MULTIVARIATE ULTRAMETRIC ROOT COUNTING 9

The equations are given by non-zero polynomials in K X1

±1

, . . . , Xn

±1

and the

unknowns are in K∗. The system F will be written (f1, . . . , fn) in order to simplify

the notation. We define the tropical prevariety Trop(F ) induced by F as

Trop(F ) = Trop(f1) ∩ · · · ∩ Trop(fn).

For any w ∈ Trop(F ) we denote by F

[w]

and inw(F ) the systems of polyno-

mial equations given by the lower polynomials

f1w],...,fnw] [ [

and the initial forms

inw(f1), . . . , inw(fn) respectively.

By Proposition 2.2, any solution x ∈ (K∗)n of F satisfies v(x) ∈ Trop(F ).

Lemma 3.1. Let F be a system of n polynomials in K X1

±1,

. . . , Xn

±1

. If w

is an isolated point of Trop(F ), then Trop(F

[w])

= {w} and all the solutions x ∈

(K∗)n of F [w] have valuation vector v(x) = w.

Proof. By Lemma 2.5, the tropical prevarieties Trop(F ) and Trop(F

[w])

coin-

cide in a neighborhood of w. In particular, there exists ε 0 such that Trop(F [w])∩

Bε(w) = {w}. On the other hand, by Lemma 2.4, the tropical prevariety Trop(F

[w])

is a cone centered at w. This implies that Trop(F

[w])

= {w}. Therefore, by Propo-

sition 2.2, all the solutions x ∈ (K∗)n of F [w] have valuation vector v(x) = w.

Definition 3.2. Consider a system F = (f1, . . . , fn) of n polynomials in

K X1

±1

, . . . , Xn

±1

, and let w ∈

Rn.

We say that F is semiregular at w if either

w ∈ Trop(F ) ∩

v(π)Zn

or inw(F ) has no degenerate zero in

(k∗)n.

We say that F

is normalized at w if tr(f1; w) = · · · = tr(fn; w) = 0.

Lemma 3.2. Let F = (f1, . . . , fn) be a system of n polynomials in

K X1

±1,

. . . , Xn

±1

semiregular at w ∈

Rn.

Then, for each zero x ∈

(K∗)n

of F

with v(x) = w, we have

v(Jac(F )(x)) = tr(f1; w) + · · · + tr(fn; w) − (w1 + · · · + wn).

Proof. In the case w ∈ Trop(F ) ∩ v(π)Zn, there are no zeros of F with

valuation w, and there is nothing to prove. Therefore, we can assume without

loss of generality that w ∈ Trop(F ) ∩ v(π)Zn. Take a zero x ∈ (K∗)n of F with

valuation v(x) = w. By Lemma 2.2, the point δ(x) ∈ (k∗)n is a zero of imw(f), and

then, by the semiregularity of F at w, we have det

∂inw(fi)

∂Xj

δ(x)

= 0. Again by

Lemma 2.2, this means that det πwj −tr(fi)

∂fi

∂Xj

x

≡ 0 mod M, and by factoring

out the powers of π of the determinant, we conclude that v(Jac(F )(x)) = tr(f1; w)+

· · · + tr(fn; w) − (w1 + · · · + wn).

The following three lemmas show how semiregularity behaves with respect to

a rescaling of variables and multiplication by monomials.

Lemma 3.3. Let F = (f1, . . . , fn) be a system of n polynomials in

K X1

±1,

. . . , Xn ±1 . Let w ∈ Rn, a1,...,an ∈ K∗, and α1,...,αn ∈ Zn. Then

F is semiregular at w if and only if

˜

F =

(a1Xα1 f1,...,anXαn

fn) is semiregular

at w.

Proof. By Item 2 of Lemma 2.3, we have that Trop(F ) = Trop(

˜),

F and since

the claim is symmetric, it is enough to prove that when w ∈ Trop(F ) ∩

v(π)Zn

and

inw(F ) has no degenerate zero in (k∗)n then also inw(

˜)

F has no degenerate zero.

