On a class of power ideals

In this paper we study the class of power ideals generated by the $k^n$ forms $(x_0+\xi^{g_1}x_1+\ldots+\xi^{g_n}x_n)^{(k-1)d}$ where $\xi$ is a fixed primitive $k^{th}$-root of unity and $0\leq g_j\leq k-1$ for all $j$. For $k=2$, by using a $\mathbb{Z}_k^{n+1}$-grading on $\mathbb{C}[x_0,\ldots,x_n]$, we compute the Hilbert series of the associated quotient rings via a simple numerical algorithm. We also conjecture the extension for $k>2$. Via Macaulay duality, those power ideals are related to schemes of fat points with support on the $k^n$ points $[1:\xi^{g_1}:\ldots:\xi^{g_n}]$ in $\mathbb{P}^n$. We compute Hilbert series, Betti numbers and Gr\"obner basis for such $0$-dimensional schemes. This explicitly determines the Hilbert series of the power ideal for all $k$: that this agrees with our conjecture for $k>2$ is supported by several computer experiments

On the Hilbert function of general fat points in P1×P1\mathbb{P}^1 \times \mathbb{P}^1

We study the bi-graded Hilbert function of ideals of general fat points with same multiplicity in $\mathbb{P}^1\times\mathbb{P}^1$. Our first tool is the multiprojective-affine-projective method introduced by the second author in previous works with A.V. Geramita and A. Gimigliano where they solved the case of double points. In this way, we compute the Hilbert function when the smallest entry of the bi-degree is at most the multiplicity of the points. Our second tool is the differential Horace method introduced by J. Alexander and A. Hirschowitz to study the Hilbert function of sets of fat points in standard projective spaces. In this way, we compute the entire bi-graded Hilbert function in the case of triple points.Comment: 25 pages; minor changes (Remark 1.7 added and Example 3.13 improved

On generic and maximal k-ranks of binary forms

In what follows, we pose two general conjectures about decompositions of homogeneous polynomials as sums of powers. The first one (suggested by G. Ottaviani) deals with the generic k-rank of complex-valued forms of any degree divisible by k in any number of variables. The second one (by the fourth author) deals with the maximal k-rank of binary forms. We settle the first conjecture in the cases of two variables and the second in the first non-trivial case of the 3-rd powers of quadratic binary forms.Comment: 17 pages, 1 figur

On the strength of general polynomials

A slice decomposition is an expression of a homogeneous polynomial as a sum of forms with a linear factor. A strength decomposition is an expression of a homogeneous polynomial as a sum of reducible forms. The slice rank and strength of a polynomial are the minimal lengths of such decompositions, respectively. The slice rank is an upper bound for the strength and the gap between these two values can be arbitrary large. However, in line with a conjecture by Catalisano et al. on the dimensions of secant varieties of the varieties of reducible forms, we conjecture that equality holds for general forms. By using a weaker version of Fr\"oberg's Conjecture on the Hilbert series of ideals generated by general forms, we show that our conjecture holds up to degree $7$ and in degree $9$.Comment: 19 pages, final revisio

Hadamard-Hitchcock decompositions: identifiability and computation

A Hadamard-Hitchcock decomposition of a multidimensional array is a decomposition that expresses the latter as a Hadamard product of several tensor rank decompositions. Such decompositions can encode probability distributions that arise from statistical graphical models associated to complete bipartite graphs with one layer of observed random variables and one layer of hidden ones, usually called restricted Boltzmann machines. We establish generic identifiability of Hadamard-Hitchcock decompositions by exploiting the reshaped Kruskal criterion for tensor rank decompositions. A flexible algorithm leveraging existing decomposition algorithms for tensor rank decomposition is introduced for computing a Hadamard-Hitchcock decomposition. Numerical experiments illustrate its computational performance and numerical accuracy.Comment: 25 pages, 3 figure

Secant non-defectivity via collisions of fat points

Secant defectivity of projective varieties is classically approached via dimensions of linear systems with multiple base points in general position. The latter can be studied via degenerations. We exploit a technique that allows some of the base points to collapse together. We deduce a general result which we apply to prove a conjecture by Abo and Brambilla: for $c \geq 3$ and $d \geq 3$, the Segre-Veronese embedding of $\mathbb{P}^m\times\mathbb{P}^n$ in bidegree $(c,d)$ is non-defective.Comment: 36 pages, 4 pages, all comments are welcome