551 research outputs found
Computing Spectral Elimination Ideals
We present here an overview of the hypermatrix spectral decomposition deduced
from the Bhattacharya-Mesner hypermatrix algebra. We describe necessary and
sufficient conditions for the existence of a spectral decomposition. We further
extend to hypermatrices the notion of resolution of identity and use them to
derive hypermatrix analog of matrix spectral bounds. Finally we describe an
algorithm for computing generators of the spectral elimination ideals which
considerably improves on Groebner basis computation suggested in
Elliptic Curves and Hyperdeterminants in Quantum Gravity
Hyperdeterminants are generalizations of determinants from matrices to
multi-dimensional hypermatrices. They were discovered in the 19th century by
Arthur Cayley but were largely ignored over a period of 100 years before once
again being recognised as important in algebraic geometry, physics and number
theory. It is shown that a cubic elliptic curve whose Mordell-Weil group
contains a Z2 x Z2 x Z subgroup can be transformed into the degree four
hyperdeterminant on a 2x2x2 hypermatrix comprising its variables and
coefficients. Furthermore, a multilinear problem defined on a 2x2x2x2
hypermatrix of coefficients can be reduced to a quartic elliptic curve whose
J-invariant is expressed in terms of the hypermatrix and related invariants
including the degree 24 hyperdeterminant. These connections between elliptic
curves and hyperdeterminants may have applications in other areas including
physics.Comment: 7 page
Ideals of varieties parameterized by certain symmetric tensors
The ideal of a Segre variety is generated by the 2-minors of a generic hypermatrix of indeterminates. We extend this result to the case of SegreVeronese varieties. The main tool is the concept of “weak generic hypermatrix” which allows us to treat also the case of projection of Veronese surfaces from a set of generic points and of Veronese varieties from a Cohen-Macaulay subvariety of codimension 2
Alternating Sign Matrices and Hypermatrices, and a Generalization of Latin Square
An alternating sign matrix, or ASM, is a -matrix where the
nonzero entries in each row and column alternate in sign. We generalize this
notion to hypermatrices: an hypermatrix is an
{\em alternating sign hypermatrix}, or ASHM, if each of its planes, obtained by
fixing one of the three indices, is an ASM. Several results concerning ASHMs
are shown, such as finding the maximum number of nonzeros of an ASHM, and properties related to Latin squares. Moreover, we
investigate completion problems, in which one asks if a subhypermatrix can be
completed (extended) into an ASHM. We show several theorems of this type.Comment: 39 page
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