4

MART´

IN

AVENDA˜

NO AND ASHRAF IBRAHIM

zeros of the initial form system inw(F ) in

(k∗)n.

In a few words, semiregularity

at w is a condition on F that reformulates the hypothesis of Hensel’s Lemma

(see [13, Pag. 48]) for zeros of valuation w and for polynomials with coeﬃcients

in K instead of A. Semiregularity at w also provides the solution of the third

problem: for each w ∈ Trop(F ) and ε ∈ Dw(F ), there is exactly one solution of F

in (K∗)n with valuation vector w and first digits ε, i.e. the set Ew,ε(F ) has only

one element. In particular, for a w-semiregular system of polynomials F , where

w ∈ Trop(F ) ∩ v(π)Zn, the first digit map δ : (K∗)n → (k∗)n provides a bijection

between roots of F with valuation w and roots of the initial form system inw(F )

in (k∗)n. The definition of semiregularity (that was obtained by keeping track

several changes of variables carefully) and the main root counting theorem (proven

by undoing all these changes of variables) are presented in detail in Section 3 and

summarized in the following statement:

Theorem 1.2. Let F = (f1, . . . , fn) be a system of polynomial equations in

K X1

±1,

. . . , Xn

±1

. Let w ∈

v(π)Zn

be an isolated point of Trop(F ). If the initial

form system inw(F ) has no degenerate zeros in

(k∗)n,

then the first digit map

induces a bijection between the set of zeros of F in

(K∗)n

with valuation w and

the set of zeros of inw(F ) in

(k∗)n.

As a consequence of the results described in the last paragraph, we derive

explicit formulas (or more precisely, an algorithm) to compute eﬃciently the number

of roots in

(K∗)n

of a large class of systems of polynomial equations. These systems,

called regular, are characterized by having a finite tropical prevariety, by being

semiregular at any point, and by having initial forms consisting only of binomials.

Our notion of regularity and the formulas for the number of roots generalize those

shown in [2, Def. 1, Thm. 4.5] to the multivariate case. All this work is done in

Section 4.

Although regularity seems to impose a very strong constraint on the system,

we prove in Section 5 that this is not actually the case: regularity occurs generically

when the residue field k has characteristic zero. The notion of genericity implicit

in the previous statement (called tropical genericity) refers to coeﬃcients whose

valuation vector do not lie in the union of certain hyperplanes. This notion is

the natural extension of the genericity in the algebraic geometry sense to tropical

geometry.

Since we have explicit formulas for the number of roots of generic polynomials

(with given support), we should be able to compute the expected number of roots in

(K∗)n

of random polynomials. The only problem is that we need a way of choosing

the coeﬃcients at random that produce tropically generic systems with probability

1. Since our root counting formula does not depend on the tail in 1 + M of the

coeﬃcients, we only need a way of selecting the valuation of the coeﬃcients and

their first digits. The approach that we use consists of choosing the valuation at

random uniformly in an interval [−M, M] and then letting M go to infinity. The

first digits are selected uniformly from

k∗

when k is a finite field, or in the case of

k = R with any probability measure that gives equal probability to R

0

and R 0.

In the case that k is algebraically closed, any selection of the first digits gives the

same number of roots, and therefore no probability measure in k is needed.

4

MART´

IN

AVENDA˜

NO AND ASHRAF IBRAHIM

zeros of the initial form system inw(F ) in

(k∗)n.

In a few words, semiregularity

at w is a condition on F that reformulates the hypothesis of Hensel’s Lemma

(see [13, Pag. 48]) for zeros of valuation w and for polynomials with coeﬃcients

in K instead of A. Semiregularity at w also provides the solution of the third

problem: for each w ∈ Trop(F ) and ε ∈ Dw(F ), there is exactly one solution of F

in (K∗)n with valuation vector w and first digits ε, i.e. the set Ew,ε(F ) has only

one element. In particular, for a w-semiregular system of polynomials F , where

w ∈ Trop(F ) ∩ v(π)Zn, the first digit map δ : (K∗)n → (k∗)n provides a bijection

between roots of F with valuation w and roots of the initial form system inw(F )

in (k∗)n. The definition of semiregularity (that was obtained by keeping track

several changes of variables carefully) and the main root counting theorem (proven

by undoing all these changes of variables) are presented in detail in Section 3 and

summarized in the following statement:

Theorem 1.2. Let F = (f1, . . . , fn) be a system of polynomial equations in

K X1

±1,

. . . , Xn

±1

. Let w ∈

v(π)Zn

be an isolated point of Trop(F ). If the initial

form system inw(F ) has no degenerate zeros in

(k∗)n,

then the first digit map

induces a bijection between the set of zeros of F in

(K∗)n

with valuation w and

the set of zeros of inw(F ) in

(k∗)n.

As a consequence of the results described in the last paragraph, we derive

explicit formulas (or more precisely, an algorithm) to compute eﬃciently the number

of roots in

(K∗)n

of a large class of systems of polynomial equations. These systems,

called regular, are characterized by having a finite tropical prevariety, by being

semiregular at any point, and by having initial forms consisting only of binomials.

Our notion of regularity and the formulas for the number of roots generalize those

shown in [2, Def. 1, Thm. 4.5] to the multivariate case. All this work is done in

Section 4.

Although regularity seems to impose a very strong constraint on the system,

we prove in Section 5 that this is not actually the case: regularity occurs generically

when the residue field k has characteristic zero. The notion of genericity implicit

in the previous statement (called tropical genericity) refers to coeﬃcients whose

valuation vector do not lie in the union of certain hyperplanes. This notion is

the natural extension of the genericity in the algebraic geometry sense to tropical

geometry.

Since we have explicit formulas for the number of roots of generic polynomials

(with given support), we should be able to compute the expected number of roots in

(K∗)n

of random polynomials. The only problem is that we need a way of choosing

the coeﬃcients at random that produce tropically generic systems with probability

1. Since our root counting formula does not depend on the tail in 1 + M of the

coeﬃcients, we only need a way of selecting the valuation of the coeﬃcients and

their first digits. The approach that we use consists of choosing the valuation at

random uniformly in an interval [−M, M] and then letting M go to infinity. The

first digits are selected uniformly from

k∗

when k is a finite field, or in the case of

k = R with any probability measure that gives equal probability to R

0

and R 0.

In the case that k is algebraically closed, any selection of the first digits gives the

same number of roots, and therefore no probability measure in k is needed.

4