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
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