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
fx1(y)
fx2(y)
0
fx0(y)  0  gx0(y)  0 
fx0(y)  gx1(y)  gx0(y) 
fx1(y)  gx2(y)  gx1(y) 
fx2(y)  0  gx2(y)

h1x0
h1x1
h2x0
h2x1
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/(xalpha) and let h2=f/(xalpha), 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 nonzero, 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
(5y^2  4y  1)
(6y^2 + 3y + 4)
0
(2y^2 + 3)  0  (y^2  4x  2)
 0 
(2y^2 + 3)  (3y^2  9y  7)  (y^2  4x  2)

(5y^2  4y  1)  (5y^2 + 7y + 5)  (3y^2
 9y  7) 
(6y^2 + 3y + 4)  0  (5y^2 + 7y + 5)
