  A sum-of-squares is a polynomial that can be expressed as a sum of squares of
other polynomials. Determining if a sum-of-squares decomposition exists for a
given polynomial is equivalent to a linear matrix inequality feasibility
problem. The computation required to solve the feasibility problem depends on
the number of monomials used in the decomposition. The Newton polytope is a
method to prune unnecessary monomials from the decomposition. This method
requires the construction of a convex hull and this can be time consuming for
polynomials with many terms. This paper presents a new algorithm for removing
monomials based on a simple property of positive semidefinite matrices. It
returns a set of monomials that is never larger than the set returned by the
Newton polytope method and, for some polynomials, is a strictly smaller set.
Moreover, the algorithm takes significantly less computation than the convex
hull construction. This algorithm is then extended to a more general
simplification method for sum-of-squares programming.
