Nonlinear Equations - UFRJ

[SEC. 6.4: CALCULUS OF POLYTOPES AND KERNELS 81
has a common root. Since facets have dimension < n, one variable
may be eliminated. Hence, systems with such a common root are
confined to a certain non-trivial Zariski closed set.
Since the number of facets is finite, the systems with a root at
toric infinity are contained in a Zariski closed set.
The proof of Bernstein’s theorem follows now exactly as for Kushnirenko’s
theorem.
Remark 6.14. We omitted many interesting mathematical developments
related to the contents of this chapter, such as isoperimetric
inequalities. A good reference is [45].
Exercise 6.1. Assume that ω is degenerate. Show that the polytopes
are all orthogonal to some direction. Show that the set of f with
common roots is a non-trivial closed Zariski set.
Exercise 6.2. Let K(x, y), L(x, y) be complex symmetric functions on
M and are linear in x, and λ, µ > 0, then
ω KL = ω K + ω L
Exercise 6.3. Let
K(x, y) = ∑ a∈A
c a e a(x+ȳ)
and L(x, y) = ∑ a∈A c ae λa(x+ȳ) . Then (ω L ) x = λ 2 (ω K ) λx .
Exercise 6.4. Complete the proof of Proposition 6.12