March 8, 2017

Graded and Filtered Rings and Modules by C. Nastasescu, F. van Oystaeyen

By C. Nastasescu, F. van Oystaeyen

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10) of K, we add the p redundant quadratic constraints 0 ≤ gm+k (x) (:= nk M 2 − x2i ), k = 1, . . 29) i∈Ik and set m = m + p, so that K is now defined by: K := {x ∈ Rn : gj (x) ≥ 0, j = 1, . . , m }. 30) Note that gm+k ∈ R[x(Ik )], for all k = 1, . . , p. 4. 30). The index set J = {1, . . , m } is partitioned into p disjoint sets Jk , k = 1, . . , p, and the collections {Ik } and {Jk } satisfy: (a) For every j ∈ Jk , gj ∈ R[x(Ik )], that is, for every j ∈ Jk , the constraint gj (x) ≥ 0 only involves the variables x(Ik ) = {xi : i ∈ Ik }.

Ym ) and (z1 , . . , zp ). 28) for some polynomials (gj ) ⊂ R[x, y], (hk ) ⊂ R[y, z], and some finite index sets Ixy , Iyz ⊂ N. Denote by Σ[x, y] (resp. Σ[y, z]) the set of sums of squares in R[x, y] (resp. R[y, z]). Let P (g) ⊂ R[x, y] and P (h) ⊂ R[y, z] be the preorderings generated by (gj )j∈Ixy and (hk )k∈Iyz , respectively, that is P (g) = P (h) =    J⊆Ixy      σJ  gj  : σJ ∈ Σ[x, y] σJ hk j∈J J⊆Iyz k∈J : σJ ∈ Σ[y, z]       . Similarly, let Q(g) ⊂ R[x, y] and Q(h) ⊂ R[y, z] be the quadratic modules     Q(g) = σ0 + σj gj : σ0 , σj ∈ Σ[x, y]   j∈Ixy July 24, 2009 15:8 World Scientific Book - 9in x 6in moments 2 Positive Polynomials 40 Q(h) =    σ0 + σk h k : k∈Iyz σ0 , σk ∈ Σ[y, z]    .

July 24, 2009 15:8 World Scientific Book - 9in x 6in moments 2 Positive Polynomials 26 We next characterize when a semi-algebraic set described by polynomial inequalities, equalities and non-equalities is empty. In order to achieve this, we need the following definition. 1. For F := {f1 , . . , fm } ⊂ R[x], and a set J ⊆ {1, . . , m} we denote by fJ ∈ R[x] the polynomial x → fJ (x) := j∈J fj (x), with the convention that f∅ = 1. The set P (f1 , . . 8) is called (by algebraic geometers) a preordering.

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