By I. R. Shafarevich (editor), V.I. Danilov, V.V. Shokurov

ISBN-10: 3540519955

ISBN-13: 9783540519959

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Extra resources for Algebraic geometry 01 Algebraic curves, algebraic manifolds and schemes

Sample text

A d- l , 1, respec­ tively, we obtain the vector vo (O, a0, a1 , . . , ad- l , 1) E A. It turns out that if K is sufficiently large compared to the ai , then v0 is the shortest nonzero vector, and moreover, every vector not much longer than vo is a rnultiple of v0. The LLL algorithm applied to A thus finds v0 , and this yields q(x). Of course, we do not know the degree of q(x), but we can test all possible degrees one by one, and the required mag­ nitude of K can be estimated from the coefficients of p(x ) .

Therefore a2 + b2 = p. D = = The fact that every prime congruent to 1 mod 4 can be written as the sum of two squares was already known to Fermat (a more rigorous proof was given by Euler) . The possibility of expressing every natural number as a sum of at most 4 squares was proved by Lagrange in 1770, as a part of his work on quadratic forms. The proof indicated in Exercise 1 below is due to Davenport. Bibliography and remarks. Exercises 1 . (Lagrange's four-square theorem) Let p be a prime. (a) Show that there exist integers a, b with a2 + b2 -1 (modp).

Theorem 2 . 1 . 1 is often called Minkowski 's first theorem. What is, then, Minkowski's second theorem? 2, where we also review a few more of the basic results in the geometry of numbers and point to some interesting connections and directions of research. Most of our exposition in this chapter follows a similar chapter in Pach and Agarwal [PA95] . Older books on the geometry of numbers are Cassels [Cas59] and Gruber and Lekkerkerker [GL87] . A pleasant but somewhat aged introduction is Siegel [Sie89] .

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Algebraic geometry 01 Algebraic curves, algebraic manifolds and schemes by I. R. Shafarevich (editor), V.I. Danilov, V.V. Shokurov

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