This paper introduces a hierarchy of convex relaxations for semialgebraic problems. For problems reducible to a finite number of polynomial equalities and inequalities, it shows how to construct a complete family of polynomially sized semidefinite programming conditions that prove infeasibility. The main tools are a semidefinite programming formulation of the sum of squares decomposition for multivariate polynomials and results from real algebraic geometry. The techniques provide a constructive approach for finding bounded degree solutions to the Positivstellensatz and are illustrated with examples from diverse application fields. The paper discusses the global nonnegativity of polynomial functions, which is a fundamental question in applied mathematics. It presents the sum of squares decomposition as a sufficient condition for nonnegativity. A brief review of semidefinite programming is given, showing how to compute sum of squares decompositions by solving a semidefinite program. Basic elements of semialgebraic geometry are described, and the Positivstellensatz is stated. The main result shows how the sum of squares decision procedure allows for the search of bounded degree solutions to the Positivstellensatz equation. The methodology is interpreted in a refutation-based way and compared with earlier related work. Computational aspects of the implementation are discussed, as are applications in different applied mathematics areas, including enhanced semidefinite relaxations for quadratic programming problems and stronger conditions for matrix copositivity. The paper is organized into sections covering the problem of global nonnegativity, semidefinite programming, semialgebraic geometry, the main result, methodology, computational aspects, and applications. The notation used is standard, with definitions for inner products and positive semidefinite matrices. The paper is based on the author's dissertation, with new examples and references added. The goal is to provide a self-contained introduction to semidefinite programming and real algebra, highlighting the potential for interaction between these fields in practical applications.This paper introduces a hierarchy of convex relaxations for semialgebraic problems. For problems reducible to a finite number of polynomial equalities and inequalities, it shows how to construct a complete family of polynomially sized semidefinite programming conditions that prove infeasibility. The main tools are a semidefinite programming formulation of the sum of squares decomposition for multivariate polynomials and results from real algebraic geometry. The techniques provide a constructive approach for finding bounded degree solutions to the Positivstellensatz and are illustrated with examples from diverse application fields. The paper discusses the global nonnegativity of polynomial functions, which is a fundamental question in applied mathematics. It presents the sum of squares decomposition as a sufficient condition for nonnegativity. A brief review of semidefinite programming is given, showing how to compute sum of squares decompositions by solving a semidefinite program. Basic elements of semialgebraic geometry are described, and the Positivstellensatz is stated. The main result shows how the sum of squares decision procedure allows for the search of bounded degree solutions to the Positivstellensatz equation. The methodology is interpreted in a refutation-based way and compared with earlier related work. Computational aspects of the implementation are discussed, as are applications in different applied mathematics areas, including enhanced semidefinite relaxations for quadratic programming problems and stronger conditions for matrix copositivity. The paper is organized into sections covering the problem of global nonnegativity, semidefinite programming, semialgebraic geometry, the main result, methodology, computational aspects, and applications. The notation used is standard, with definitions for inner products and positive semidefinite matrices. The paper is based on the author's dissertation, with new examples and references added. The goal is to provide a self-contained introduction to semidefinite programming and real algebra, highlighting the potential for interaction between these fields in practical applications.