Resultant Elimination Techniques
Resultant elimination techniques are used to solve multipolynomial
systems by constructing a univariate polynomial expression such that
all of the roots of the original multipolynomial system are
represented as roots of the constructed univariate polynomial.
In other words, given a system of three functions f1(x,y,z), f2(x,y,z)
and f3(x,y,z), resultant techniques can formulate a new univariate
system Q(z) such that the z' values of all of the common roots
(x',y',z') of the original system (f1,f2,f3) are also roots of the
univariate system Q(z).
Resultant techniques are based upon the Nullstellenstatz observation,
as follows:
Suppose f1,f2, and f3 are linear in x and y such that
f1(x,y,z),f2(x,y,z),f3(x,y,z) can be written in linear algebra form
as:
|
f1_x(z) | f1_y(z) | f1_1(z) |
f2_x(z) | f2_y(z) | f2_1(z) |
f3_x(z) | f3_y(z) | f3_1(z)
|
where f1(x,y,z) = f1_x(z) x + f1_y(z) y + f1_1(z),
f2(x,y,z) = f2_x(z) x + f2_y(z) y + f2_1(z),
f3(x,y,z) = f3_x(z) x + f3_y(z) y + f3_1(z).
Now consider any common roots (x',y',z') of the three equations
(f1,f2,f3). Since the linear algebra representation exactly
characterizes f1,f2,f3, it must be the case that the linear algebra
matrix-vector is also (0,0,0) at the common roots. We know that the
column vector (x,y,1) is non-zero since it includes 1. Consequently,
from linear algebra theory, we can infer that the determinant of the
|
f1_x(z) | f1_y(z) | f1_1(z) |
f2_x(z) | f2_y(z) | f2_1(z) |
f3_x(z) | f3_y(z) | f3_1(z)
|
matrix must be 0.
Notice that the determinant of the matrix is an expression solely in
z. This is how we construct a univariate expression Q(z) =
Det(matrix) such that every roots (x',y',z') of the original system
corresponds to a root z' of Q(z)
Resultant formulations include: