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The first author with B. Sturmfels studied the variety of matrices with eigenvectors in a given linear subspace, called Kalman variety. We extend that study from matrices to symmetric tensors, proving in the tensor setting the irreducibility of the Kalman variety and computing its codimension and degree. Furthermore we consider the Kalman variety of tensors having singular t-ples with the first component in a given linear subspace and we prove analogous results, which are new even in the case of matrices. Main techniques come from Algebraic Geometry, using Chern classes for enumerative computations., Comment: 11 pages and one table

Let $q$ be a nondegenerate quadratic form on $V$. Let $X\subset V$ be invariant for the action of a Lie group $G$ contained in $SO(V,q)$. For any $f\in V$ consider the function $d_f$ from $X$ to $C$ defined by $d_f(x)=q(f-x)$. We show that the critical points of $d_f$ lie in the subspace orthogonal to ${\mathfrak g}\cdot f$, that we call critical space. In particular any closest point to $f$ in $X$ lie in the critical space. This construction applies to singular t-ples for tensors and to flag varieties and generalizes a previous result of Draisma, Tocino and the author. As an application, we compute the Euclidean Distance degree of a complete flag variety., 9 pages

We present a fine-structure entanglement classification under stochastic local operation and classical communication (SLOCC) for multiqubit pure states. To this end, we employ specific algebraic-geometry tools that are SLOCC invariants, secant varieties, to show that for $n$-qubit systems there are $\lceil\frac{2^{n}}{n+1}\rceil$ entanglement families. By using another invariant, $\ell$-multilinear ranks, each family can be further split into a finite number of subfamilies. Not only does this method facilitate the classification of multipartite entanglement, but it also turns out to be operationally meaningful as it quantifies entanglement as a resource., Comment: 11 pages, 2 figures, Minor changes, Published version

For any (real) algebraic variety $X$ in a Euclidean space $V$ endowed with a nondegenerate quadratic form $q$, we introduce a polynomial $\mathrm{EDpoly}_{X,u}(t^2)$ which, for any $u\in V$, has among its roots the distance from $u$ to $X$. The degree of $\mathrm{EDpoly}_{X,u}$ is the {\em Euclidean Distance degree} of $X$. We prove a duality property when $X$ is a projective variety, namely $\mathrm{EDpoly}_{X,u}(t^2)=\mathrm{EDpoly}_{X^\vee,u}(q(u)-t^2)$ where $X^\vee$ is the dual variety of $X$. When $X$ is transversal to the isotropic quadric $Q$, we prove that the ED polynomial of $X$ is monic and the zero locus of its lower term is $X\cup(X^\vee\cap Q)^\vee$., 24 pages, 4 figures, accepted for publication in Computer Aided Geometric Design

Two fundamental invariants attached to a projective variety are its classical algebraic degree and its Euclidean Distance degree (ED degree). In this paper, we study the asymptotic behavior of these two degrees of some Segre products and their dual varieties. We analyze the asymptotics of degrees of (hypercubical) hyperdeterminants, the dual hypersurfaces to Segre varieties. We offer an alternative viewpoint on the stabilization of the ED degree of some Segre varieties. Although this phenomenon was incidentally known from Friedland-Ottaviani's formula expressing the number of singular vector tuples of a general tensor, our approach provides a geometric explanation. Finally, we establish the stabilization of the degree of the dual variety of a Segre product $X\times Q_{n}$, where $X$ is a projective variety and $Q_n\subset \mathbb{P}^{n+1}$ is a smooth quadric hypersurface., Comment: 24 pages

We define tensors, corresponding to cubic polynomials, which have the same exponent $\omega$ as the matrix multiplication tensor. In particular, we study the symmetrized matrix multiplication tensor $sM_n$ defined on an $n\times n$ matrix $A$ by $sM_n(A)=trace(A^3)$. The use of polynomials enables the introduction of additional techniques from algebraic geometry in the study of the matrix multiplication exponent $\omega$., Comment: 14 pages + appendix of 3 pages with numerical decompositions

