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 |α − 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

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 |α − 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