A NULLSTELLENSATZ FOR AMOEBAS

408 KEVIN PURBHOO
defined at the point z = (z 1 ,...,z r ) by
Log(z) = (log |z 1 |,...,log |z r |).
We denote the amoeba of V by either A V or A I .IfV = Z f is a hypersurface,
the zero locus of a single function f , we also use the notation A f . We refer the
reader to Mikhalkin’s survey article [M] for a broad discussion of amoebas and their
applications.
In this article, we address the following fundamental question: given a point
a ∈ R r and an ideal I ⊂ C[z 1 ,z −1
1 ,...,z r,zr
−1 ],whenisa ∈ A I ? This problem was
previously studied by Theobald [T], who gave a practical answer for certain families
of amoebas. Here we give a general answer to this question. We first consider the case
where I =〈f 〉 is the ideal of a hypersurface. From this, we deduce a characterisation
theorem for arbitrary ideals which is the analytic counterpart to a fundamental theorem
for tropical varieties.
Consider f ∈ C[z 1 ,z −1
1 ,...,z r,zr
−1 ], and consider a point a ∈ R r . Write f as a
sum of monomials f (z) = m 1 (z) +···+m d (z).Definef {a} to be the list of positive
real numbers
f {a} := {∣ ∣ m1
(
Log −1 (a) )∣ ∣ ,...,
∣ ∣md
(
Log −1 (a) )∣ ∣ } .
Note that since the m i are monomials, this is well defined, even though Log is not
injective.
Definition 1.2
We say that a list of positive numbers is lopsided if one of the numbers is greater than
the sum of all the others.
Equivalently, a list of numbers {b 1 ,...,b d } is not lopsided if it is possible to choose
complex phases φ i (|φ i |=1), so that ∑ φ i b i = 0. This follows from the triangle
inequality. We also define
LA f := { a ∈ R r ∣ ∣ f {a} is not lopsided
}
.
One can easily see that if a ∈ A f ,thenf {a} cannot be lopsided; in other words,
LA f ⊃ A f .Indeed,iff (z) = 0, thenm 1 (z) +···+m d (z) = 0, soitisgivinga
way to assign complex phases to the list {|m 1 (z)|,...,|m d (z)|} = f {Log(z)} such
that the sum is zero. Thus one can think of LA f as a crude approximation to the
amoeba A f .
Example 1.3
Suppose that f (z 1 ,z 2 ) = 1 + z 1 z 2 + z2 2, and let a ∈ R2 . For any complex
phases φ 1 ,φ 2 , there exist (z 1 ,z 2 ) ∈ Log −1 (a) such that φ 1 |z 1 z 2 | = z 1 z 2 and