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

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

17