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We show that if F is a topological field, then there is a transitive, free and continuous action of a natural quotient of GLk(F)×GLk+1(F) on the set Mk(F) of 2×k×(k+1) hypermatrices over F with non-zero hyperdeterminant. We use this action to study the homotopy type of Mk(C) and Mk(R) and count elements of Mk(Fq) (generalizing an unpublished result of Lewis and Sam).

We observe that biquadratic potentials admit non-trivial flat directions when the determinant of the quartic coupling matrix of the scalar fields vanishes. This consideration suggests a new approach to the problem of finding flat directions in scale-invariant theories, noticeably simplifying the study of scalar potentials involving many fields. The method generalizes to arbitrary quartic potentials by requiring that the hyperdeterminant of the tensor of scalar couplings be zero. We demonstrate our approach with detailed examples pertaining to common scalar extensions of the Standard Model., Comment: 6 pages, 1 figure

Linde, Moore, and Nordahl introduced a generalisation of the honeycomb dimer model to higher dimensions. The purpose of this article is to describe a number of structural properties of this generalised model. First, it is shown that the samples of the model are in one-to-one correspondence with the perfect matchings of a hypergraph. This leads to a generalised Kasteleyn theory: the partition function of the model equals the Cayley hyperdeterminant of the adjacency hypermatrix of the hypergraph. Second, we prove an identity which relates the covariance matrix of the random height function directly to the random geometrical structure of the model. This identity is known in the planar case but is new for higher dimensions. It relies on a more explicit formulation of Sheffield's cluster swap which is made possible by the structure of the honeycomb dimer model. Finally, we use the special properties of this explicit cluster swap to give a new and simplified proof of strict convexity of the surface tension in this case., Comment: 32 pages, 4 figures; optimised presentation

Minkowski's second theorem can be stated as an inequality for $n$-dimensional flat Finsler tori relating the volume and the minimal product of the lengths of closed geodesics which form a homology basis. In this paper we show how this fundamental result can be promoted to a principle holding for a larger class of Finsler manifolds. This includes manifolds for which first Betti number and dimension do no necessarily coincide, a prime example being the case of surfaces. This class of manifolds is described by a non-vanishing condition for the hyperdeterminant reduced modulo $2$ of the multilinear map induced by the fundamental class of the manifold on its first ${\mathbb Z}_2$-cohomology group using the cup product., Comment: 24 pages

In this paper, we first present an explicit factorization expression of hyperdeterminants of Cohen even functions defined on cartesian powers of meet-semilattices, which is a unified generalization of results in previous literatures. Then applying this expression, we evaluate a kind of hyperdeterminants associated with coprime decomposable meet-semilattices. This gives a complete solution of a problem raised in [Hong S, Li M, and Wang B. Hyperdeterminants associated with multiple even functions, Ramanujan J. 2014;34:265–281].

In this paper, a new product of tensors is given, which is a generalization of the product of tensors introduced by Gelfand, Kapranov, Zelevinsky. This product satisfies the associative law. Using the associative law, it is concluded that the product of tensors given by Bu and others and the general product of tensors introduced by Shao can be represented by this new tensors product. As an application of this new tensors product and the associative law, the hyperdeterminant of the product of a kind of tensors is studied, i.e., some formulas for the hyperdeterminant of boundary formats product are shown.

The paper links four conjectures: (1) (Rota's bases conjecture): For any system A = ( A 1 , … , A n ) of non-singular real valued matrices the multiset of all columns of matrices in A can be decomposed into n independent systems of representatives of A . (2) (Alon–Tarsi): For even n , the number of even n × n Latin squares differs from the number of odd n × n Latin squares. (3) (Stones–Wanless, Kotlar): For all n , the number of even n × n Latin squares with the identity permutation as first row and first column differs from the number of odd n × n Latin squares of this type. (4) (Aharoni–Berger): Let M and N be two matroids on the same vertex set, and let A 1 , … , A n be sets of size n + 1 belonging to M ∩ N . Then there exists a set belonging to M ∩ N meeting all A i . Huang and Rota [8] and independently Onn [11] proved that for any n (2) implies (1). We prove equivalence between (2) and (3). Using this, and a special case of (4), we prove the Huang–Rota–Onn theorem for n odd and a restricted class of input matrices: assuming the Alon–Tarsi conjecture for n − 1 , Rota's conjecture is true for any system of non-singular real valued matrices where one of them is non-negative and the remaining have non-negative inverses.

