A NULLSTELLENSATZ FOR AMOEBAS

418 KEVIN PURBHOO
The idea behind the proof of Theorem 1 is to look at the family of ˜f n (z) and
interpret this as a function of a single variable z i . At the point ζ = (ζ 1 ,...,ζ r ) ∈ C r ,
we define
˜f i,ζ
n (z) := ˜f n (ζ 1 ,...,ζ i−1 ,z,ζ i+1 ,...,ζ n ).
We apply Lemma 2.2 to these and find a single dominant term in this polynomial of
one variable. Then by an averaging argument, we show that this implies that ˜f n has a
single dominant term.
First, however, we need a few simple observations.
PROPOSITION 3.1
We have A f = A ˜f n
.
Proof
The cyclic resultant ˜f n (z) is a product of terms g u1 ,...,u r
(z) = f (u 1 z 1 ,...,u r z r ), where
u n i = 1.Since|u i |=1, A gu1 ,...,ur
= A f , and so A ˜f n
= ⋃ A gu1 ,...,ur
= A f .
We also need to know some information about the number and degree of the terms
which appear in ˜f n . First, note the following important fact.
PROPOSITION 3.2
The only monomials that appear in ˜f n are of the form cz nk 1
1 ···z nk r
r
. In particular, the
only terms appearing in ˜f
n i,ζ(z)
are of the form cznk .
Proof
Let C n denote the cyclic group of roots of z n −1.Now, ˜f n is manifestly invariant under
the group action of (C n ) r acting on C[z 1 ,z −1
1 ,...,z r,zr −1 ] by (u 1 ,...,u n ) · g(z) =
g(u 1 z 1 ,...,u n z n ). Thus each monomial of ˜f n must be invariant under this action.
The only monomials with this property are of the form cz nk 1
1 ···z nk r
r
.
The statement about ˜f
n i,ζ (z) follows immediately.
Recall that if g ∈ C[z 1 ,z −1
1 ,...,z r,zr
−1 ], then its Newton polytope, denoted (g),is
the subset of R r defined as the convex hull of the exponent vectors of the monomials
which appear in g.
For any polytope , letd() be any upper bound on (#{Z r ∩ m})/m r .In
general, it is not easy to find a tight upper bound for this number. If one can compute
the Ehrhart polynomial of explicitly, then an easy upper bound is the sum of the
positive coefficients. Otherwise, it is possible to bound the coefficients of the Ehrhart
polynomial in terms of the volume of (see [BM]). Using these estimates, for each r
one can compute constants A and B such that (#{Z r ∩ m})/m r