Publications

On the ranks of the third secant variety of SegreVeronese embeddings 2019
Ballico, Edoardo; Bernardi, Alessandra, "On the ranks of the third secant variety of SegreVeronese embeddings" in LINEAR & MULTILINEAR ALGEBRA, v. 2019, n. Vol 67, NO. 3 (2019), p. 583597.  URL: https://doi.org/10.1080/03081087.2018.1430117 .  DOI: 10.1080/03081087.2018.1430117
We give an upper bound for the rank of the border rank 3 partially symmetric tensors. In the special case of border rank 3 tensors T∈V1⊗⋯⊗Vk (Segre case) we can show that all ranks among 3 and k−1 arise and if dimVi≥3 for all i's, then also all the ranks between k and 2k−1 arise.
2019 journal paper

Quantum Physics and Geometry 2019
Ballico, Edoardo; Bernardi, Alessandra; Carusotto, Iacopo; Mazzucchi, Sonia; Moretti, Valter (edited by), "Quantum Physics and Geometry", by Edoardo Ballico, Alessandra Bernardi, Iacopo Carusotto, Sonia Mazzucchi, Valter Moretti, Luca Chiantini, Frédéric Holweck, M. Joseph Landsberg, F.M. Ciaglia, Alberto Ibort, Giuseppe Marmo, Davide Pastorello, Bassano Vacchini, Switzerland: Springer, 2019, 172 p.  (LECTURE NOTES OF THE UNIONE MATEMATICA ITALIANA).  ISBN: 9783030061210.  DOI: 10.1007/9783030061227
This book collects independent contributions on current developments in quantum information theory, a very interdisciplinary field at the intersection of physics, computer science and mathematics, which makes intense use of the most advanced concepts from each discipline. In each contribution, the authors give pedagogical introductions to the main concepts underlying their present research and present a personal perspective on some of the most exciting open problems.Keeping this diverse audience in mind, special efforts have been made to ensure that the basic concepts underlying quantum information are covered in an understandable way for mathematical readers, who can find new open challenges for their research. At the same time, the volume will also be of use to physicists wishing to learn advanced mathematical tools, especially those of a differential and algebraic geometric nature.
2019 book

On the partially symmetric rank of tensor products of Wstates and other symmetric tensors 2019
Ballico, Edoardo; Bernardi, Alessandra; Christandl, Matthias; Gesmundo, Fulvio, "On the partially symmetric rank of tensor products of Wstates and other symmetric tensors" in ATTI DELLA ACCADEMIA NAZIONALE DEI LINCEI. RENDICONTI LINCEI. MATEMATICA E APPLICAZIONI, v. 2019, n. 30 (2019), p. 93124.  URL: http://www.emsph.org/journals/journal.php?jrn=rlm .  DOI: 10.4171/RLM/837
Given tensors $T$ and $T'$ of order $k$ and $k'$ respectively, the tensor product $T \otimes T'$ is a tensor of order $k+k'$. It was recently shown that the tensor rank can be strictly submultiplicative under this operation ([ChristandlJensenZuiddam]). We study this phenomenon for symmetric tensors where additional techniques from algebraic geometry are available. The tensor product of symmetric tensors results in a partially symmetric tensor and our results amount to bounds on the partially symmetric rank. Following motivations from algebraic complexity theory and quantum information theory, we focus on the socalled \emph{$W$states}, namely monomials of the form $x^{d1}y$, and on products of such. In particular, we prove that the partially symmetric rank of $x^{d_1 1}y \ootimes x^{d_k1} y$ is at most $2^{k1}(d_1+ \cdots +d_k)$.
2019 journal paper

Introduction 2019
Ballico, Edoardo; Bernardi, Alessandra; Carusotto, Iacopo; Mazzucchi, Sonia; Moretti, Valter, "Introduction" in "Quantum Physics and Geometry", by Edoardo Ballico, Alessandra Bernardi, Iacopo Carusotto, Sonia Mazzucchi, Valter Moretti, edited by Edoardo Ballico, Alessandra Bernardi, Iacopo Carusotto, Sonia Mazzucchi, Valter Moretti, M. Joseph Landsberg, Davide Pastorello, Bassano Vacchini, Frédéric Holweck, Luca Chiantini, F. M. Ciaglia, A. Ibort, G. Marmo, Switzerland: Springer, Cham, 2019, p. 14.  URL: https://link.springer.com/chapter/10.1007/9783030061227_1 .  DOI: 10.1007/9783030061227_1
2019 part of book

