Determinants as Combinatorial Summation Formulas over an Algebra with a Unique $n$-ary Operation
Since the late 1980s the author has published a number of results on matrix functions, which were obtained using the generating functions, mixed discriminants (mixed volumes in $\mathbb R^n$), and the well-known polarization theorem (the most general version of this theorem is published in "...
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| Format: | Article |
| Language: | English |
| Published: |
Irkutsk State University
2018-12-01
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| Series: | Известия Иркутского государственного университета: Серия "Математика" |
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| Online Access: | http://mathizv.isu.ru/en/article/file?id=1286 |
| Summary: | Since the late 1980s the author has published a number of results on matrix functions, which were obtained
using the generating functions, mixed discriminants (mixed volumes in $\mathbb R^n$), and
the well-known polarization theorem (the most general version of this theorem is published in
"The Bulletin of Irkutsk State University. Series Mathematics'' in 2017). The polarization theorem allows us to obtain a set of computational formulas (polynomial identities) containing a family of free variables
for polyadditive and symmetric functions. In 1979-1980, the author has found the first polynomial identity for permanents over a commutative
ring, and, in 2013, the polynomial identity of a new type for determinants over a noncommutative ring with associative powers.
In this paper we give a general definition for determinant (the $e$-determinant) over an algebra
with a unique $n$-ary $f$-operation. This definition is different from the well-known definition
of the noncommutative Gelfand determinant. It is shown that under natural restrictions on the
$f$-operation the $e$-determinant keeps the basic properties of classical determinants over the
field $\mathbb{R}$. A family of polynomial identities for the $e$-determinants is obtained. We are convinced
that the task of obtaining similar polynomial identities for Schur
functions, the mixed determinants, resultants and other matrix functions over various algebraic
systems is quite interesting. And an answer to the following question is especially
interesting: {\it{for which $n$-ary $f$-operations a fast quantum computers based calculation of $e$-determinants is
possible?}} |
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| ISSN: | 1997-7670 2541-8785 |