Alessandra Bernardi - Academia.edu (original) (raw)

Papers by Alessandra Bernardi

Research paper thumbnail of The Hitchhiker Guide to: Secant Varieties and Tensor Decomposition

Mathematics, 2018

We consider here the problem, which is quite classical in Algebraic geometry, of studying the sec... more 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.

Research paper thumbnail of High order singular value decomposition for plant diversity estimation

Bollettino dell'Unione Matematica Italiana

We propose a new method to estimate plant diversity with Rényi and Rao indexes through the so cal... more We propose a new method to estimate plant diversity with Rényi and Rao indexes through the so called High Order Singular Value Decomposition (HOSVD) of tensors. Starting from NASA multi-spectral images we evaluate diversity and we compare original diversity estimates with those realized via the HOSVD compression methods for big data. Our strategy turns out to be extremely powerful in terms of memory storage and precision of the outcome. The obtained results are so promising that we can support the efficiency of our method in the ecological framework.

Research paper thumbnail of Waring, tangential and cactus decompositions

Journal de Mathématiques Pures et Appliquées

Research paper thumbnail of Foreword: Special issue of JSC on the occasion of MEGA 2019

Journal of Symbolic Computation

Research paper thumbnail of Geometric conditions for strict submultiplicativity of rank and border rank

Annali di Matematica Pura ed Applicata (1923 -)

Research paper thumbnail of Skew Symmetric Tensor Decomposition

Communications in Contemporary Mathematics

We introduce the “skew apolarity lemma” and we use it to give algorithms for the skew-symmetric r... more We introduce the “skew apolarity lemma” and we use it to give algorithms for the skew-symmetric rank and the decompositions of tensors in [Formula: see text] with [Formula: see text] and [Formula: see text]. New algorithms to compute the rank and a minimal decomposition of a tritensor are also presented.

Research paper thumbnail of Bounds on the tensor rank

Annali di Matematica Pura ed Applicata (1923 -)

We give a sufficient criterion for a lower bound of the cactus rank of a tensor. Then we refine t... more We give a sufficient criterion for a lower bound of the cactus rank of a tensor. Then we refine that criterion in order to be able to give an explicit sufficient condition for a non-redundant decomposition of a tensor to be minimal and unique.

Research paper thumbnail of A new class of non-identifiable skew-symmetric tensors

Annali di Matematica Pura ed Applicata (1923 -)

We prove that the generic element of the fifth secant variety σ 5 (Gr(P 2 , P 9)) ⊂ P(3 C 10) of ... more We prove that the generic element of the fifth secant variety σ 5 (Gr(P 2 , P 9)) ⊂ P(3 C 10) of the Grassmannian of planes of P 9 has exactly two decompositions as a sum of five projective classes of decomposable skew-symmetric tensors. We show that this, together with Gr(P 3 , P 8), is the only non-identifiable case among the non-defective secant varieties σ s (Gr(P k , P n)) for any n < 14. In the same range for n, we classify all the weakly defective and all tangentially weakly defective secant varieties of any Grassmannians. We also show that the dual variety (σ 3 (Gr(P 2 , P 7))) ∨ of the variety of 3-secant planes of the Grassmannian of P 2 ⊂ P 7 is σ 2 (Gr(P 2 , P 7)) the variety of bi-secant lines of the same Grassmannian. The proof of this last fact has a very interesting physical interpretation in terms of measurement of the entanglement of a system of 3 identical fermions, the state of each of them belonging to a 8-th dimensional "Hilbert" space.

Research paper thumbnail of A uniqueness result on the decompositions of a bi-homogeneous polynomial

Linear and Multilinear Algebra

In the first part of this paper we give a precise description of all the minimal decompositions o... more In the first part of this paper we give a precise description of all the minimal decompositions of any bi-homogeneous polynomial p (i.e. a partially symmetric tensor of S d 1 V 1 ⊗ S d 2 V 2 where V 1 , V 2 are two complex, finite dimensional vector spaces) if its rank with respect to the Segre-Veronese variety S d 1 ,d 2 (V 1 , V 2) is at most min{d 1 , d 2 }. Such a polynomial may not have a unique minimal decomposition as p = r i=1 λ i p i with p i ∈ S d 1 ,d 2 (V 1 , V 2) and λ i coefficients, but we can show that there exist unique p 1 , .

