MULTIVARIATE ULTRAMETRIC ROOT COUNTING 17
5. Tropical genericity of regular systems
Definition 5.1. Consider a proposition P : (K∗)n {True, False}. We say
that P is true for any generic x (K∗)n if and only if P −1(False) is contained
in an algebraic hypersurface of (K∗)n. Similarly, a proposition P : (K∗)n
{True, False} is said to be true for any tropically generic x (K∗)n if and only if
v(P −1(False)) is contained in a finite union of hyperplanes of Rn.
Note that genericity implies tropical genericity: if a statement P is true for
generic x
(K∗)n,
then there is a hypersurface ZK (G)
(K∗)n
that contains
P
−1(False),
and therefore, the tropical hypersurface Trop(G), which is contained
in a finite union of hyperplanes of
Rn,
contains v(P
−1(False)).
Let A1,..., An
Zn
be nonempty finite sets. Consider a system of polynomials
F = (f1, . . . , fn) in K X1
±1,
. . . , Xn
±1
with undetermined (non-zero) coefficients
and Supp(fi) = Ai for all i = 1, . . . , n. Let N = |A1| + · · · + |An| be the number
of coefficients in F . Once these supports have been fixed, we can speak about
propositions for generic or tropically generic systems F in the sense of Definition 5.1:
the domain of the propositions is understood to be the coefficient space
(K∗)N
of
the systems.
Theorem 5.2. Any tropically generic system F = (f1, . . . , fn) in
K X1
±1,
. . . , Xn ±1 has finite tropical prevariety Trop(F ) and its lower polynomials
fi[w]
are binomials for all w Trop(F ) and i = 1, . . . , n.
Proof. Write fi =

α∈Ai
aα)Xα (i
for i = 1, . . . , n. Assume first that Trop(F )
is an infinite set. We will show that the vector μ =
v(aα))1≤i≤n, (i
α∈Ai
RN lies on
a finite union of hyperplanes H
RN
that depends only on the sets A1,..., An.
By Lemma 3.3, there are αi,βi Ai for i = 1, . . . , n such that the system of linear
equations
v(aα1
(1))
+ α1 · w = v(aβ1
(1)
) + β1 · w
.
.
.
.
.
. (5.1)
v(aαn)) (n
+ αn · w =
v(aβn)) (
n
+ βn · w
has infinitely many solutions w
Rn.
This means that the determinant of the
matrix whose rows are αi βi for i = 1, . . . , n is zero and that
v(aα1
(1))
v(aβ1
(1)
), . . . ,
v(aαn)) (n

v(aβn)) (
n
αi βi : i = 1, . . . , n
Since the vectors αi βi for i = 1, . . . , n are R-linearly dependent (the determinant
of the matrix is zero), the subspace at the right side of the condition above has
codimension one (or more) in Rn. This translates into a condition that says that
μ belongs to some hyperplane of RN that depends only on α1,β1,...,αn,βn. We
conclude by taking H as the union of these hyperplanes for all possible choice of
α1,β1 A1,...,αn,βn An such that {αi βi : i = 1, . . . , n} is a R-linearly
dependent set.
Now assume that μ H, and in particular Trop(F ) is finite, but
fi[w]
has
three or more terms for some i = 1, . . . , n and w Trop(F ). We will show that
there is a finite union of hyperplanes H RN , that depends only on A1,..., An,
such that μ H . It is enough to consider the case where the polynomial with
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