MULTIVARIATE ULTRAMETRIC ROOT COUNTING 13

4. Regularity.

Definition 4.1. A system F of n polynomials in K X1

±1,

. . . , Xn ±1 is regular

if Trop(F ) is finite, F

[w]

consists solely of binomials and F is semiregular at w for

all w ∈ Trop(F ).

For this kind of system, we can provide an explicit formula for the number of

roots in

(K∗)n.

We will also give a different characterization of regularity that is

easier to check. First of all, the notion of regularity is well-behaved under monomial

changes of variables.

Lemma 4.1. Let F = (f1, . . . , fn) be a system of polynomials in K X1

±1,

. . . , Xn

±1

.

Let a1,...,an ∈

K∗,

b1,...,bn ∈

K∗,

and α1,...,αn ∈

Zn.

The following three

statements are equivalent.

(1) F is regular.

(2)

(a1Xα1 f1,...,anXαn

fn) is regular.

(3) (f1(b1X1, . . . , bnXn),...,fn(b1X1,...,bnXn)) is regular.

Proof. A consequence of Lemmas 2.3, 3.3 and 3.4.

The problem of deciding whether a system is regular or not can be reduced to

the case of binomial systems: in Definition 4.1, the condition F is semiregular at w

can be replaced, according to Lemma 3.5, by the condition F [w] is semiregular at

w. The following lemma and proposition characterize semiregularity for binomial

systems.

Lemma 4.2. Consider a binomial system

B = (a1Xα1 − b1Xβ1 , . . . , anXαn − bnXβn )

with coeﬃcients a = (a1, . . . , an) ∈ (K∗)n, let b = (b1, . . . , bn) ∈ (K∗)n, and let

M ∈ Zn×n be the matrix whose i-th row is αi − βi for i = 1, . . . , n. Then

Trop(B) = {w ∈

Rn

: Mw = v(b) − v(a)}.

In particular, Trop(B) is finite (and non-empty) if and only if det(M) = 0.

Proof. By Lemma 2.1, the tropical hypersurface of the i-th binomial is

Trop(aiXαi − biXβi ) = {w ∈ Rn : v(ai) + αi · w = v(bi) + βi · w}. This equation

corresponds with the i-th row of Mw = v(b) − v(a).

For any vector x = (x1, . . . , xn) with non-zero entries and any matrix M =

(mij)1≤i,j≤n ∈

Zn×n,

we write

xM

= (x1

m11

· · · xn

m1n

, . . . , x1

mn1

· · · xn

mnn

).

Note that if P, Q ∈

Zn×n,

then

xPQ

=

(xQ)P

.

Proposition 4.2. Consider the binomial system

B =

(a1Xα1

−

b1Xβ1

, . . . ,

anXαn

−

bnXβn

)

in K X1

±1,

. . . , Xn

±1

. Let a = (a1, . . . , an) and b = (b1, . . . , bn). Assume that

the matrix M ∈ Zn×n, whose i-th row is αi − βi for i = 1, . . . , n, has non-zero

determinant. Let M = PDQ be the Smith Normal Form of M, i.e. P, Q ∈ Zn×n

are invertible and D = diag(d1, . . . , dn) with d1 | d2 | · · · | dn positive integers. Then

B is semiregular at w = M −1(v(b) − v(a)) if and only if either:

(1) w ∈ v(π)Zn.

(2) char(k) det(M).

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4. Regularity.

Definition 4.1. A system F of n polynomials in K X1

±1,

. . . , Xn ±1 is regular

if Trop(F ) is finite, F

[w]

consists solely of binomials and F is semiregular at w for

all w ∈ Trop(F ).

For this kind of system, we can provide an explicit formula for the number of

roots in

(K∗)n.

We will also give a different characterization of regularity that is

easier to check. First of all, the notion of regularity is well-behaved under monomial

changes of variables.

Lemma 4.1. Let F = (f1, . . . , fn) be a system of polynomials in K X1

±1,

. . . , Xn

±1

.

Let a1,...,an ∈

K∗,

b1,...,bn ∈

K∗,

and α1,...,αn ∈

Zn.

The following three

statements are equivalent.

(1) F is regular.

(2)

(a1Xα1 f1,...,anXαn

fn) is regular.

(3) (f1(b1X1, . . . , bnXn),...,fn(b1X1,...,bnXn)) is regular.

Proof. A consequence of Lemmas 2.3, 3.3 and 3.4.

The problem of deciding whether a system is regular or not can be reduced to

the case of binomial systems: in Definition 4.1, the condition F is semiregular at w

can be replaced, according to Lemma 3.5, by the condition F [w] is semiregular at

w. The following lemma and proposition characterize semiregularity for binomial

systems.

Lemma 4.2. Consider a binomial system

B = (a1Xα1 − b1Xβ1 , . . . , anXαn − bnXβn )

with coeﬃcients a = (a1, . . . , an) ∈ (K∗)n, let b = (b1, . . . , bn) ∈ (K∗)n, and let

M ∈ Zn×n be the matrix whose i-th row is αi − βi for i = 1, . . . , n. Then

Trop(B) = {w ∈

Rn

: Mw = v(b) − v(a)}.

In particular, Trop(B) is finite (and non-empty) if and only if det(M) = 0.

Proof. By Lemma 2.1, the tropical hypersurface of the i-th binomial is

Trop(aiXαi − biXβi ) = {w ∈ Rn : v(ai) + αi · w = v(bi) + βi · w}. This equation

corresponds with the i-th row of Mw = v(b) − v(a).

For any vector x = (x1, . . . , xn) with non-zero entries and any matrix M =

(mij)1≤i,j≤n ∈

Zn×n,

we write

xM

= (x1

m11

· · · xn

m1n

, . . . , x1

mn1

· · · xn

mnn

).

Note that if P, Q ∈

Zn×n,

then

xPQ

=

(xQ)P

.

Proposition 4.2. Consider the binomial system

B =

(a1Xα1

−

b1Xβ1

, . . . ,

anXαn

−

bnXβn

)

in K X1

±1,

. . . , Xn

±1

. Let a = (a1, . . . , an) and b = (b1, . . . , bn). Assume that

the matrix M ∈ Zn×n, whose i-th row is αi − βi for i = 1, . . . , n, has non-zero

determinant. Let M = PDQ be the Smith Normal Form of M, i.e. P, Q ∈ Zn×n

are invertible and D = diag(d1, . . . , dn) with d1 | d2 | · · · | dn positive integers. Then

B is semiregular at w = M −1(v(b) − v(a)) if and only if either:

(1) w ∈ v(π)Zn.

(2) char(k) det(M).

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