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Session S07 - Differential operators in algebraic geometry and commutative algebra

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

Maral Mostafazadehfard

Federal University of Rio de Janeiro, Brazil   -   This email address is being protected from spambots. You need JavaScript enabled to view it.

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.

Joint work with Rainelly Cunha (Federal institute of education, science, and technology of Rio Grande do Norte), Zaqueu Ramos (Universidade Federal de Sergipe) and Aron Simis (Universidade Federal de Pernambuco).

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