Constructions of several multidimensional determinants for matrices in dimensions n = 2m are described. With one exception, the constructions are facilitated by isomorphisms with rings of commutative hypercomplex numbers. Properties shared by the Cayley hyperdeterminant, such as the multiplicative property |AB| = |A||B|, are established and the degrees and homogeneity of the determinants are discussed. These considerations emphasize the close relationship between order 2 multidimensional matrices in dimension m and hypercomplex systems in dimension n = 2m.

Using vertex operator we study Macdonald symmetric functions of rectangular shapes and their connection with the q-Dyson Laurent polynomial. We find a vertex operator realization of Macdonald functions and thus give a generalized Frobenius formula for them. As byproducts of the realization, we find a q-Dyson constant term orthogonality relation which generalizes a conjecture due to Kadell in 2000, and we generalize Matsumoto's hyperdeterminant formula for rectangular Jack functions to Macdonald functions., 25 pages, Main results reported on The 13th national Conference on Lie Algebra, Sichuan,China (July 22, 2013)

We briefly review previous work on the invariant theory of 3 × 3 × 3 arrays. We then recall how to generate arrays of arbitrary size $${m_1 \times \cdots \times m_k}$$ with hyperdeterminant 0. Our main result is an explicit formula for the 3 × 3 × 3 hyperdeterminant as a polynomial in the fundamental invariants I 6, I 9, I 12 for the action of the Lie group $${SL_3(\mathbb{C}) \times SL_3(\mathbb{C}) \times SL_3(\mathbb{C})}$$ . We apply our calculations to Nurmiev’s classification of normal forms for 3 × 3 × 3 arrays.

In this paper, we introduce a new method to give the formulae for the hyperdeterminants of higher-dimensional matrices associated with multiple even function (modr) on the gcd-closed sets. Our result generalizes the result of Yamasaki obtained in 2010 and also extends the results of Haukkanen and Hong obtained in 1992 and 2002, respectively.

The aim of this review is to present and analyze the probabilistic models of mathematical phylogenetics which have been intensively used in recent years in biology as the cornerstone of attempts to infer and reconstruct the ancestral relationships between species. We outline the development of theoretical phylogenetics, from the earliest studies based on morphological characters, through to the use of molecular data in a wide variety of forms. We bring the lens of mathematical physics to bear on the formulation of theoretical models, focussing on the applicability of many methods from the toolkit of that tradition -- techniques of groups and representations to guide model specification and to exploit the multilinear setting of the models in the presence of underlying symmetries; extensions to coalgebraic properties of the generators associated to rate matrices underlying the models, in relation to the graphical structures (trees and networks) which form the search space for inferring evolutionary trees. Aspects presented, include relating model classes to relevant matrix Lie algebras, as well as manipulations with group characters to enumerate various natural polynomial invariants, for identifying robust, low-parameter quantities for use in inference. Above all, we wish to emphasize the many features of multipartite entanglement which are shared between descriptions of quantum states on the physics side, and the multi-way tensor probability arrays arising in phylogenetics. In some instances, well-known objects such as the Cayley hyperdeterminant (the `tangle') can be directly imported into the formalism -- for models with binary character traits, and triplets of taxa. In other cases new objects appear, such as the remarkable quintic `squangle' invariants for quartet tree discrimination and DNA data, with their own unique interpretation in the phylogenetic modeling context., Comment: 51 pages, LaTeX, 3 figures. Minor clarifications added and typos corrected

