6

MART´

IN

AVENDA˜

NO AND ASHRAF IBRAHIM

Proof. Sort all the t monomials of f according to their valuation at x.

li1 (f; v(x)) ≤ li2 (f; v(x)) ≤ · · · ≤ lit (f; v(x))

Since the sum of all the monomials at x is zero, the first two valuations in this list

must coincide. We conclude from Lemma 2.1 that v(x) ∈ Trop(f).

Definition 2.3. Let f ∈ K X1

±1,

. . . , Xn ±1 be a polynomial with t non-zero

terms f =

∑t

i=1

aiXαi

and let w ∈

Rn.

We define the lower polynomial f

[w]

of f

with respect to the valuation vector w as

f

[w]

=

i : li(f;w)=tr(f;w)

aiXαi

∈ K X1

±1,

. . . , Xn

±1

.

We also define the initial form inw(f) of f with respect to w as

inw(f) =

i : li(f;w)=tr(f;w)

δ(ai)Xαi

∈ k X1

±1,

. . . , Xn

±1

.

Note that, according to Lemma 2.1, w ∈ Trop(f) if and only if inw(f) has

at least two terms. This can be taken as an alternative definition of the tropical

hypersurface. A key property of the initial forms is that if x ∈

(K∗)n

is a solution

of f with v(x) = w, then δ(x) ∈

(k∗)n

is a solution of inw(f), as shown in the

following lemma.

Lemma 2.2. Let f ∈ K X1

±1,

. . . , Xn

±1

, let w ∈

Rn,

let x ∈

(K∗)n

with

v(x) = w, and let 1 ≤ j ≤ n. Then:

(1)

π−tr(f;w)/v(π)f(x)

∈ A.

(2)

π−tr(f;w)/v(π)f(x)

≡ inw(f)(δ(x)) mod M.

(3)

π(wj −tr(f;w))/v(π)

∂f

∂Xj

(x) ∈ A.

(4) π(wj −tr(f;w))/v(π)

∂f

∂Xj

(x) ≡

∂inw(f)

∂Xj

(δ(x)) mod M.

Proof. Let f =

∑t

i=1

aiXαi

∈ K X1

±1,

. . . , Xn

±1

. The valuation of the i-

th term of f(x) is li(f; w) and the minimum of all these valuations is tr(f; w).

This proves that π−tr(f;w)/v(π)f(x) ∈ A. Moreover, if li(f; w) tr(f; w), then

the i-term of f(x) multiplied by

π−tr(f;w)/v(π)

reduces to zero modulo M, so

π−tr(f;w)/v(π)f(x)

≡

π−tr(f;w)/v(π)f [w](x)

mod M. Besides, all the terms in

π−tr(f;w)/v(π)f [w](x) have valuation zero, so reducing it modulo M is the same

as adding the first digit of each term. This proves that

π−tr(f;w)/v(π)f(x)

≡

inw(f)(δ(x)) mod M. The partial derivative of f with respect to Xj is ∂f/∂Xj =

∑t

i=1

aiαi,jXαi−ej

, where {e1,...,en} is the standard basis of

Rn.

The valuation of

the i-th term of ∂f/∂Xj (x) is li(f; w) − wj + v(αi,j ) and thus

π(wj −tr(f;w))/v(π)∂f/∂Xj (x) ∈ A. Finally, in the reduction of

π(wj −tr(f;w))/v(π)∂f/∂Xj (x) modulo M, all the terms with li(f; w) tr(f; w) − wj

dissapear, as well as the terms with v(αi,j ) 0. The remaining terms have all

valuation zero, and their first digits coincide with those of ∂inw(f)/∂Xj(δ(x)).

The following lemma shows that the notions of tropicalization, tropical

hypersurface, lower polynomial, and initial form, behave well under rescaling of

the variables and multiplication by monomials.

Lemma 2.3. Let f ∈ K X1

±1,

. . . , Xn ±1 , a ∈ K∗, b = (b1, . . . , bn) ∈ (K∗)n,

α ∈ Zn and w ∈ Rn.

