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:
|
f(x,y) | g(x,y) |
g(alpha,y) | g(alpha,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.