## 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)