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

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

10