Though algebraic geometry over $\mathbb C$ is often used to describe the closure of the tensors of a given size and complex rank, this variety includes tensors of both smaller and larger rank. Here we focus on the $n\times n\times n$ tensors of rank $n$ over $\mathbb C$, which has as a dense subset the orbit of a single tensor under a natural group action. We construct polynomial invariants under this group action whose non-vanishing distinguishes this orbit from points only in its closure. Together with an explicit subset of the defining polynomials of the variety, this gives a semialgebraic description of the tensors of rank $n$ and multilinear rank $(n,n,n)$. The polynomials we construct coincide with Cayley's hyperdeterminant in the case $n=2$, and thus generalize it. Though our construction is direct and explicit, we also recast our functions in the language of representation theory for additional insights. We give three applications in different directions: First, we develop basic topological understanding of how the real tensors of complex rank $n$ and multilinear rank $(n,n,n)$ form a collection of path-connected subsets, one of which contains tensors of real rank $n$. Second, we use the invariants to develop a semialgebraic description of the set of probability distributions that can arise from a simple stochastic model with a hidden variable, a model that is important in phylogenetics and other fields. Third, we construct simple examples of tensors of rank $2n-1$ which lie in the closure of those of rank $n$., 31 pages, 1 figure

We study an upper bound of the ranks of n-tensors with format \((2,2,\ldots ,2)\) over the complex and real number fields. We consider Cayley’s hyperdeterminant and give the necessary and sufficient condition for a tensor of format (2, 2, 2) to have maximal rank 3. Brylinski showed that the maximal rank of complex tensors with format (2, 2, 2, 2) is 4, and Kong and Jiang showed that the maximal rank of real tensors with format (2, 2, 2, 2) is less than or equal to 5. Catalisano, Geramita, and Gimigliano showed that the generic rank of complex n-tensors with format \((2,2,\ldots ,2)\) is equal to \(\lceil 2^n/(n+1)\rceil \), and then, the maximal rank of real n-tensors with format \((2,2,\ldots ,2)\) is greater than or equal to \(\lceil 2^n/(n+1)\rceil \). The maximal rank of real n-tensors with format \((2,2,\ldots ,2)\) is less than or equal to \(2^{n-1}+1\). Bleckherman and Teitler showed that the maximal rank is less than or equal to twice the generic rank, which gives a more strict upper bound of the maximal rank of n-tensors with format \((2,2,\ldots ,2)\) over the real number field.

The real rank two locus of an algebraic variety is the closure of the union of all secant lines spanned by real points. We seek a semi-algebraic description of this set. Its algebraic boundary consists of the tangential variety and the edge variety. Our study of Segre and Veronese varieties yields a characterization of tensors of real rank two., Comment: 20 pages

The hyperdeterminant of a polynomial (interpreted as a symmetric tensor) factors into several irreducible factors with multiplicities. Using geometric techniques these factors are identified along with their degrees and their multiplicities. The analogous decomposition for the \mu-discriminant of polynomial is found., Comment: some typos corrected

As a first step in the classification of nonsingular 2×2×2×2 hypercubes up to equivalence, we resolve the case where the base field is finite and the hypercubes can be written as a product of two 2×2×2 hypercubes. (Nonsingular hypercubes were introduced by D. Knuth in the context of semifields. Where semifields are related to hypercubes of dimension 3, this paper considers the next case, i.e., hypercubes of dimension 4.) We define the notion of ij-rank (with 1 ≤ i < j ≤ 4) and prove that a hypercube is the product of two 2×2×2 hypercubes if and only if its 12-rank is at most 2. We derive a ‘standard form’ for nonsingular 2×2×2×2 hypercubes of 12-rank less than 4 as a first step in the classification of such hypercubes up to equivalence. Our main result states that the equivalence class of a nonsingular 2×2×2×2 hypercube M of 12-rank 2 depends only on the value of an invariant δ0(M) which derives in a natural way from the Cayley hyperdeterminant det0M and another polynomial invariant det M of degree 4. As a corollary we prove that the number of equivalence classes is (q + 1)/2 or q/2 depending on whether the order q of the field is odd or even.

We use the representation theory of Lie algebras and computational linear algebra to determine the simplest non-constant invariant polynomial in the entries of a general 2 × 2 × 3 array. This polynomial is homogeneous of degree 6 and has 66 terms with coefficients ±1, ±2 in the 12 indeterminates x ijk where i, j = 1, 2 and k = 1, 2, 3. This invariant can be regarded as a natural generalization of Cayley's hyperdeterminant for 2 × 2 × 2 arrays.

