SOS approximation of polynomials nonnegative on an algebraic set

Let V ⊂R
ⁿ
be a real algebraic set described by finitely many polynomials equations g
j
(x)=0, j∈J, and let f be a real polynomial, nonnegative on V. We show that for every ∈>0, there exist nonnegative scalars {λ
j
}
j∈J
such that, for all r sufficiently large, f
∈r
+∑
j∈J
λ
j
g
²
j
, is a sum of squares, for some polynomial f
∈r
with a simple and explicit form in terms of f and the parameters ∈>0, r∈N, and such that ||f-f
∈r
||
1
→0 as ∈→0. This representation is an obvious certificate of nonnegativity of f
∈r
on V, and valid with no assumption on V. In addition, this representation is also useful from a computational point of view, as we can define semidefinite programming relaxations to approximate the global minimum of f on a real algebraic set V, or a basic closed semi-algebraic set K, and again, with no assumption on V or K.