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.
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