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algebra Algebra () is one of the areas of mathematics, broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathem ...
, the hyperdeterminant is a generalization of the
determinant In mathematics, the determinant is a scalar value that is a function of the entries of a square matrix. It characterizes some properties of the matrix and the linear map represented by the matrix. In particular, the determinant is nonzero if ...
. Whereas a determinant is a
scalar Scalar may refer to: *Scalar (mathematics), an element of a field, which is used to define a vector space, usually the field of real numbers *Scalar (physics), a physical quantity that can be described by a single element of a number field such a ...
valued
function Function or functionality may refer to: Computing * Function key, a type of key on computer keyboards * Function model, a structured representation of processes in a system * Function object or functor or functionoid, a concept of object-orie ...
defined on an ''n'' × ''n''
square matrix In mathematics, a square matrix is a matrix with the same number of rows and columns. An ''n''-by-''n'' matrix is known as a square matrix of order Any two square matrices of the same order can be added and multiplied. Square matrices are ofte ...
, a hyperdeterminant is defined on a multidimensional array of numbers or
tensor In mathematics, a tensor is an algebraic object that describes a multilinear relationship between sets of algebraic objects related to a vector space. Tensors may map between different objects such as vectors, scalars, and even other tens ...
. Like a determinant, the hyperdeterminant is a
homogeneous polynomial In mathematics, a homogeneous polynomial, sometimes called quantic in older texts, is a polynomial whose nonzero terms all have the same degree. For example, x^5 + 2 x^3 y^2 + 9 x y^4 is a homogeneous polynomial of degree 5, in two variables; ...
with
integer An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign ( −1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language ...
coefficient In mathematics, a coefficient is a multiplicative factor in some term of a polynomial, a series, or an expression; it is usually a number, but may be any expression (including variables such as , and ). When the coefficients are themselves ...
s in the components of the tensor. Many other properties of determinants generalize in some way to hyperdeterminants, but unlike a determinant, the hyperdeterminant does not have a simple geometric interpretation in terms of
volume Volume is a measure of occupied three-dimensional space. It is often quantified numerically using SI derived units (such as the cubic metre and litre) or by various imperial or US customary units (such as the gallon, quart, cubic inch). Th ...
s. There are at least three definitions of hyperdeterminant. The first was discovered by
Arthur Cayley Arthur Cayley (; 16 August 1821 – 26 January 1895) was a prolific British mathematician who worked mostly on algebra. He helped found the modern British school of pure mathematics. As a child, Cayley enjoyed solving complex maths problems ...
in 1843 presented to the
Cambridge Philosophical Society The Cambridge Philosophical Society (CPS) is a scientific society at the University of Cambridge. It was founded in 1819. The name derives from the medieval use of the word philosophy to denote any research undertaken outside the fields of l ...
. A. Cayley, "On the theory of determinants", ''Trans. Camb. Philos. Soc.'', 1-16 (1843) https://archive.org/details/collectedmathem01caylgoog It is in two parts and Cayley's first hyperdeterminant is covered in the second part. It is usually denoted by det0. The second Cayley hyperdeterminant originated in 1845 A. Cayley, "On the Theory of Linear Transformations", ''Cambridge Math. J.'', vol 4, 193–209, (1845), https://archive.org/details/collectedmathem01caylgoog and is often denoted "Det". This definition is a
discriminant In mathematics, the discriminant of a polynomial is a quantity that depends on the coefficients and allows deducing some properties of the roots without computing them. More precisely, it is a polynomial function of the coefficients of the ori ...
for a singular point on a scalar valued
multilinear map In linear algebra, a multilinear map is a function of several variables that is linear separately in each variable. More precisely, a multilinear map is a function :f\colon V_1 \times \cdots \times V_n \to W\text where V_1,\ldots,V_n and W ar ...
. Cayley's first hyperdeterminant is defined only for
hypercube In geometry, a hypercube is an ''n''-dimensional analogue of a square () and a cube (). It is a closed, compact, convex figure whose 1-skeleton consists of groups of opposite parallel line segments aligned in each of the space's dimensions ...
s having an
even Even may refer to: General * Even (given name), a Norwegian male personal name * Even (surname) * Even (people), an ethnic group from Siberia and Russian Far East ** Even language, a language spoken by the Evens * Odd and Even, a solitaire ga ...
number of dimensions (although variations exist in
odd Odd means unpaired, occasional, strange or unusual, or a person who is viewed as eccentric. Odd may also refer to: Acronym * ODD (Text Encoding Initiative) ("One Document Does it all"), an abstracted literate-programming format for describing X ...
dimensions). Cayley's second hyperdeterminant is defined for a restricted range of hypermatrix formats (including the hypercubes of any dimensions). The third hyperdeterminant, most recently defined by Glynn, occurs only for
fields Fields may refer to: Music *Fields (band), an indie rock band formed in 2006 *Fields (progressive rock band), a progressive rock band formed in 1971 * ''Fields'' (album), an LP by Swedish-based indie rock band Junip (2010) * "Fields", a song by ...
of
prime A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only way ...
characteristic A characteristic is a distinguishing feature of a person or thing. It may refer to: Computing * Characteristic (biased exponent), an ambiguous term formerly used by some authors to specify some type of exponent of a floating point number * Charact ...
''p''. It is denoted by det''p'' and acts on all hypercubes over such a field. David G. Glynn,"The modular counterparts of Cayley's hyperdeterminants", ''Bulletin of the Australian Mathematical Society'', vol. 57(3) 479 (1998). Only the first and third hyperdeterminants are "multiplicative," except for the second hyperdeterminant in the case of "boundary" formats. The first and third hyperdeterminants also have closed formulae as polynomials and therefore their degrees are known, whereas the second one does not appear to have a closed formula or degree in all cases that are known. The notation for determinants can be extended to hyperdeterminants without change or ambiguity. Hence the hyperdeterminant of a hypermatrix ''A'' may be written using the vertical bar notation as , ''A'', or as ''det''(''A''). A standard modern textbook on Cayley's second hyperdeterminant Det (as well as many other results) is "Discriminants, Resultants and Multidimensional Determinants" by Gel'fand, Kapranov and
Zelevinsky Andrei Vladlenovich Zelevinsky (; 30 January 1953 – 10 April 2013) was a Russian-American mathematician who made important contributions to algebra, combinatorics, and representation theory, among other areas. Biography Zelevinsky graduated i ...
. Their notation and terminology is followed in the next section.


