The strong Lefschetz property conjecture for zero-dimensional complete intersections

Let $S=\mathbb{C}[x_1,\ldots,x_n]$ with $n\geq 3$, and let $f_j\in S_{d_j}$ be homogeneous polynomials of degrees $d_j\geq 2$ such that

$$J({\bf f})=(f_1,\ldots,f_n)$$

is a zero-dimensional complete intersection. Set $M({\bf f})=S/J({\bf f})$, and let

$$T=\sum_{j=1}^n d_j-n$$

is its socle degree. Strong Lefschetz property conjecture. For any zero-dimensional homogeneous complete intersection ideal $J({\bf f})\subset S$, any integer $k\in[0,T/2)$, and any generic linear form $\ell\in S_1$, the induced multiplication map

$$\ell^{T-2k}:M({\bf f})_k\to M({\bf f})_{T-k}$$

is an isomorphism. The strong Lefschetz property is a central question about the multiplication structure of Artinian Gorenstein algebras; the paper investigates this statement through the discriminant of the associated Macaulay inverse system.