S.o.s. approximation of polynomials nonnegative on a real algebraic set

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With every real polynomial f, we associate a family {fεᵣ}ε, ᵣ of real polynomials, in explicit form in terms of f and the parameters ε>0,r∈ N, and such that f-fεᵣ₁→ 0 as ε→ 0. Let V⊂ Rⁿ be a real algebraic set described by finitely many polynomials equations gⱼ(x)=0,j∈ J, and let f be a real polynomial, nonnegative on V. We show that for every ε>0, there exist nonnegative scalars {λⱼ(ε)}ⱼ∈ ⱼ such that, for all r sufficiently large, fεᵣ+∑ⱼ∈ ⱼ λⱼ(ε) gⱼ², is a sum of squares. This representation is an obvious certificate of nonnegativity of fεᵣ on V, and very specific in terms of the gⱼ that define the set V. In particular, it is valid with no assumption on V. In addition, this representation is also useful from a computation point of view, as we can define semidefinite programing relaxations to approximate the global minimum of f on a real algebraic set V, or a semi-algebraic set K, and again, with no assumption on V or K.