Cayley’s hyperdeterminant is a homogeneous polynomial of degree 4 in the 8 entries of a 2 × 2 × 2 array. It is the simplest (nonconstant) polynomial which is invariant under changes of basis in three directions. We use elementary facts about representations of the 3-dimensional simple Lie algebra s l 2 ( C ) to reduce the problem of finding the invariant polynomials for a 2 × 2 × 2 array to a combinatorial problem on the enumeration of 2 × 2 × 2 arrays with non-negative integer entries. We then apply results from linear algebra to obtain a new proof that Cayley’s hyperdeterminant generates all the invariants. In the last section we discuss the application of our methods to general multidimensional arrays.

Let $X$ be a smooth irreducible nondegenerate projective variety and let $X^*$ denote its dual variety. It is well known that $\sigma_2(X)^*$, the dual of the 2-secant variety of $X$, is a component of the singular locus of $X^*$. The locus of bitangent hyperplanes, i.e. hyperplanes tangent to at least two points of $X$, is a component of the sigular locus of $X^*$. In this paper we provide a sufficient condition for this component to be of maximal dimension and show how it can be used to determine which dual varieties of Grassmannians are normal. That last part may be compared to what has been done for hyperdeterminants by J. Weyman and A. Zelevinski (1996)., Comment: 14 pages, appeared in Journal of Algebra (2011)

We solve a number of problems in quantum computing by applying genetic algorithms. We use the bitset class of C ++ to represent any data type for genetic algorithms. Thus we have a flexible way to solve any optimization problem. The Bell-CHSH inequality and entanglement measures are studied using genetic algorithms. Entangled states form the backbone for teleportation. The C ++ code is also provided.

A formula for Glynn's hyperdeterminant $\det_p$ ($p$ prime) of a square matrix shows that the number of ways to decompose any integral doubly stochastic matrix with row and column sums $p-1$ into $p-1$ permutation matrices with even product, minus the number of ways with odd product, is 1 (mod $p$). It follows that the number of even Latin squares of order $p-1$ is not equal to the number of odd Latin squares of that order. Thus Rota's basis conjecture is true for a vector space of dimension $p-1$ over any field of characteristic zero or $p$, and all other characteristics except possibly a finite number. It is also shown where there is a mistake in a published proof that claimed to multiply the known dimensions by powers of two, and that also claimed that the number of even Latin squares is greater than the number of odd Latin squares. Now, 26 is the smallest unknown case where Rota's basis conjecture for vector spaces of even dimension over a field is unsolved.

As well as arising naturally in the study of non-intersecting random paths, random spanning trees, and eigenvalues of random matrices, determinantal point processes (sometimes also called fermionic point processes) are relatively easy to simulate and provide a quite broad class of models that exhibit repulsion between points. The fundamental ingredient used to construct a determinantal point process is a kernel giving the pairwise interactions between points: the joint distribution of any number of points then has a simple expression in terms of determinants of certain matrices defined from this kernel. In this paper we initiate the study of an analogous class of point processes that are defined in terms of a kernel giving the interaction between 2M points for some integer M. The role of matrices is now played by 2M-dimensional “hypercubic” arrays, and the determinant is replaced by a suitable generalization of it to such arrays—Cayley’s first hyperdeterminant. We show that some of the desirable features of determinantal point processes continue to be exhibited by this generalization.

We derive a Jacobi-Trudi type formula for Jack functions of rectangular shapes. In this formula, we make use of a hyperdeterminant, which is Cayley's simple generalization of the determinant. In addition, after developing the general theory of hyperdeterminants, we give summation formulae for Schur functions involving hyperdeterminants, and evaluate Toeplitz-type hyperdeterminants by using Jack function theory., 18 pages. The title of 3rd version is ''Hyperdeterminant expressions ...''. The title of 1st and 2nd versions was ``Summation formulas for Schur functions involving hyperdeterminants''

In this paper, we present an eigenvalue method for testing positive definiteness of a multivariate form. This problem plays an important role in the stability study of nonlinear autonomous systems via Lyapunov's direct method in automatic control. At first we apply the D'Andrea-Dickenstein version of the classical Macaulay formulas of the resultant to compute the symmetric hyperdeterminant of an even order supersymmetric tensor. By using the supersymmetry property, we give detailed computation procedures for the Bezoutians and specified ordering of monomials in this approach. We then use these formulas to calculate the characteristic polynomial of a fourth order three dimensional supersymmetric tensor and give an eigenvalue method for testing positive definiteness of a quartic form of three variables. Some numerical results of this method are reported.

