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Graham and Pollak showed in 1971 that the determinant of a tree’s distance matrix depends only on its number of vertices, and, in particular, it is always nonzero. The Steiner distance of a collection of kk vertices in a graph is the fewest number of edges in any connected subgraph containing those vertices; for k=2k=2, this reduces to the ordinary definition of graphical distance. Here we show that the hyperdeterminant of the kkth order Steiner distance hypermatrix is always nonzero if kk is even, extending their result beyond k=2k=2. Previously, the authors showed that the kk-Steiner distance hyperdeterminant is always zero for kk odd, so together this provides a generalization to all kk. We conjecture that not just the vanishing, but the value itself, of the kk-Steiner distance hyperdeterminant of an nn-vertex tree depends only on kk and nn.

Cayley's first hyperdeterminant is a straightforward generalization of determinants for tensors. We prove that nonzero hyperdeterminants imply lower bounds on some types of tensor ranks. This result applies to the slice rank introduced by Tao and more generally to partition ranks introduced by Naslund. As an application, we show upper bounds on some generalizations of colored sum-free sets based on constraints related to order polytopes. [ABSTRACT FROM AUTHOR]

We study quantum circuits composed of a sequence of CNOT gates between distant qubits of a n-qubit system. We present some results concerning two important topics related to these circuits: circuit optimization and emergence of entanglement. Regarding the optimization problem, we first describe the group structure underlying quantum circuits generated by CNOT gates (group isomorphism, group presentation), then we apply these mathematical results to the description of heuristics to reduce the number of gates in these circuits and we also propose an optimization algorithm for circuits of a few qubits. Concerning entanglement, we show how to create some useful entangled states when a circuit of CNOT gates acts on a fully factorized state. In the case of a 3-qubit system, we prove that a CNOT circuit acting on a fully factorized state can create all types of entanglement and we propose a method to evaluate the reliability of the implementation of a SLOCC -equivalent to the state | W 3 ⟩ in a quantum machine by computing the value of the hyperdeterminant. In the case of a 4-qubit system, we propose a circuit to compute a generic entangled state and a (computer-assisted) proof of the impossibility of creating a SLOCC -equivalent to the state | W 4 ⟩ from a circuit of CNOT gates acting on a factorized state [ABSTRACT FROM AUTHOR]

We describe quantum circuits generating four-qubit maximally entangled states, the amount of entanglement being quantified by using the absolute value of the Cayley hyperdeterminant as an entanglement monotone. More precisely we show that this type of four-qubit entangled states can be obtained by the action of a family of $\mathtt{CNOT}$ circuits on some special states of the LU orbit of the state $|0000\rangle$. [ABSTRACT FROM AUTHOR]

We study embeddings J → G of simple linear algebraic groups with the following property: the simple components of the J module Lie (G) / Lie (J) are all minuscule representations of J. One family of examples occurs when the group G has roots of two different lengths and J is the subgroup generated by the long roots. We classify all such embeddings when J = SL 2 and J = SL 3 , show how each embedding implies the existence of exceptional algebraic structures on the graded components of Lie (G) , and relate properties of those structures to the existence of various twisted forms of G with certain relative root systems. [ABSTRACT FROM AUTHOR]

We discuss the existence of zero coefficients in the powers of the determinant polynomial of order n. D. G. Glynn proved that the coefficients of the m th power of the determinant polynomial are all nonzero, if m = p − 1 with a prime p. We show that the converse also holds, if n ≥ 3. The proof is quite elementary. [ABSTRACT FROM AUTHOR]

Graham and Pollak showed that the determinant of the distance matrix of a tree T depends only on the number of vertices of T. Graphical distance, a function of pairs of vertices, can be generalized to "Steiner distance" of sets S of vertices of arbitrary cardinality, by defining it to be the minimum number of edges in a connected subgraph containing all the vertices of S. Here, we show that the same is true for trees' Steiner distance hypermatrix of all odd orders, whereas the theorem of Graham-Pollak concerns order 2. We conjecture that the statement holds for all even orders as well. • Expansion on Graham-Pollak tree theorem to calculate the hyperdeterminant of Steiner distance hypermatrices. • Odd order Steiner distance hypermatrices have a hyperdeterminant of zero. • The hyperdeterminant of even order Steiner distance hypermatrices seems to only depend on the number of vertices in the tree. • The codimension of an order-3 Steiner nullvariety of a tree is 2. [ABSTRACT FROM AUTHOR]

To every projective variety X , we associate a list of hypersurfaces in different Grassmannians, called the coisotropic hypersurfaces of X. These include the Chow form and the Hurwitz form of X. Gel'fand, Kapranov and Zelevinsky characterized coisotropic hypersurfaces by a rank one condition on tangent spaces. We present a new and simplified proof of that result. We show that the coisotropic hypersurfaces of X equal those of its projectively dual variety, and that their degrees are the polar degrees of X. Coisotropic hypersurfaces of Segre varieties are defined by hyperdeterminants, and all hyperdeterminants arise in that manner. We generalize Cayley's differential characterization of coisotropy and derive new equations for the Cayley variety which parametrizes all coisotropic hypersurfaces of given degree in a fixed Grassmannian. We provide a Macaulay2 package for transitioning between X and its coisotropic hypersurfaces. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

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]. [ABSTRACT FROM PUBLISHER]

