Related papers: Finiteness of Hadamard ranks

For tensors of fixed order, we establish three types of upper bounds for the geometric rank in terms of the subrank. Firstly, we prove that, under a mild condition on the characteristic of the base field, the geometric rank of a tensor is…

We exhibit, for each even degree, a ternary form of rank strictly greater than the maximum rank of monomials. Together with an earlier result in the odd case, this gives the lower bound…

We study the arithmetical ranks and the cohomological dimensions of an infinite class of Cohen-Macaulay varieties of minimal degree. Among these we find, on the one hand, infinitely many set-theoretic complete intersections, on the other…

There are close relations between tripartite tensors with bounded geometric ranks and linear determinantal varieties with bounded codimensions. We study linear determinantal varieties with bounded codimensions, and prove upper bounds of the…

We consider here the problem, which is quite classical in Algebraic geometry, of studying the secant varieties of a projective variety $X$. The case we concentrate on is when $X$ is a Veronese variety, a Grassmannian or a Segre variety. Not…

We give an upper bound for the cactus rank of any multi-homogeneous polynomial.

We prove (with a mild restriction on the multidegrees) that all secant varieties of Segre-Veronese varieties with $k>2$ factors, $k-2$ of them being $\mathbb{P}^1$, have the expected dimension. This is equivalent to compute the dimension of…

The real rank two locus of an algebraic variety is the closure of the union of all secant lines spanned by real points. We seek a semi-algebraic description of this set. Its algebraic boundary consists of the tangential variety and the edge…

We extend to characteristic $2$ and $3$ the classification of projective homogeneous varieties of Picard group isomorphic to $\mathbf{Z}$, corresponding to parabolic subgroup schemes with maximal reduced subgroup. The latter are all…

We show the validity of select existence results for a vector optimization problem, and a variational inequality. More generally, we consider generalized vector quasi-variational inequalities, as well as, fixed point problems on genuine…

We determine normal forms and ranks of tensors of border rank at most three. We present a differential-geometric analysis of limits of secant planes in a more general context. In particular there are at most four types of points on limiting…

We survey variety theory for modules of finite dimensional Hopf algebras, recalling some definitions and basic properties of support and rank varieties where they are known. We focus specifically on properties known for classes of examples…

Provably finding stationary points on bounded-rank tensors turns out to be an open problem [E. Levin, J. Kileel, and N. Boumal, Math. Program., 199 (2023), pp. 831--864] due to the inherent non-smoothness of the set of bounded-rank tensors.…

This is a survey primarily about determining the border rank of tensors, especially those relevant for the study of the complexity of matrix multiplication. This is a subject that on the one hand is of great significance in theoretical…

The Hadamard product of two power series $\sum a_n z^n$ and $\sum b_n z^n$ is the power series $\sum a_n b_n z^n$. We define the (Hadamard) grade of a power series $A$ to be the least number (finite or infinite) of algebraic power series,…

We determine the lower rank of the direct product of finitely many hereditarily just infinite profinite groups of finite lower rank.

We show that in a finite tensor category, the tensor product property holds for support varieties if and only if it holds between indecomposable periodic objects. We apply this to certain Hopf algebras in the form of skew group algebras. In…

In many cases (e.g. for many Segre or Segre embeddings of multiprojective spaces) we prove that a hypersurface of the $b$-secant variety of $X\subset \mathbb {P}^r$ has $X$-rank $>b$. We prove it proving that the $X$-rank of a general point…

Matroid varieties are the closures in the Grassmannian of sets of points defined by specifying which Pl\"ucker coordinates vanish and which don't --- the set of nonvanishing Pl\"ucker coordinates forms a well-studied object called a…

Classes of algebraic structures that are defined by equational laws are called varieties or equational classes. A variety is finitely generated if it is defined by the laws that hold in some fixed finite algebra. We show that every…