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
Previous Page Next Page