An Introduction to the Geometry of Numbers by J. W. S. Cassels (auth.)

By J. W. S. Cassels (auth.)

Reihentext + Geometry of Numbers From the reports: "The paintings is thoroughly written. it's good influenced, and fascinating to learn, whether it isn't constantly easy... ancient fabric is included... the writer has written a great account of an attractive subject." (Mathematical Gazette) "A well-written, very thorough account ... one of the issues are lattices, relief, Minkowski's Theorem, distance features, packings, and automorphs; a few purposes to quantity idea; first-class bibliographical references." (The American Mathematical Monthly)

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For later reference we formulate a typical result as a Lemma. LEMMA 4. The following four statements about a number x are eqttivalent, where q;(X) = X~ + ... + X~ - X~+l - ... - X!. ( i) In every lattice A there is a vector A +0 with (ii) In every lattice A of determinant 1 there is a vector A +0 with 1q;(A)1 ~x. 4+0 in d(A)~y'-n/2 there is a vector Iq;(A)1 ~ 1- (iv) For every quadratic form L fii is an integer vector a 0 such that + Xi Xi of signature (r, n-r) there I/(a)1 ~xldet(/;iWI". That (i), (ii) and (iii) are equivalent follows from homogeneity, since q;(tX) =t2q;(X) and since the set tA of all tX (XEA) is a lattice tA of d (A); and we may choose t so that t" d (A) = 1.

Since lo(u) is an integer we have 10(u)~1. Since D(/o) =i, this shows that the equality signs are required for 10' Part A of the theorem follows from the rest. Hence we need only prove Part B and that equality in B occurs only for multiples of 10' Following GAUSS (1831a) we distinguish two cases. Suppose first that Then after a substitution X j -+ ± x; we may suppose without loss of generality that 112~O, Write 123~o, 131~o. (I;i Then = Ii;)' (1) {}i; ~ 0 for all i =t= j since I is reduced. For example 1(1, -1, 0) ~ 1(1, 0, 0) gives {}2l ~ o.

An example may make this clearer. O)=1. B are to be investigated. 1) ;;;:;-1. 1) ;S+1 . Fig. -1) ;S +1. is xi + Xl x 2 - x~. as the reader will easily verify. Hence any other form with I (i. 1);S+1. 1);;;:;-1. 1(2. -1);;;:;-1. The form X~+XIX2-X~ is thus in a strong sense isolated from all other forms (1) with M(f) = 1. B are plotted as cartesian coordinates for the form I. a condition I/{u l • u2)1;S 1 excludes a strip of the plane between two parallel lines. 1)I;S 1. 1)1;S 1. 1/(2. -1)I;S 1 exclude three strips.

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