In the tensor space $\mathrm{Sym}^d {\mathbb R}^2$ of binary forms we study the best rank $k$ approximation problem. The critical points of the best rank $1$ approximation problem are the eigenvectors and it is known that they span a hyperplane. We prove that the critical points of the best rank $k$ approximation problem lie in the same hyperplane., 9 pages

We prove that the general symmetric tensor in $S^d {\mathbb C}^{n+1}$ of rank r is identifiable, provided that r is smaller than the generic rank. That is, its Waring decomposition as a sum of r powers of linear forms is unique. Only three exceptional cases arise, all of which were known in the literature. Our original contribution regards the case of cubics ($d=3$), while for $d\ge 4$ we rely on known results on weak defectivity by Ballico, Ciliberto, Chiantini, and Mella., 20 pages, two M2 files as ancillary files; v2 contains some changes in section 5

Given a tensor f in a Euclidean tensor space, we are interested in the critical points of the distance function from f to the set of tensors of rank at most k, which we call the critical rank-at-most-k tensors for f. When f is a matrix, the critical rank-one matrices for f correspond to the singular pairs of f. The critical rank-one tensors for f lie in a linear subspace $$H_f$$ H f , the critical space of f. Our main result is that, for any k, the critical rank-at-most-k tensors for a sufficiently general f also lie in the critical space $$H_f$$ H f . This is the part of Eckart–Young Theorem that generalizes from matrices to tensors. Moreover, we show that when the tensor format satisfies the triangle inequalities, the critical space $$H_f$$ H f is spanned by the complex critical rank-one tensors. Since f itself belongs to $$H_f$$ H f , we deduce that also f itself is a linear combination of its critical rank-one tensors.

This is the first in a series of papers on rank decompositions of the matrix multiplication tensor. In this paper we: establish general facts about rank decompositions of tensors, describe potential ways to search for new matrix multiplication decompositions, give a geometric proof of the theorem of Burichenko's theorem establishing the symmetry group of Strassen's algorithm, and present two particularly nice subfamilies in the Strassen family of decompositions., 9 pages

In applications where the tensor rank decomposition arises, one often relies on its identifiability properties for interpreting the individual rank-$1$ terms appearing in the decomposition. Several criteria for identifiability have been proposed in the literature, however few results exist on how frequently they are satisfied. We propose to call a criterion effective if it is satisfied on a dense, open subset of the smallest semi-algebraic set enclosing the set of rank-$r$ tensors. We analyze the effectiveness of Kruskal's criterion when it is combined with reshaping. It is proved that this criterion is effective for both real and complex tensors in its entire range of applicability, which is usually much smaller than the smallest typical rank. Our proof explains when reshaping-based algorithms for computing tensor rank decompositions may be expected to recover the decomposition. Specializing the analysis to symmetric tensors or forms reveals that the reshaped Kruskal criterion may even be effective up to the smallest typical rank for some third, fourth and sixth order symmetric tensors of small dimension as well as for binary forms of degree at least three. We extended this result to $4 \times 4 \times 4 \times 4$ symmetric tensors by analyzing the Hilbert function, resulting in a criterion for symmetric identifiability that is effective up to symmetric rank $8$, which is optimal., Comment: 31 pages, 2 Macaulay2 codes

The Chow form of the essential variety in computer vision is calculated. Our derivation uses secant varieties, Ulrich sheaves and representation theory. Numerical experiments show that our formula can detect noisy point correspondences between two images., Comment: 27 pages, 1 figure, 6 Macaulay2 ancillary files. v2: edits to Theorem 1.1, references, acknowledgements

The Alexander-Hirschowitz theorem says that a general collection of $k$ double points in ${\bf P}^n$ imposes independent conditions on homogeneous polynomials of degree $d$ with a well known list of exceptions. We generalize this theorem to arbitrary zero-dimensional schemes contained in a general union of double points. We work in the polynomial interpolation setting. In this framework our main result says that the affine space of polynomials of degree $\le d$ in $n$ variables, with assigned values of any number of general linear combinations of first partial derivatives, has the expected dimension if $d\neq 2$ with only five exceptional cases. If $d=2$ the exceptional cases are fully described., 34 pages, 2 tables, revised version: different proof of Theorem 4.1, Section 4 significantly changed, Appendix added

