MULTIVARIATE ULTRAMETRIC ROOT COUNTING 21
have that
K[X]A
NB,i(f) df = IwIδIe, where
Iw =
([−M,M]∩v(π)Z)t
χ
K[X]A
B
(f)χ
v(π)Z
wβi+1 wβi
βi+1 βi
dw,
=
(k∗)t
Zk(δβi+1
Xβi+1
+ δβi
Xβi
) dδ,
Ie =
(1+M)t
1 de.
It is clear that Ie = 1 and also = Ek(βi+1 βi,D1) by definition. The integral
defining Iw is in fact a finite sum over a lattice: if we write N = [M/v(π)] and
= wα/v(π), then
Iw = (2N +
1)−t
−N≤v1,...,vt≤N
χ
K[X]B
A
(f)χ
Z
vβi+1 vβi
βi+1 βi
=
= (2N +
1)−t
−N≤v1,...,vt≤N
βi+1−βi|vβi+1 −vβi
χ
K[X]B
A
(f).
The expression χK[X]A
B
(f) in the last sum is a function of vα1 , . . . , vαt that test
whether the Newton Polygon of the set of points {(vα,α) : α A} is supported at
B, i.e. is equal to χ
S(B/A)
(vα1 , . . . , vαt ). Since the set S(B/A) is invariant under
rescaling and translations, then
Iw = (2N +
1)−t
−N≤v1,...,vt≤N
βi+1−βi|vβi+1 −vβi
χ
S(B/A)
N + vα1
2N + 1
, . . . ,
N + vαt
2N + 1
.
Without the condition βi+1 βi|vβi+1 vβi , the expression is exactly a Riemman
sum of χ
S(B/A)
, with a partition of [0,
1]t
corresponding to the lattice {0, 1/(2N +
1), . . . ,
1}t.
Adding this extra condition is equivalent to taking a sublattice of order
βi+1 βi, so limM→∞ Iw = P (B/A)(βi+1
βi)−1.
This shows that
lim
M→∞
K[X]A
NB,i(f) df = P (B/A)
Ek(βi+1 βi,D1)
βi+1 βi
.
Going back to our formula for E(A,D1,D2,K), we get
E(A,D1,D2,K) =
{α1,α|A|}⊆B⊆A
P (B/A)
|B|−1
i=1
Ek(βi+1 βi,D1)
βi+1 βi
.
To conclude the proof, note that the right term does not depend on the probability
distribution D2, and then we can safely write E(A,D1,K), as claimed.
We conclude this section with an analysis of the case where the residue field
is algebraically closed. In this case, we have Ek(γ, D1) = γ, regardless of the
probability distribution D1, so the formula of Theorem 6.1 reduces to
E(A,D1,K) =
{α1,α|A|}⊆B⊆A
P (B/A)(|B| 1) = 1 +
t
i=2
Pi,
21
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