The first published proof is due to Euler in The proof given is a simple consequence of Minkowski's theorem. Although some mathematicians' devised ways to find solutions, Lagrange was the first to publish a proof that solutions always exist, use Minkowski's theorem to prove solutions exist.
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Search all titles. Search all titles Search all collections. Your Account Logout. It only takes a minute to sign up. Geometry of numbers was initiated by Hermann Minkowski roughly a hundred years ago.
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At its heart is the relation between lattices the group, not the poset and convex bodies. While there was a lot of activity in the field until at least , it seems that in recent decades not so many people are working on it anymore. One of the reasons could be that the field is somewhat stuck, or in Gruber's more polite words "It seems that fundamental advance in the future will require new ideas and additional tools from other areas. I would like to know more about current research trends in the geometry of numbers.
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What are hot topics right now? In which areas was recently considerable progress achieved? Did maybe even the "fundamental advance", that Gruber mentions, take place? There has indeed been exciting recent work in this area, by Bhargava and Shankar see this Bourbaki expose by Poonen and also by Bhargava and Gross. Section 4 of the quite readable write-up by Poonen explains why I refer to these results as recent advances in the geometry of numbers: both of these results boil down to subtle!
It's worth noting that the work of Bhargava and Shankar does not use adelic language, and so is more obviously related to the "classical" geometry of numbers.
THE GEOMETRY OF NUMBERS
I can't say that what I'll relate is fundamental, but it does fit into the new ideas category. Since I and my collaborator Florent Balacheff have given talks on the subject and the paper will be in the ArXiv in a few days I feel free to comment on it.
This post is an annoucement of joint work with Florent Balacheff and Kroum Tzanev. It is easy to see that Minkowski's theorem fails completely, but that's because one is not thinking symplectically. By using some Hamiltonian dynamics of the sort Balacheff and I used to study isosystolic inequalities in this paper , we guessed that the "right" result should be the following:. In other words, one should have a sort of uncertainty principle: if the origin is localized as the unique integer point inside a convex body, the dual body cannot be too small.
Another formulation of the conjecture that seems more elementary goes as follows:.
In fact, this result is equivalent to Bourgain-Milman. Moreover, it easily implies the asymptotic version of a conjecture of Ehrhart:. I just need a definition:. Introduction to the geometry of complex numbers. Introduction to the Geometry of Complex Numbers.
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Elements of Algebra: Geometry, Numbers, Equations. Elements of algebra: Geometry, numbers, equations.
Complex Numbers and Geometry Spectrum Series. Development of the Minkowski Geometry of Numbers Volume 2. Development of the Minkowski Geometry of Numbers Volume 1. Algebraic Geometry over the Complex Numbers. Book of Numbers. Book Of Numbers.