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## On the Partial derivatives of determinant of square Hankel matrix and its degenerations

The generic Hankel matrix is the "super"-symmetric matrix of the following form. This type of matrix is the main member of the family of $1$-generic matrices $$\left( \begin{matrix} x_1&x_2&\ldots &x_{m-1} & x_{m}\\ x_2&x_3&\ldots &x_{m}& x_{m+1}\\ \vdots &\vdots &\ldots &\vdots \\ x_{m-1}&x_{m}&\ldots &x_{2m-3} &x_{2m-2}\\ x_{m}&x_{m+1}&\ldots &x_{2m-2}&x_{2m-1} \\ \end{matrix} \right). \;$$ By degeneration of the Hankel matrix, we mean to set all the last $r$ variables zero, whenever $r$ is varying from $1$ to $m-2$.
Suppose that $f$ is the determinant of the Hankel matrix or its degenerations. We consider the polar map defined by $f$ and study the properties of this map through the Hessian matrix and ideal of sub-maximal minors. Homaloidalness is our target.
Throughout one deals with the effect of the degenerateness on the numerical invariants and ideal theoretic properties of the gradient ideal of $f$. Among others are Reduction Number, minimal reduction, codimension, Cohen-Macaulayness, and Normality.