10
MART´
IN
AVENDA˜
NO AND ASHRAF IBRAHIM
By Item 4 of Lemma 2.3, we have that
inw(
˜)
F =
(δ(a1)Xα1
inw(f1), . . . ,
δ(an)Xαn
inw(fn)),
and in particular, inw(F ) and inw(
˜)
F have the same zeros in (k∗). Let x (k∗)n
be one of these zeros, which by assumption is a non-degenerate zero of inw(F ).
We have to show that x is also a non-degenerate zero of inw(
˜).
F The Jacobian of
inw(
˜)
F is given by the following expression.
Jac(inw(
˜))
F = det
δ(ai)αijXαi−ej
inw(fi) +
δ(ai)Xαi
∂inw(fi)
∂Xj
1≤i,j≤n
Evaluating at X = x we get Jac(inw(
˜))(x)
F = δ(a1 · · ·
an)xα1+...+αn
Jac(inw(F ))(x),
which is not zero in
k∗,
since x is a non-degenerate zero of inw(F ).
Lemma 3.4. Let F = (f1, . . . , fn) be a system of n polynomials in
K X1
±1
, . . . , Xn
±1
. Let w
Rn
and b = (b1, . . . , bn)
(K∗)n.
Then F is semireg-
ular at w if and only if the system with rescaled variables
˜
F = (f1(b1X1, . . . , bnXn),...,fn(b1X1,...,bnXn))
is semiregular at w v(b).
Proof. By Item 6 of Lemma 2.3, we have that Trop(
˜)
F = Trop(F ) v(b).
Since v(b)
v(π)Zn,
then w Trop(F )
v(π)Zn
if and only if w Trop(
˜)
F
v(π)Zn.
By the symmetry of the claim, it is enough to show that when w
Trop(F
)∩v(π)Zn
and inw(F ) has no degenerate zero in
(k∗)n,
then also inw−v(b)(
˜)
F
has no degenerate zero. By Item 8 of Lemma 2.3, we have that inw−v(b)(
˜)
F =
(inw(f1)(δ(b)X), . . ., inw(fn)(δ(b)X)), and in particular, if x (k∗)n is a zero of
inw−v(b)(
˜),
F then y = δ(b)x is a zero of inw(F ). A simple computation using the
chain rule shows that Jac(inw−v(b)(
˜))(x)
F = δ(b1 · · · bn)Jac(inw(F ))(y). Since the
right hand side does not vanish at any zero y of inw(F ), then the zeros of inw−v(b)(
˜)
F
are all non-degenerate.
Lemma 3.5. Let F be a system of n polynomials in K X1
±1,
. . . , Xn ±1 and let
w Trop(F ). Then F is semiregular (resp. normalized) at w if and only if F [w] is
semiregular (resp. normalized) at w.
Proof. The claim that F is normalized at w if and only F
[w]
is normalized at
w follows from the fact that tr(fi; w) =
tr(fi[w];
w) for all i = 1, . . . , n. The claim
about semiregularity is immediate from inw(F ) = inw(F
[w]).
At this point we have all the necessary ingredients for the main result of this
section, which is a reformulation of Hensel’s Lemma in the language of Defini-
tion 3.2. For pedagogical reasons, we start with the classical statement, and then,
we reformulate it progressively until we arrive to the final version in Corollary 3.5.
Lemma 3.6 (Hensel). Let F be a system of n polynomials in A X1
±1,
. . . , Xn
±1
and denote by F the system reduced modulo M. Let x
(k∗)n
be a solution of F
such that Jac(F )(x) = 0. Then there exists a unique solution x (A \ M)n of F
such that x = x mod M.
Proof. See [4, Prop. 2.11].
Lemma 3.7. Let F be a system of n polynomials in K X1
±1,
. . . , Xn ±1 such
that 0 Trop(F ). Assume also that F is normalized and semiregular at 0. Then
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