Introduction to the Hyperdeterminant and to the Rank of Multidimensional Matrices

Giorgio Ottaviani ()
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Giorgio Ottaviani: Università di Firenze, Dipartimento di Matematica

A chapter in Commutative Algebra, 2013, pp 609-638 from Springer

Abstract: Abstract 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.

Keywords: Vector Bundle; Dual Variety; Boundary Format; Zariski Closure; Secant Variety (search for similar items in EconPapers)
Date: 2013
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Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-1-4614-5292-8_20

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DOI: 10.1007/978-1-4614-5292-8_20

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