# Higher order Jacobians, Hessians and Milnor algebras

## TL;DR

This paper introduces higher order algebraic objects related to hypersurfaces, exploring their properties, relationships to Gorenstein algebras, and implications for Hilbert functions and Lefschetz properties.

## Contribution

It defines and studies higher order Jacobian ideals, Hessians, polar maps, and Milnor algebras, connecting them to classical algebraic structures and properties.

## Key findings

- Higher order Jacobian ideals and Hessians are systematically defined and analyzed.
- Connections established between higher order Milnor algebras and Artinian Gorenstein algebras.
- Results on Hilbert functions and Lefschetz properties for these higher order structures.

## Abstract

We introduce and study higher order Jacobian ideals, higher order and mixed Hessians, higher order polar maps, and higher order Milnor algebras associated to a reduced projective hypersurface. We relate these higher order objects to some standard graded Artinian Gorenstein algebras, and we study the corresponding Hilbert functions and Lefschetz properties.