Sylvester Resultant Formulation
The Sylvester
resultant formulation only handles the case of two polynomials:
f(x1,x2), g(x1,x2).
The sylvester resultant is based upon the following observation: we
can symbolically multiply f by another polynomial h1, and multiply g
by another polynomial h2 using linear algebra. Furthermore, if h1 has
degree 1 less than g and h2 has degree one less than f, then f and g
must share a common root. The reason we make the "degree 1 less than"
requirement is that we are going to compute the difference
f(x,y)h1(x,y) - g(x,y)h2(x,y) and assume it is equal to 0; but, this
sum is only interesting if h1 has degree less than g and h2 has degree
less than f because otherwise, we could simply let h1 be equal to (or
a product of) g and h2 be equal to (or a product of) f, and achieve
the condition that f(x,y)h1(x,y) - g(x,y)h2(x,y) is 0.
Consider the following two polynomials:
f(x,y) = x^2 (6y^2 + 3y + 4) + x (5y^2 - 4y - 1) + (2y^2 + 3)
g(x,y) = x^2 (5y^2 + 7y + 5) + x (3y^2 - 9y - 7) + (y^2 - 4x - 2)
which we can rewrite as:
f(x,y) = x^2 fx2 + x fx1 + fx0
g(x,y) = x^2 gx2 + x gx1 + gx0
Consider two functions h1 = x h1x1 + h1x0 and h2 = x h2x1 + h2x0,
which are functions solely of x (not y). Furthermore, consider the
difference of their products f(x,y)h1(x,y) - g(x,y)h2(x,y) which can
be written in linear algebra form as
fx0(y) | 0 | gx0(y) | 0 |
fx1(y) fx0(y) | gx1(y) | gx0(y) |
fx2(y) fx1(y) | gx2(y) | gx1(y) |
0 fx2(y) | 0 | gx2(y)
|
If f(x',y')=g(x',y')=0, then, for some y, f(x') and g(x') share a
common root, let's call alpha. Since f(x') and g(x') share a common
root, then let h1=g/(x-alpha) and let h2=f/(x-alpha), and in this
case, the linear algebra product f(x,y)h1(x,y) - g(x,y)h2(x,y) will be
[0, 0, 0, 0].
Again, by Nullstellenstatz, if the linear algebra product is [0, 0, 0,
0], and since the column vector is non-zero, it must be the case that
the determinant of the matrix is 0. Consequently, we have constructed
a univariate polynomial (Det(Matrix(y))) whose roots (y') satisfy the
property that any common root of (f(x,y),g(x,y)) is also a root of Q(y).
In the above example, the matrix would be
(2y^2 + 3) | 0 | (y^2 - 4x - 2)
| 0 |
(5y^2 - 4y - 1) (2y^2 + 3) | (3y^2 - 9y - 7) | (y^2 - 4x - 2)
|
(6y^2 + 3y + 4) (5y^2 - 4y - 1) | (5y^2 + 7y + 5) | (3y^2
- 9y - 7) |
0 (6y^2 + 3y + 4) | 0 | (5y^2 + 7y + 5)
|