Computational approaches to representation theorems for finitely generate real algebras

Schmüdgen's Representation Theorem states that if a compact set S in $Rn$ is defined by finitely many polynomial inequalities, then any polynomial which is strictly positive on S can be written in terms of the defining polynomials for S and sums of squares of polynomials. Recently, Schweighofer found a constructive approach to this representation theorem which depends on having representations of polynomials of the form N ± x_i for 1 ≤ i ≤ n.; In this dissertation, we use Schweighofer's approach to give a complete algorithm for Schmüdgen's Theorem in one variable. Then we use the ideas of Schweighofer to give a constructive approach to the Kadison-Dubois Theorem which gives a representation in terms of the polynomials for S, but not sums of squares. Finally, we apply this to give a constructive approach to a result of M. Marshall which gives a representation theorem in certain cases where S is non-compact.