Research paper thumbnail of Tensor decomposition and homotopy continuation

Differential Geometry and its Applications

A computationally challenging classical elimination theory problem is to compute polynomials whic... more A computationally challenging classical elimination theory problem is to compute polynomials which vanish on the set of tensors of a given rank. By moving away from computing polynomials via elimination theory to computing pseudowitness sets via numerical elimination theory, we develop computational methods for computing ranks and border ranks of tensors along with decompositions. More generally, we present our approach using joins of any collection of irreducible and nondegenerate projective varieties X1,. .. , X k ⊂ P N defined over C. After computing ranks over C, we also explore computing real ranks. A variety of examples are included to demonstrate the numerical algebraic geometric approaches.

Research paper thumbnail of Computing the cactus rank of a general form

Using Macaulay's correspondence we study the family of Artinian Gorenstein local algebras with fi... more Using Macaulay's correspondence we study the family of Artinian Gorenstein local algebras with fixed symmetric Hilbert function decomposition. As an application we give a new lower bound for cactus varieties of the third Veronese embedding. We discuss the case of cubic surfaces, where interesting phenomena occur.

Research paper thumbnail of Curvilinear schemes and maximum rank of forms

We define the \emph{curvilinear rank} of a degree ddd form PPP in n+1n+1n+1 variables as the minimum ... more We define the \emph{curvilinear rank} of a degree ddd form PPP in n+1n+1n+1 variables as the minimum length of a curvilinear scheme, contained in the ddd-th Veronese embedding of mathbbPn\mathbb{P}^nmathbbPn, whose span contains the projective class of PPP. Then, we give a bound for rank of any homogenous polynomial, in dependance on its curvilinear rank.

Research paper thumbnail of Higher Secant Varieties of ℙ n × ℙ1 Embedded in Bi-Degree (a, b)

Http Dx Doi Org 10 1080 00927872 2011 595748, Sep 11, 2012

Research paper thumbnail of Tensor rank on tangent developable of Segre varieties

Research paper thumbnail of Varieties parameterizing forms and their secant varieties

Research paper thumbnail of Schemi 0-dimensionali e forme canoniche di polinomi omogenei

Research paper thumbnail of Varietà che parametrizzano forme e loro varietà delle secanti

Bollettino Dell Unione Matematica Italiana Sezione a La Matematica Nella Societa E Nella Cultura, Aug 1, 2007

Research paper thumbnail of Higher secant varieties of Pn x Pm embedded in bi-degree (1,d)

Journal of Pure and Applied Algebra, 2011

Research paper thumbnail of Decomposition of homogeneous polynomials over an algebraically closed field

Let F be a homogeneous polynomial of degree d in m + 1 variables defined over an algebraically cl... more Let F be a homogeneous polynomial of degree d in m + 1 variables defined over an algebraically closed field of characteristic 0 and suppose that F belongs to the s-th secant variety of the d-uple Veronese embedding of P m into P (m+d d)−1 but that its minimal decomposition as a sum of d-th powers of linear forms M 1 ,. .. , M r is F = M d 1 +• • •+M d r with r > s. We show that if s+r ≤ 2d+1 then such a decomposition of F can be split in two parts: one of them is made by linear forms that can be written using only two variables, the other part is uniquely determined once one has fixed the first part. We also obtain a uniqueness theorem for the minimal decomposition of F if r is at most d and a mild condition is satisfied.

Research paper thumbnail of Some defective secant varieties to osculating varieties of Veronese surfaces

Collectanea Mathematica

We consider the k-osculating varieties O k,d to the Veronese d−uple embeddings of P 2. By studyin... more We consider the k-osculating varieties O k,d to the Veronese d−uple embeddings of P 2. By studying the Hilbert function of certain zero-dimensional schemes Y ⊂ P 2 , we find the dimension of O s k,d , the (s − 1) th secant varieties of O k,d , for 3 ≤ s ≤ 6 and s = 9, and we determine whether those secant varieties are defective or not. 0. Introduction. The problem of determining the dimension of the higher secant varieties of a projective variety is a classical subject of study. In the present paper we are concerned with the (s − 1) th higher secant varieties of O k,V n,d , where O k,V n,d is the k-osculating variety to the Veronese embedding V n,d of P n into P N (N = d+n n −1) via the complete linear system R d , where R = K[x 0 ,. .. , x n ], and K is an algebraically closed field of characteristic zero. This matter has been dealt with by several authors in the last few years (see [2], [3], [4], [5], [7]). We wish to mention E.Ballico and C.Fontanari. In [2] and [3] they study the higher secant varieties of O k,V n,d for n = 2 and k = 1, 2, and they prove the following results: * All authors supported by MIUR funds.

