Selecting a monomial basis for sums of squares programming over a quotient ring

In this paper we describe a method for choosing a “good” monomial basis for a sums of squares (SOS) program formulated over a quotient ring. It is known that the monomial basis need only include standard monomials with respect to a Groebner basis. We show that in many cases it is possible to use a reduced subset of standard monomials by combining Groebner basis techniques with the well-known Newton polytope reduction. This reduced subset of standard monomials yields a smaller semidefinite program for obtaining a certificate of non-negativity of a polynomial on an algebraic variety.