A Resultant System as the Set of Coefficients of a Single Resultant

Explicit expressions for polynomials forming a homogeneous resultant system of a set of $m+1$ homogeneous polynomial equations in $n+1<m+1$ variables are given. These polynomials are obtained as coefficients of a homogeneous resultant for an appropriate system of $n+1$ equations in $n+1$ variables, which is explicitly constructed from the initial system. Similar results are obtained for mixed resultant systems of sets of $n+1$ sections of line bundles on a projective variety of dimension $n<m$. As an application, an algorithm determining whether one of the orbits under an action of an affine irreducible algebraic group on a quasi-affine variety is contained in the closure of another orbit is described.