The principal minors of a symmetric $n{\times}n$-matrix form a vector of length $2^n$. We characterize these vectors in terms of algebraic equations derived from the $ 2{\times}2{\times}2$-hyperdeterminant., Comment: 12 pages; referee's suggestions incorporated, new section "Invariance and the Quartic Generation Conjecture" added

It has been recently shown that Nambu-Goto action can be re-expressed in terms of Cayley's hyperdeterminant with the manifest SL(2,R) X SL(2,R) X SL(2,R) symmetry. In the present paper, we show that the same feature is shared by Green-Schwarz sigma-model for N=2 superstring whose target space-time is D=2+2. When its zweibein field is eliminated from the action, it contains the Nambu-Goto action which is nothing but the square root of Cayley's hyperdeterminant of the pull-back in superspace \sqrt{\hyperdet(\Pi_{i \a\Dot\a})} manifestly invariant under SL(2,R) X SL(2,R) X SL(2,R). The target space-time D=2+2 can accommodate self-dual supersymmetric Yang-Mills theory. Our action has also fermionic kappa-symmetry, satisfying the criterion for its light-cone equivalence to Neveu-Schwarz-Ramond formulation for N=2 superstring., Comment: 14 pages, no figures

We study the entanglement of a pure state of a composite quantum system consisting of several subsystems with $d$ levels each. It can be described by the R\'enyi-Ingarden-Urbanik entropy $S_q$ of a decomposition of the state in a product basis, minimized over all local unitary transformations. In the case $q=0$ this quantity becomes a function of the rank of the tensor representing the state, while in the limit $q \to \infty$ the entropy becomes related to the overlap with the closest separable state and the geometric measure of entanglement. For any bipartite system the entropy $S_1$ coincides with the standard entanglement entropy. We analyze the distribution of the minimal entropy for random states of three and four-qubit systems. In the former case the distributions of $3$-tangle is studied and some of its moments are evaluated, while in the latter case we analyze the distribution of the hyperdeterminant. The behavior of the maximum overlap of a three-qudit system with the closest separable state is also investigated in the asymptotic limit., Comment: 19 pages

In this paper, we define the symmetric hyperdeterminant, eigenvalues and E-eigenvalues of a real supersymmetric tensor. We show that eigenvalues are roots of a one-dimensional polynomial, and when the order of the tensor is even, E-eigenvalues are roots of another one-dimensional polynomial. These two one-dimensional polynomials are associated with the symmetric hyperdeterminant. We call them the characteristic polynomial and the E-characteristic polynomial of that supersymmetric tensor. Real eigenvalues (E-eigenvalues) with real eigenvectors (E-eigenvectors) are called H-eigenvalues (Z-eigenvalues). When the order of the supersymmetric tensor is even, H-eigenvalues (Z-eigenvalues) exist and the supersymmetric tensor is positive definite if and only if all of its H-eigenvalues (Z-eigenvalues) are positive. An mth-order n-dimensional supersymmetric tensor where m is even has exactly n(m-1)^n^-^1 eigenvalues, and the number of its E-eigenvalues is strictly less than n(m-1)^n^-^1 when m>=4. We show that the product of all the eigenvalues is equal to the value of the symmetric hyperdeterminant, while the sum of all the eigenvalues is equal to the sum of the diagonal elements of that supersymmetric tensor, multiplied by (m-1)^n^-^1. The n(m-1)^n^-^1 eigenvalues are distributed in n disks in C. The centers and radii of these n disks are the diagonal elements, and the sums of the absolute values of the corresponding off-diagonal elements, of that supersymmetric tensor. On the other hand, E-eigenvalues are invariant under orthogonal transformations.

Let A, B be multi-dimensional matrices of boundary format ∏i=0p(ki+1), ∏j=0q(lj+1), respectively. Assume that kp=l0 so that the convolution A∗B is defined. We prove that Det (A∗B)= Det (A) α · Det (B) β where α=l0!/(l1!…lq!), β=(k0+1)!/(k1!…kp−1!(kp+1)!), and Det is the hyperdeterminant. When A, B are square matrices, this formula is the usual Binet–Cauchy Theorem computing the determinant of the product A·B. It follows that A∗B is nondegenerate if and only if A and B are both nondegenerate. We show by a counterexample that the assumption of boundary format cannot be dropped.