A note on the cactus rank for SegreVeronese varieties 2019
Ballico, Edoardo; Bernardi, Alessandra; Gesmundo, Fulvio, "A note on the cactus rank for SegreVeronese varieties" in JOURNAL OF ALGEBRA, v. 2019, n. 526 (2019), p. 611.  URL: https://www.sciencedirect.com/science/article/pii/S002186931930078X .  DOI: 10.1016/j.jalgebra.2019.01.027
We give an upper bound for the cactus rank of any multihomogeneous polynomial.
2019 journal paper

On real typical ranks 2018
Grigoriy, Blekherman; Ottaviani, Giorgio; Bernardi, Alessandra, "On real typical ranks" in BOLLETTINO DELLA UNIONE MATEMATICA ITALIANA, v. 2018, n. 11, 3 (2018), p. 293307.  URL: https://link.springer.com/article/10.1007/s4057401701340?wt_mc=Internal.Event.1.SEM.ArticleAuthorOnlineFirst .  DOI: 10.1007/s4057401701340
We study typical ranks with respect to a real variety X. Examples of such are tensor rank (X is the Segre variety) and symmetric tensor rank (X is the Veronese variety). We show that any rank between the minimal typical rank and the maximal typical rank is also typical. We investigate typical ranks of nvariate symmetric tensors of order d, or equivalently homogeneous polynomials of degree d in n variables, for small values of n and d. We show that 4 is the unique typical rank of real ternary cubics, and quaternary cubics have typical ranks 5 and 6 only. For ternary quartics we show that 6 and 7 are typical ranks and that all typical ranks are between 6 and 8. For ternary quintics we show that the typical ranks are between 7 and 13.
2018 journal paper

On the dimension of contact loci and the identifiability of tensors 2018
Ballico, Edoardo; Bernardi, Alessandra; Chiantini, Luca, "On the dimension of contact loci and the identifiability of tensors" in ARKIV FÖR MATEMATIK, v. 2018, n. Vol. 56, No. 2 (2018), p. 265283.  URL: http://www.intlpress.com/site/pub/pages/journals/items/arkiv/content/vols/0056/0002/a004/index.html .  DOI: 10.4310/ARKIV.2018.v56.n2.a4
Let X⊂ℙr be an integral and nondegenerate variety. Set n:=dim(X). We prove that if the (k+n−1)secant variety of X has (the expected) dimension (k+n−1)(n+1)−1
2018 journal paper

Typical and Admissible ranks over fields 2018
Ballico, Edoardo; Bernardi, Alessandra, "Typical and Admissible ranks over fields" in RENDICONTI DEL CIRCOLO MATEMATICO DI PALERMO, v. 2018, n. Volume 67, Issue 1 (2018), p. 115128.  URL: https://link.springer.com/article/10.1007/s1221501702995?wt_mc=Internal.Event.1.SEM.ArticleAuthorAssignedToIssue .  DOI: 10.1007/s1221501702995
Let $X(\mathbb{R})$ be a geometrically connected variety defined over $\mathbb{R}$ such that the set of all its complex points $X(\mathbb{C})$ is nondegenerate. We introduce the notion of \emph{admissible rank} of a point $P$ with respect to $X$ to be the minimal cardinality of a set $S\subset X(\mathbb{C})$ of points such that $S$ spans $P$ and $S$ is stable under conjugation. Any set evincing the admissible rank can be equipped with a \emph{label} keeping track of the number of its complex and real points. We show that, in the case of generic identifiability, there is an open dense euclidean subset of points with certain admissible rank for any possible label. Moreover we show that if $X$ is a rational normal curve then there always exists a label for the generic element. We present two examples in which either the label doesn't exist or the admissible rank is strictly bigger than the usual complex rank.
2018 journal paper

