14
MART´
IN
AVENDA˜
NO AND ASHRAF IBRAHIM
(3) the i-th entry of (δ(b1/a1), . . .,
δ(bn/an))P
−1
is not a di-th power in
k∗
for
some i = 1, . . . , n.
In this case, if (1) and (3) do not hold, then the number of solutions of the system
B in
(K∗)n
is |ZK (B)| =
n
i=1
|{ξ
k∗
:
ξdi
= 1}|. Otherwise B has no solutions
in
(K∗)n.
Proof. By Lemma 4.1, we have w Trop(B). In case that w v(π)Zn, then
B is semiregular at w by definition, B has no solutions in
(K∗)n
since there are no
elements in
(K∗)n
with valuation w, and the proposition is proven. Now assume
that w
v(π)Zn.
By Lemma 3.3, the system B is semiregular at w if and only if
the system
XM
= b/a is semiregular at w. The initial form system is
XM
= δ(b/a).
Any solution x (K∗)n of this system satisfies (xQ)D = (δ(b/a))P
−1
and then the
condition of item 3 is not met. In other words, if the system satisfies the third con-
dition, then the initial form system (and also B) has no solution, B is automatically
semiregular at w, and the proposition in proven. So we can assume without loss of
generality that B does not satisfy items 1 and 3. In this case, there exist y (k∗)n
such that
yD
=
(δ(b/a))P
−1
, and then x =
yQ−1

(k∗)n
is a zero of
XM
= δ(b/a).
The Jacobian of this system is J = det([mijX1
mi1
· · · Xj
mij −1
· · · Xn
min
]1≤i,j≤n),
which, after factoring out Xj
−1
from the j-th column, and then X1
mi1
· · · Xn
min
from
the i-th row, becomes a single term with coefficient det(M). In particular, a solu-
tion x (k∗)n of XM = δ(b/a) is non-degenerate if and only if char(k) det(M).
This shows the equivalence between semiregularity of B at w and item 2. Finally,
the number of solutions of XM = δ(b/a) is equal to the number of solutions of
Y
D
=
(δ(b/a))P
−1
, since the map x
xQ
is a bijection. We know already that
there is a solution y
(k∗)n,
and it is clear that all other solution can be obtained
by multiplying the i-th entry of y by a di-th root of unity in
k∗.
This proves the
formula for the number of zeros of B.
A system of polynomials F is regular if and only if Trop(F ) is finite and F
[w]
is a binomial system that satisfies the assumptions of Proposition 4.2 for all w
Trop(F ). In this case, an explicit formula for the number of roots of F in
(K∗)n
can be obtained from Corollary 3.6 and Proposition 4.2. The following algorithm
summarizes this procedure.
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