By I. R. Shafarevich (editor), V.I. Danilov, V.V. Shokurov
"... To sum up, this ebook is helping to profit algebraic geometry very quickly, its concrete type is pleasing for college kids and divulges the wonderful thing about mathematics." --Acta Scientiarum Mathematicarum
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Extra resources for Algebraic geometry 01 Algebraic curves, algebraic manifolds and schemes
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] .
Algebraic geometry 01 Algebraic curves, algebraic manifolds and schemes by I. R. Shafarevich (editor), V.I. Danilov, V.V. Shokurov