9

The equations are given by non-zero polynomials in K X1

±1

, . . . , Xn

±1

and the

unknowns are in K∗. The system F will be written (f1, . . . , fn) in order to simplify

the notation. We define the tropical prevariety Trop(F ) induced by F as

Trop(F ) = Trop(f1) ∩ · · · ∩ Trop(fn).

For any w ∈ Trop(F ) we denote by F

[w]

and inw(F ) the systems of polyno-

mial equations given by the lower polynomials

f1w],...,fnw] [ [

and the initial forms

inw(f1), . . . , inw(fn) respectively.

By Proposition 2.2, any solution x ∈ (K∗)n of F satisfies v(x) ∈ Trop(F ).

Lemma 3.1. Let F be a system of n polynomials in K X1

±1,

. . . , Xn

±1

. If w

is an isolated point of Trop(F ), then Trop(F

[w])

= {w} and all the solutions x ∈

(K∗)n of F [w] have valuation vector v(x) = w.

Proof. By Lemma 2.5, the tropical prevarieties Trop(F ) and Trop(F

[w])

coin-

cide in a neighborhood of w. In particular, there exists ε 0 such that Trop(F [w])∩

Bε(w) = {w}. On the other hand, by Lemma 2.4, the tropical prevariety Trop(F

[w])

is a cone centered at w. This implies that Trop(F

[w])

= {w}. Therefore, by Propo-

sition 2.2, all the solutions x ∈ (K∗)n of F [w] have valuation vector v(x) = w.

Definition 3.2. Consider a system F = (f1, . . . , fn) of n polynomials in

K X1

±1

, . . . , Xn

±1

, and let w ∈

Rn.

We say that F is semiregular at w if either

w ∈ Trop(F ) ∩

v(π)Zn

or inw(F ) has no degenerate zero in

(k∗)n.

We say that F

is normalized at w if tr(f1; w) = · · · = tr(fn; w) = 0.

Lemma 3.2. Let F = (f1, . . . , fn) be a system of n polynomials in

K X1

±1,

. . . , Xn

±1

semiregular at w ∈

Rn.

Then, for each zero x ∈

(K∗)n

of F

with v(x) = w, we have

v(Jac(F )(x)) = tr(f1; w) + · · · + tr(fn; w) − (w1 + · · · + wn).

Proof. In the case w ∈ Trop(F ) ∩ v(π)Zn, there are no zeros of F with

valuation w, and there is nothing to prove. Therefore, we can assume without

loss of generality that w ∈ Trop(F ) ∩ v(π)Zn. Take a zero x ∈ (K∗)n of F with

valuation v(x) = w. By Lemma 2.2, the point δ(x) ∈ (k∗)n is a zero of imw(f), and

then, by the semiregularity of F at w, we have det

∂inw(fi)

∂Xj

δ(x)

= 0. Again by

Lemma 2.2, this means that det πwj −tr(fi)

∂fi

∂Xj

x

≡ 0 mod M, and by factoring

out the powers of π of the determinant, we conclude that v(Jac(F )(x)) = tr(f1; w)+

· · · + tr(fn; w) − (w1 + · · · + wn).

The following three lemmas show how semiregularity behaves with respect to

a rescaling of variables and multiplication by monomials.

Lemma 3.3. Let F = (f1, . . . , fn) be a system of n polynomials in

K X1

±1,

. . . , Xn ±1 . Let w ∈ Rn, a1,...,an ∈ K∗, and α1,...,αn ∈ Zn. Then

F is semiregular at w if and only if

˜

F =

(a1Xα1 f1,...,anXαn

fn) is semiregular

at w.

Proof. By Item 2 of Lemma 2.3, we have that Trop(F ) = Trop(

˜),

F and since

the claim is symmetric, it is enough to prove that when w ∈ Trop(F ) ∩

v(π)Zn

and

inw(F ) has no degenerate zero in (k∗)n then also inw(

˜)

F has no degenerate zero.

9