Nonlinear Equations - UFRJ

78 [CH. 6: EXPONENTIAL SUMS AND SPARSE POLYNOMIAL SYSTEMS
Proof. The implicit function theorem guarantees that (v t ) is defined
for some interval (0, τ). Take τ maximal with that property. If τ < 1
and v t converges for t → τ, then we could apply the implicit function
theorem at t = τ and increase τ. Therefore v t diverges, and since the
first projection is smooth π 2 (v t ) diverges.
It would be convenient to have a compact M. Recall that in the
Kushnirenko setting, M can be thought as a subset of P(F A ) (while
F = FA n ). More precisely,
K : M → F,
x ↦→ K(·, ¯x)
is an embedding and an isometry into P(F A ). Let ¯M be the ordinary
closure of K(M). In this setting, it is the same as the Zariski closure.
The set ¯M is an example of a toric variety.
Can we then replace M by ¯M in the theory? The answer is not
always.
Example 6.10. Let
A = {0, e 1 , e 2 , e 3 , e 1 + e 2 } ⊂ Z 3
Then ¯M has a singularity at (0 : 0 : 0 : 1 : 0) and hence is not a
manifold.
This phenomenon can be averted if the polytope A satisfies a
geometric-combinatorial condition [34]. Here, however, we need to
proceed in a more general setting to prove theorems 1.6 and 1.9.
Let B be a facet of A, that is the set of maxima of linear functional
0 ≠ ω B : R n → R while restricted to A. Let B = A ∩ B be the set of
corresponding exponents.
We say that P ∈ ¯M is a zero at B-infinity for f if and only if,
P ⊥ f in F A and moreover, P = lim K(·, x j with m xj → B. A zero
at toric infinity is a zero at B-infinity for some facet B.
Toric varieties are manifolds if and only if they satisfy a certain
condition on their vertices [34]. In view of this example, we will not
assume this condition. Instead,
Lemma 6.11. The set of f ∈ FA
n with a zero at toric infinity is
contained in a non-trivial Zariski-closed set of F A .