Cayley's second hyperdeterminant Det

In the special case of a 2 × 2 × 2 hypermatrix the hyperdeterminant is known as Cayley's hyperdeterminant after the British mathematician Arthur Cayley who discovered it. The quartic expression for the Cayley's hyperdeterminant of hypermatrix ''A'' with components ''a''''ijk'', ∊ is given by :Det(''A'') = ''a''0002''a''1112 + ''a''0012''a''1102 + ''a''0102''a''1012 + ''a''1002''a''0112 :: − 2''a''000''a''001''a''110''a''111 − 2''a''000''a''010''a''101''a''111 − 2''a''000''a''011''a''100''a''111 − 2''a''001''a''010''a''101''a''110 − 2''a''001''a''011''a''110''a''100 − 2''a''010''a''011''a''101''a''100 + 4''a''000''a''011''a''101''a''110 + 4''a''001''a''010''a''100''a''111. This expression acts as a discriminant in the sense that it is zero ''if and only if'' there is a non-zero solution in six unknowns ''x''''i'', ''y''''i'', ''z''''i'', (with superscript ''i'' = 0 or 1) of the following system of equations :''a''000''x''0''y''0 + ''a''010''x''0''y''1 + ''a''100''x''1''y''0 + ''a''110''x''1''y''1 = 0 :''a''001''x''0''y''0 + ''a''011''x''0''y''1 + ''a''101''x''1''y''0 + ''a''111''x''1''y''1 = 0 :''a''000''x''0''z''0 + ''a''001''x''0''z''1 + ''a''100''x''1''z''0 + ''a''101''x''1''z''1 = 0 :''a''010''x''0''z''0 + ''a''011''x''0''z''1 + ''a''110''x''1''z''0 + ''a''111''x''1''z''1 = 0 :''a''000''y''0''z''0 + ''a''001''y''0''z''1 + ''a''010''y''1''z''0 + ''a''011''y''1''z''1 = 0 :''a''100''y''0''z''0 + ''a''101''y''0''z''1 + ''a''110''y''1''z''0 + ''a''111''y''1''z''1 = 0. The hyperdeterminant can be written in a more compact form using the
Einstein convention In mathematics, especially the usage of linear algebra in Mathematical physics, Einstein notation (also known as the Einstein summation convention or Einstein summation notation) is a notational convention that implies summation over a set of ...
for summing over indices and the
Levi-Civita symbol In mathematics, particularly in linear algebra, tensor analysis, and differential geometry, the Levi-Civita symbol or Levi-Civita epsilon represents a collection of numbers; defined from the sign of a permutation of the natural numbers , for s ...
which is an alternating tensor density with components ε''ij'' specified by ε00 = ε11 = 0, ε01 = −ε10 = 1: :''b''''kn'' = (1/2)ε''il''ε''jm''''a''''ijk''''a''''lmn'' :Det(''A'') = (1/2)ε''il''ε''jm''''b''''ij''''b''''lm''. Using the same conventions we can define a
multilinear form In abstract algebra and multilinear algebra, a multilinear form on a vector space V over a field K is a map :f\colon V^k \to K that is separately ''K''-linear in each of its ''k'' arguments. More generally, one can define multilinear forms on ...
:''f''(x,y,z) = ''a''''ijk'' ''x''''i''''y''''j''''z''''k'' Then the hyperdeterminant is zero if and only if there is a non-trivial point where all
partial derivative In mathematics, a partial derivative of a function of several variables is its derivative with respect to one of those variables, with the others held constant (as opposed to the total derivative, in which all variables are allowed to vary). Pa ...
s of ''f'' vanish.


