Publications

The decomposition locus of a tensor is the set of all rank-one tensors appearing in a minimal tensor-rank decomposition of the tensor. In this paper, we compute decomposition loci of tensors living in tensor spaces with finitely many orbits with respect to the action of product of general linear groups.

We define and explicitly construct schemes evinced by generalized additive decompositions (GADs) of a given d-homogeneous polynomial F. We employ GADs to investigate the regularity of 0-dimensional schemes apolar to F, focusing on those satisfying some minimality conditions. We show that irredundant schemes to F need not be d-regular, unless they are evinced by special GADs of F. Instead, we prove that tangential decompositions of minimal length are always d-regular, as well as irredundant apolar schemes of length at most 2d + 1.

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.

We prove the existence of ternary forms admitting apolar sets of points of cardinality equal to the Waring rank, but having different Hilbert function and different regularity. This is done exploiting liaison theory and Cayley-Bacharach properties for sets of points in the projective plane

We consider the inverse problem for the polynomial map which sends an m-tuple of quadratic forms in n variables to the sum of their d-th powers. This map captures the moment problem for mixtures of m centered n-variate Gaussians. In the first non-trivial case d = 3, we show that for any n ∈ N, this map is generically one-to-one (up to permutations of q1,...,qm and third roots of unity) in two ranges: m ≤ {n \choose 2}+1 for n ≤ 16 and m ≤ {n+5 \choose 6} / {n+1 \choose 2} − {n+1 \choose 2} − 1 for n > 16, thus proving generic identifiability for mixtures of centered Gaussians from their (exact) moments of degree at most 6. The first result is obtained by the explicit geometry of the tangential contact locus of the variety of sums of cubes of quadratic forms, while the second result is accomplished using the link between secant non-defectivity with identifiability. The latter approach generalises also to sums of d-th powers of k-forms for d ≥ 3 and k

The strength of a homogeneous polynomial (or form) $f$ is the smallest length of an additive decomposition expressing it whose summands are reducible forms. We show that the set of forms with bounded strength is not always Zariski-closed. In particular, if the ground field has characteristic 0, we prove that the set of quartics with strength ⩽ 3 is not Zariski-closed for a large number of variables.

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 the strength of a polynomial are the minimal lengths of such decompositions, respectively. The slice rank is an upper bound for the strength and we observe that the gap between these two values can be arbitrarily large. However, in line with a conjecture by Catalisano et al. on the dimensions of the secant varieties of reducible forms, we conjecture that equality holds for general forms. By using a weaker version of Fröberg’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.

The X-rank of a point p in projective space is the minimal number of points of an algebraic variety X whose linear span contains p. This notion is naturally submultiplicative under tensor product. We study geometric conditions that guarantee strict submultiplicativity. We prove that in the case of points of rank two, strict submultiplicativity is entirely characterized in terms of the trisecant lines to the variety. Moreover, we focus on the case of curves: we prove that for curves embedded in an even-dimensional projective space, there are always points for which strict submultiplicativity occurs, with the only exception of rational normal curves.

We use an algebraic approach to construct minimal decompositions of symmetric tensors with low rank. This is done by using Apolarity Theory and by studying minimal sets of reduced points apolar to a given symmetric tensor, namely, whose ideal is contained in the apolar ideal associated to the tensor. In particular, we focus on the structure of the Hilbert function of these ideals of points. We give a procedure which produces a minimal set of points apolar to any symmetric tensor of rank at most 5. This procedure is also implemented in the algebra software Macaulay2.

We study the bi-graded Hilbert function of ideals of general fat points with same multiplicity in ℙ1×ℙ1. 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.

We introduce the monic rank of a vector relative to an affine-hyperplane section of an irreducible Zariski-closed affine cone X. We show that the monic rank is finite and greater than or equal to the usual X-rank. We describe an algorithmic technique based on classical invariant theory to determine, in concrete situations, the maximal monic rank. Using this technique, we establish three new instances of a conjecture due to B. Shapiro which states that a binary form of degree d⋅e is the sum of d d-th powers of forms of degree e. Furthermore, in the case where X is the cone of highest weight vectors in an irreducible representation---this includes the well-known cases of tensor rank and symmetric rank---we raise the question whether the maximal rank equals the maximal monic rank. We answer this question affirmatively in several instances.

