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Alessandro Oneto

Publications and source records attributed to Alessandro Oneto.

36 records · Page 2Linked to original sources
arXiv2019

Partially symmetric variants of Comon's problem via simultaneous rank

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. By exploiting algebraic tools such as apolarity theory, 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. This approach aims to understand to what extent the symmetries of a tensor affect its rank. We apply this to the special cases of binary forms, ternary and quaternary cubics, monomials, and elementary symmetric polynomials.

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arXiv2019

The monic rank

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 $d\cdot e$ 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.

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arXiv2019

Border Waring Rank via Asymptotic Rank

We investigate an extension of a lower bound on the Waring (cactus) rank of homogeneous forms due to Ranestad and Schreyer. We show that for particular classes of homogeneous forms, for which a generalization of this method applies, the lower bound extends to the level of border (cactus) rank. The approach is based on recent results on tensor asymptotic rank.

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arXiv2018

The Hitchhiker guide to: Secant Varieties and Tensor Decomposition

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.

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arXiv2018

On minimal decompositions of low rank symmetric tensors

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.

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arXiv2018

Tangential varieties of Segre-Veronese surfaces are never defective

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.

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arXiv2018

Unexpected curves arising from special line arrangements

In a recent paper arXiv:1602.02300v2, Cook II, Harbourne, Migliore and Nagel related the splitting type of a line arrangement in the projective plane to the number of conditions imposed by a general fat point of multiplicity $j$ to the linear system of curves of degree $j+1$ passing through the configuration of points dual to the given arrangement. If the number of conditions is less than the expected, we say 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.

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arXiv2017

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

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.

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arXiv2017

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.

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arXiv2017

Waring-like decompositions of polynomials - 1

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 $d^{th}$ 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.

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arXiv2017

Waring loci and the Strassen conjecture

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.

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arXiv2016

On the real rank of monomials

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.

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arXiv2015

On the quantum periods of del Pezzo surfaces with $\frac{1}{3}(1,1)$ singularities

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 $\mathbb{Q}$-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 $\frac{1}{3}(1,1)$ 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.

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arXiv2015

Mirror Symmetry and the Classification of Orbifold del Pezzo Surfaces

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.

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arXiv2014

On a class of power ideals

In this paper we study the class of power ideals generated by the $k^n$ forms $(x_0+ξ^{g_1}x_1+\ldots+ξ^{g_n}x_n)^{(k-1)d}$ where $ξ$ 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 $[1:ξ^{g_1}:\ldots:ξ^{g_n}]$ in $\mathbb{P}^n$. 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.

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arXiv2013

Monomials as sum of k-th powers of forms

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.

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