The aim of my talk is to introduce the border apolarity idea toegether with the tools necessary for its proof. I will recall the setting of border apolarity as it was done my joint paper with Jarek Buczynski. There we have formulated a version of apolarity lemma for a toric variety embedded via very ample line bundle and have proved it in the characteristic zero case. The main tool is to use...

The cactus variety of a projective variety X is a version of the secant variety, where we take into account the linear spans of all finite subschemes of bounded length, not only the smooth ones or smoothable ones. I will discuss the definitions and basic properties of cactus rank and cactus varieties, with a particular focus on why they are relevant as an obstruction to study secant varieties....

The first step in lower bounding the border rank of a tensor or polynomial

with border apolarity is to enumerate all ideals contained in the

annihilator with Hilbert series equal to the Hilbert series of an ideal of

general points. The second step requires determining whether any such ideal

may be deformed to an ideal of points. Typically, one simplifies these

questions by asking if there...

The slice technique is a tool which let use to translate the question about rank (or border rank) of a tensor in to the analogue question about the subspace spanned by tensors of a smaller order. The technique works in the case of a rank and border rank, but not for cactus and border cactus rank. Gesmundo, Oneto and Ventura gave an example of a family of forms such that their simultaneous...

In this talk and the next one we will revise the algorithm for polynomial

decomposition originally proposed by Brachat-Comon-Mourrain-Tsidgaridas and

we will show how we can improve it. Then we will see how certain

modifications to the algorithm can lead to a cactus decomposition.

Some of the classical tensor decomposition algorithms are based on the ability of solving particular zero-dimensional polynomial system, defining the set of points of the decomposition. Generalized eigenvalue methods can be used for this task, and their complexity is controlled by the regularity of certain associated ideals, which are often non-saturated. We determine these regularity values...

One measure of the complexity of a tensor is its border rank.

Finding the border rank of a tensor, or even bounding it, is a difficult problem that is currently an area of active research, as several problems in theoretical computer science come down to determining the border ranks of certain tensors.

For a class of $GL(V)$-invariant tensors lying in a $GL(V)$-invariant space $V\otimes...

Let $C_n(M_d)$ denote the affine variety of all $n$-tuples of commuting $d\times d$ matrices. The ADHM construction relates these varieties to Quot schemes, and in particular to Hilbert schemes. On the more applied side, varieties $C_n(M_d)$ are directly connected to the question whether a tensor has minimal border rank. Although $C_n(M_d)$ is usually reducible for $n>2$ and $d>3$, very few...

I will present the study of minimal cactus rank with respect to Veronese variety, Segre variety, and Segre-Veronese variety using an approach complementary to the one taken by Blaeser and Lysikov, and Jelisiejew, Pal, and Landsberg. I will analyze the case of 14th cactus variety in more detail.

I will present some necessary conditions for a point of a multigraded Hilbert scheme corresponding to r points in general position in a smooth projective complex toric variaty to be in the Slip component. These criteria can be used to classify irreducible multigraded Hilbert schemes corresponding to points in general position in the product of projective spaces.

Determining the rank of the powers of quadratic forms is a classical problem. Many examples of special decompositions appear in the literature. We analyze this problem from a modern point of view and we give an estimate of the value of the rank. Moreover, we determine its smoothable rank and its border rank.