The paper introduces a hierarchy of convex relaxations for semialgebraic problems, focusing on polynomial equalities and inequalities. It presents a method to construct a complete family of polynomially sized semidefinite programming conditions that can prove infeasibility. The main tools used are the semidefinite programming formulation of the sum of squares decomposition for multivariate polynomials and results from real algebraic geometry. These techniques provide a constructive approach to finding bounded degree solutions to the Positivstellensatz and are illustrated with examples from various application fields. The paper also reviews existing approaches, discusses the computational aspects, and presents applications in areas such as quadratic programming and matrix copositivity. The notation and key concepts are defined, and the problem of global nonnegativity of polynomial functions is introduced, highlighting its importance in applied mathematics.The paper introduces a hierarchy of convex relaxations for semialgebraic problems, focusing on polynomial equalities and inequalities. It presents a method to construct a complete family of polynomially sized semidefinite programming conditions that can prove infeasibility. The main tools used are the semidefinite programming formulation of the sum of squares decomposition for multivariate polynomials and results from real algebraic geometry. These techniques provide a constructive approach to finding bounded degree solutions to the Positivstellensatz and are illustrated with examples from various application fields. The paper also reviews existing approaches, discusses the computational aspects, and presents applications in areas such as quadratic programming and matrix copositivity. The notation and key concepts are defined, and the problem of global nonnegativity of polynomial functions is introduced, highlighting its importance in applied mathematics.