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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