Research paper thumbnail of The Hitchhiker Guide to: Secant Varieties and Tensor Decomposition

Mathematics, 2018

We consider here the problem, which is quite classical in Algebraic geometry, of studying the sec... more 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.

Research paper thumbnail of High order singular value decomposition for plant diversity estimation

Bollettino dell'Unione Matematica Italiana

We propose a new method to estimate plant diversity with Rényi and Rao indexes through the so cal... more We propose a new method to estimate plant diversity with Rényi and Rao indexes through the so called High Order Singular Value Decomposition (HOSVD) of tensors. Starting from NASA multi-spectral images we evaluate diversity and we compare original diversity estimates with those realized via the HOSVD compression methods for big data. Our strategy turns out to be extremely powerful in terms of memory storage and precision of the outcome. The obtained results are so promising that we can support the efficiency of our method in the ecological framework.

Research paper thumbnail of Waring, tangential and cactus decompositions

Journal de Mathématiques Pures et Appliquées

Research paper thumbnail of Foreword: Special issue of JSC on the occasion of MEGA 2019

Journal of Symbolic Computation

Research paper thumbnail of Geometric conditions for strict submultiplicativity of rank and border rank

Annali di Matematica Pura ed Applicata (1923 -)

Research paper thumbnail of Skew Symmetric Tensor Decomposition

Communications in Contemporary Mathematics

We introduce the “skew apolarity lemma” and we use it to give algorithms for the skew-symmetric r... more We introduce the “skew apolarity lemma” and we use it to give algorithms for the skew-symmetric rank and the decompositions of tensors in [Formula: see text] with [Formula: see text] and [Formula: see text]. New algorithms to compute the rank and a minimal decomposition of a tritensor are also presented.

Research paper thumbnail of Bounds on the tensor rank

Annali di Matematica Pura ed Applicata (1923 -)

We give a sufficient criterion for a lower bound of the cactus rank of a tensor. Then we refine t... more We give a sufficient criterion for a lower bound of the cactus rank of a tensor. Then we refine that criterion in order to be able to give an explicit sufficient condition for a non-redundant decomposition of a tensor to be minimal and unique.

Research paper thumbnail of A new class of non-identifiable skew-symmetric tensors

Annali di Matematica Pura ed Applicata (1923 -)

We prove that the generic element of the fifth secant variety σ 5 (Gr(P 2 , P 9)) ⊂ P(3 C 10) of ... more We prove that the generic element of the fifth secant variety σ 5 (Gr(P 2 , P 9)) ⊂ P(3 C 10) of the Grassmannian of planes of P 9 has exactly two decompositions as a sum of five projective classes of decomposable skew-symmetric tensors. We show that this, together with Gr(P 3 , P 8), is the only non-identifiable case among the non-defective secant varieties σ s (Gr(P k , P n)) for any n < 14. In the same range for n, we classify all the weakly defective and all tangentially weakly defective secant varieties of any Grassmannians. We also show that the dual variety (σ 3 (Gr(P 2 , P 7))) ∨ of the variety of 3-secant planes of the Grassmannian of P 2 ⊂ P 7 is σ 2 (Gr(P 2 , P 7)) the variety of bi-secant lines of the same Grassmannian. The proof of this last fact has a very interesting physical interpretation in terms of measurement of the entanglement of a system of 3 identical fermions, the state of each of them belonging to a 8-th dimensional "Hilbert" space.

Research paper thumbnail of A uniqueness result on the decompositions of a bi-homogeneous polynomial

Linear and Multilinear Algebra

In the first part of this paper we give a precise description of all the minimal decompositions o... more In the first part of this paper we give a precise description of all the minimal decompositions of any bi-homogeneous polynomial p (i.e. a partially symmetric tensor of S d 1 V 1 ⊗ S d 2 V 2 where V 1 , V 2 are two complex, finite dimensional vector spaces) if its rank with respect to the Segre-Veronese variety S d 1 ,d 2 (V 1 , V 2) is at most min{d 1 , d 2 }. Such a polynomial may not have a unique minimal decomposition as p = r i=1 λ i p i with p i ∈ S d 1 ,d 2 (V 1 , V 2) and λ i coefficients, but we can show that there exist unique p 1 , .