We find the generating set of \(\mathop{\mathrm{SL}}\nolimits\)-invariant polynomials in four qubits that are also invariant under permutations of the qubits. The set consists of four polynomials of degrees 2, 6, 8, and 12, for which we find an elegant expression in the space of critical states. These invariants are the degrees if the basic invariants of the invariants for F 4, and in fact, the group plays an important role in this note. In addition, we show that the hyperdeterminant in four qubits is the only \(\mathop{\mathrm{SL}}\nolimits\)-invariant polynomial (up to powers of itself) that is non-vanishing precisely on the set of generic states.

The classical theory of determinants was placed on a solid basis by Cayley in 1843. A few years later, Cayley himself elaborated a generalization to the multidimensional setting in two different ways. There are indeed several ways to generalize the notion of determinant to multidimensional matrices. Cayley’s second attempt has a geometric flavor and was very fruitful. This invariant constructed by Cayley is named today hyperdeterminant and reduces to the determinant in the case of square matrices. The explicit computation of the hyperdeterminant presented from the very beginning exceptional difficulties. In 1992, thanks to a fundamental paper by Gelfand, Kapranov and Zelevinsky, the theory was placed in the modern language and many new results have been found. In this survey we introduce the hyperdeterminants and some of its properties from scratch. Our aim is to provide elementary arguments, when they are available. The main tools we use are the biduality theorem and the language of vector bundles.

The absolute value of the hyperdeterminant of four qubits is a useful measure of genuine entanglement. We prove a recent conjecture of Gour and Wallach describing the pure maximally entangled four-qubit states with respect to this measure., Comment: 6 pages, minor revision

Alice seeks an information-theoretically secure source of private random data. Unfortunately, she lacks a personal source and must use remote sources controlled by other parties. Alice wants to simulate a coin flip of specified bias α, as a function of data she receives from p sources; she seeks privacy from any coalition of r of them. We show: If p/2≤r

Many of the stochastic models used in inference of phylogenetic trees from biological sequence data have polynomial parameterization maps. The image of such a map --- the collection of joint distributions for a model --- forms the model space. Since the parameterization is polynomial, the Zariski closure of the model space is an algebraic variety which is typically much larger than the model space, but has been usefully studied with algebraic methods. Of ultimate interest, however, is not the full variety, but only the model space. Here we develop complete semialgebraic descriptions of the model space arising from the k-state general Markov model on a tree, with slightly restricted parameters. Our approach depends upon both recently-formulated analogs of Cayley's hyperdeterminant, and the construction of certain quadratic forms from the joint distribution whose positive (semi-)definiteness encodes information about parameter values. We additionally investigate the use of Sturm sequences for obtaining similar results., Comment: 29 pages, 0 figures; Mittag-Leffler Institute, Spring 2011

Diophantine m-tuples with property D(n), for n an integer, are sets of m positive integers such that the product of any two of them plus n is a square. Triples and quadruples with this property can be classed as regular or irregular according to whether they satisfy certain polynomial identities. Given any such m-tuple, a symmetric integer matrix can be formed with the elements of the set placed in the diagonal and with corresponding roots off-diagonal. In the case of quadruples, Jacobi's theorem for the minors of the adjugate matrix can be used to show that up to eight new Diophantine quadruples can be formed from the adjugate matrices with various combinations of signs for the roots. We call these adjugate quadruples.

We prove the discrete triality invariance of the N=2 NSR superstring moving in a D=2+2 target space. We find that triality holds also in the Siegel-Berkovits formulation of the selfdual superstring. A supersymmetric generalization of Cayley's hyperdeterminant, based on a quartic invariant of the SL(2|1)^3 superalgebra, is presented., Comment: 11 pages, 1 figure, LaTeX. Misprints corrected in the supermatrices of eq. (5.2)

It has recently been shown that there is a connection between Cayley's hypdeterminant and the principal minors of a symmetric matrix. With an eye towards characterizing the entropy region of jointly Gaussian random variables, we obtain three new results on the relationship between Gaussian random variables and the hyperdeterminant. The first is a new (determinant) formula for the 2times2times2 hyperdeterminant. The second is a new (transparent) proof of the fact that the principal minors of an ntimesn symmetric matrix satisfy the 2 times 2 times .... times 2 (n times) hyperdeterminant relations. The third is a minimal set of 5 equations that 15 real numbers must satisfy to be the principal minors of a 4 times 4 symmetric matrix.