A new class of nonidentifiable skew symmetric tensors 2018
Bernardi, Alessandra; Vanzo, Davide, "A new class of nonidentifiable skew symmetric tensors" in ANNALI DI MATEMATICA PURA ED APPLICATA, v. 2018, n. Volume 197, Issue 5 (2018), p. 14991510.  URL: https://link.springer.com/article/10.1007/s102310180734z?wt_mc=alerts.TOCjournals&utm_source=toc&utm_medium=email&utm_campaign=toc_10231_197_5 .  DOI: 10.1007/s102310180734z
We prove that the generic element of the fifth secant variety $\sigma_5(Gr(\mathbb{P}^2,\mathbb{P}^9)) \subset \mathbb{P}(\bigwedge^3 \mathbb{C}^{10})$ of the Grassmannian of planes of $\mathbb{P}^9$ has exactly two decompositions as a sum of five projective classes of decomposable skewsymmetric tensors. {We show that this, {together with $Gr(\mathbb{P}^3, \mathbb{P}^8)$, is the only nonidentifiable case} among the nondefective secant varieties $\sigma_s(Gr(\mathbb{P}^k, \mathbb{P}^n))$ 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 $(\sigma_3(Gr(\mathbb{P}^2,\mathbb{P}^7)))^{\vee}$ of the variety of 3secant planes of the Grassmannian of $\mathbb{P}^2\subset \mathbb{P}^7$ is $\sigma_2(Gr(\mathbb{P}^2,\mathbb{P}^7))$ the variety of bisecant 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 8th dimensional ``Hilbert'' space.
2018 journal paper

Bounds on the tensor rank 2018
Ballico, Edoardo; Bernardi, Alessandra; Chiantini, Luca; Guardo, Elena, "Bounds on the tensor rank" in ANNALI DI MATEMATICA PURA ED APPLICATA, v. 2018, n. 197(6) (2018), p. 17711785.  URL: http://link.springer.com/article/10.1007/s1023101807486 .  DOI: 10.1007/s1023101807486
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 nonredundant decomposition of a tensor to be minimal and unique.
2018 journal paper

The Hitchhiker Guide to: Secant Varieties and Tensor Decomposition 2018
Bernardi, Alessandra; Carlini, Enrico; Virginia Catalisano, Maria; Gimigliano, Alessandro; Oneto, Alessandro, "The Hitchhiker Guide to: Secant Varieties and Tensor Decomposition" in MATHEMATICS, v. 2018, n. 6 (2018), p. 314.1314.86.  URL: https://www.mdpi.com/22277390/6/12/314/htm .  DOI: 10.3390/math6120314
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, skewsymmetric 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.
2018 journal paper

On polynomials with given Hilbert function and applications 2018
Bernardi, Alessandra; Jelisiejew, Joachim; Macias Marques, Pedro; Ranestad, Kristian, "On polynomials with given Hilbert function and applications" in COLLECTANEA MATHEMATICA, v. 2018, n. Volume 69, Issue 1 (2018), p. 3964.  URL: https://link.springer.com/article/10.1007/s1334801601902 .  DOI: 10.1007/s1334801601902
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 the dimension of cactus varieties of the third Veronese embedding. We discuss the case of cubic surfaces, where interesting phenomena occur.
2018 journal paper

Singularities of plane rational curves via projections 2018
Bernardi, Alessandra; Gimigliano, Alessandro; Idà, Monica, "Singularities of plane rational curves via projections" in JOURNAL OF SYMBOLIC COMPUTATION, v. 86, n. May–June (2018), p. 189214.  URL: http://dx.doi.org/10.1016/j.jsc.2017.05.003 .  DOI: 10.1016/j.jsc.2017.05.003
We consider the parameterization ${\mayhbf{f}}=(f_0:,f_1:f_2)$ of a plane rational curve $C$ of degree $n$, and we study the singularities of $C$ via such parameterization. We use the projection from the rational normal curve $C_n\subset\mathbb{P}^n$ to $C$ and its interplay with the secant varieties to $C_n$. IN particular, we define via $\mathbf{f}$ certain 0dimensioal schemes $X_k\subset \mathbb{P}^k$, $2\leq k \leq (n1)$, which encode all information on the singularities of multiplicity $\geq k$ of $C$ (e.g. using $X_2$ we can give a criterion to determine whether $C$ is a cuspidal curve or has only ordinary singularities). We give a series of algorithms which allow one to obtain information about the singularities from such schemes.
2018 journal paper

