# Publications

## Publications

• #### A note on the cactus rank for Segre-Veronese varieties 2019

Ballico, Edoardo; Bernardi, Alessandra; Gesmundo, Fulvio, "A note on the cactus rank for Segre-Veronese varieties" in JOURNAL OF ALGEBRA, v. 526, (2019), p. 6-11. - 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 multi-homogeneous polynomial.

2019 journal paper

• #### On the partially symmetric rank of tensor products of W-states and other symmetric tensors 2019

Ballico, Edoardo; Bernardi, Alessandra; Christandl, Matthias; Gesmundo, Fulvio, "On the partially symmetric rank of tensor products of W-states and other symmetric tensors" in ATTI DELLA ACCADEMIA NAZIONALE DEI LINCEI. RENDICONTI LINCEI. MATEMATICA E APPLICAZIONI, v. 30, n. 1 (2019), p. 93-124. - URL: http://www.ems-ph.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 ([Christandl-Jensen-Zuiddam]). 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 so-called emph{$W$-states}, namely monomials of the form $x^{d-1}y$, and on products of such. In particular, we prove that the partially symmetric rank of $x^{d_1 -1}y ootimes x^{d_k-1} y$ is at most $2^{k-1}(d_1+ cdots +d_k)$.

2019 journal paper

• #### On the ranks of the third secant variety of Segre-Veronese embeddings 2019

Ballico, Edoardo; Bernardi, Alessandra, "On the ranks of the third secant variety of Segre-Veronese embeddings" in LINEAR & MULTILINEAR ALGEBRA, v. 2019, 67, n. 3 (2019), p. 583-597. - URL: https://www.tandfonline.com/doi/abs/10.1080/03081087.2018.1430117 . - DOI: 10.1080/03081087.2018.1430117

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). - ISMN: 978-3-030-06122-7. - URL: https://www.springer.com/gp/book/9783030061210 . - DOI: 10.1007/978-3-030-06122-7

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

• #### 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. 4-9. - URL: https://link.springer.com/chapter/10.1007/978-3-030-06122-7_1 . - DOI: 10.1007/978-3-030-06122-7_1

2019 part of book

• #### 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. 6, n. 12 (2018), p. 31401-31486. - URL: https://www.mdpi.com/2227-7390/6/12/314/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, 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.

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. 189-214. - 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 $C_nsubsetmathbb{P}^n$ to $C$ and its interplay with the secant varieties to $C_n$. IN particular, we define via $mathbf{f}$ certain 0-dimensioal schemes $X_ksubset mathbb{P}^k$, $2leq k leq (n-1)$, 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

• #### A new class of non-identifiable skew symmetric tensors 2018

Bernardi, Alessandra; Vanzo, Davide, "A new class of non-identifiable skew symmetric tensors" in ANNALI DI MATEMATICA PURA ED APPLICATA, v. 2018, n. Volume 197, Issue 5 (2018), p. 1499-1510. - URL: https://link.springer.com/article/10.1007/s10231-018-0734-z?wt_mc=alerts.TOCjournals&utm_source=toc&utm_medium=email&utm_campaign=toc_10231_197_5 . - DOI: 10.1007/s10231-018-0734-z

We prove that the generic element of the fifth secant variety $sigma_5(Gr(mathbb{P}^2,mathbb{P}^9)) subset mathbb{P}(igwedge^3 mathbb{C}^{10})$ of the Grassmannian of planes of $mathbb{P}^9$ has exactly two decompositions as a sum of five projective classes of decomposable skew-symmetric tensors. {We show that this, {together with $Gr(mathbb{P}^3, mathbb{P}^8)$, is the only non-identifiable case} among the non-defective 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 $(sigma_3(Gr(mathbb{P}^2,mathbb{P}^7)))^{ee}$ of the variety of 3-secant planes of the Grassmannian of $mathbb{P}^2subset mathbb{P}^7$ is $sigma_2(Gr(mathbb{P}^2,mathbb{P}^7))$ 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.

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. 39-64. - URL: https://link.springer.com/article/10.1007/s13348-016-0190-2 . - DOI: 10.1007/s13348-016-0190-2

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

• #### 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. 67, n. 1 (2018), p. 115-128. - URL: https://link.springer.com/article/10.1007/s12215-017-0299-5?wt_mc=Internal.Event.1.SEM.ArticleAuthorAssignedToIssue . - DOI: 10.1007/s12215-017-0299-5

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 non-degenerate. We introduce the notion of emph{admissible rank} of a point $P$ with respect to $X$ to be the minimal cardinality of a set $Ssubset X(mathbb{C})$ 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

• #### 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. 56, n. 2 (2018), p. 265-283. - 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 non-degenerate 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

• #### 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, 197, n. 6 (2018), p. 1771-1785. - URL: http://link.springer.com/article/10.1007/s10231-018-0748-6 . - DOI: 10.1007/s10231-018-0748-6

2018 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, 11, n. 3 (2018), p. 293-307. - URL: https://link.springer.com/article/10.1007/s40574-017-0134-0 . - DOI: 10.1007/s40574-017-0134-0

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. 72, n. 1 (2017), p. 137-144. - 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

