Bezout Resultant Formulation

The Bezout resultant handles polynomial systems of any size: f1(x1,x2,...,xn), f2(x1,x2,...,xn),...,fn(x1,x2,...,xn)
The Bezout resultant technique involves constructing a multivariate expression and decomposing it into a linear algebra term so that we can use Nullstellenstatz. For the Bezout resultant, the expression involves the symbolic determinant of a specifically crafted matrix divided by another value; for two equations f(x,y),g(x,y) in two variables, that matrix is: g(alpha,y)
f(x,y) g(x,y)
and the other value is (x-alpha). The ratio delta(x,alpha)/(x-alpha) (where delta refers to the determinant of the symbolic matrix) is an n-1 degree polynomial in alpha and symmetric at x and alpha. It vanishes at every common zero of f(x,y) and g(x,y) no matter what value alpha has. Consequently, we can write the ratio as a linear algebra row-matrix-column product where the x monomials are in the right column and the alpha monomials are in the left row and the matrix is a function solely of y (the hidden variable).
Since the row-matrix-column product will be zero for arbitrary values of alpha at the correct (x,y) position, by Nullstellenstatz, the center matrix's determinant must be zero.