Curvilinear schemes and maximum rank of forms 2017
Ballico, E; Bernardi, Alessandra, "Curvilinear schemes and maximum rank of forms" in LE MATEMATICHE, v. LXXII (2017), n. Fasc. I (2017), p. 137144.  URL: https://lematematiche.dmi.unict.it/index.php/lematematiche/article/view/1360/1015 .  DOI: 10.4418/2017.72.1.10
We define the \emph{curvilinear rank} of a degree $d$ form $P$ in $n+1$ variables as the minimum length of a curvilinear scheme, contained in the $d$th Veronese embedding of $\mathbb{P}^n$, whose span contains the projective class of $P$. Then, we give a bound for rank of any homogenous polynomial, in dependance on its curvilinear rank.
2017 journal paper

A uniqueness result on the decompositions of a bihomogeneous polynomial 2017
Ballico, Edoardo; Bernardi, Alessandra, "A uniqueness result on the decompositions of a bihomogeneous polynomial" in LINEAR & MULTILINEAR ALGEBRA, v. 2017, 65, n. 4 (2017), p. 677698.  URL: https://www.tandfonline.com/doi/full/10.1080/03081087.2016.1202182 .  DOI: 10.1080/03081087.2016.1202182
In the first part of this paper we give a precise description of all the minimal decompositions of any bihomogeneous polynomial $p$ (i.e. a partially symmetric tensor of $S^{d_1}V_1\otimes S^{d_2}V_2$ where $V_1,V_2$ are two complex, finite dimensional vector spaces) if its rank with respect to the SegreVeronese 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 $p=\sum_{i=1}^r\lambda_i p_i$ with $p_i\in S_{d_1,d_2}(V_1,V_2)$ and $\lambda_i$ coefficients, but we can show that there exist unique $p_1, \ldots , p_{r'}$, $p_{1}', \ldots , p_{r''}'\in S_{d_1,d_2}(V_1,V_2) $, two unique linear forms $l\in V_1^*$, $l'\in V_2^*$, and two unique bivariate polynomials $q\in S^{d_2}V_2^*$ and $q'\in S^{d_1}V_1^*$ such that either $p=\sum_{i=1}^{r'} \lambda_i p_i+l^{d_1}q $ or $ p= \sum_{i=1}^{r''}\lambda'_i p_i'+l'^{d_2}q'$, ($\lambda_i, \lambda'_i$ being appropriate coefficients). In the second part of the paper we focus on the tangential variety of the SegreVeronese varieties. We compute the rank of their tensors (that is valid also in the case of SegreVeronese of more factors) and we describe the structure of the decompositions of the elements in the tangential variety of the twofactors SegreVeronese varieties.
2017 journal paper

Tensor decomposition and homotopy continuation 2017
Noah, Daleo; Jonathan, Hauenstein; Bernard, Mourrain; Bernardi, Alessandra, "Tensor decomposition and homotopy continuation" in DIFFERENTIAL GEOMETRY AND ITS APPLICATIONS, v. 2017, n. 55 (2017), p. 78105.  URL: https://authors.elsevier.com/c/1W51~3IcLpLWC .  DOI: 10.1016/j.difgeo.2017.07.009
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 $X_1, \ldots , X_k \subset \mathbb{P}^N$ defined over $\mathbb{C}$. After computing ranks over , we also explore computing real ranks. A variety of examples are included to demonstrate the numerical algebraic geometric approaches.
2017 journal paper

On parameterizations of plane rational curves and their syzygies 2016
Bernardi, Alessandra; Gimigliano, A.; Idà, M., "On parameterizations of plane rational curves and their syzygies" in MATHEMATISCHE NACHRICHTEN, v. 56, (2016), p. 537545.  URL: http://www3.interscience.wiley.com/journal/60500208/home .  DOI: 10.1002/mana.201500264
2016 journal paper

A Note on plane rational curves and the associated Poncelet Surfaces 2015
Bernardi, Alessandra; Gimigliano, A; Monica, I., "A Note on plane rational curves and the associated Poncelet Surfaces" in RENDICONTI DELL'ISTITUTO DI MATEMATICA DELL'UNIVERSITÀ DI TRIESTE, v. 47, (2015), p. 16
2015 journal paper

