A NULLSTELLENSATZ FOR AMOEBAS

A NULLSTELLENSATZ FOR AMOEBAS 409
φ 2 |z2 2|=z2 2 . Thus a ∈ A f
A f = LA f .
if and only if {1, |z 1 z 2 |, |z2 2 |} is nonlopsided; that is,
In the above example, we have enough freedom to choose the phases of the monomials
m i (z) for z ∈ Log −1 (a) so that LA f = A f . However, this works only because f
has very few nonzero terms. In general, LA f can be quite different from A f (see
Figure 1). Nevertheless, we show that for a suitable multiple of f , we can use this
lopsidedness test to get very good approximations for A f .
Let n be a positive integer. We consider the polynomials
˜f n (z) =
∏n−1
k 1 =0
These ˜f n are cyclic resultants
···
∏n−1
k r =0
f (e 2πi k 1/n z 1 ,...,e 2πi k r /n z r ).
˜f n (z) = Res ( Res(...Res(f (u 1 z 1 ,...,u r z r ),u n 1 − 1) ...,un r−1 − 1),un r − 1)
and as such can be practically computed. When n = 2 k , this can be done reasonably
efficiently as follows. Define polynomials h i by h 0 := f ,andlet
h i (z 1 ,...,z 2 [i] ,...,z r):= h i−1 (z 1 ,...,z [i] ,...,z r ) h i−1 (z 1 ,...,−z [i] ,...,z r ),
where [i] ∼ = i (mod r). Then ˜f n (z) = h kr (z1 n,...,zn r
), and the recursion computes
this in O(n 2(r2−r) )-time, which is proportional to the square of the number of terms of
˜f n .
Our main result for amoebas of hypersurfaces is roughly the following. The
precise version is stated and proved in Section 3.2.
THEOREM 1 (Rough version)
As n →∞, the family LA ˜f n
converges uniformly to A f . There exists an integer N
such that to compute A f to within ε, it suffices to compute LA ˜f n
for any n ≥ N.
Moreover, N depends only on ε and the Newton polytope (or degree) of f and can be
computed explicitly from these data.
This leads us to the following characterisation of the amoeba of a general subvariety
of (C ∗ ) r .
THEOREM 2
Let I ⊂ C[z 1 ,z −1
1 ,...,z r,zr
−1 ] be an ideal. A point a ∈ R r is in the amoeba A I if
and only if for every g ∈ I, g{a} is not lopsided.
Phrased another way, if a point a is outside the amoeba A I , a polynomial f ∈ I may
witness this fact by being lopsided at a. Theorem 2 then states that there is always
a witness. We actually show something slightly stronger in both Theorems 1 and 2.