MULTIVARIATE ULTRAMETRIC ROOT COUNTING 7

(1)

tr(aXαf;

w) = tr(f; w) + v(a) + α · w.

(2)

Trop(aXαf) = Trop(f).

(3)

(aXαf)[w]

=

aXαf [w].

(4)

inw(aXαf)

=

δ(a)Xαinw(f).

(5) tr(f(b1X1, . . . , bnXn); w) = tr(f; w + v(b)).

(6) Trop(f(b1X1, . . . , bnXn)) = Trop(f) − (v(b1), . . . , v(bn)).

(7)

f(b1X1,...,bnXn)[w]

= f

[w+v(b)](b1X1,...,bnXn).

(8) inw(f(b1X1, . . . , bnXn)) = inw+v(b)(f)(δ(b1)X1, . . . , δ(bn)Xn).

Proof. Items 1 and 5 follow immediately from the identities

li(aXαf;

w) =

li(f; w)+ v(a)+ α · w and li(f(b1X1,...,bnXn); w) = li(f; w + v(b)). Items 2 and 6

are consequences of the previous two and the definition of tropical hypersurface.

The indices of the monomials of f that are in

(aXαf)[w]

correspond with the in-

dices that minimize the value of

li(aXαf;

w). Since v(a)+ α · w is a constant, these

indices also minimize

li(f; w), i.e. they correspond with the monomials of f in

f

[w].

Therefore

(aXαf)[w]

=

aXαf [w].

Similarly, the indices of the terms of f in

f(b1X1,...,bnXn)[w] minimize the expression li(f(b1X1,...,bnXn); w), and there-

fore, coincide with the same indices of the monomials in f

[w+v(b)](b1X1,...,bnXn).

This proves that

f(b1X1,...,bnXn)[w]

= f

[w+v(b)](b1X1,...,bnXn).

Finally, items 4

and 8 follow from 3 and 7 by taking the first digit of all the terms.

In the next two lemmas, we show the relation between Trop(f) and Trop(f

[w])

for any w ∈

Rn.

It is clear that if w ∈ Trop(f), then f

[w]

is a single mono-

mial, and therefore Trop(f [w]) = ∅. Otherwise, when w ∈ Trop(f), we have

w ∈ Trop(f

[w])

and tr(f; w) = tr(f

[w];

w). We will prove next that the tropical

hypersurface Trop(f

[w])

is a cone centered at w, that coincides with Trop(f) in a

neighborhood of w. This completely characterizes Trop(f [w]) in terms of Trop(f).

Lemma 2.4. Let f ∈ K X1

±1,

. . . , Xn ±1 and let w ∈ Trop(f). Then, for any

w ∈ Trop(f

[w]),

the ray w + λ(w − w) with λ ≥ 0 is contained in Trop(f

[w]).

Proof. Let t be the number of terms of f. Write the lower polynomial of f at w

as f [w] = ai1 Xαi1 +· · ·+air Xαir where 1 ≤ i1 i2 · · · ir ≤ t are all the indices

that minimize the linear functions li(f; w). The s-th term of f

[w]

is the is-th term of

f. In particular, we have that ls(f

[w];

w) = lis (f; w) = tr(f; w) for all s = 1, . . . , r.

Since w ∈ Trop(f [w]) we have, by Lemma 2.1, two indices 1 ≤ n m ≤ r such that

ln(f

[w];

w ) = lm(f

[w];

w ) ≤ ls(f

[w];

w ) for all s = 1, . . . , r. Subtracting tr(f; w),

multiplying by λ ≥ 0 and then adding tr(f; w) to these (in)equalities we get

ln(f

[w]

; w + λ(w − w)) = lm(f

[w]

; w + λ(w − w)) ≤ ls(f

[w]

; w + λ(w − w))

for all s = 1, . . . , r. This implies, by Lemma 2.1, that w+λ(w −w) is in Trop(f [w]).

Lemma 2.5. Let f ∈ K X1

±1,

. . . , Xn

±1

and let w ∈ Trop(f). Then there exists

ε 0 such that Trop(f) ∩ Bε(w) = Trop(f [w]) ∩ Bε(w).

Proof. Let t be the number of terms of f. Let I = {1 ≤ i ≤ t : li(f; w) =

tr(f; w)} be the set of indices of the monomials of f in f [w]. Note that li(f; w)

lk(f; w) for all i ∈ I and k ∈ I. Since li(f; ·) : Rn → R are continuous functions,

there exists ε 0 such that

(2.1) li(f; w ) lk(f; w ) ∀ w ∈ Bε(w), ∀ i ∈ I, ∀ k ∈ I.

