22
MART´
IN
AVENDA˜
NO AND ASHRAF IBRAHIM
where Pi =

{α1,αi,α|A|}⊆B⊆A
P (B/A) is the probability that αi is in the support
of the Newton Polygon. The value of Pi can be written in terms of integrals, as
shown in the following formula:
Pi =
1
0
· · ·
1
0
min
1≤jik≤t
vj
αi αk
αj αk
+ vk
αj αi
αj αk
dv1 · · · dvi · · · dvt.
The estimations
Pi
1
0
· · ·
1
0
max(min(v1, . . . , vi−1), min(vi+1, . . . , vt))dv1 · · · dvi · · · dvt,
Pi
1
0
· · ·
1
0
min(v1, . . . , vi,...,vt)dv1 · · · dvi · · · dvt,
show that
1
t
Pi
1
i
+
1
t−i+1

1
t
, and therefore
2
2
t
E(A,D1,K) 2
t
i=2
1
i
2 ln(t).
Acknowledgements
We would like to thank J. Maurice Rojas and Bernd Sturmfels for several
fruitful discussions about regularity and semiregularity, and for encouraging us to
publish these results.
References
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