In a recent paper, Cook et al. (Compos Math 154:2150–2194, 2018) used the splitting type of a line arrangement in the projective plane to study the number of conditions imposed by a general fat point of multiplicity j on the linear system of curves of degree 𝑗+1j+1 passing through the configuration of points dual to the given arrangement. If the number of conditions is less than the expected, they said that the configuration of points admits unexpected curves. In this paper, we characterize supersolvable line arrangements whose dual configuration admits unexpected curves and we provide other infinite families of line arrangements with this property.

We compute the dimensions of all the secant varieties to the tangential varieties of all Segre–Veronese surfaces. We exploit the typical approach of computing the Hilbert function of special 0-dimensional schemes on projective plane by using a new degeneration technique.

A symmetric tensor may be regarded as a partially symmetric tensor in several different ways. These produce different notions of rank for the symmetric tensor which are related by chains of inequalities. We show how the study of the simultaneous symmetric rank of partial derivatives of the homogeneous polynomial associated to the symmetric tensor can be used to prove equalities among different partially symmetric ranks. We apply this to the special cases of binary forms, ternary and quaternary cubics, monomials, and elementary symmetric polynomials.

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.

In earlier joint work with our collaborators Akhtar, Coates, Corti, Heuberger, Kasprzyk, Prince and Tveiten, we gave a conjectural classification of a broad class of orbifold del Pezzo surfaces, using Mirror Symmetry. We proposed that del Pezzo surfaces X with isolated cyclic quotient singularities such that X admits a ℚ-Gorenstein toric degeneration correspond under Mirror Symmetry to maximally mutable Laurent polynomials f in two variables, and that the quantum period of such a surface X, which is a generating function for Gromov-Witten invariants of X, coincides with the classical period of its mirror partner f. In this paper, we prove a large part of this conjecture for del Pezzo surfaces with 13(1,1) singularities, by computing many of the quantum periods involved. Our tools are the Quantum Lefschetz theorem and the Abelian/non-Abelian Correspondence; our main results are contingent on, and give strong evidence for, conjectural generalizations of these results to the orbifold setting.

Let F be a homogeneous form of degree d in n variables. A Waring decomposition of F is a way to express F as a sum of dth powers of linear forms. In this paper we consider the decompositions of a form as a sum of expressions, each of which is a fixed monomial evaluated at linear forms.

The Waring locus of a form F is the collection of the degree one forms appearing in some minimal sum of powers decomposition of F. In this paper, we give a complete description of Waring loci for several family of forms, such as quadrics, monomials, binary forms and plane cubics. We also introduce a Waring loci version of Strassen's Conjecture, which implies the original conjecture, and we prove it in many cases.

In this paper we study the real rank of monomials and we give an upper bound for the real rank of all monomials. We show that the real and the complex ranks of a monomial coincide if and only if the least exponent is equal to one.

We state a number of conjectures that together allow one to classify a broad class of del Pezzo surfaces with cyclic quotient singularities using mirror symmetry. We prove our conjectures in the simplest cases. The conjectures relate mutation-equivalence classes of Fano polygons with Q-Gorenstein deformation classes of del Pezzo surfaces.

In this paper we study a class of power ideals, i.e., ideals generated by powers of linear forms. We compute the Hilbert series of the associated quotient rings via a simple numerical algorithm. Via Macaulay duality, those power ideals are related to  a scheme of fat points. We compute Hilbert series, Betti numbers and Gröbner 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.

Motivated by recent results on the Waring problem for polynomial rings and representation of monomial as sum of powers of linear forms, we consider the problem of presenting monomials of degree kd as sums of k-th powers of forms of degree d. We produce a general bound on the number of summands for any number of variables which we refine in the two variables case. We completely solve the k=3 case for monomials in two and three variables.

We study Minkowski sums and Hadamard products of algebraic varieties. Specifically we explore when these are varieties and examine their properties in terms of those of the original varieties.

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 only these varieties are among the ones that have been most classically studied, but a strong motivation in taking them into consideration is the fact that they parameterize, respectively, symmetric, skew-symmetric and general tensors, which are decomposable, and their secant varieties give a stratification of tensors via tensor rank. We collect here most of the known results and the open problems on this fascinating subject.

We present a number of questions in commutative algebra posed on the problem solving seminar in algebra at Stockholm University during the period Fall 2014 - Spring 2017.