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I will say that a homogeneous polynomial $P(X)\in\mathbb{R}[X]$ ($X=(X_1,\ldots,X_n)$) is elliptic if its zero locus in $\mathbb{R}^n$ is $\{0\}$.

I will say that the zero locus of a homogeneous polynomial $P(X)\in\mathbb{R}[X]$ in $\mathbb{C}^n$ is self-intersecting (on a semi-real line) if for some $x,y\in\mathbb{R}^n$ the polynomial $P(x+\lambda y)$ of complex variable $\lambda\in\mathbb{C}$ has a root of multiplicity higher than one.

Question: What is an example of a real elliptic homogeneous polynomial $P(X_1,\ldots,X_n)\in\mathbb{R}[X_1,\ldots,X_n]$, irreducible over $\mathbb{C}$, such that its zero locus in $\mathbb{C}^n$ is self-intersecting?

Discussion: If I remove the condition of ellipticity then $P(X,Y,Z)=Y^2Z-X^2Z-X^3$ gives a self-intersection along the real line $(0,0,1)+\lambda(0,1,0)$.

I have zero background in algebraic geometry, so please, be as specific as possible. Thank you.

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    $\begingroup$ As a user for several years, you should already know that cross-posting to MO within 24 hours of posting to MSE is probably not long enough to give the MSE users a chance to answer. FWIW, as I am not an algebraic geometer, if the question had only been posted on MO I would not necessarily have voted to move it to MSE; but since it started on MSE, it seems more appropriate to wait a while there before opening a duplicate on MO $\endgroup$
    – Yemon Choi
    Commented May 25, 2020 at 3:14
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    $\begingroup$ As a user for several years, I can bet that no-one is going to answer any of my questions in MSE. I am posting there first only as a tribute to the formal notion that the question may be indeed very elementary for a specialist in the subject, and they may prefer to answer it in MSE rather than in MO. But that is only a theoretical possibility and has never happened. If you answer this question on MSE, you are welcome to close it on MO. But as long as there is no answer in any of them, I see no reasoning in closing a question. $\endgroup$
    – Bedovlat
    Commented May 25, 2020 at 3:54
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    $\begingroup$ Fair enough. As it happens, I haven't downvoted or voted to close; my comment was prompted just by seeing the question come up in the "review votes to close" queue $\endgroup$
    – Yemon Choi
    Commented May 25, 2020 at 20:38

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