Nonlinear Equations - UFRJ

[SEC. 1.3: SPARSE POLYNOMIAL SYSTEMS 9
+ 1 2
=
0 0
0
Figure 1.1: Minkowski linear combination.
Definition 1.7 (Minkowski linear combinations). (See fig.1.1) Given
convex sets A 1 , . . . , A n and fixed coefficients λ 1 , . . . , λ n , the linear
combination λ 1 A 1 + · · · + λ n A n is the set of all
where a i ∈ A i .
λ 1 a 1 + · · · + λ n a n
The reader will show in the exercises that
Proposition 1.8. Let A 1 , . . . , A s be compact convex subsets of R n .
Let λ 1 , . . . , λ s > 0. Then,
Vol(λ 1 A 1 + · · · + λ s A s )
is a homogeneous polynomial of degree s in λ 1 , . . . , λ s .
Theorem 1.9 (Bernstein [17]). Let A 1 , . . . , A n ⊂ Z n be finite sets.
Let A i be the convex hull of A i . Let B be the coefficient of λ 1 . . . λ n
in the polynomial
Vol(λ 1 A 1 + · · · + λ n A n ).
Then, for a generic choice of coefficients f ia ∈ C, the system of