Let V (is contained in) R~(n) be a real algebraic set describedby finitely many polynomials equations g_(j)(x) velence 0, j E J, and let f be a real polynomial, nonnegative on V. We show that for every E > 0, there exist nonnegative scalars {lambda_(j)}_(jEJ) such that, for all r sufficiently large, f_(Er) + sum from jEJ of lambda_(j) g_(j)~(2), is a sum of squares, for some polynomial f_(Er) with a simple and explicit form in terms of f and the parameters E > 0, r E N, and such that ||f - f_(Er)||_(1) -> 0 as E -> 0. This representation is an obvious certificate of nonnegativity of f_(Er) 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.
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