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MART´
IN
AVENDA˜
NO AND ASHRAF IBRAHIM
K[X]A
B
the set of polynomials f K[X]A with Newton Polygon supported at B.
By definition, we have that
E(A,D1,D2,M,K) =
K[X]A
|ZK (f)| df =
{α1,αt}⊆B
B⊆A
K[X]AB
|ZK (f)| df
and also
E(A,D1,D2,K) =
{α1,αt}⊆B
B⊆A
lim
M→∞
K[X]B
A
|ZK (f)| df.
For any f K[X]A,
B
define
N(f) =
|B|−1
i=1
χ
v(π)Z
v(aβi+1 ) v(aβi )
βi+1 βi
Zk(δ(aβi+1
)Xβi+1
+ δ(aβi
)Xβi
) ,
where χS(·) represents the characteristic function of the set S. This gives a function
N : K[X]A N0 that, by Proposition 4.2, coincides with |ZK (f)| for any f
K[X]A regular. Moreover, the difference N(f) |ZK (f)| is bounded on K[X]A.
By Theorem 5.2, the probability of the set of non-regular polynomials approaches
0 as M goes to infinity, and then we can also write
E(A,D1,D2,K) =
{α1,αt}⊆B
B⊆A
lim
M→∞
K[X]AB
N(f) df =
=
{α1,αt}⊆B
B⊆A
lim
M→∞
K[X]A
χ
K[X]B
A
(f)N(f) df =
=
{α1,αt}⊆B
B⊆A
|B|−1
i=1
lim
M→∞
K[X]A
NB,i(f) df
where NB,i(f) is the expression
χ
K[X]B
A
(f)χ
v(π)Z
v(aβi+1 ) v(aβi )
βi+1 βi
Zk(δ(aβi+1
)Xβi+1
+ δ(aβi
)Xβi
) .
Any polynomial f K[X]A correspond with a unique point (w, δ, e) ([−M, M]
v(π)Z)t ×(k∗)t ×(1+M)t. Using this representation, we can write
K[X]A
NB,i(f) df
as the triple integral
χ
K[X]A
B
(f)χv(π)Z
wβi+1 wβi
βi+1 βi
Zk(δβi+1
Xβi+1
+ δβi
Xβi
) de dw.
Since the function χ
K[X]B
A
(f)χ
v(π)Z
((wβi+1 wβi )/(βi+1 βi)) depends only on w,
and the function |Zk(δβi+1 Xβi+1 + δβi Xβi )| depends only on δ, the triple integral
above can be splitted as a product of three simple integrals. More precisely, we
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