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Apolarity, Hessian and Macaulay polynomials. (English) Zbl 1262.14069
Let \(f\in R = C[x_0, \dots , x_n]\) be a homogeneous polynomial. The ideal \(J(f)\) generated by the partial derivatives of \(f\) is called the Jacobian ideal, or the gradient ideal, of \(f\). In the smooth case this ideal contains a power of the irrelevant ideal, so it has maximum depth in the coordinates ring and it is generated by a regular sequence. The associated ring \(R(f) = R/J(f)\), the so-called Jacobian ring of \(f\), is an Artinian Gorenstein graded ring.
Apolarity allows us to associate an Artinian Gorenstein graded ring to a form. This nice property is described in a classical theorem due to Macaulay (Theorem 2.2): there exists a homogeneous polynomial (the Macaulay poly nomial) \(g\) such that \(J(f)\) is equal to \(g^\perp\), where \(g^\perp\subset T =C[\frac{\delta}{\delta x_0} , \dots, \frac{\delta}{\delta x_n}]\) (upon identifying \(x_i\) with \(\frac{\delta}{\delta x_i}\) ). It is not so immediate to compute by hand the Macaulay polynomial associated to a given Artinian Gorenstein graded ring, but in the case of the Jacobian ring it seems natural to look at the Hessian polynomial \(\mathrm{Hess}(f)\) of \(f\) since it has the right degree. It immediately turns out that if \(f\) is a Fermat polynomial, then \(\mathrm{Hess}(f)\) and the Macaulay polynomial associated to \(R(f)\) coincide, up to scalars (see Example 3.1). Therefore the authors ask if \(\mathrm{Hess}(f)\) is always the Macaulay polynomial (up to scalar multiplication) associated to \(f\).
In Section 4 the authors study the question for binary forms, giving a complete answer for forms of degree 3 and 4. In Section 5 they completely answer the question in the plane cubics case. Since any smooth cubic in \(P^2\) is projectively equivalent to a cubic in the Hasse pencil \(f_a(x, y, z) = x^3 + y^3 + z^3- 3axyz\) for certain \(a\in C,a^3\neq 1,\) we can reduce the problem to the analysis of the cubics in the Hasse pencil.
In Section 6, the authors give some examples about the relationship between the Hessian and the Macaulay polynomials for plane quartics. In Section 7 they use the computer algebra system CoCoA to attack this problem.

14N15 Classical problems, Schubert calculus
14J70 Hypersurfaces and algebraic geometry
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