Let $Gr(k,n)$ be the Pl\"ucker embedding of the Grassmann variety of projective $k$-planes in $\P n$. For a projective variety $X$, let $\sigma_s(X)$ denote the variety of its $s-1$ secant planes. More precisely, $\sigma_s(X)$ denotes the Zariski closure of the union of linear spans of $s$-tuples of points lying on $X$. We exhibit two functions $s_0(n)\le s_1(n)$ such that $\sigma_s(Gr(2,n))$ has the expected dimension whenever $n\geq 9$ and either $s\le s_0(n)$ or $s_1(n)\le s$. Both $s_0(n)$ and $s_1(n)$ are asymptotic to $\frac{n^2}{18}$. This yields, asymptotically, the typical rank of an element of $\wedge^{3} 1pt {\mathbb C}^{n+1}$. Finally, we classify all defective $\sigma_s(Gr(k,n))$ for $s\le 6$ and provide geometric arguments underlying each defective case., Comment: 17 pages

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 $13$., 13 pages, 1 figure

In this note we discuss an analog of the classical Waring problem for C[x_0, x_1,...,x_n]. Namely, we show that a general homogeneous polynomial p \in C[x_0,x_1,...,x_n] of degree divisible by k\ge 2 can be represented as a sum of at most k^n k-th powers of homogeneous polynomials in C[x_0, x_1,...,x_n]. Noticeably, k^n coincides with the number obtained by naive dimension count., Comment: 6 pages

Let T be a general complex tensor of format $(n_1,...,n_d)$. When the fraction $\prod_in_i/[1+\sum_i(n_i-1)]$ is an integer, and a natural inequality (called balancedness) is satisfied, it is expected that T has finitely many minimal decomposition as a sum of decomposable tensors. We show how homotopy techniques allow us to find all the decompositions of T, starting from a given one. Computationally, this gives a guess regarding the total number of such decompositions. This guess matches exactly with all cases previously known, and predicts several unknown cases. Some surprising experiments yielded two new cases of generic identifiability: formats (3,4,5) and (2,2,2,3) which have a unique decomposition as the sum of 6 and 4 decomposable tensors, respectively. We conjecture that these two cases together with the classically known matrix pencils are the only cases where generic identifiability holds, i.e., the only identifiable cases. Building on the computational experiments, we use algebraic geometry to prove these two new cases are indeed generically identifiable., 21 pages, two Macaulay2 codes as ancillary files. Conjecture 1.4 in v1 is now Theorem 1.4 by Galuppi and Mella

We propose a new sufficient condition for verifying whether generic rank-r complex tensors of arbitrary order admit a unique decomposition as a linear combination of rank-1 tensors. A practical algorithm is proposed for verifying this condition, with which it was established that in all spaces of dimension less than 15000, with a few known exceptions, listed in the paper, generic identifiability holds for ranks up to one less than the generic rank of the space. This is the largest possible rank value for which generic identifiability can hold, except for spaces with a perfect shape. The algorithm can also verify the identifiability of a given specific rank-r decomposition, provided that it can be shown to correspond to a nonsingular point of the r-th order secant variety. For sufficiently small rank, which nevertheless improves upon the known bounds for specific identifiability, some local equations of this variety are known, allowing us to verify this property. As a particular example of our approach, we prove the identifiability of a specific 5x5x5 tensor of rank 7, which cannot be handled by the conditions recently provided in [I. Domanov and L. De Lathauwer, On the Uniqueness of the Canonical Polyadic Decomposition of third-order tensors--Part II: Uniqueness of the overall decomposition, SIAM J. Matrix Anal. Appl. 34(3), 2013]. Finally, we also present a surprising new class of weakly-defective Segre varieties that nevertheless turns out to admit a generically unique decomposition., 22 pages, one C++ code and two Macaulay2 codes as ancillary files

