Notations
We use the customary symbol s for sets of number s N C Z C Q C R C C, lettin g
N** C Zd Q - indicate tf-tuples a = (a
1
, ... , a
d
) o f suc h numbers.W e let (R d)+ denot e
the a R
d
wit h af. 0 for all i, an d we let (Q
d)+
an d
(ZJ)+
b e the corresponding subset s
of Q
d
an d R
d.
W e notationally distinguis h th e dual vector space (R
d)*
fro m R
d,
lettin g
ev ... , e
d
an d e x, ... , e
d
b e the canonical base s for R
d
an d (R
d)*,
respectively.W e give
(Rd)*
th e dua l ordering
(Rd):={Z«/e,:«,o},
and we le t R d an d (R J)* have the dual norms
ItE^JL
= maxla/l
P
a/e/||j =
Ste/I-
Wecall A , = { / (R*)*
:
/ 0 , \\f\\x = 1} th e fundamental d-simplex.
We say that a n r x s matri x \p = [fc
f
.A , K,.. £ R is positive i f fc
f
- 0 fo r all /, /, an d we
then write y 0 . Equivalently , if we regard y a s a linear map Rs Rr, we have a 0 im-
plies that if{a) 0 (a similar statement applie s to the dua l spaces).W e denote the transposed
matrix by p* and we identify i t with a matrix
(Rr)* —y
(R*) * in the usual way.
Finally we let GL(r , Z) denote the group of r x r matrices with entrie s in Z and de-
terminant ± 1, and we let G\Jj, Z)
+
b e the correspondin g matrice s with entries in Z
+.
I n
general y £ GL(r , Z)
+
doe s not impl y ip~
l
G GL(r, Z)
+.
http://dx.doi.org/10.1090/cbms/046/02
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