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We consider the global minimization of a multivariate polynomial on a semi-algebraic set Ω defined with polynomial inequalities. We then compare two hierarchies of relaxations, namely, LP relaxations based on products of the original constraints, in the spirit of the RLT procedure of Sherali and Adams (1990), and recent semidefinite programming (SDP) relaxations introduced by the author. The comparison is analyzed in light of recent results in real algebraic geometry on various representations of polynomials, positive on a compact semi-algebraic set.