The identifiability of parameters in a probabilistic model is a crucial notion in statistical inference. We prove that a general tensor of rank 8 in C^3\otimes C^6\otimes C^6 has at least 6 decompositions as sum of simple tensors, so it is not 8-identifiable. This is the highest known example of balanced tensors of dimension 3, which are not k-identifiable, when k is smaller than the generic rank., Comment: 7 pages, one Macaulay2 script as ancillary file, two references added

Structured low-rank approximation is the problem of minimizing a weighted Frobenius distance to a given matrix among all matrices of fixed rank in a linear space of matrices. We study exact solutions to this problem by way of computational algebraic geometry. A particular focus lies on Hankel matrices, Sylvester matrices and generic linear spaces., Comment: 22 pages; theorem numbering fits with the journal version

We prove that the general tensor of size 2^n and rank k has a unique decomposition as the sum of decomposable tensors if k, 12 pages, three Macaulay2 scripts as ancillary files. v3: final version to appear in Annali di Matematica Pura e Applicata

The nearest point map of a real algebraic variety with respect to Euclidean distance is an algebraic function. For instance, for varieties of low rank matrices, the Eckart-Young Theorem states that this map is given by the singular value decomposition. This article develops a theory of such nearest point maps from the perspective of computational algebraic geometry. The Euclidean distance degree of a variety is the number of critical points of the squared distance to a generic point outside the variety. Focusing on varieties seen in applications, we present numerous tools for exact computations., Comment: to appear in Foundations of Computational Mathematics

A Waring decomposition of a (homogeneous) polynomial f is a minimal sum of powers of linear forms expressing f. Under certain conditions, such a decomposition is unique. We discuss some algorithms to compute the Waring decomposition, which are linked to the equation of certain secant varieties and to eigenvectors of tensors. In particular we explicitly decompose a general cubic polynomial in three variables as the sum of five cubes (Sylvester Pentahedral Theorem)., 32 pages; three Macaulay2 files as ancillary files. Revised with referee's suggestions. Accepted JSC

We determine the rank of a general real binary form of degree d=4 and d=5. In the case d=5, the possible values of the rank of such general forms are 3,4,5. The existence of three typical ranks was unexpected. We prove that a real binary form of degree d with d real roots has rank d., Comment: 12 pages, 2 figures

In this paper we discuss the notion of singular vector tuples of a complex valued $d$-mode tensor of dimension m_1 x ... x m_d. We show that a generic tensor has a finite number of singular vector tuples, viewed as points in the corresponding Segre product. We give the formula for the number of singular vector tuples. We show similar results for tensors with partial symmetry. We give analogous results for the homogeneous pencil eigenvalue problem for cubic tensors, i.e. m_1=...=m_d. We show uniqueness of best approximations for almost all real tensors in the following cases: rank one approximation; rank one approximation for partially symmetric tensors (this approximation is also partially symmetric); rank-(r_1,...,r_d) approximation for $d$-mode tensors., Comment: 32 pages, 1 figure, 2 tables

We introduce an inductive method for the study of the uniqueness of decompositions of tensors, by means of tensors of rank 1. The method is based on the geometric notion of weak defectivity. For three-dimensional tensors of type (a, b, c), a\le b\le c, our method proves that the decomposition is unique (i.e. k-identifiability holds) for general tensors of rank k, as soon as k\le (a+1)(b+1)/16. This improves considerably the known range for identifiability. The method applies also to tensor of higher dimension. For tensors of small size, we give a complete list of situations where identifiability does not hold. Among them, there are 4\times4\times4 tensors of rank 6, an interesting case because of its connection with the study of DNA strings., Comment: 19 pages; two Macaulay2 files as ancillary files

We prove that r independent homogeneous polynomials of the same degree d become dependent when restricted to any hyperplane if and only if their inverse system parameterizes a variety whose (d-1)-osculating spaces have dimension smaller than expected. This gives an equivalence between an algebraic notion (called Weak Lefschetz Property) and a differential geometric notion, concerning varieties which satisfy certain Laplace equations. In the toric case, some relevant examples are classified and as byproduct we provide counterexamples to Ilardi's conjecture., Comment: 21 pages, 4 figures