6

MART´

IN

AVENDA˜

NO AND ASHRAF IBRAHIM

Proof. Sort all the t monomials of f according to their valuation at x.

li1 (f; v(x)) ≤ li2 (f; v(x)) ≤ · · · ≤ lit (f; v(x))

Since the sum of all the monomials at x is zero, the first two valuations in this list

must coincide. We conclude from Lemma 2.1 that v(x) ∈ Trop(f).

Definition 2.3. Let f ∈ K X1

±1,

. . . , Xn ±1 be a polynomial with t non-zero

terms f =

∑t

i=1

aiXαi

and let w ∈

Rn.

We define the lower polynomial f

[w]

of f

with respect to the valuation vector w as

f

[w]

=

i : li(f;w)=tr(f;w)

aiXαi

∈ K X1

±1,

. . . , Xn

±1

.

We also define the initial form inw(f) of f with respect to w as

inw(f) =

i : li(f;w)=tr(f;w)

δ(ai)Xαi

∈ k X1

±1,

. . . , Xn

±1

.

Note that, according to Lemma 2.1, w ∈ Trop(f) if and only if inw(f) has

at least two terms. This can be taken as an alternative definition of the tropical

hypersurface. A key property of the initial forms is that if x ∈

(K∗)n

is a solution

of f with v(x) = w, then δ(x) ∈

(k∗)n

is a solution of inw(f), as shown in the

following lemma.

Lemma 2.2. Let f ∈ K X1

±1,

. . . , Xn

±1

, let w ∈

Rn,

let x ∈

(K∗)n

with

v(x) = w, and let 1 ≤ j ≤ n. Then:

(1)

π−tr(f;w)/v(π)f(x)

∈ A.

(2)

π−tr(f;w)/v(π)f(x)

≡ inw(f)(δ(x)) mod M.

(3)

π(wj −tr(f;w))/v(π)

∂f

∂Xj

(x) ∈ A.

(4) π(wj −tr(f;w))/v(π)

∂f

∂Xj

(x) ≡

∂inw(f)

∂Xj

(δ(x)) mod M.

Proof. Let f =

∑t

i=1

aiXαi

∈ K X1

±1,

. . . , Xn

±1

. The valuation of the i-

th term of f(x) is li(f; w) and the minimum of all these valuations is tr(f; w).

This proves that π−tr(f;w)/v(π)f(x) ∈ A. Moreover, if li(f; w) tr(f; w), then

the i-term of f(x) multiplied by

π−tr(f;w)/v(π)

reduces to zero modulo M, so

π−tr(f;w)/v(π)f(x)

≡

π−tr(f;w)/v(π)f [w](x)

mod M. Besides, all the terms in

π−tr(f;w)/v(π)f [w](x) have valuation zero, so reducing it modulo M is the same

as adding the first digit of each term. This proves that

π−tr(f;w)/v(π)f(x)

≡

inw(f)(δ(x)) mod M. The partial derivative of f with respect to Xj is ∂f/∂Xj =

∑t

i=1

aiαi,jXαi−ej

, where {e1,...,en} is the standard basis of

Rn.

The valuation of

the i-th term of ∂f/∂Xj (x) is li(f; w) − wj + v(αi,j ) and thus

π(wj −tr(f;w))/v(π)∂f/∂Xj (x) ∈ A. Finally, in the reduction of

π(wj −tr(f;w))/v(π)∂f/∂Xj (x) modulo M, all the terms with li(f; w) tr(f; w) − wj

dissapear, as well as the terms with v(αi,j ) 0. The remaining terms have all

valuation zero, and their first digits coincide with those of ∂inw(f)/∂Xj(δ(x)).

The following lemma shows that the notions of tropicalization, tropical

hypersurface, lower polynomial, and initial form, behave well under rescaling of

the variables and multiplication by monomials.

Lemma 2.3. Let f ∈ K X1

±1,

. . . , Xn ±1 , a ∈ K∗, b = (b1, . . . , bn) ∈ (K∗)n,

α ∈ Zn and w ∈ Rn.

6