In quantum information theory, it is well known that the tripartite entanglement of three qubits is described by the group [SL(2,C)]^3 and that the entanglement measure is given by Cayley's hyperdeterminant. This has provided an analogy with certain N=2 supersymmetric black holes in string theory, whose entropy is also given by the hyperdeterminant. In this paper, we extend the analogy to N=8. We propose that a particular tripartite entanglement of seven qubits, encoded in the Fano plane, is described by the exceptional group E_7(C) and that the entanglement measure is given by Cartan's quartic E_7 invariant., Minor improvements. 15 page latex

The four-dimensional N=2 STU model of string compactification is invariant under an SL(2,Z)_S x SL(2,Z)_T x SL(2,Z)_U duality acting on the dilaton/axion S, complex Kahler form T and the complex structure fields U, and also under a string/string/string triality S-T-U. The model admits an extremal black hole solution with four electric and four magnetic charges whose entropy must respect these symmetries. It is given by the square root of the hyperdeterminant introduced by Cayley in 1845. This also features in three-qubit quantum entanglement., Minor improvements. 9 pages latex

The hyperdeterminant of format 2 x 2 x 2 x 2 is a polynomial of degree 24 in 16 unknowns which has 2894276 terms. We compute the Newton polytope of this polynomial and the secondary polytope of the 4-cube. The 87959448 regular triangulations of the 4-cube are classified into 25448 D-equivalence classes, one for each vertex of the Newton polytope. The 4-cube has 80876 coarsest regular subdivisions, one for each facet of the secondary polytope, but only 268 of them come from the hyperdeterminant., Comment: 30 pages, 6 figures; An author's name changed, typos fixed

There has been continued interest in seeking a theorem describing optimal low-rank approximations to tensors of order 3 or higher, that parallels the Eckart-Young theorem for matrices. In this paper, we argue that the naive approach to this problem is doomed to failure because, unlike matrices, tensors of order 3 or higher can fail to have best rank-r approximations. The phenomenon is much more widespread than one might suspect: examples of this failure can be constructed over a wide range of dimensions, orders and ranks, regardless of the choice of norm (or even Bregman divergence). Moreover, we show that in many instances these counterexamples have positive volume: they cannot be regarded as isolated phenomena. In one extreme case, we exhibit a tensor space in which no rank-3 tensor has an optimal rank-2 approximation. The notable exceptions to this misbehavior are rank-1 tensors and order-2 tensors. In a more positive spirit, we propose a natural way of overcoming the ill-posedness of the low-rank approximation problem, by using weak solutions when true solutions do not exist. In our work we emphasize the importance of closely studying concrete low-dimensional examples as a first step towards more general results. To this end, we present a detailed analysis of equivalence classes of 2-by-2-by-2 tensors, and we develop methods for extending results upwards to higher orders and dimensions. Finally, we link our work to existing studies of tensors from an algebraic geometric point of view. The rank of a tensor can in theory be given a semialgebraic description; i.e., can be determined by a system of polynomial inequalities. In particular we make extensive use of the 2-by-2-by-2 hyperdeterminant., Comment: To appear in the SIAM Journal on Matrix Analysis and Applications. Includes another 7 pages of additional materials not in the earlier Tech Report version

We organize the eight variables of the four-dimensional bosonic string ({\dot X}^{\mu}, X'^{\mu}) into a 2 x 2 x 2 hypermatrix a_{AA'A''} and show that in signature (2,2) the Nambu-Goto Lagrangian is given by \sqrt{Det a} where Det is Cayley's hyperdeterminant. This is invariant not only under [SL(2,R)]^{3} but also under interchange of the indices A, A' and A''. This triality reveals hitherto hidden discrete symmetries of the Nambu-Goto action., Comment: Version appearing in Phys. Lett. B . 5 pages latex

