# Relationship between determinant and determinant of Hessian of determinant

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 $$n = 3$$.

• 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.
– user91684
Jul 4 '20 at 16:12
• Done, hope it is clearer. I posed this question back when I was studying in college, so the wording not good the first time. Jul 4 '20 at 19:42