Hyperbolic Fourth-R Quadratic Equation and Holomorphic Fourth-R Polynomials

MSC 2010: 30C10, 32A30, 30G35

The algebra R(1; j; j2; j3), j4 = ¡1 of the fourth-R numbers, or in other words the algebra of the double-complex numbers C(1; j) and the corresponding functions, were studied in the papers of S. Dimiev and al. (see [1], [2], [3], [4]). The hyperbolic fourth-R numbers form other similar to C(1; j) algebra with zero divisors. In this note the square roots of hyperbolic fourth-R numbers and hyperbolic complex numbers are found. The quadratic equation with hyperbolic fourth-R coefficients and variables is solved. The Cauchy-Riemann system for holomorphicity of fourth-R functions is recalled. Holomorphic homogeneous polynomials of fourth-R variables are listed.