How to convert $x^3+y^3-x^2+y-1=0$ to homogenous form using the variables $X,Y,Z$

Question

I'm to figure out how to convert algebraic curve such as $x^3+y^3-x^2+y-1=0$ to homogenous form using the variables $X,Y,Z$.

Any help will be appreciated!

Answer 1

If you start with $x = X$, $y = Y$, i.e. simply renaming the variables, then trivially

$$ X^3+Y^3−X^2+Y−1 = 0 $$

Now you multiply each term by either $Z^3, Z^2, Z^1$ or $Z^0 = 1$ so that the sum of the exponents of each term is $3$. So $X^3$ and $Y^3$ gets multiplied by $1$, $X^2$ gets multiplied by $Z$ and so on. You end up with

$$ X^3+Y^3−ZX^2+Z^2Y−Z^3 = 0 $$