As a tensor expression

The above determinant can be written in terms of a generalisation of the
Levi-Civita symbol In mathematics, particularly in linear algebra, tensor analysis, and differential geometry, the Levi-Civita symbol or Levi-Civita epsilon represents a collection of numbers; defined from the sign of a permutation of the natural numbers , for s ...
: :\mathrm(A) = f^f^f^a_a_a_a_ where ''f'' is a generalisation of the Levi-Civita symbol which allows two indices to be the same: :f^=f^=f^=f^ = -1/2 :f^=f^=1 where the ''f'' satisfy: :f^ + f^+f^+f^ + f^+f^ = 0.


As a discriminant

For symmetric 2 × 2 × 2 × ⋯ hypermatrices, the hyperdeterminant is the
discriminant In mathematics, the discriminant of a polynomial is a quantity that depends on the coefficients and allows deducing some properties of the roots without computing them. More precisely, it is a polynomial function of the coefficients of the ori ...
of a polynomial. For example, :a_=a :a_=a_=a_ = b :a_=a_=a_ = c :a_=d Then Det(''A'') is the discriminant of ax^3 + 3bx^2 + 3cx + d.


Other general hyperdeterminants related to Cayley's Det


Definitions

In the general case a hyperdeterminant is defined as a discriminant for a multilinear map ''f'' from
finite-dimensional In mathematics, the dimension of a vector space ''V'' is the cardinality (i.e., the number of vectors) of a basis of ''V'' over its base field. p. 44, §2.36 It is sometimes called Hamel dimension (after Georg Hamel) or algebraic dimension to d ...
vector space In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called '' scalars''. Scalars are often real numbers, but ...
s ''V''''i'' to their underlying
field Field may refer to: Expanses of open ground * Field (agriculture), an area of land used for agricultural purposes * Airfield, an aerodrome that lacks the infrastructure of an airport * Battlefield * Lawn, an area of mowed grass * Meadow, a grass ...
''K'' which may be \mathbb or \mathbb. :f: V_1 \otimes V_2 \otimes \cdots \otimes V_r \to K ''f'' can be identified with a tensor in the tensor product of each
dual space In mathematics, any vector space ''V'' has a corresponding dual vector space (or just dual space for short) consisting of all linear forms on ''V'', together with the vector space structure of pointwise addition and scalar multiplication by con ...
''V''*''i'' :f \in V^*_1 \otimes V^*_2 \otimes \cdots \otimes V^*_r By definition a hyperdeterminant ''Det''(''f'') is a polynomial in components of the tensor ''f'' which is zero if and only if the map ''f'' has a non-trivial point where all
partial derivative In mathematics, a partial derivative of a function of several variables is its derivative with respect to one of those variables, with the others held constant (as opposed to the total derivative, in which all variables are allowed to vary). Pa ...
s with respect to the components of its vector arguments vanish (a non-trivial point means that none of the vector arguments are zero.) The vector spaces ''V''''i'' need not have the same dimensions and the hyperdeterminant is said to be of format if the dimension of each space ''V''''i'' is It can be shown that the hyperdeterminant exists for a given format and is unique up to a scalar factor, if and only if the largest number in the format is less than or equal to the sum of the other numbers in the format. This definition does not provide a means to construct the hyperdeteriminant and in general this is a difficult task. For hyperdeterminants with formats where the number of terms is usually too large to write out the hyperdeterminant in full. For larger ''r'' even the degree of the polynomial increases rapidly and does not have a convenient general formula.


