\n<\/p>\n
Mathematicians just made a big leap forward on one of the field\u2019s all-time favorite problems.<\/p>\n
Curves\u2014squiggly lines through space, such as a comet\u2019s trajectory or a stock market trend\u2014are some of math\u2019s simplest objects. But even though they have been studied for thousands of years, mathematicians still have some basic questions about them left unanswered.<\/p>\n
Number theorists have particularly sought special points on a curve with coordinates on an x-y grid that are either whole numbers or fractions. These rarified points are often interrelated in complicated and meaningful ways. \u201cWe\u2019re mathematicians, and we care about structure,\u201d says Barry Mazur, Gerhard Gade University Professor at Harvard University. That structure can sometimes be useful; the rational points on so-called elliptic curves gave birth to a whole branch of cryptography, for instance.<\/p>\n
If you’re enjoying this article, consider supporting our award-winning journalism by subscribing. By purchasing a subscription you are helping to ensure the future of impactful stories about the discoveries and ideas shaping our world today.<\/p>\n
But there\u2019s a vast menagerie of curves out there, composed of numerous infinite families, and each has its own structure of rational points. Number theorists have dreamed of finding a concrete mathematical rule that applies to every curve. But such a unilateral formula has long eluded them.<\/p>\n
That changed a few weeks ago. In a preprint paper posted on February 2, three Chinese mathematicians placed the first ever hard upper limit on the number of rational points any curve can have. The mathematical consequences are limitless.<\/p>\n
\u201cThis really is an amazing result that sets a new standard for what to expect,\u201d says Hector Pasten, a mathematician at the Pontifical Catholic University of Chile, who wasn\u2019t involved in the work.<\/p>\n
Curves are mathematically represented by simple equations called polynomials. They\u2019re essentially a handful of variables multiplied and added together.<\/p>\n
Think of the equation x2 + y2 = 1. If x and y are the two axes of a coordinate plane, this equation represents a circle. Every point on the circle corresponds to a different solution to this equation. For instance, the point x = 1 and y = 0, written as the coordinate pair (1, 0), is on the circle: if you put those values for x and y into the equation, you get 1 = 1, which is a valid solution.<\/p>\n
Some solutions, including (1, 0) and (3\u20445, 4\u20445), are \u201crational,\u201d meaning both x and y are either whole numbers or ratios of whole numbers. Other solutions, such as (1\u2044\u221a2, 1\u2044\u221a2), are \u201cirrational.\u201d Plug in these values for x and y, and you get a valid solution to the equation\u2014the coordinates land right on the circle. But you can never express them in terms of whole numbers and their ratios.<\/p>\n
Ancient Greek mathematicians were obsessed with finding rational points along curves. They wondered how many of these special points a given curve has. It\u2019s one of the simplest questions in math, but it has vexed mathematicians for millennia. \u201cThese problems sit at the heart of number theory,\u201d says Shenxuan Zhou, a mathematician at the Toulouse Mathematics Institute who co-authored the new result.<\/p>\n
The circle\u2014a particular kind of curve\u2014has infinitely many rational points. The same is true for any other curve where neither x nor y is raised to a power bigger than 2. These \u201cdegree 2\u201d equations always either have no rational points at all or infinitely many. The number of rational points on a curve that is one degree higher, degree 3, is sometimes infinite and sometimes finite.<\/p>\n
But in 1922 Louis Mordell made a famous conjecture that indicated the situation sharply changes for higher-degree equations. It stated that when the degree of a curve is 4 or more, there will always be a finite number of rational points.<\/p>\n
Sixty-one years later Gerd Faltings proved Mordell right; he was rewarded with a Fields Medal, math\u2019s highest honor. But Mordell\u2019s conjecture, now called Faltings\u2019s theorem, says nothing about how many points these curves have.<\/p>\n
Since then, mathematicians have sought a formula to answer this question. \u201cWe just know that there is a formula,\u201d Pasten says. \u201cIt\u2019s somewhere out there, and that\u2019s good, but we want it.\u201d<\/p>\n
That\u2019s where the new proof comes in. Its authors present a formula that can be applied to any curve in the mathematical universe, whatever its degree. It doesn\u2019t say precisely how many rational points that curve has, but it gives an upper limit on what that number can be.<\/p>\n
Previous formulas of this kind either didn\u2019t apply to all curves or depended on the specific equation used to define them. The new formula is something mathematicians have hoped for since Faltings\u2019s proof, a \u201cuniform\u201d statement that applies to all curves without depending on the coefficients in their equations. \u201cThis one statement gives us a broad sweep of understanding,\u201d Mazur says.<\/p>\n
It depends on only two things. The first is the degree of the polynomial that defines the curve\u2014the higher the degree is, the weaker the statement becomes. The second thing the formula depends on is called the \u201cJacobian variety,\u201d a special surface that can be constructed from any curve. Jacobian varieties are interesting in their own right, and the formula offers a tantalizing path for studying them as well.<\/p>\n
The new result is a first step toward knowing how many points curves have, not just whether or not they have an infinite number of points. \u201cThere are more questions on the horizon,\u201d Pasten says. \u201cWe can get more ambitious now.\u201d<\/p>\n
Curves are also just a first foothold on the mathematical world of shapes carved out by equations. Polynomial equations with additional variables besides x and y can generate more complicated objects, such as surfaces or their higher-dimensional analogues, called \u201cmanifolds.\u201d Manifolds are central to modern mathematics, as well as theoretical physics, where they\u2019re used to map out space and time.<\/p>\n
All these questions about rational points matter for those higher-dimensional objects, too. Pasten and mathematician Jerson Caro placed an upper bound on the number of rational points for certain surfaces in a 2023 paper, for example. The new result gives Pasten hope for further progress in this far broader quest.<\/p>\n
This finding is one of several recent new results about rational points on curves. Taken together, the surge might signify a new chapter in this millennia-old saga.<\/p>\n
\u201cThis is an exciting, fast-moving area,\u201d Mazur says. \u201cThere\u2019s something big happening right now.\u201d<\/p>\n","protected":false},"excerpt":{"rendered":"
Mathematicians just made a big leap forward on one of the field\u2019s all-time favorite problems. Curves\u2014squiggly lines through space, such as a comet\u2019s trajectory or a stock market trend\u2014are some of math\u2019s simplest objects. But even though they have been studied for thousands of years, mathematicians still have some basic questions about them left unanswered.<\/p>\n","protected":false},"author":1,"featured_media":45138,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[50],"tags":[17206,5236,23302,8203,1651],"class_list":{"0":"post-45137","1":"post","2":"type-post","3":"status-publish","4":"format-standard","5":"has-post-thumbnail","7":"category-environment","8":"tag-2000yearold","9":"tag-breakthrough","10":"tag-curves","11":"tag-mathematicians","12":"tag-problem"},"_links":{"self":[{"href":"https:\/\/naijaglobalnews.org\/index.php?rest_route=\/wp\/v2\/posts\/45137","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/naijaglobalnews.org\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/naijaglobalnews.org\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/naijaglobalnews.org\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/naijaglobalnews.org\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=45137"}],"version-history":[{"count":0,"href":"https:\/\/naijaglobalnews.org\/index.php?rest_route=\/wp\/v2\/posts\/45137\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/naijaglobalnews.org\/index.php?rest_route=\/wp\/v2\/media\/45138"}],"wp:attachment":[{"href":"https:\/\/naijaglobalnews.org\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=45137"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/naijaglobalnews.org\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=45137"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/naijaglobalnews.org\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=45137"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}