MULTIVARIATE ULTRAMETRIC ROOT COUNTING 3

Let K be a complete field with respect to a discrete non-archimedian valuation

v : K → R ∪ {∞}. Let A = {x ∈ K : v(x) ≥ 0} be the valuation ring of K. The

ring A is local with maximal ideal M = {x ∈ K : v(x) 0}, which is principal

M = πA since v is discrete. We denote by k = A/M the residue field of K with

respect to v. We denote the first digit of x ∈

K∗

by δ(x) =

π−v(x)/v(π)x

mod M.

The map δ :

K∗

→

k∗

is a homomorphism, that can be seen as the composition of

the homomorphisms

K∗

→ Z ×

A∗

→

A∗

→

k∗,

where the first map is the isomorphism x → (v(x)/v(π),

π−v(x)/v(π)x),

the second

arrow is the projection on the second factor, and the third arrow is the reduction

modulo M.

Fix a set Δ ⊆ A \ M of representatives of the first digit map. For any x ∈ K∗,

we write Δ(x) the representative corresponding to δ(x). Any element in x ∈ K∗ can

be factorized as x =

πv(x)/v(π)Δ(x)e(x)

where e(x) =

xπ−v(x)/v(π)Δ(x)−1

∈ 1+M.

Moreover, this is the only possible factorization of x as the product of a power

of π, an element in Δ, and an element in 1 + M. This implies that the map

K∗

→ v(π)Z ×

k∗

× (1 + M) given by x → (v(x), δ(x),e(x)) is a bijection. The

spirit behind most of our results is this bijection: we compute/count the solutions

of systems of polynomials by first looking at the valuation, then the first digit, and

then the tail in 1 + M. Our notions of genericity and randomness are also based

on the bijection.

Consider a square system F = (f1, . . . , fn) of n polynomials in K X1

±1,

. . . , Xn

±1

.

Denote by ZK (F ) the set of solutions of F in

(K∗)n.

The study of the set ZK (F )

that we do in this paper is based on the following program:

(1) Study the set S(F ) = {v(x) : x ∈ ZK (F )} ⊆

v(π)Zn.

(2) For each w ∈ S(F ) study the set

Dw(F ) = {δ(x) : x ∈ ZK (F ), v(x) = w} ⊆

(k∗)n.

(3) For each w ∈ S(F ) and ε ∈ Dw(F ) study the set

Ew,ε(F ) = {e(x) : x ∈ ZK (F ), v(x) = w, δ(x) = ε} ⊆ (1 +

M)n.

A similar program was successfully used by B. Sturmfels and D. Speyer in [15],

working on the field of Puisseax series C{{t}}, to give a simple proof of Kapranov’s

Theorem: item 1 correspond with their Theorem 2.1 and item 2 with Corollary 2.2.

Our approach for the first problem requires us to work only with the valuations

of the coeﬃcients and the exponent vectors of the monomials of F . We will prove

that S(F ) ⊆ Trop(F ) ∩

v(π)Zn,

where the set Trop(F ) = Trop(f1) ∩ · · · ∩ Trop(fn)

is the tropical prevariety induced by F . Recall that for a given polynomial f =

∑

t

i=1

aiXαi

∈ K X1

±1,

. . . , Xn

±1

, the set Trop(f) is defined as the set of all possible

w ∈

Rn

such that v(ai) + w · αi for i = 1, . . . , t reaches its minimum value at least

twice. For any w ∈

Rn,

the initial form inw(f) ∈ k X1

±1,

. . . , Xn

±1

is defined as

the sum of

δ(ai)Xαi

, but including only the terms that minimize v(ai)+ w · αi. All

the notions of tropical geometry used in this paper are defined in Section 2 and can

also be found in the literature in [15, 12, 6].

