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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). In practice, one fixes an a priori (much smaller) degree bound and solves the corresponding semidefinite program. If the latter has a feasible solution then one obtains a certificate g, h ∈ P (f1 , . . , fm ). However, such a certificate is numerical and so can be obtained only up to some machine precision, because of numerical inaccuracies inherent to semidefinite programming solvers.

S. 3) α∈Γd |α| − 2d min[0, gα ] α∈Γd |α| , 2d ∀ i = 1, . . , n. 4) are only sufficient but on the other hand, they define a convex polyhedron in the space Rs(2d) of coefficients (gα ) of g ∈ R[x]. s. polynomials of degree at most d. s. s. s. but not whether it is only nonnegative. There are two kinds of results for that comparison, depending on whether or not one keeps the degree fixed. s. polynomials increases and is unbounded. Namely, let P[x]d (resp. H[x]d ) denote the cone of homogeneous and nonnegative (resp.

Given r ∈ N arbitrary, let Θr , θr ∈ R[x] be the polynomials n n x2r i ; Θr (x) := 1 + i=1 r θr (x) := i=1 k=0 x2k i . k! 6) Given f ∈ R[x] let f 1 := α∈Nn |fα | if f = (fα ) is the vector of coefficients of f . Next, with > 0, we define f 1r := f + Θr ; f 2r := f + θr . 4. (a) If f ∈ R[x] is nonnegative on [−1, 1]n , then for every > 0 there exists r1 such that f 1r ∈ Σ[x] for all r ≥ r 1 and f − f 1r 1 →0 as ↓ 0 (and r ≥ r1 ). (b) If f ∈ R[x] is nonnegative, then for every > 0 there exists r 2 such that f 2r ∈ Σ[x] for all r ≥ r 2 and f − f 2r 1 →0 as ↓ 0 (and r ≥ r 2 ).

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An Introduction To Linear Algebra by Kuttler

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