Research paper thumbnail of Tensor decomposition and homotopy continuation

Differential Geometry and its Applications

A computationally challenging classical elimination theory problem is to compute polynomials whic... more A computationally challenging classical elimination theory problem is to compute polynomials which vanish on the set of tensors of a given rank. By moving away from computing polynomials via elimination theory to computing pseudowitness sets via numerical elimination theory, we develop computational methods for computing ranks and border ranks of tensors along with decompositions. More generally, we present our approach using joins of any collection of irreducible and nondegenerate projective varieties X1,. .. , X k ⊂ P N defined over C. After computing ranks over C, we also explore computing real ranks. A variety of examples are included to demonstrate the numerical algebraic geometric approaches.

Research paper thumbnail of Computing the cactus rank of a general form

Using Macaulay's correspondence we study the family of Artinian Gorenstein local algebras with fi... more Using Macaulay's correspondence we study the family of Artinian Gorenstein local algebras with fixed symmetric Hilbert function decomposition. As an application we give a new lower bound for cactus varieties of the third Veronese embedding. We discuss the case of cubic surfaces, where interesting phenomena occur.

Research paper thumbnail of Curvilinear schemes and maximum rank of forms

We define the \emph{curvilinear rank} of a degree ddd form PPP in n+1n+1n+1 variables as the minimum ... more We define the \emph{curvilinear rank} of a degree ddd form PPP in n+1n+1n+1 variables as the minimum length of a curvilinear scheme, contained in the ddd-th Veronese embedding of mathbbPn\mathbb{P}^nmathbbPn, whose span contains the projective class of PPP. Then, we give a bound for rank of any homogenous polynomial, in dependance on its curvilinear rank.

Research paper thumbnail of Higher Secant Varieties of ℙ n × ℙ1 Embedded in Bi-Degree (a, b)

Http Dx Doi Org 10 1080 00927872 2011 595748, Sep 11, 2012

Research paper thumbnail of Tensor rank on tangent developable of Segre varieties

Research paper thumbnail of Varieties parameterizing forms and their secant varieties

Research paper thumbnail of Schemi 0-dimensionali e forme canoniche di polinomi omogenei

Research paper thumbnail of Varietà che parametrizzano forme e loro varietà delle secanti

Bollettino Dell Unione Matematica Italiana Sezione a La Matematica Nella Societa E Nella Cultura, Aug 1, 2007

Research paper thumbnail of Higher secant varieties of Pn x Pm embedded in bi-degree (1,d)

Journal of Pure and Applied Algebra, 2011

Research paper thumbnail of Decomposition of homogeneous polynomials over an algebraically closed field

Let F be a homogeneous polynomial of degree d in m + 1 variables defined over an algebraically cl... more Let F be a homogeneous polynomial of degree d in m + 1 variables defined over an algebraically closed field of characteristic 0 and suppose that F belongs to the s-th secant variety of the d-uple Veronese embedding of P m into P (m+d d)−1 but that its minimal decomposition as a sum of d-th powers of linear forms M 1 ,. .. , M r is F = M d 1 +• • •+M d r with r > s. We show that if s+r ≤ 2d+1 then such a decomposition of F can be split in two parts: one of them is made by linear forms that can be written using only two variables, the other part is uniquely determined once one has fixed the first part. We also obtain a uniqueness theorem for the minimal decomposition of F if r is at most d and a mild condition is satisfied.

Research paper thumbnail of Some defective secant varieties to osculating varieties of Veronese surfaces

Collectanea Mathematica

We consider the k-osculating varieties O k,d to the Veronese d−uple embeddings of P 2. By studyin... more We consider the k-osculating varieties O k,d to the Veronese d−uple embeddings of P 2. By studying the Hilbert function of certain zero-dimensional schemes Y ⊂ P 2 , we find the dimension of O s k,d , the (s − 1) th secant varieties of O k,d , for 3 ≤ s ≤ 6 and s = 9, and we determine whether those secant varieties are defective or not. 0. Introduction. The problem of determining the dimension of the higher secant varieties of a projective variety is a classical subject of study. In the present paper we are concerned with the (s − 1) th higher secant varieties of O k,V n,d , where O k,V n,d is the k-osculating variety to the Veronese embedding V n,d of P n into P N (N = d+n n −1) via the complete linear system R d , where R = K[x 0 ,. .. , x n ], and K is an algebraically closed field of characteristic zero. This matter has been dealt with by several authors in the last few years (see [2], [3], [4], [5], [7]). We wish to mention E.Ballico and C.Fontanari. In [2] and [3] they study the higher secant varieties of O k,V n,d for n = 2 and k = 1, 2, and they prove the following results: * All authors supported by MIUR funds.