12
MART´
IN
AVENDA˜
NO AND ASHRAF IBRAHIM
Algorithm 1 Decide whether a system F = (f1, . . . , fn) of n polynomials in
K X1
±1,
. . . , Xn
±1
is semiregular at a given point w = (w1, . . . , wn)
Rn.
In
case of semiregularity, print the number of solutions in
(K∗)n
with valuation vector
w.
1:
if w
v(π)Zn
then
2:
print the system has no solutions in
(K∗)n
with valuation w
3:
return YES
4:
end if
5:
for i = 1, . . . , n do
6:
˜
f
i
inw(fi)
7:
if
˜
f
i
is a monomial then
8:
print the system has no solutions in
(K∗)n
with valuation w
9:
return YES
10:
end if
11:
end for
12:
Jac(
˜)
F det(∂
˜
f
i
/∂Xj)
13:
if there is a solution of
˜
f 1(x) = · · · =
˜
fn(x) = Jac(
˜)(x)
F = 0 in
(k∗)n
then
14:
return NO
15:
end if
16:
s number of solutions of
˜
f
1
(x) = · · · =
˜
fn(x) = 0 in (k∗)n
17:
print the system has s solutions in (K∗)n with valuation w
18:
return YES
In case that only an estimation for the number of zeros is needed, the following
statement might be useful.
Corollary 3.6. Let F be a system of n polynomials in K X1
±1,
. . . , Xn
±1
. If
Trop(F ) is finite and F is semiregular at any w Trop(F ), then the number of
solutions of F in
(K∗)n
is
|ZK (F )| =
w∈Trop(F )∩v(π)Zn
|Zk(inw(F )| |Trop(F )
v(π)Zn|
·
|k∗|n
|Trop(F )| ·
|k∗|n.
Note that when Trop(F ) is a finite set, then it has at most
n
i=1
(ti)
2
points,
where ti is the number of monomials of fi. Each Trop(fi) is contained in the union
of
(
ti
2
)
hyperplanes (see Lemma 2.1), and the intersection of n of these hyperplanes
(one in each Trop(fi)) determines at most one point in Trop(F ). In a
system F that satisfies the hypothesis of Corollary 3.6 has at most
(
t1
2
)particular,
· · ·
(
tn
2
)
|k∗|n
roots in (K∗)n, and all these roots are non-degenerate.
We conclude this section with a discussion of the univariate case. Consider
f =
∑t
i=1
aiXαi K[X]. In section 2, we showed that the tropical hypersurface of
f is the set of minus the slope of the segments of the lower hull of NP(f). For each
of these w Trop(f), the lower polynomial f
[w]
and initial form inw(f) are simply
the polynomials obtained by keeping only the terms with (αi, v(ai)) lying on the
segment of slope −w. For each w Trop(f), semiregularity at w means that either
w v(π)Z, in which case f has no solutions in
K∗
with valuation w, or inw(f)
has no degenerate zeros in
k∗.
In case of semiregularity at w Trop(f) v(π)Z,
our main result says that the number of roots of f in K∗ with valuation w and the
number of roots of inw(f) in k∗ coincide.
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