Algebra and Trigonometry (International Textbooks in by Alvin K Bettinger

By Alvin K Bettinger

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The set P (f1 , . . 8) is called (by algebraic geometers) a preordering. It is also a convex cone of R[x]; see Appendix A. 11 (Stengle). Let k be a real closed field, and let F := (fi )i∈I1 , G := (gi )i∈I2 , H := (hi )i∈I3 ⊂ k[x] be finite families of polynomials. Let (a) P (F ) be the preordering generated by the family F , (b) M (G) be the set of all finite products of the gi ’s, i ∈ I2 (the empty product being the constant polynomial 1), and (c) I(H) be the ideal generated by H. Consider semi-algebraic set K = {x ∈ k n : fi (x) ≥ 0, hi (x) = 0, ∀ i ∈ I1 ; ∀ i ∈ I3 }.

S. polynomials. For any two real symmetric matrices A, B, recall that A, B stands for trace(AB). Finally, n for a multi-index α ∈ Nn , let |α| := i=1 αi . Consider the vector vd (x) = (xα )|α|≤d = (1, x1 , . . , xn , x21 , x1 x2 , . . , xn−1 xn , x2n , . . , 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 .

0 ≤ f 2r (x)→f (x)). 4(b), observe that in addition to the l1 -norm convergence f − f 2r 1 →0, the convergence is also uniform on compact sets. s. s. approximation f 2r . s. approximation f ≈ f 1r is not uniform on compact sets, and is really more appropriate for polynomials nonnegative on [−1, 1]n only (and indeed the approximation f ≈ f 1r is uniform on [−1, 1]n ). 4(a) the integer r 1 does not depend on the explicit choice of the polynomial f but only on: (a) and the dimension n, (b) the degree and the size of the coefficients of f .

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