Download Algebraists' Homage: Papers in Ring Theory and Related by S. A. Amitsur, D. J. Saltman, George B. Seligman PDF
By S. A. Amitsur, D. J. Saltman, George B. Seligman
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Extra resources for Algebraists' Homage: Papers in Ring Theory and Related Topics
Xd1 , . . , xdn ) , of all the monomials xα of degree less than or equal to d, which has dimension s(d) := n+d d . Those monomials form the canonical basis of the vector space R[x]d of polynomials of degree at most d. 1. ) if and only if there exists a real symmetric and positive semidefinite matrix Q ∈ Rs(d)×s(d) such that g(x) = vd (x) Qvd (x), for all x ∈ Rn . Proof. Suppose there exists a real symmetric s(d) × s(d) matrix Q 0 for which g(x) = vd (x) Qvd (x), for all x ∈ Rn . Then Q = HH for some s(d) × k matrix H, and thus, k (H vd (x))2i , g(x) = vd (x) HH vd (x) = i=1 ∀x ∈ Rn .
F ≥ 0 on K if and only if there exists ∈ N, and g, h ∈ P (f1 , . . , fm ) such that f g = f 2 + h. (b) Positivstellensatz. f > 0 on K if and only if there exist g, h ∈ P (f1 , . . , fm ) such that f g = 1 + h. (c) Nullstellensatz. f = 0 on K if and only if there exists ∈ N, and g ∈ P (f1 , . . , fm ) such that f 2 + g = 0. s. 8) of g, h ∈ P (f1 , . . , fm ). This bound depends only on the dimension n and on the degree of the polynomials (f, f1 , . . , fm ). 12(a)-(c) requires solving a single semidefinite program (but of huge size).
If Q ⊂ R[x] is a maximal proper quadratic module then Q ∪ −Q = R[x]. Proof. Let f ∈ R[x] and assume that f ∈ Q ∪ −Q. As Q is maximal, the quadratic modules Q + f Σ[x] and Q − f Σ[x] are not proper; that is there exists g1 , g2 ∈ Q and s1 , s2 ∈ Σ[x] such that −1 = g1 + s1 f and −1 = g2 − s2 f . Multiplying the first by s2 and the second by s1 yield s1 + s2 + s1 g1 + s2 g2 = 0. Therefore s1 , s2 ∈ I := Q ∩ −Q. But then s1 f ∈ I because I is an ideal and so we obtain the contradiction −1 = g1 +s1 f ∈ Q.