U-Resultant Formulation

The U-resultant resultant is a way to homogenize a multipolynomial system so that one can use the Macaulay resultant formulation to handle polynomial systems of any size: f1(x1,x2,...,xn), f2(x1,x2,...,xn),...,fn(x1,x2,...,xn)
The U-Resultant technique involves homogenizing the functions by adding an extra variable to all the given equations and also adding an extra equation. The extra variable, which we call homogenizingVariable, is intended to be exactly 1, so that we can multiply any of the monomial terms in any of the functions by a necessary power of homogenizingVariable in order to achieve a homogeneous system. The added equation is f(x1,...,xn+1) = x1 - homogenizingVariable*u (where u is another additional variable - remember, elimination uses n+1 variables for n homogeneous equations)
For example, suppose we're given f1(x,y)=x^2+y^2-1,f2(x,y)=(x-y-1)*(x-y+1), we construct a new system of three variables in three unknowns: f1(x,y,homogenizingVariable)=x^2+y^2-homogenizingVariable^2, f2(x,y,homogenizingVariable)=(x-y-homogenizingVariable) * (x-y+homogenizingVariable), f3(x,y,homogenizingVariable)=homogenizingVariable*u-x.
Now, we can use the Macaulay construction on the system of three equations (f1,f2,f3) in four variables (x,y,u,homogenizingVariable) to generate an expression solely in u (which is equal to x, since homogenizingVariable is intended to be unity)