• #### Tensor decomposition and homotopy continuation 2017

Bernardi, Alessandra; Noah, Daleo; Jonathan, Hauenstein; Bernard, Mourrain, "Tensor decomposition and homotopy continuation" in DIFFERENTIAL GEOMETRY AND ITS APPLICATIONS, v. 2017, n. 55 (2017), p. 78-105. - URL: https://www.sciencedirect.com/science/article/pii/S0926224517301055?via=ihub . - 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

• #### A uniqueness result on the decompositions of a bi-homogeneous polynomial 2017

Ballico, Edoardo; Bernardi, Alessandra, "A uniqueness result on the decompositions of a bi-homogeneous polynomial" in LINEAR & MULTILINEAR ALGEBRA, v. 2017, 65, n. 4 (2017), p. 677-698. - 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 bi-homogeneous 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 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 $p=\sum_{i=1}^r\lambda_i p_i$ 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 Segre-Veronese varieties. We compute the rank of their tensors (that is valid also in the case of Segre-Veronese of more factors) and we describe the structure of the decompositions of the elements in the tangential variety of the two-factors Segre-Veronese varieties.

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. 289, n. 5-6 (2016), p. 537-545. - URL: http://www3.interscience.wiley.com/journal/60500208/home . - DOI: 10.1002/mana.201500264

Let $C$ be a plane rational curve of degree $d$ and $p: ilde C ightarrow C$ its normalization. We are interested in the {it splitting type} $(a,b)$ of $C$, where $mathcal{O}_{mathbb{P}^1}(-a-d)oplus mathcal{O}_{mathbb{P}^1}(-b-d)$ gives the syzigies of the ideal $(f_0,f_1,f_2)subset K[s,t]$, and , $(f_0,f_1,f_2)$ is a parameterization of $C$. We want to describe in which cases $(a,b)=(k,d-k)$ ($2kleq d)$, via a geometric description; namely we show that $(a,b)=(k,d-k)$ if and only if $C$ is the projection of a rational curve on a rational normal surface in $PP^{k+1}$.

2016 journal paper

• #### A Note on plane rational curves and the associated Poncelet Surfaces 2015

Bernardi, Alessandra; Gimigliano, A; Idà, M., "A Note on plane rational curves and the associated Poncelet Surfaces" in RENDICONTI DELL'ISTITUTO DI MATEMATICA DELL'UNIVERSITÀ DI TRIESTE, v. Vol. 47, (2015), p. 59-64. - URL: https://rendiconti.dmi.units.it/node/4/6300 . - DOI: 10.13137/0049-4704/11219

2015 journal paper

• #### A comparison of different notions of ranks of symmetric tensors 2014

Bernardi, Alessandra; Jérôme, Brachat; Bernard, Mourrain, "A comparison of different notions of ranks of symmetric tensors" in LINEAR ALGEBRA AND ITS APPLICATIONS, v. 460, (2014), p. 205-230. - 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

• #### On the cactus rank of cubic forms 2013

Bernardi, Alessandra; Ranestad, K., "On the cactus rank of cubic forms" in JOURNAL OF SYMBOLIC COMPUTATION, v. 50, (2013), p. 291-297. - URL: http://dx.doi.org/10.1016/j.jsc.2012.08.001 . - DOI: 10.1016/j.jsc.2012.08.001

We prove that the smallest degree of an apolar 0-dimensional scheme of a general cubic form in n + 1 variables is at most 2n + 2, when n >= 8, and therefore smaller than the rank of the form. For the general reducible cubic form the smallest degree of an apolar subscheme is n + 2, while the rank is at least 2n.

2013 journal paper

• #### Real and complex rank for real symmetric tensors with low ranks 2013

Ballico, Edoardo; Bernardi, Alessandra, "Real and complex rank for real symmetric tensors with low ranks" in ALGEBRA, v. 2013, (2013), p. 794054-1-794054-5. - DOI: 10.1155/2013/794054

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. 219-224. - URL: http://journals.impan.pl/ap/Inf/108-3-2.html . - DOI: 10.4064/ap108-3-2

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 d-uple Veronese embedding of P-m 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 = M-1(d) + ... + M-t(d) + Q, where M-1, ... , M-t are linear forms with t <= (d - 1)/2, and Q is a binary form such that Q = Sigma(q)(i=1) l(i)(d-di)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

• #### Minimal decomposition of binary forms with respect to tangential projections 2013

Ballico, Edoardo; Bernardi, Alessandra, "Minimal decomposition of binary forms with respect to tangential projections" in JOURNAL OF ALGEBRA AND ITS APPLICATIONS, v. 12, n. 6 (2013), p. 1350010-1-1350010-8. - DOI: 10.1142/S0219498813500102

2013 journal paper

• #### Stratification of the fourth secant variety of Veronese varieties via the symmetric rank 2013

Ballico, Edoardo; Bernardi, Alessandra, "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. 215-250. - DOI: 10.1515/apam-2013-0015

2013 journal paper

• #### General tensor decomposition, moment matrices and applications 2013

Bernardi, Alessandra; Jerome, Brachat; Pierre, Comon; Bernard, Mourrain, "General tensor decomposition, moment matrices and applications" in JOURNAL OF SYMBOLIC COMPUTATION, v. 52, (2013), p. 51-71. - 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 Quasi-Hankel 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