Gruber P. Convex and Discrete Geometry

362 Geometry of Numbers
(ii) Given a basis {b1,...,bd} of L, there is a basis {c1,...,cd} of M such that (1)
holds.
Proof. (i) This is an immediate consequence of the case i = d of Proposition (3) in
the proof of Theorem 21.2.
(ii) We first show that
(2) uL ⊆ M with a suitable positive u ∈ Z.
Consider bases {b1,...,bd} of L and {c1,...,cd} of M such that (1) holds. The
inverse of the integer lower triangular matrix (uik), with determinant u = u11 ···
udd �= 0, is a lower triangular matrix, the entries of which are of the form vik/u,
where vik ∈ Z. Hence
ub1 = v11c1
ub2 = v21c1 + v22c2
............................
ubd = vd1c1 +······+vddcd.
From this it follows that uL is a sub-lattice of M, concluding the proof of (2).
To prove (ii), let {b1,...,bd} be a basis of L. Choose u as in (2). Then
{ub1,...,ubd} is a basis of uL. An application of (i) to the sub-lattice uL of M
thus yields a basis {d1,...,dd} of M such that
(3)
ub1 = w11d1
ub2 = w21d1 + w22d2
.............................
ubd = wd1d1 +······+wdddd
where wik ∈ Z,wii �= 0.
Solving (3) for d1,...,dd, we see that we may express d1,...,dd in the form
(1) with d1,...,dd instead of c1,...,cd, but where the uik are rational. Since
{b1,...,bd} is a basis of L and M a sub-lattice, the uik are, in fact, integers. ⊓⊔
Primitive Points and Bases
If {b1,...,bd} is a basis of L and b = u1b1 +···+udbd, then b is primitive if and
only if 1 is the greatest common divisor of u1,...,ud.
Corollary 21.1. Let b be a primitive point of a lattice L in E d . Then there are points
b2,...,bd ∈ L, such that {b, b2,...,bd} is a basis of L.
Proof. Choose c2,...,cd ∈ L, such that b, c2,...,cd are linearly independent. By
Proposition (i) of the above theorem, there is a basis {b1,...,bd} of L, such that
b = u11b1
c2 = u21b1 + u22b2
...........................
cd = ud1b1 +······+uddbd
where uik ∈ Z, uii �= 0.
Since b is primitive, this can hold only if u11 =±1orb =±b1. Hence {b, b2,...,bd}
is also a basis of L. ⊓⊔