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
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