By Lam T.Y., Magid A.R. (eds.)

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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 }.

F 2r ) such that f − f 1r 1 →0 (resp. f − f 2r 1 →0). It also provides a certificate of nonnegativity of f on [−1, 1]n (resp. on Rn ). Indeed fix 22 2 Positive Polynomials x ∈ [−1, 1]n (resp. x ∈ Rn ) and let →0 to obtain that 0 ≤ f 1r (x)→f (x) (resp. 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 ).

3. Let K = {(x, y, z)| 1 − x2 − y 2 ≥ 0, 1 − y 2 − z 2 ≥ 0} and f (x, y, z) = 1 + x2 y 2 − x2 y 4 + y 2 + z 2 − y 2 z 4 . 27(b) allows the sparse representation f = 1 + x4 y 2 + y 4 z 2 + x2 y 2 (1 − x2 − y 2 ) + y 2 z 2 (1 − y 2 − z 2 ), with p0 (x, y) = 1 + x4 y 2 , p1 (x, y) = x2 y 2 , q0 (y, z) = y 4 z 2 and q1 (y, z) = y 2 z 2 . Note that p0 , p1 ∈ Σ(x, y) and q0 , q1 ∈ Σ(y, z). 27 to more general sparsity patterns, when f and gj satisfy certain sparsity conditions themselves. With R[x] = R[x1 , .