On Hyperdeterminants And Equations Over Noncommutative Rings

Abstract

The main goal of the paper is to investigate some features of polynomials and hyperdeterminants over noncommutative rings, namely over quaternion skew-field and division rings with involution; these results generalized well-known results of A. Cayley, I. Gelfand, M. Kapranov, A. Zelevinsky, X. Zhao, Y. Zhang and others. Main results are: the estimation of number of roots of canonical polynomials over quaternions – they are infinite unlike number of roots of polynomials over real field and generalization of results of X. Zhao, Y. Zhang on resultants and its features of polynomials over quaternions to polynomials over division rings with involution (Theorems 2-5). Also, in last paragraph of the paper is hypothesized what form should it be the cubical hyperdeterminant of order three over division ring with involution.