MULTIVARIATE ULTRAMETRIC ROOT COUNTING 5
Let A = {α1 α2 · · · αt} Z be a finite set (t 2) and consider an
univariate polynomial f K[X] with supp(f) = A and random coefficients (chosen
as explained above). Let E(A,K) be the limit of the expected number of roots of
f in K∗ as M goes to infinity. Our main result of section 6, is a general formula for
E(A,K). As a particular case, we have the following result that it is interesting in
itself, and simple enough to be stated in this introduction:
Theorem 1.3. Let A = {α1 α2 · · · αt} Z be a finite set with t 2.
If k is algebraically closed with char(k) = 0 or char(k) maxα,β∈A β|, then
2
2
t
E(A,K) 2 ln(t).
A previous estimation for the expected number of roots of random polynomials
with p-adic coefficients, although for a different distribution (related to the Haar
measure on Zp) was obtained by S. Evans in [8].
2. Tropical hypersurface induced by a Laurent polynomial
The main goal of this section is to introduce the reader the notions of tropical
geometry used in the rest of the paper.
Definition 2.1. Let f K X1
±1
, . . . , Xn
±1
be a polynomial with t non-zero
terms f =
∑t
i=1
aiXαi where ai K∗ and αi = (αi1, . . . , αin) Zn for all i =
1, . . . , t. We define the tropicalization of f as the piecewise linear function tr(f; w) =
min{li(f; w) i = 1, . . . , t} where li(f; w) = v(ai)+ αi · w. The tropical hypersurface
induced by f is the set
Trop(f) = {w0
Rn
: tr(f; w) is not differentiable at w0}.
The value of li(f; w) is usually referred in the literature as the w-weight of the
i-th term of f.
Lemma 2.1. Let f K X1
±1,
. . . , Xn
±1
be a polynomial with t terms and let
w0
Rn.
Then w0 Trop(f) if and only if there are indices 1 i j t such
that li(f; w0) = lj(f; w0) lk(f; w0) for all k = 1, . . . , t.
Proof. (⇐) Assume first that li(f; w0) lk(f; w0) for all k = i. Since the
functions
li(f; w) are continuous, all these inequalities remain valid in a neighbor-
hood U of w0, and then tr(f; w) coincides with the linear function li(f; w) in U. In
particular, tr(f; w) is differentiable at w0, i.e. w0 Trop(f).
(⇒) Now take w0 Trop(f). Since tr(f; w) is differentiable at w0, then the
linear function l(w) = tr(f; w0) + ∇tr(f; w0) · (w w0) approximates tr(f; w) with
order two near w0, and since tr(f; w) is piecewise linear, then tr(f; w) = l(w) =
li(f; w) for some 1 i t in a neighborhood U of w0. Therefore, for any other
index k = i, we have that tr(f; w) = li(f; w) lk(f; w) in U, or equivalently,
li(f; w0) lk(f; w0) (αk αi) · (w w0) in U. The right hand side of this
inequality can be made strictly negative by selecting w w0 a vector with the
direction of αi αk, hence li(f; w0) lk(f; w0) for all k = i.
Note that for any x
(K∗)n,
the valuation of the i-th term of f at x is given
by li(f; v(x)).
Proposition 2.2. Let f K X1
±1,
. . . , Xn
±1
and let x
(K∗)n
be a zero of f.
Then v(x) Trop(f).
5
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