Recently I have been studying projective geometry. When investigating a polynomial it can be continued to the projective space by homogenizing it. So for example:
$$ F(x, y) = x^2+xy+1 \quad \longrightarrow \quad \bar{F}(x_0, x_1, x_2) = x_1^2 + x_1x_2 + x_0^2 $$
I understand that this is very useful, because it preserves the roots.
Anyway I am kind of puzzled by the following definition:
$$ \bar{F}(x_0,x_1,x_2) := x_0^d F(\frac{x_1}{x_0},\frac{x_2}{x_0}) $$ where d denotes the degree of the polynomial.
Isn't this definition invalid for $ x_0 = 0$ or am I missing an essential part?
When you define $\overline F(x_0,x_1,x_2)$, the $x_i$s are indeterminates, which should be denoted $X_i$ to avoid confusion, they're not elements of the base field. Hence it is meaningless to set $x_0=0$.
But setting values to replace them yields (projective) points of the projective curve. Denoting the values with lowercase letters, if $x_0\ne 0$, there corresponds to the point $[x_0:x_1:x_2]$ satisfying the homogeneous equation $\overline F(x_0,x_1,x_2)=0$, a point in the affine space: $(y_1,y_2)=\Bigl(\dfrac{x_1}{x_0},\dfrac{x_2}{x_0}\Bigr)$, satisfying the equation $F(y_1,y_2)=0$.
If $x_0=0$, you obtain the ‘points at $\infty$ on the affine curve’. For instance the affine curve $x^2+xy+1=0$ has two points at $\infty$, defined by the homogeneous part of $F$ of highest degree: $x^2+xy=x(x+y)=0$. One is the point at $\infty$ on the $y$-axis ($x=0$), the other is the point at $\infty$ on the line $x+y=0$.