Examples

The case of formats with ''r'' = 1 deals with vectors of length In this case the sum of the other format numbers is zero and ''k''1 is always greater than zero so no hyperdeterminants exist. The case of ''r'' = 2 deals with
matrices Matrix most commonly refers to: * ''The Matrix'' (franchise), an American media franchise ** ''The Matrix'', a 1999 science-fiction action film ** "The Matrix", a fictional setting, a virtual reality environment, within ''The Matrix'' (franchis ...
. Each format number must be greater than or equal to the other, therefore only
square matrices In mathematics, a square matrix is a matrix with the same number of rows and columns. An ''n''-by-''n'' matrix is known as a square matrix of order Any two square matrices of the same order can be added and multiplied. Square matrices are ofte ...
''S'' have hyperdeterminants and they can be identified with the determinant det(''S''). Applying the definition of the hyperdeterminant as a discriminant to this case requires that det(''S'') is zero when there are vectors ''X'' and ''Y'' such that the matrix equations and have solutions for non-zero ''X'' and ''Y''. For there are hyperdeterminants with different formats satisfying the format inequality. For example, Cayley's hyperdeterminant has format (1, 1, 1) and a hyperdeterminant of format also exists. However a hyperdeterminant would have format but so it does not exist.


Degree

Since the hyperdeterminant is homogeneous in its variables it has a well-defined degree that is a function of the format and is written ''N''(''k''1, ..., ''k''''r''). In special cases we can write down an expression for the degree. For example, a hyperdeterminant is said to be of boundary format when the largest format number is the sum of the others and in this case we have :N(k_2 + \cdots + k_r, k_2, \ldots, k_r) = \frac. For hyperdeterminants of dimensions 2''r'', a convenient generating formula for the degrees ''N''''r'' is :\sum_^\infty N_r \frac = \frac. In particular for ''r'' = 2,3,4,5,6 the degree is respectively and then grows very rapidly. Three other special formulae for computing the degree of hyperdeterminants are given in for 2 × ''m'' × ''m'' use ''N''(1, ''m'' − 1, ''m'' − 1) = 2''m''(''m'' − 1) for 3 × ''m'' × ''m'' use ''N''(2, ''m'' − 1, ''m'' − 1) = 3''m''(''m'' − 1)2 for 4 × ''m'' × ''m'' use ''N''(3, ''m'' − 1, ''m'' − 1) = (2/3)''m''(''m'' − 1)(''m'' − 2)(5''m'' − 3) A general result that follows from the hyperdeterminants product rule and invariance properties listed below is that the
least common multiple In arithmetic and number theory, the least common multiple, lowest common multiple, or smallest common multiple of two integers ''a'' and ''b'', usually denoted by lcm(''a'', ''b''), is the smallest positive integer that is divisible by bo ...
of the dimensions of the vector spaces on which the
linear map In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping V \to W between two vector spaces that pr ...
acts
divides In mathematics, a divisor of an integer n, also called a factor of n, is an integer m that may be multiplied by some integer to produce n. In this case, one also says that n is a multiple of m. An integer n is divisible or evenly divisible by ...
the degree of the hyperdeterminant, that is, :lcm(''k''1 + 1, ..., ''k''r + 1) , ''N''(''k''1, ..., ''k''''r'').


