# Publications

## Publications

• #### 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. 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 $S\subset X(\mathbb{C})$ 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

• #### On the ranks of the third secant variety of Segre-Veronese embeddings 2018

Ballico, Edoardo; Bernardi, Alessandra, "On the ranks of the third secant variety of Segre-Veronese embeddings" in LINEAR & MULTILINEAR ALGEBRA, v. 2018, (2018). - 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.

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

• #### Singularities of plane rational curves via projections 2017

Bernardi, Alessandra; Gimigliano, Alessandro; Idà, Monica, "Singularities of plane rational curves via projections" in JOURNAL OF SYMBOLIC COMPUTATION, v. 2018, n. 86 (2017), 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 $C_n\subset\mathbb{P}^n$ to $C$ and its interplay with the secant varieties to $C_n$. IN particular, we define via $\mathbf{f}$ certain 0-dimensioal schemes $X_k\subset \mathbb{P}^k$, $2\leq k \leq (n-1)$, 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.

2017 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. 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

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. 78-105. - 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

• #### 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, n. 65, 4 (2017), p. 677-698. - URL: http://dx.doi.org/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 real typical ranks 2017

Grigoriy, Blekherman; Ottaviani, Giorgio; Bernardi, Alessandra, "On real typical ranks" in BOLLETTINO DELLA UNIONE MATEMATICA ITALIANA, v. 2017, (2017). - URL: https://link.springer.com/article/10.1007/s40574-017-0134-0?wt_mc=Internal.Event.1.SEM.ArticleAuthorOnlineFirst . - DOI: 10.1007/s40574-017-0134-0

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 n-variate 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.

2017 journal paper

• #### On parameterizations of plane rational curves and their syzygies 2016

Bernardi, A; Gimigliano, A.; Idà, M., "On parameterizations of plane rational curves and their syzygies" in MATHEMATISCHE NACHRICHTEN, v. 5-6, (2016), p. 537-545. - 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. 1-6

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. 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

• #### Unique decomposition for a polynomial of low rank 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. 1350010-1-1350010-8. - 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 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

• #### 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. 215-250. - DOI: 10.1515/apam-2013-0015

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. 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

• #### 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. 794054-1-794054-5. - DOI: 10.1155/2013/794054

2013 journal paper

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

Bernardi A; 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

• #### 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. 881-894. - DOI: 10.1080/03081087.2012.716430

2013 journal paper

• #### On the X-rank with respect to linearly normal curves 2013

E. Ballico; A. Bernardi, "On the X-rank with respect to linearly normal curves" in COLLECTANEA MATHEMATICA, v. 64, n. 2 (2013), p. 141-154. - DOI: 10.1007//s13348-011-0058-4

2013 journal paper

• #### Decomposition of homogeneous polynomials with low rank 2012

E. Ballico; A. Bernardi, "Decomposition of homogeneous polynomials with low rank" in MATHEMATISCHE ZEITSCHRIFT, v. 271, (2012), p. 1141-1149. - DOI: 10.1007/s00209-011-0907-6

Let F be a homogeneous polynomial of degree d in m + 1 variables definedover an algebraically closed field of characteristic 0 and suppose that F belongs to the s-thsecant variety of the d-uple Veronese embedding of Pm into PN, N= \binom{m+d}{m}-1.but that its minimaldecomposition as a sum of d-th powers of linear forms requires r>s d-powers of linear forms.We show that if s+r ≤ 2d+1, then such a decomposition of F can be split in twoparts: one of them is made by linear forms that can be written using only two variables, theother part is uniquely determined once one has fixed the first part. We also obtain a uniquenesstheorem for the minimal decomposition of F if r is at most d and a mild condition issatisfied.

