Homogeneous Polynomial or Projective Model¶
Definition¶
A homogeneous polynomial is a polynomial in which every term has the same total degree. For example, \(x^2 + 2xy + y^2\) is a homogeneous polynomial of degree 2.
To homogenize a polynomial:
\[
F^*(X,Y,Z) = Z^d F\left(\frac{X}{Z}, \frac{Y}{Z}\right) \quad \text{where } d = \text{deg}(F)
\]
Example¶
Given \(F(x, y) = 3x^3 - 2xy + 5y^2\):
- \(d = \text{deg}(F) = 3\)
- \(F^*(x, y, z) = z^d F\left(\frac{x}{z}, \frac{y}{z}\right)\)
Substitute the values:
\[
F^*=
z^3\left[3\left(\frac{x}{z}\right)^3 - 2\left(\frac{x}{z}\right)\left(\frac{y}{z}\right) + 5\left(\frac{y}{z}\right)^2\right]
\]
Simplify:
\[
F^*= 3x^3 - 2xyz + 5y^2z
\]