= Solution
Homogenizing the affine equation gives the <projective closure>
$$
X=V(F)\subset\mathbb P^2,
\qquad
F(X,Y,Z)=Y^3Z-X^4-Z^4.
$$
Its partial derivatives are
$$
F_X=-4X^3,
\qquad
F_Y=3Y^2Z,
\qquad
F_Z=Y^3-4Z^3.
$$
If all three vanished at a point of $X$, then $X=0$ and either $Y=0$ or $Z=0$. If $Y=0$, the equation $F=0$ forces $Z=0$; if $Z=0$, the equation $F_Z=0$ forces $Y=0$. Either conclusion gives the forbidden zero triple of <homogeneous coordinates>. The <Jacobian criterion> therefore shows that $X$ is a <smooth projective curve>; more specifically, it is the <projective closure of y cubed equals x to the fourth plus one>, a <smooth plane quartic>.
Back to article page
