a conceptual representation of mathematics on a blackboard.
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After the German mathematician Gerd Faltings proved Mordell’s conjecture in 1983, He was awarded the Fields MedalIt is often described as the “Nobel Prize of Mathematics”. The conjecture describes the conditions under which a polynomial exists the equation in two variables (eg x2 + y4 = 4) is guaranteed to have only a finite number of solutions that can be written as fractions.
Faltings’ evidence answered a question that had been open since the early 1900s. It also opened new mathematical doors to other unanswered questions, many of which are still being explored by researchers today. In recent years, mathematicians have made tremendous progress in understanding these outbursts and their consequences basic mathematics.
A proof of Mordell’s conjecture corresponds to this situation: Suppose a polynomial equation in two variables defines a curved line. The question at the heart of Mordell’s conjecture is: What is the relationship between the genus of the curve and the number of rational solutions for the polynomial equation that defines it? Genus is a property related to the highest exponent of the polynomial equation that describes the curve. It is an invariant property, that is, it remains the same even if certain operations or transformations are applied to the curve.
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The answer Mordell’s conjectureThe main question of this is that if an algebraic curve is of genus two or higher, there will be a finite number of rational solutions to the polynomial equation. (This number excludes solutions that are multiples of other solutions.) For curves of genus zero or genus one, there can be infinitely many rational solutions.
“A little over 100 years ago, Mordell figured out that this genus controlled the infinity or infinity of rational points on one of these curves,” says University of Cambridge mathematician Holly Krieger. Consider a point (x, y). If both x and y are numbers that can be written as fractions, then (x, y) is a rational point. For example, (1⁄33) is a rational point, but (√2, 3) is not. Mordell’s idea meant that “if your gender was big enough, your curve is somehow geometrically complicated,” says Krieger. he gave one invited lecture On the history of Mordell’s conjecture and some subsequent work at the 2024 Joint Mathematics Meetings.
Faulting’s evidence raised new possibilities for exploring questions that extend Mordell’s conjecture. One such exciting question—the Uniform Mordell-Lang conjecture—was raised in 1986, the same year he was awarded the Faltings Fields Medal.
The Mordell-Lang Uniform conjecture, which was formalized by Harvard University’s Barry Mazur, “was proven in a series of papers completed in 2021,” says Krieger. The work of four mathematicians—Vesselin Dimitrov of the California Institute of Technology, Ziyang Gao of the University of California, Los Angeles, and Philipp Habegger of the University of Basel in Switzerland, who were collaborators, and Lars Kühne of University College Dublin—. he worked one by one—he took this conjecture to prove.
For the Mordell-Lang Uniform Conjecture, mathematicians are asking: What if you extend the mathematical discussion to include high-dimensional objects? What can be said, then, about the relationship between the gender of a mathematical object and the number of associated rational points? The answer is that the upper bound—meaning the largest possible number—of rational points associated with a curve or a high-dimensional object such as a surface depends on the gender of that object. For the surface, the gender corresponds to the number of holes on the surface.
There is an important caveat, however, according to Dimitrov, Gao, and Habegger. “Geometric objects (curves, surfaces, triples, etc.) must exist within a very special environment space, called an abelian variety,” they wrote in an email. American scientific. “An abelian variety itself is ultimately defined by polynomial equations, but it comes equipped with a group structure. Abelian varieties have many surprising properties and it is a kind of miracle that they even exist.’
The proof of the Mordell-Lang Uniform Conjecture “is not just the resolution of a problem that has been open for 40 years,” says Krieger. “It goes to the heart of the fundamental question in mathematics.” These questions focus on finding rational solutions (which can be written as fractions) of polynomial equations. Such questions are often called Diophantine problems.
Mordell’s conjecture “is a kind of example of what geometry means to determine arithmetic,” says Habegger. The team’s contribution to proving the uniform Mordell-Lang conjecture showed that “the number of (rational) points is fundamentally limited by geometry,” he says. Therefore, proving the Mordell-Lang Uniform does not give mathematicians an exact number of how many rational solutions there will be for a given genus. But he tells them as many solutions as possible.
The 2021 proof is certainly not the final chapter on the problems stemming from Mordell’s conjecture. “The beauty of Mordell’s original conjecture is that it opens up a world of more questions,” says Mazur. According to Habegger, “the main open question is to prove Mordell effective” – a consequence of the original conjecture. Solving this problem would mean entering another mathematical field, in which to identify exactly how many rational solutions there are for a given scenario.
There is a significant gap between the information provided between proving the Uniform Mordell-Lang conjecture and solving the efficient Mordell problem. Knowing how many rational solutions there are for a given situation “doesn’t really help you” determine what those solutions are, Habegger says.
“Let’s say you know that the number of solutions is at most one million. And if you only find two solutions, you’ll never know there are more,” he says. If mathematicians can solve Mordell efficiently, that will bring them incredibly close to being able to use a computer algorithm to quickly find all the rational solutions, instead of having to tediously search for them one by one.