Properties of hyperdeterminants

Hyperdeterminants generalise many of the properties of determinants. The property of being a discriminant is one of them and it is used in the definition above.


Multiplicative properties

One of the most familiar properties of determinants is the multiplication rule which is sometimes known as the Binet-Cauchy formula. For square ''n'' × ''n'' matrices ''A'' and ''B'' the rule says that : det(''AB'') = det(''A'')det(''B'') This is one of the harder rules to generalize from determinants to hyperdeterminants because generalizations of products of hypermatrices can give hypermatrices of different sizes. The full domain of cases in which the product rule can be generalized is still a subject of research. However, there are some basic instances that can be stated. Given a multilinear form ''f''(x1, ..., x''r'') we can apply a linear transformation on the last argument using an ''n'' × ''n'' matrix ''B'', y''r'' = ''B'' x''r''. This generates a new multilinear form of the same format, :''g''(x1, ..., xr) = ''f''(x1, ..., y''r'') In terms of hypermatrices this defines a product which can be written ''g'' = ''f''.''B'' It is then possible to use the definition of the hyperdeterminant to show that : det(''f''.''B'') = det(''f'')det(''B'')''N''/''n'' where ''n'' is the degree of the hyperdeterminant. This generalises the product rule for matrices. Further generalizations of the product rule have been demonstrated for appropriate products of hypermatrices of boundary format. Cayley's first hyperdeterminant det0 is multiplicative in the following sense. Let ''A'' be a ''r''-dimensional ''n'' × ... × ''n'' hypermatrix with elements ''a''''i'', ..., ''k'', ''B'' be a ''s''-dimensional ''n'' × ... × ''n'' hypermatrix with elements ''b''..., and ''C'' be a (''r'' + ''s'' − 2)-dimensional ''n'' × ... × ''n'' hypermatrix with elements ''c''... such that (using
Einstein notation In mathematics, especially the usage of linear algebra in Mathematical physics, Einstein notation (also known as the Einstein summation convention or Einstein summation notation) is a notational convention that implies summation over a set of ...
) : ''c''''i'', ..., ''j'', ''l'', ..., ''m'' = ''a''''i'', ..., ''j''''k''''b''''k'', ''l'', ..., ''m'', then : det0(C) = det0(A) det0(B).


Invariance properties

A determinant is not usually considered in terms of its properties as an
algebraic invariant Invariant theory is a branch of abstract algebra dealing with actions of groups on algebraic varieties, such as vector spaces, from the point of view of their effect on functions. Classically, the theory dealt with the question of explicit desc ...
but when determinants are generalized to hyperdeterminants the invariance is more notable. Using the multiplication rule above on the hyperdeterminant of a hypermatrix ''H'' times a matrix ''S'' with determinant equal to one gives :det(''H''.''S'') = det(''H'') In other words, the hyperdeterminant is an algebraic invariant under the action of the
special linear group In mathematics, the special linear group of degree ''n'' over a field ''F'' is the set of matrices with determinant 1, with the group operations of ordinary matrix multiplication and matrix inversion. This is the normal subgroup of the gen ...
SL(''n'') on the hypermatrix. The transformation can be equally well applied to any of the vector spaces on which the multilinear map acts to give another distinct invariance. This leads to the general result, : The hyperdeterminant of format (k_1,\ldots,k_r) is an invariant under an action of the
group A group is a number of persons or things that are located, gathered, or classed together. Groups of people * Cultural group, a group whose members share the same cultural identity * Ethnic group, a group whose members share the same ethnic ide ...
\mathrm(k_1+1) \otimes \cdots \otimes \mathrm(k_r+1) For example, the determinant of an ''n'' × ''n'' matrix is an SL(''n'')2 invariant and Cayley's hyperdeterminant for a 2 × 2 × 2 hypermatrix is an SL(2)3 invariant. A more familiar property of a determinant is that if you add a multiple of a row (or column) to a different row (or column) of a square matrix then its determinant is unchanged. This is a special case of its invariance in the case where the special linear transformation matrix is an identity matrix plus a matrix with only one non-zero
off-diagonal element In geometry, a diagonal is a line segment joining two vertices of a polygon or polyhedron, when those vertices are not on the same edge. Informally, any sloping line is called diagonal. The word ''diagonal'' derives from the ancient Greek δ ...
. This property generalizes immediately to hyperdeterminants implying invariance when you add a multiple of one slice of a hypermatrix to another parallel slice. A hyperdeterminant is not the only polynomial algebraic invariant for the group acting on the hypermatrix. For example, other algebraic invariants can be formed by adding and multiplying hyperdeterminants. In general the invariants form a
ring Ring may refer to: * Ring (jewellery), a round band, usually made of metal, worn as ornamental jewelry * To make a sound with a bell, and the sound made by a bell :(hence) to initiate a telephone connection Arts, entertainment and media Film and ...
algebra and it follows from Hilbert's basis theorem that the ring is finitely generated. In other words, for a given hypermatrix format, all the polynomial algebraic invariants with integer coefficients can be formed using addition, subtraction and multiplication starting from a finite number of them. In the case of a 2 × 2 × 2 hypermatrix, all such invariants can be generated in this way from Cayley's second hyperdeterminant alone, but this is not a typical result for other formats. For example, the second hyperdeterminant for a hypermatrix of format 2 × 2 × 2 × 2 is an algebraic invariant of degree 24 yet all the invariants can be generated from a set of four simpler invariants of degree 6 and less.


