Quantitative Local Analysis of Nonlinear Systems - University of ...

The Positivstellensatz, a central theorem from real algebraic geometry, provides generalizations
of Lemma 2.3.2. For its statement, a few definitions are needed.
Definition 2.3.1. Given {g 1 , . . . , g t } ∈ R[x], the multiplicative monoid generated by g j ’s is
the set of all finite products of g j ’s, including 1 (i.e. the empty product). It is denoted as
M(g 1 , . . . , g t ). For completeness define M(∅) := 1.
⊳
Definition 2.3.2. Given {f 1 , . . . , f r } ∈ R[x], the cone generated by f i ’s is
{
}
m∑
P(f 1 , . . . , f r ) := s 0 + s i b i : m ∈ Z + , s i ∈ Σ[x], b i ∈ M(f 1 , . . . , f r ) .
i=1
⊳
Definition 2.3.3. Given {h 1 , . . . , h u } ∈ R[x], the ideal generated by h k ’s is
{∑ }
I(h 1 , . . . , h u ) := hk p k : p k ∈ R[x] .
⊳
With these definitions, we can state the following theorem from [12, Theorem 4.2.2]:
Theorem 2.3.1 (Positivstellensatz). Given polynomials {f 1 , . . . , f r }, {g 1 , . . . , g t }, and
{h 1 , . . . , h u } in R[x], the following are equivalent:
i. The set below is empty:
⎧
⎪⎨
x ∈ R n :
⎪⎩
f 1 (x) ≥ 0, . . . , f r (x) ≥ 0,
g 1 (x) ≠ 0, . . . , g t (x) ≠ 0,
h 1 (x) = 0, . . . , h u (x) = 0
⎫
⎪⎬
⎪⎭
ii. There exist polynomials f ∈ P(f 1 , . . . , f r ), g ∈ M(g 1 , . . . , g t ), h ∈ I(h 1 , . . . , h u ) such
that
f + g 2 + h = 0.
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