A comparison of different notions of ranks of symmetric tensors 2014
Alessandra Bernardi; Jérôme Brachat; Bernard Mourrain, "A comparison of different notions of ranks of symmetric tensors" in LINEAR ALGEBRA AND ITS APPLICATIONS, v. 460, (2014), p. 205230.  URL: http://www.sciencedirect.com/science/article/pii/S002437951400487X .  DOI: 10.1016/j.laa.2014.07.036
We introduce various notions of rank for a high order symmetric tensor taking values over the complex numbers, namely: rank, border rank, catalecticant rank, generalized rank, scheme length, border scheme length, extension rank and smoothable rank. We analyze the stratification induced by these ranks. The mutual relations between these stratifications allow us to describe the hierarchy among all the ranks. We show that strict inequalities are possible between rank, border rank, extension rank and catalecticant rank. Moreover we show that scheme length, generalized rank and extension rank coincide.
2014 journal paper

Minimal decomposition of binary forms with respect to tangential projections 2013
E. Ballico; A. Bernardi, "Minimal decomposition of binary forms with respect to tangential projections" in JOURNAL OF ALGEBRA AND ITS APPLICATIONS, v. 12, n. 6 (2013), p. 1350010113500108.  DOI: 10.1142/S0219498813500102
2013 journal paper

Stratification of the fourth secant variety of Veronese varieties via the symmetric rank 2013
E. Ballico; A. Bernardi, "Stratification of the fourth secant variety of Veronese varieties via the symmetric rank" in ADVANCES IN PURE AND APPLIED MATHEMATICS, v. 4, n. 2 (2013), p. 215250.  DOI: 10.1515/apam20130015
2013 journal paper

General tensor decomposition, moment matrices and applications 2013
Alessandra Bernardi; Jerome Brachat; Pierre Comon; Bernard Mourrain, "General tensor decomposition, moment matrices and applications" in JOURNAL OF SYMBOLIC COMPUTATION, v. 52, (2013), p. 5171.  URL: http://dx.doi.org/10.1016/j.jsc.2012.05.012 .  DOI: 10.1016/j.jsc.2012.05.012
In the paper, we address the important problem of tensor decompositions which can be seen as a generalisation of Singular Value Decomposition for matrices. We consider general multilinear and multihomogeneous tensors. We show how to reduce the problem to a truncated moment matrix problem and give a new criterion for flat extension of QuasiHankel matrices. We connect this criterion to the commutation characterisation of border bases. A new algorithm is described which applies for general multihomogeneous tensors, extending the approach of J.J. Sylvester on binary forms. An example illustrates the algebraic operations involved in this approach and how the decomposition can be recovered from eigenvector computation.
2013 journal paper

Real and complex rank for real symmetric tensors with low ranks 2013
E. Ballico; A. Bernardi, "Real and complex rank for real symmetric tensors with low ranks" in ALGEBRA, v. 2013, (2013), p. 79405417940545.  DOI: 10.1155/2013/794054
2013 journal paper

Tensor ranks on tangent developable of Segre varieties 2013
E. Ballico; A. Bernardi, "Tensor ranks on tangent developable of Segre varieties" in LINEAR & MULTILINEAR ALGEBRA, n.s., v. 61, n. 7 (2013), p. 881894.  DOI: 10.1080/03081087.2012.716430
2013 journal paper

Unique decomposition for a polynomial of low rank 2013
Ballico, Edoardo; Bernardi, Alessandra, "Unique decomposition for a polynomial of low rank" in ANNALES POLONICI MATHEMATICI, v. 108, (2013), p. 219224.  URL: http://journals.impan.pl/ap/Inf/10832.html .  DOI: 10.1142/S0219498813500102
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 sth secant variety of the duple Veronese embedding of Pm into P((m+d)(d))(1) but that its minimal decomposition as a sum of dth powers of linear forms requires more than s summands. We show that if s <= d then F can be uniquely written as F = M1(d) + ... + Mt(d) + Q, where M1, ... , Mt are linear forms with t <= (d  1)/2, and Q is a binary form such that Q = Sigma(q)(i=1) l(i)(ddi)m(i) with l(i)'s linear forms and m(i)'s forms of degree d(i) such that Sigma(d(i) + 1) = s  t.
2013 journal paper