History and applications

The second hyperdeterminant was invented and named by Arthur Cayley in 1845, who was able to write down the expression for the 2 × 2 × 2 format, but Cayley went on to use the term for any algebraic invariant and later abandoned the concept in favour of a general theory of polynomial forms which he called "quantics". For the next 140 years there were few developments in the subject and hyperdeterminants were largely forgotten until they were rediscovered by Gel'fand, Kapranov and Zelevinsky in the 1980s as an offshoot of their work on generalized hypergeometric functions. This led to them writing their textbook in which the hyperdeterminant is reintroduced as a discriminant. Indeed, Cayley's first hyperdeterminant is more fundamental than his second, since it is a straightforward generalization the ordinary determinant, and has found recent applications in the Alon-Tarsi conjecture. Since then the hyperdeterminant has found applications over a wide range of disciplines including algebraic geometry,
number theory Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and integer-valued functions. German mathematician Carl Friedrich Gauss (1777–1855) said, "Math ...
,
quantum computing Quantum computing is a type of computation whose operations can harness the phenomena of quantum mechanics, such as superposition, interference, and entanglement. Devices that perform quantum computations are known as quantum computers. Thou ...
and string theory. In ''algebraic geometry'' the second hyperdeterminant is studied as a special case of an X-discriminant. A principal result is that there is a correspondence between the vertices of the
Newton polytope In mathematics, the Newton polytope is an integral polytope associated with a multivariate polynomial. It can be used to analyze the polynomial's behavior when specific variables are considered negligible relative to the others. Specifically, given ...
for hyperdeterminants and the "triangulation" of a cube into
simplices In geometry, a simplex (plural: simplexes or simplices) is a generalization of the notion of a triangle or tetrahedron to arbitrary dimensions. The simplex is so-named because it represents the simplest possible polytope in any given dimension. ...
. In ''quantum computing'' the invariants on hypermatrices of format 2''N'' are used to study the entanglement of ''N''
qubits In quantum computing, a qubit () or quantum bit is a basic unit of quantum information—the quantum version of the classic binary bit physically realized with a two-state device. A qubit is a two-state (or two-level) quantum-mechanical system, ...
. In ''string theory'' the hyperdeterminant first surfaced in connection with string dualities and black hole entropy.


References


Sources

* * * * * * * * * * *


Further reading

For other historical developments not contained in the book from Gel'fand, Kapranov and Zelevinsky, see: * * *{{cite book, last1=Pascal, first1=E., title=I Determinanti, url=https://archive.org/details/ideterminantite00pascgoog, date=1897, publisher=Hoepli, location=Milan (also translated in into German: "Die Determinanten", H. Leitzmann, Halle, 1900.) There is a short section about hyperdeterminants and their history up to 1900. Multilinear algebra