Let $K$ be a field, $\overline{K}$ an algebraic closure of $K$ and $F \in K[X,Y,Z]$ a homogeneous polynomial of degree $d$. Let $F'_X, F'_Y, F'_Z$ denote the partial derivatives of $F$ and let $I=\langle F, F'_X, F'_Y, F'_Z \rangle$.
Let's suppose that there is an exponent $e$ such that $X^e, Y^e, Z^e \in I$. What are the common zeroes of $F$ and its partial derivatives in $\overline{K}\times \overline{K} \times \overline{K}$? I thank you in advance for any suggestions.
For an ideal $J \subseteq K[X,Y,Z]$, let $Z(J)$ denote the zero-set of $J$, i.e. $Z(J):=\{ x \in \overline{K}^3: f(x)=0 \forall f \in J\}$. Then for ideals $J_1, J_2$ such that $J_1 \subseteq J_2$, we clearly have $Z(J_2)\subseteq Z(J_1)$. In our case, note that $J:=\langle X^e, Y^e, Z^e\rangle$ satisfies $Z(J)=\{(0,0,0)\}$ and $J\subseteq I$, so we get $Z(I) \subseteq \{(0,0,0)\}$.
As KReiser pointed out, both possibilities $Z(I) = \emptyset$ and $Z(I)=\{(0,0,0)\}$ do occur.