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Higher order Jacobians, Hessians and Milnor algebras. (English) Zbl 1448.13003
Let $$f \in R = \mathbb C[x_0,\dots,x_n]$$ be a reduced form. Associated to $$f$$ are a number of objects studied in this paper. The Jacobian ideal is $$J(f) = (\frac{\partial f}{\partial x_0},\dots,\frac{\partial f}{\partial x_n})$$. The associated hypersurface is $$V(f) \subset \mathbb P^n$$. The generators define the polar map $$\varphi_X : \mathbb P^n \dashrightarrow \mathbb P^n$$, and $$Z_X := \varphi(\mathbb P^n)$$ is the polar image of $$X$$. The restriction $$\varphi_X | X = \mathcal G_X$$ is the Gauss map of $$X$$. The Milnor algebra of $$f$$ is $$R/J(F)$$. Via Macaulay-Matlis duality the annihilator of $$f$$ is a homogeneous Gorenstein artinian algebra $$A(f)$$ in the dual ring $$Q = \mathbb C[X_0,\dots,X_n]$$. The Jacobian matrix associated to the polar map is the Hessian matrix of $$f$$. In this paper the authors generalize these objects to higher order, such as the $$k$$-th order Milnor algebra $$M^k(f) := R/J^k$$, and describe some geometry associated to them. They then study $$A(f)$$ and $$M^k(f)$$, for different choices of $$f$$ and $$k$$. In particular, they study the Hilbert function and Lefschetz properties of $$A(f)$$ and $$M^k(f)$$. They paint a nice picture of how these things all fit together.
##### MSC:
 13A02 Graded rings 05E40 Combinatorial aspects of commutative algebra 13D40 Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series 13E10 Commutative Artinian rings and modules, finite-dimensional algebras
CoCoA
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