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
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