I've got a very simple question about a homogenous polynomial, for which I cannot see neatly how to proceed (probably due to my limitations in algebraic geometry though). Any help would be greatly appreciated.


So, let $p:\mathbb R^4\to \mathbb R$ be a homogeneous polynomial of degree 4, and assume that $p$ has a non-trivial zero in $\mathbb R^4$. Is it true then that there exists $x\_0\in \mathbb R^4$ such that $p(x\_0)=0$ but $\nabla\_{x\_0}p$ is not the zero vector?


Thank you very much for all your replies.


**Edit:** Ok I need one more hypothesis that I missed: the polynomial $p$ takes both signs, otherwise any square would be a trivial counterexample, as pointed out in the comments.



Homogeneous polynomial in 4 variable with non degenerate zero
This is indeed true. One can drop the assumption on $p$ to be homogeneous and dimension $4$ is not relevant, though the condition $\deg(p)=4$ is crucial. The statement follows from the following facts.


**Fact 1.** Let $q(x)$ be a polynomial in one variable $x$ such that $\deg(q)=4$ and $q$ changes sign. Then there is a root $x\_0$ of $q$ such that $q'(x\_0)\ne 0$.


*Proof.* Let us take the root $y\_0$ of $q$ at which $q$ changes sign. If $q'(y\_0)\ne 0$ we are done. If not then $y\_0$ is a root of multiplicity $3$. It follows that $q$ has a different root of multiplicity $1$, and we are done. QED


**Fact 2.** Let $p$ be a polynomial of degree $d$ on $\mathbb R^n$.
Then any line $L$ in $\mathbb R^n$ can be slightly perturbed so that the restriction of $P$ to it has degree $d$.


*Proof.* This follows from the fact that the subset of lines in $\mathbb R^n$ to which $p$ restricts as a polynomial of degree $<d$ belongs to a hypersurface. (if needed I can give more details). QED


Now, let us deduce the statement. By the assumptions we know that there exists two points $x\_1$ and $x\_2$ in $\mathbb R^4$ such that $p(x\_1)>0>p(x\_2)$. Let $L$ be the line through $x\_1,\,x\_2$. By Fact 2 we can perturb it a bit to a line $L'$ so that $p$ restricts to $L$ as a degree $4$ polynomial, which changes sing. Now apply Fact 1. 