We recall the notion of hyperdeterminant of a multidimensional matrix (tensor). We prove that if we restrict the hyperdeterminant to a skew-symmetric tensor ⋀ p V ⊆ V ⊗ p with p ≥ 3 then it vanishes. The hyperdeterminant also vanishes when we restrict it to the space Γ λ ⊗ S λ V ⊂ V ⊗ p where λ is a Young diagram with p boxes and λ 2 ≥ 2 or λ 3 ≥ 1 . [ABSTRACT FROM AUTHOR]

We present an efficient algorithm to compute permanents, mixed discriminants and hyperdeterminants of structured matrices and multidimensional arrays (tensors). We describe the sparsity structure of an array in terms of a graph, and we assume that its treewidth, denoted as ω , is small. Our algorithm requires O ˜ ( n 2 ω ) arithmetic operations to compute permanents, and O ˜ ( n 2 + n 3 ω ) for mixed discriminants and hyperdeterminants. We finally show that mixed volume computation continues to be hard under bounded treewidth assumptions. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a new method to give the formulae for the hyperdeterminants of higher-dimensional matrices associated with multiple even function (mod r) 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. [ABSTRACT FROM AUTHOR]

Though algebraic geometry over 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 × n × n tensors of rank n over 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 nonvanishing 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 pathconnected 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. [ABSTRACT FROM AUTHOR]

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 δ( M) which derives in a natural way from the Cayley hyperdeterminant det M 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. [ABSTRACT FROM AUTHOR]

Algebraic statistics for binary random variables is concerned with highly structured algebraic varieties in the space of 2×2×···×2-tensors. We demonstrate the advantages of representing such varieties in the coordinate system of binary cumulants. Our primary focus lies on hidden subset models. Parametrizations and implicit equations in cumulants are derived for hyperdeterminants, for secant and tangential varieties of Segre varieties, and for certain context-specific independence models. Extending work of Rota and collaborators, we explore the polynomial inequalities satisfied by cumulants. [ABSTRACT FROM AUTHOR]

Abstract: 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 -discriminant of polynomial is found. [Copyright &y& Elsevier]

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. [ABSTRACT FROM AUTHOR]

Abstract: Let be a smooth irreducible nondegenerate projective variety and let denote its dual variety. The locus of bitangent hyperplanes, i.e. hyperplanes tangent to at least two points of X, is a component of the singular locus of . 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. Zelevinsky (1996) in . [Copyright &y& Elsevier]

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. [ABSTRACT FROM AUTHOR]

A formula for Glynn's hyperdeterminant detp (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. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

Abstract: The image of the principal minor map for -matrices is shown to be closed. In the 19th century, Nanson and Muir studied the implicitization problem of finding all relations among principal minors when . We complete their partial results by constructing explicit polynomials of degree 12 that scheme-theoretically define this affine variety and also its projective closure in . The latter is the main component in the singular locus of the -hyperdeterminant. [Copyright &y& Elsevier]

Abstract: We review the recently established relationships between black hole entropy in string theory and the quantum entanglement of qubits and qutrits in quantum information theory. The first example is provided by the measure of the tripartite entanglement of three qubits (Alice, Bob and Charlie), known as the 3-tangle, and the entropy of the 8-charge black hole of supergravity, both of which are given by the invariant hyperdeterminant, a quantity first introduced by Cayley in 1845. Moreover the classification of three-qubit entanglements is related to the classification of supersymmetric black holes. There are further relationships between the attractor mechanism and local distillation protocols and between supersymmetry and the suppression of bit flip errors. At the microscopic level, the black holes are described by intersecting -branes whose wrapping around the six compact dimensions provides the string-theoretic interpretation of the charges and we associate the three-qubit basis vectors, , with the corresponding 8 wrapping cycles. The black hole/qubit correspondence extends to the 56 charge black holes and the tripartite entanglement of seven qubits where the measure is provided by Cartan’s invariant. The qubits are naturally described by the seven vertices of the Fano plane, which provides the multiplication table of the seven imaginary octonions, reflecting the fact that has a natural structure of an -graded algebra. This in turn provides a novel imaginary octonionic interpretation of the charges of : the NS–NS charges correspond to the three imaginary quaternions and the R–R to the four complementary imaginary octonions. We contrast this approach with that based on Jordan algebras and the Freudenthal triple system. black holes (or black strings) in five dimensions are also related to the bipartite entanglement of three qutrits (3-state systems), where the analogous measure is Cartan’s invariant. Similar analogies exist for magic supergravity black holes in both four and five dimensions. Despite the ubiquity of octonions, our analogy between black holes and quantum information theory is based on conventional quantum mechanics but for completeness we also provide a more exotic one based on octonionic quantum mechanics. Finally, we note some intriguing, but still mysterious, assignments of entanglements to cosets, such as the 4-way entanglement of eight qubits to . [Copyright &y& Elsevier]

Abstract: 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 formulas for Schur functions involving hyperdeterminants, and evaluate Toeplitz type hyperdeterminants by using Jack function theory. [Copyright &y& Elsevier]

Abstract: The principal minors of a symmetric -matrix form a vector of length . We characterize these vectors in terms of algebraic equations derived from the -hyperdeterminant. [Copyright &y& Elsevier]

We present algorithms for the k-Matroid Intersection Problem and for the Matroid k-Parity Problem when the matroids are represented over the field of rational numbers and k > 2. The computational complexity of the algorithms is linear in the cardinality and singly exponential in the rank of the matroids. As an application, we describe new polynomially solvable cases of the k-Dimensional Assignment Problem and of the k-Dimensional Matching Problem. The algorithms use some new identities in multilinear algebra including the generalized Binet-Cauchy formula and its analogue for the Pfaffian. These techniques extend known methods developed earlier for k = 2. [ABSTRACT FROM AUTHOR]