8

MART´

IN

AVENDA˜

NO AND ASHRAF IBRAHIM

Take w ∈ Trop(f) ∩ Bε(w). By Lemma 2.1, there are indices 1 ≤ i j ≤ t such

that

li(f; w ) = lj(f; w ) ≤ lk(f; w ) for all k = 1, . . . , t. By the inequalities (2.1),

we conclude that i, j ∈ I. Therefore, by Lemma 2.1, w ∈ Trop(f [w]).

Now take w ∈ Trop(f

[w])

∩ Bε(w). By Lemma 2.1 we have two different

indices i, j ∈ I such that li(f; w ) = lj(f; w ) ≤ lk(f; w ) for all k ∈ I. By (2.1), this

inequality holds also for k ∈ I. This means, by Lemma 2.1, that w ∈ Trop(f).

Lemma 2.1 gives a simple procedure to compute tropical hypersurfaces that

requiere to solve systems of linear equations and inequalities. The following is a

simple geometric interpretation of that using polyhedra.

Definition 2.4. Let f =

∑t

i=1

aiXαi

∈ K X1

±1,

. . . , Xn

±1

. The Newton

Polytope of f, denoted NP(f), is the convex hull of the set

{(αi,v(ai)) : i = 1, . . . , t} ⊆

Rn+1.

A hyperplane H ⊆

Rn+1,

not parallel to the line x1 = · · · = xn = 0, is a supporting

hyperplane of the Newton Polytope of f if NP(f) is included in the upper half-

space2

determined by H and NP(f) ∩ H = ∅.

Lemma 2.6. Let f ∈ K X1

±1

, . . . , Xn

±1

. Then Trop(f) is the set of all w ∈

Rn

such that (w, 1) ∈ Rn+1 is the normal vector of a supporting hyperplane H of NP(f)

with |H ∩ NP(f)| 1.

Proof. Write f =

∑t

i=1

aiXαi . (⊆) Take w ∈ Trop(f). By Lemma 2.1, there

are two indices 1 ≤ i j ≤ t such that li(f; w) = lj(f; w) ≤ lk(f; w) for all

k = 1, . . . , t. This is equivalent to say that the hyperplane

H = {x ∈

Rn+1

: (w, 1) · x = tr(f; w)},

with normal vector (w, 1), contains the points (αi, v(ai)) and (αj, v(aj)), and the

upper half-space

H+

determined by H contains all the points (αk, v(ak)). Since

H+

is convex, then NP(f) ⊆

H+.

(⊇) Now assume that H is a supporting hyperplane

with normal vector (w, 1) that contains at least two points of the Newton Polytope

of f. Since NP(f) is a polyhedron, then H contains at least two vertices (αi, v(ai))

and (αj, v(aj)). The remaining vertices are contained in the upper half-space de-

termined by H. This means that αi · w + v(ai) = αj · w + v(aj) ≤ αk · w + v(ak)

for all k = 1, . . . , t, and by Lemma 2.1, that w ∈ Trop(f).

In the case of an univariate polynomial f ∈ K[X], Lemma 2.6 says that Trop(f)

is the set of minus the slope of the segments of the lower hull of NP(f).

3. Semiregular systems of polynomial equations.

Definition 3.1. Consider a system F of n equations in n variables.

F =

⎧

⎪

⎨

⎪

⎩

f1(X1,...,Xn) = 0

.

.

.

fn(X1,...,Xn) = 0

2Up

and down is understood with respect to the variable xn+1. The upper half-space of H

is well-defined since H is not parallel to the vertical axis.

8

MART´

IN

AVENDA˜

NO AND ASHRAF IBRAHIM

Take w ∈ Trop(f) ∩ Bε(w). By Lemma 2.1, there are indices 1 ≤ i j ≤ t such

that

li(f; w ) = lj(f; w ) ≤ lk(f; w ) for all k = 1, . . . , t. By the inequalities (2.1),

we conclude that i, j ∈ I. Therefore, by Lemma 2.1, w ∈ Trop(f [w]).

Now take w ∈ Trop(f

[w])

∩ Bε(w). By Lemma 2.1 we have two different

indices i, j ∈ I such that li(f; w ) = lj(f; w ) ≤ lk(f; w ) for all k ∈ I. By (2.1), this

inequality holds also for k ∈ I. This means, by Lemma 2.1, that w ∈ Trop(f).

Lemma 2.1 gives a simple procedure to compute tropical hypersurfaces that

requiere to solve systems of linear equations and inequalities. The following is a

simple geometric interpretation of that using polyhedra.

Definition 2.4. Let f =

∑t

i=1

aiXαi

∈ K X1

±1,

. . . , Xn

±1

. The Newton

Polytope of f, denoted NP(f), is the convex hull of the set

{(αi,v(ai)) : i = 1, . . . , t} ⊆

Rn+1.

A hyperplane H ⊆

Rn+1,

not parallel to the line x1 = · · · = xn = 0, is a supporting

hyperplane of the Newton Polytope of f if NP(f) is included in the upper half-

space2

determined by H and NP(f) ∩ H = ∅.

Lemma 2.6. Let f ∈ K X1

±1

, . . . , Xn

±1

. Then Trop(f) is the set of all w ∈

Rn

such that (w, 1) ∈ Rn+1 is the normal vector of a supporting hyperplane H of NP(f)

with |H ∩ NP(f)| 1.

Proof. Write f =

∑t

i=1

aiXαi . (⊆) Take w ∈ Trop(f). By Lemma 2.1, there

are two indices 1 ≤ i j ≤ t such that li(f; w) = lj(f; w) ≤ lk(f; w) for all

k = 1, . . . , t. This is equivalent to say that the hyperplane

H = {x ∈

Rn+1

: (w, 1) · x = tr(f; w)},

with normal vector (w, 1), contains the points (αi, v(ai)) and (αj, v(aj)), and the

upper half-space

H+

determined by H contains all the points (αk, v(ak)). Since

H+

is convex, then NP(f) ⊆

H+.

(⊇) Now assume that H is a supporting hyperplane

with normal vector (w, 1) that contains at least two points of the Newton Polytope

of f. Since NP(f) is a polyhedron, then H contains at least two vertices (αi, v(ai))

and (αj, v(aj)). The remaining vertices are contained in the upper half-space de-

termined by H. This means that αi · w + v(ai) = αj · w + v(aj) ≤ αk · w + v(ak)

for all k = 1, . . . , t, and by Lemma 2.1, that w ∈ Trop(f).

In the case of an univariate polynomial f ∈ K[X], Lemma 2.6 says that Trop(f)

is the set of minus the slope of the segments of the lower hull of NP(f).

3. Semiregular systems of polynomial equations.

Definition 3.1. Consider a system F of n equations in n variables.

F =

⎧

⎪

⎨

⎪

⎩

f1(X1,...,Xn) = 0

.

.

.

fn(X1,...,Xn) = 0

2Up

and down is understood with respect to the variable xn+1. The upper half-space of H

is well-defined since H is not parallel to the vertical axis.

8