For the second problem, we introduce the notion of w-semiregularity at a given

w ∈ Trop(F ) ∩ v(π)Zn, that guarantees that Dw(F ) coincides with the set of

3

Let K be a complete field with respect to a discrete non-archimedian valuation

v : K → R ∪ {∞}. Let A = {x ∈ K : v(x) ≥ 0} be the valuation ring of K. The

ring A is local with maximal ideal M = {x ∈ K : v(x) 0}, which is principal

M = πA since v is discrete. We denote by k = A/M the residue field of K with

respect to v. We denote the first digit of x ∈

K∗

by δ(x) =

π−v(x)/v(π)x

mod M.

The map δ :

K∗

→

k∗

is a homomorphism, that can be seen as the composition of

the homomorphisms

K∗

→ Z ×

A∗

→

A∗

→

k∗,

where the first map is the isomorphism x → (v(x)/v(π),

π−v(x)/v(π)x),

the second

arrow is the projection on the second factor, and the third arrow is the reduction

modulo M.

Fix a set Δ ⊆ A \ M of representatives of the first digit map. For any x ∈ K∗,

we write Δ(x) the representative corresponding to δ(x). Any element in x ∈ K∗ can

be factorized as x =

πv(x)/v(π)Δ(x)e(x)

where e(x) =

xπ−v(x)/v(π)Δ(x)−1

∈ 1+M.

Moreover, this is the only possible factorization of x as the product of a power

of π, an element in Δ, and an element in 1 + M. This implies that the map

K∗

→ v(π)Z ×

k∗

× (1 + M) given by x → (v(x), δ(x),e(x)) is a bijection. The

spirit behind most of our results is this bijection: we compute/count the solutions

of systems of polynomials by first looking at the valuation, then the first digit, and

then the tail in 1 + M. Our notions of genericity and randomness are also based

on the bijection.

Consider a square system F = (f1, . . . , fn) of n polynomials in K X1

±1,

. . . , Xn

±1

.

Denote by ZK (F ) the set of solutions of F in

(K∗)n.

The study of the set ZK (F )

that we do in this paper is based on the following program:

(1) Study the set S(F ) = {v(x) : x ∈ ZK (F )} ⊆

v(π)Zn.

(2) For each w ∈ S(F ) study the set

Dw(F ) = {δ(x) : x ∈ ZK (F ), v(x) = w} ⊆

(k∗)n.

(3) For each w ∈ S(F ) and ε ∈ Dw(F ) study the set

Ew,ε(F ) = {e(x) : x ∈ ZK (F ), v(x) = w, δ(x) = ε} ⊆ (1 +

M)n.

A similar program was successfully used by B. Sturmfels and D. Speyer in [15],

working on the field of Puisseax series C{{t}}, to give a simple proof of Kapranov’s

Theorem: item 1 correspond with their Theorem 2.1 and item 2 with Corollary 2.2.

Our approach for the first problem requires us to work only with the valuations

of the coeﬃcients and the exponent vectors of the monomials of F . We will prove

that S(F ) ⊆ Trop(F ) ∩

v(π)Zn,

where the set Trop(F ) = Trop(f1) ∩ · · · ∩ Trop(fn)

is the tropical prevariety induced by F . Recall that for a given polynomial f =

∑

t

i=1

aiXαi

∈ K X1

±1,

. . . , Xn

±1

, the set Trop(f) is defined as the set of all possible

w ∈

Rn

such that v(ai) + w · αi for i = 1, . . . , t reaches its minimum value at least

twice. For any w ∈

Rn,

the initial form inw(f) ∈ k X1

±1,

. . . , Xn

±1

is defined as

the sum of

δ(ai)Xαi

, but including only the terms that minimize v(ai)+ w · αi. All

the notions of tropical geometry used in this paper are defined in Section 2 and can

also be found in the literature in [15, 12, 6].

For the second problem, we introduce the notion of w-semiregularity at a given

w ∈ Trop(F ) ∩ v(π)Zn, that guarantees that Dw(F ) coincides with the set of

3