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.
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