7

(1)

tr(aXαf;

w) = tr(f; w) + v(a) + α · w.

(2)

Trop(aXαf) = Trop(f).

(3)

(aXαf)[w]

=

aXαf [w].

(4)

inw(aXαf)

=

δ(a)Xαinw(f).

(5) tr(f(b1X1, . . . , bnXn); w) = tr(f; w + v(b)).

(6) Trop(f(b1X1, . . . , bnXn)) = Trop(f) − (v(b1), . . . , v(bn)).

(7)

f(b1X1,...,bnXn)[w]

= f

[w+v(b)](b1X1,...,bnXn).

(8) inw(f(b1X1, . . . , bnXn)) = inw+v(b)(f)(δ(b1)X1, . . . , δ(bn)Xn).

Proof. Items 1 and 5 follow immediately from the identities

li(aXαf;

w) =

li(f; w)+ v(a)+ α · w and li(f(b1X1,...,bnXn); w) = li(f; w + v(b)). Items 2 and 6

are consequences of the previous two and the definition of tropical hypersurface.

The indices of the monomials of f that are in

(aXαf)[w]

correspond with the in-

dices that minimize the value of

li(aXαf;

w). Since v(a)+ α · w is a constant, these

indices also minimize

li(f; w), i.e. they correspond with the monomials of f in

f

[w].

Therefore

(aXαf)[w]

=

aXαf [w].

Similarly, the indices of the terms of f in

f(b1X1,...,bnXn)[w] minimize the expression li(f(b1X1,...,bnXn); w), and there-

fore, coincide with the same indices of the monomials in f

[w+v(b)](b1X1,...,bnXn).

This proves that

f(b1X1,...,bnXn)[w]

= f

[w+v(b)](b1X1,...,bnXn).

Finally, items 4

and 8 follow from 3 and 7 by taking the first digit of all the terms.

In the next two lemmas, we show the relation between Trop(f) and Trop(f

[w])

for any w ∈

Rn.

It is clear that if w ∈ Trop(f), then f

[w]

is a single mono-

mial, and therefore Trop(f [w]) = ∅. Otherwise, when w ∈ Trop(f), we have

w ∈ Trop(f

[w])

and tr(f; w) = tr(f

[w];

w). We will prove next that the tropical

hypersurface Trop(f

[w])

is a cone centered at w, that coincides with Trop(f) in a

neighborhood of w. This completely characterizes Trop(f [w]) in terms of Trop(f).

Lemma 2.4. Let f ∈ K X1

±1,

. . . , Xn ±1 and let w ∈ Trop(f). Then, for any

w ∈ Trop(f

[w]),

the ray w + λ(w − w) with λ ≥ 0 is contained in Trop(f

[w]).

Proof. Let t be the number of terms of f. Write the lower polynomial of f at w

as f [w] = ai1 Xαi1 +· · ·+air Xαir where 1 ≤ i1 i2 · · · ir ≤ t are all the indices

that minimize the linear functions li(f; w). The s-th term of f

[w]

is the is-th term of

f. In particular, we have that ls(f

[w];

w) = lis (f; w) = tr(f; w) for all s = 1, . . . , r.

Since w ∈ Trop(f [w]) we have, by Lemma 2.1, two indices 1 ≤ n m ≤ r such that

ln(f

[w];

w ) = lm(f

[w];

w ) ≤ ls(f

[w];

w ) for all s = 1, . . . , r. Subtracting tr(f; w),

multiplying by λ ≥ 0 and then adding tr(f; w) to these (in)equalities we get

ln(f

[w]

; w + λ(w − w)) = lm(f

[w]

; w + λ(w − w)) ≤ ls(f

[w]

; w + λ(w − w))

for all s = 1, . . . , r. This implies, by Lemma 2.1, that w+λ(w −w) is in Trop(f [w]).

Lemma 2.5. Let f ∈ K X1

±1,

. . . , Xn

±1

and let w ∈ Trop(f). Then there exists

ε 0 such that Trop(f) ∩ Bε(w) = Trop(f [w]) ∩ Bε(w).

Proof. Let t be the number of terms of f. Let I = {1 ≤ i ≤ t : li(f; w) =

tr(f; w)} be the set of indices of the monomials of f in f [w]. Note that li(f; w)

lk(f; w) for all i ∈ I and k ∈ I. Since li(f; ·) : Rn → R are continuous functions,

there exists ε 0 such that

(2.1) li(f; w ) lk(f; w ) ∀ w ∈ Bε(w), ∀ i ∈ I, ∀ k ∈ I.

7