2012 journal paper

• #### Algebraic geometry tools for the study of entanglement: an application to spin squeezed states 2012

Bernardi, Alessandra; Carusotto, Iacopo, "Algebraic geometry tools for the study of entanglement: an application to spin squeezed states" in JOURNAL OF PHYSICS. A, MATHEMATICAL AND THEORETICAL, v. 45, (2012), p. 105304-105316. - URL: http:// http://iopscience.iop.org/1751-8121/45/10/105304/pdf/1751-8121_45_10_105304.pdf . - DOI: 10.1088/1751-8113/45/10/105304

A short review of algebraic geometry tools for the decomposition of tensors and polynomials is given from the point of view of applications to quantum and atomic physics. Examples of application to assemblies of indistinguishable two-level bosonic atoms are discussed using modern formulations of the classical Sylvester algorithm for the decomposition of homogeneous polynomials in two variables. In particular, the symmetric rank and symmetric border rank of spin squeezed states are calculated as well as their Schrödinger-cat-like decomposition as the sum of macroscopically different coherent spin states; Fock states provide an example of states for which the symmetric rank and the symmetric border rank are different

2012 journal paper

• #### A partial stratification of secant varieties of Veronese variatiesvia curvilinear subschemes 2012

E. Ballico; A. Bernardi, "A partial stratification of secant varieties of Veronese variatiesvia curvilinear subschemes" in SARAJEVO JOURNAL OF MATHEMATICS, n.s., v. 8, n. 1 (2012), p. 33-52

2012 journal paper

• #### Higher secant varieties of P^n\times P^1 embedded in bi-degree(a,b) 2012

E. Ballico; A. Bernardi; M. V. Catalisano, "Higher secant varieties of P^n\times P^1 embedded in bi-degree(a,b)" in COMMUNICATIONS IN ALGEBRA, v. 40, n. 10 (2012), p. 3822-3840. - DOI: 10.1080/00927872.2011.595748

2012 journal paper

• #### Grassmann secants, identifiability, and linear systems of tensors 2012

E. Ballico; A. Bernardi, "Symmetric tensor rank with a tangent vector: a generic uniqueness theorem" in PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY, v. 140, n. 10 (2012), p. 3377-3384. - DOI: 10.1090/S0002-9939-2012-11191-8

2012 journal paper

• #### Higher secant varieties of embedded in bi-degree 2011

Bernardi, Alessandra; Enrico, Carlini; Maria Virginia Catalisano,, "Higher secant varieties of embedded in bi-degree" in JOURNAL OF PURE AND APPLIED ALGEBRA, v. 215, (2011), p. 2853-2858. - URL: http://www.sciencedirect.com/science/article/pii/S002240491100096X . - DOI: 10.1016/j.jpaa.2011.04.005

Let $X^{(n,m)}_{(1,d)}$ denote the Segre-Veronese embedding of $\mathbb{P}^n \times \mathbb{P}^m$ via the sections of the sheaf $\mathcal{O}(1,d)$. We study the dimensions of higher secant varieties of $X^{(n,m)}_{(1,d)}$ and we prove that there is no defective $s^{th}$ secant variety, except possibly for $n$ values of $s$. Moreover when ${m+d \choose d}$ is multiple of $(m+n+1)$, the $s^{th}$ secant variety of $X^{(n,m)}_{(1,d)}$ has the expected dimension for every $s$.

2011 journal paper

• #### Computing symmetric rank for symmetric tensors 2011

Bernardi A; A. Gimigliano; M. Idà, "Computing symmetric rank for symmetric tensors" in JOURNAL OF SYMBOLIC COMPUTATION, v. 46, n. 1 (2011), p. 34-53. - URL: http://www.sciencedirect.com/science/article/pii/S0747717110001240 . - DOI: 10.1016/j.jsc.2010.08.001

We consider the problem of determining the symmetric tensor rank for symmetric tensors with an algebraic geometry approach. We give algorithms for computing the symmetric rank for $2\times ... \times 2$ tensors and for tensors of small border rank. From a geometric point of view, we describe the symmetric rank strata for some secant varieties of Veronese varieties.

2011 journal paper