Determinants as Combinatorial Summation Formulas over an Algebra with a Unique $n$-ary Operation

• Main Author: G.P. Egorychev
• Format: Article
• Language: English
• Published: Irkutsk State University 2018-12-01
• Series: Известия Иркутского государственного университета: Серия "Математика"
• Subjects: determinants and permanents; noncommutative and multioperator algebras; polarization and inclusion-conclusion theorems; quantum computers
• Online Access: http://mathizv.isu.ru/en/article/file?id=1286

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?}}