The Kalman variety of a linear subspace in a vector space consists of all endomorphism that possess an eigenvector in that subspace. We study the defining polynomials and basic geometric invariants of the Kalman variety., 13 pages, added five references

The hypersurface of Luroth quartic curves inside the projective space of plane quartics has degree 54. We give a proof of this fact along the lines outlined in a paper by Morley, published in 1919. Another proof has been given by Le Potier and Tikhomirov in 2001, in the setting of moduli spaces of vector bundles on the projective plane. Morley's proof uses the description of plane quartics as branch curves of Geiser involutions and gives new geometrical interpretations of the 36 planes associated to the Cremona hexahedral representations of a nonsingular cubic surface., Comment: Final version to appear in Michigan Math. Journal. The last section of v1 has been removed and expanded in the paper "On singular Luroth quartics", arXiv:0911.2101v1

Plane quartics containing the ten vertices of a complete pentalateral and limits of them are called L\"uroth quartics. The locus of singular L\"uroth quartics has two irreducible components, both of codimension two in $\P^{14}$. We compute the degree of them and we discuss the consequences of this computation on the explicit form of the L\"uroth invariant. One important tool are the Cremona hexahedral equations of the cubic surface. We also compute the class in $\overline{M}_3$ of the closure of the locus of nonsingular L\"uroth quartics., Comment: Enlarged version. Computation of the class of the locus of Luroth quartics in the moduli space added

We compute the equation of the 7-secant variety to the Veronese variety (P^4,O(3)), its degree is 15. This is the last missing invariant in the Alexander-Hirschowitz classification. It gives the condition to express a homogeneous cubic polynomial in 5 variables as the sum of 7 cubes (Waring problem). The interesting side in the construction is that it comes from the determinant of a matrix of order 45 with linear entries, which is a cube. The same technique allows to express the classical Aronhold invariantof plane cubics as a pfaffian., 11 pages

This paper studies the dimension of secant varieties to Segre varieties. The problem is cast both in the setting of tensor algebra and in the setting of algebraic geometry. An inductive procedure is built around the ideas of successive specializations of points and projections. This reduces the calculation of the dimension of the secant variety in a high dimensional case to a sequence of calculations of partial secant varieties in low dimensional cases. As applications of the technique: We give a complete classification of defective $t$-secant varieties to Segre varieties for t < 7. We generalize a theorem of Catalisano-Geramita-Gimigliano on non-defectivity of tensor powers of P^n. We determine the set of p for which unbalanced Segre varieties have defective p-secant varieties. In addition, we show that the Segre varieties P^1 x P^1 x P^n x P^n and P^2 x P ^3 x P^3 are deficient and completely describe the dimensions of their secant varieties. In the final section we propose a series of conjectures about defective Segre varieties., the reference to [CGG1] in example 5.6 is expanded

We describe the cohomology groups of a homogeneous vector bundle $E$ on any Hermitian symmetric variety $X=G/P$ of ADE type as the cohomology of a complex explicitly described. The main tool is the equivalence between the category of homogeneous bundles and the category of representations of a certain quiver ${\cal Q}_X$ with relations, whose vertices are the dominant weights of the reductive part of $P$. This equivalence was found in some cases by Bondal, Kapranov and Hille and we find the appropriate relations for any Hermitian symmetric variety, computing them explicitly for Grassmannians., Comment: 41 pages, 2 figures, computation of cohomology works on any Hermitian symmetric variety of ADE type, the relations are explicitly computed for Grassmannians

In this note we prove that the moduli space of rank $2n$ symplectic instanton bundles on ${\PP^{2n+1}}$, defined from the well known monad condition, is affine. This result was not known even in the case $n=1$, where the real instanton bundles correspond to self dual Yang Mills $Sp(1)$-connections over the 4-dimensional sphere. The result is proved as a consequence of the existence of an invariant of the multidimensional matrices representing the instanton bundles., Comment: 9 pages, Latex 2e