MULTIVARIATE ULTRAMETRIC ROOT COUNTING 11
all the coefficients of F are in the valuation ring A. Moreover, the reduction map
mod M :
An kn induces a bijection between the set of zeros of F in (K∗)n with
valuation vector 0 (i.e. in (A \
M)n)
and the set of zeros of F in
(k∗)n.
Proof. Suppose that F = (f1, . . . , fn). Since the system is normalized at 0,
we have tr(fi; 0) = 0 for all i = 1, . . . , n. Since tr(fi; 0) is the minimum valuation
of the coefficients of fi, then all the coefficents of fi have valuation at least 0, i.e.
fi A X1
±1,
. . . , Xn
±1
. Moreover, the terms of fi that are kept in in0(fi) are those
with coefficients in A\M. For these terms, reducing modulo M or taking first digit
is exaclty the same, so fi = in0(fi). In particular, we have that F = in0(F ) has
no degenerate solutions in (k∗)n. It is clear that the reduction modulo M maps
zeros of F in
(K∗)n
with valuation 0 to zeros of F in
(k∗)n.
We only have to show
that the map is a bijection. For the surjectivity, take a zero of F in
(k∗)n.
The
semiregularity at 0 guarantees that it is non-degenerate zero, and Lemma 3.6 shows
that it can be lifted to a zero of F in (A \
M)n,
i.e. to a zero of F with valuation 0.
The injectivity follows from the uniqueness of the lifting in Hensel’s Lemma.
Definition 3.3. For any system of polynomials F in K X1
±1,
. . . , Xn ±1 , the
set of roots of F in (K∗)n is denoted by ZK (F ), and the set of zeros of F with
valuation w is written ZK w (F ).
Theorem 3.4. Let F be a system of n polynomials in K X1
±1,
. . . , Xn ±1 . Let
w Trop(F )∩v(π)Zn and suppose that F is semiregular at w. The first digit maps
δ : ZK
w
(F ) Zk(inw(F )) and δ : ZK
w
(F
[w])
Zk(inw(F )) are bijections (and are
well-defined between these sets of roots).
Proof. The case w = 0 and F normalized at 0 follows immediately from
Lemmas 3.7 and 3.5 and the fact that the reductions of F and F
[0]
modulo M
coincide with in0(F ). Note that the assumption that F is normalized at 0 can
be easily removed by pre-multiplying each equation in F by a suitable constant
in
K∗.
We can also reduce the general case to w = 0 by a simple change of vari-
ables. Define
ˆ
F = F (πw1/v(π)X1, . . . , πwn/v(π)Xn). By Lemma 3.4, the system
ˆ
F
is semiregular at 0. It is clear that the first digit preserving map (x1, . . . , xn)
(πw1/v(π)x1,
. . . ,
πwn/v(π)xn)
is a bijection between the set of solutions of
ˆ
F with
valuation vector 0 and the zeros of F with valuation w. Moreover, by Item 8 of
Lemma 2.3, we have inw(F ) = in0(
ˆ),
F and by Item 7 we have
F [w](πw1/v(π)X1,...,πwn/v(π)Xn) =
ˆ[0].
F This provides the reduction to the case
w = 0.
Although the previous result contains all the substance of this section, the
following corollary is the way Theorem 3.4 is intended to be used in practice.
Corollary 3.5. Let F be a system of n polynomials in K X1
±1,
. . . , Xn ±1 .
Assume that F is semiregular at w. Then there is a unique bijection between the sets
ZK w (F ) and ZK w (F [w]) that preserves first digits. If w Trop(F ) or w v(π)Zn,
then these sets are empty. Otherwise, the first digit map gives bijections from
ZK
w
(F ) and ZK
w
(F
[w])
to Zk(inw(F )).
A more computational point of view is shown in the following algorithm.
11
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