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.

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.

14