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Consider $n^2$ variables $x_1, \dots, x_{n^2}$ and let $p$ be the the determinant of the matrix $n \times n$ matrix formed by laying these variables in a square matrix, so that $p \in \mathbb{Z}[x_1, \dots, x_{n^2}]$ is a polynomial. (i.e., for $n=2$, $p = \det (\begin{smallmatrix}x_1 & x_2 \\ x_3 & x_4\end{smallmatrix}) = x_1 x_4 - x_2 x_3$).

We can also consider $p = p(x_1, \dots, x_{n^2})$ has a polynomial function. Let $H$ be the Hessian of $p(x_1, \dots, x_{n^2})$ (so $H$ is a $n^2 \times n^2$ matrix) and let $q$ be the determinant of $H$. Then $q$ is also an element of $\mathbb{Z}[x_1, \dots, x_{n^2}]$.

Question: what is the relationship between $p$ and $q$?

The conjecture is that $q$ is (up to constant) a power of $q$ in the polynomial ring. For example, I have checked that $q = -2p^3$ for $n = 3$.

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  • $\begingroup$ Really, what you write is incomprehensible. No one will answer your question if you do not clearly state your problem. Is there a hidden matrix? Its entries are in what? What are the variables you consider to calculate the Hessian? You have to rewrite everything. $\endgroup$
    – user91684
    Jul 4 '20 at 16:12
  • $\begingroup$ Done, hope it is clearer. I posed this question back when I was studying in college, so the wording not good the first time. $\endgroup$
    – Kevin
    Jul 4 '20 at 19:42

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