A general elliptic $N\times N$ matrix Lax scheme is presented, leading to two classes of elliptic lattice systems, one which we interpret as the higher-rank analogue of the Landau-Lifschitz equations, while the other class we characterize as the higher-rank analogue of the lattice Krichever-Novikov equation (or Adler's lattice). We present the general scheme, but focus mainly of the latter type of models. In the case $N=2$ we obtain a novel Lax representation of Adler's elliptic lattice equation in its so-called 3-leg form. The case of rank $N=3$ is analysed using Cayley's hyperdeterminant of format $2\times2\times2$, yielding a multi-component system of coupled 3-leg quad-equations.

An intriguing correspondence between four-qubit systems and simple singularity of type $D_4$ is established. We first consider an algebraic variety $X$ of separable states within the projective Hilbert space $\mathbb{P}(\mathcal{H})=\mathbb{P}^{15}$. Then, cutting $X$ with a specific hyperplane $H$, we prove that the $X$-hypersurface, defined from the section $X\cap H\subset X$, has an isolated singularity of type $D_4$; it is also shown that this is the "worst-possible" isolated singularity one can obtain by this construction. Moreover, it is demonstrated that this correspondence admits a dual version by proving that the equation of the dual variety of $X$, which is nothing but the Cayley hyperdeterminant of type $2\times 2\times 2\times 2$, can be expressed in terms of the SLOCC invariant polynomials as the discriminant of the miniversal deformation of the $D_4$-singularity., Comment: 20 pages, 5 tables

Consider three qubits A, B, and C which may be entangled with each other. We show that there is a trade-off between A's entanglement with B and its entanglement with C. This relation is expressed in terms of a measure of entanglement called the "tangle," which is related to the entanglement of formation. Specifically, we show that the tangle between A and B, plus the tangle between A and C, cannot be greater than the tangle between A and the pair BC. This inequality is as strong as it could be, in the sense that for any values of the tangles satisfying the corresponding equality, one can find a quantum state consistent with those values. Further exploration of this result leads to a definition of the "three-way tangle" of the system, which is invariant under permutations of the qubits., Comment: 13 pages LaTeX; references added, derivation of Eq. (11) simplified

A multiplication for a specific nested collection of multidimensional matrices is defined by association with a system of n = 2m-dimensional hypercomplex numbers. A totally symmetric and multiplicative determinant is then derived from the system which extends the Cayley hyperdeterminant to these higher dimensions. The determinant is related to the zero divisors of the system of hypercomplex numbers. Properties of the determinant are then discussed.

The notion of determinant slowly emerged during a period of one hundred and twenty years, from the pioneering work of Leibniz to the celebrated 1812 memoir of Cauchy, which contained all the fundamental properties. The subsequent forty years were devoted to a further systematization of the theory, so that its importance came to be recognized by the mathematical community. A sign of that recognition was the appearance of the first complete treatise on determinants, written by Brioschi (Pavia 1854, French and German translations 1856).1

We reconsider the sub-leading quantum perturbative corrections to N=2 cubic special Kaehler geometries. Imposing the invariance under axion-shifts, all such corrections (but the imaginary constant one) can be introduced or removed through suitable, lower unitriangular symplectic transformations, dubbed Peccei-Quinn (PQ) transformations. Since PQ transformations do not belong to the d=4 U-duality group G4, in symmetric cases they generally have a non-trivial action on the unique quartic invariant polynomial I4 of the charge representation R of G4. This leads to interesting phenomena in relation to theory of extremal black hole attractors; namely, the possibility to make transitions between different charge orbits of R, with corresponding change of the supersymmetry properties of the supported attractor solutions. Furthermore, a suitable action of PQ transformations can also set I4 to zero, or vice versa it can generate a non-vanishing I4: this corresponds to transitions between "large" and "small" charge orbits, which we classify in some detail within the "special coordinates" symplectic frame. Finally, after a brief account of the action of PQ transformations on the recently established correspondence between Cayley's hyperdeterminant and elliptic curves, we derive an equivalent, alternative expression of I4, with relevant application to black hole entropy., Comment: 1+39 pages