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 Rn), 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 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: for which n-ary f-operations a fast quantum computers based calculation of e-determinants is possible?
Georgy P. Egorychev, Dr. Sci. (Phys.–Math.), Prof., Siberian Federal University, 79 Svobodny pr., 660041, Krasnoyarsk, Russian Federation, e-mail: firstname.lastname@example.org
Egorychev G.P. Determinants as Combinatorial Summation Formulas over an Algebra with a Unique n-ary Operation. The Bulletin of Irkutsk State University. Series Mathematics, 2018, vol. 26, pp. 121-127. (in Russian) https://doi.org/10.26516/1997-7670.2018.26.121
- Egorychev G.P. New formulas for the permanent. Dokl. Acad. Nauk USSR, 1980, vol. 265, no. 4, pp. 784–787. (in Russian)
- Egorychev G.P. Discrete mathematics. Permanents. Krasnoyarsk, Siberian Federal Univ., 2007, 272 p. (in Russian)
- Egorychev G.P. New polynomial identities for determinants over commutative rings. The Bulletin of Irkutsk State University. Series Mathematics, 2012, vol. 5, no. 4, pp. 16–20. (in Russian)
- Egorychev G.P. The polarization theorem and polynomial Identities for matrix functions. The Bulletin of Irkutsk State University. Series Mathematics, 2017, vol. 21, no. 4, pp. 16–20. (in Russian) https://doi.org/10.26516/1997-7670.2017.21.77
- Kochergin V.V. About complexity of computation one-terms and powers. Discrete Analysis, Novosibirsk, Sobolev Institute of mathematics SB RAS Publ., 1994, vol. 27, pp. 94–107. (in Russian)
- Kurosh A.G. Multioperator rings and algebras. Uspechi Math. Nauk, 1969, vol. 24, issue 1(145), pp. 3–15. (in Russian)
- Pozhidaev A.P. A simple factor-algebras and subalgebras of Jacobians algebras. Sibirsk. Math. Zh., 1998, vol. 39, no. 3, pp. 593–599. (in Russian)
- Filippov V.T. On the n-Lie algebras of Jacobians. Sibirsk. Math. Zh., 1998, vol. 39, no. 3, pp. 660–669.(in Russian) https://doi.org/10.1007/BF02673915
- Arvind V., Srinivasan S. On the hardness of noncommutative dеterminants. Electronic Colloquium on Computational Complexity. Report no. 103, 2009, pp. 1–18.
- Barvinok A.I. New permanent estimators via non-commutative determinants. Preprint arXiv: math. /0007153, 2000, pp. 1–13.
- Burago Yu.D., Zalgaller V.A. Geometric Inequalities. N.Y., Springer Verlag, 1988, 334 p.
- Chakhmakhchyan L., Cerf N.J., Garcia-Patron R. Guantum inspired algorithm for estimating the permanent of positive semideﬁnite matrices. Preprint arXiv: quant-ph./1609.02416, 2017, pp. 1–9.
- Gelfand I.M., Retakh V.S. Determinants of matrices over noncommutative rings. Funct. Anal. Appl., 1991, vol. 25, no. 2, pp.91–102.
- Krattenthaler C. Advanced determinant calculus: A complement. Linear Algebra and Its Applications, 2005, vol. 411, pp. 68–166. https://doi.org/10.1016/j.laa.2005.06.042
- Krob D., Leclerc B. Minor identities for quasi-determinants and quantum determinants. Commun. Math. Phys. 1995, vol. 169, pp. 1–23.
- Мuir Т. The theory of determinants in the historical order of development. Vol. 1, part 1. London, 1890.
- Zeilberger D. Proof of the alternating sign matrix conjecture. arXiv: math./9407211v, 1994, pp. 1–84.