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

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