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Hyperbolic fourth-\(\mathbb R\) quadratic equation and holomorphic fourth-\(\mathbb R\) polynomials. (English) Zbl 1291.30023

Summary: the algebra \(\mathbb R(1,j,j^2,j^3)\), \(j^4=-1\) of the fourth-\(\mathbb R\) numbers, or in other words the algebras of the double-complex numbers \(\mathbb C(1,j)\) and the corresponding functions, were studied in the paper of S. Dimiev et al. The hyperbolic fourth-\(\mathbb R\) numbers form other similar to \(\mathbb C(1,j)\) algebra with zero divisors. In this note the square roots of hyperbolic fourth-\(\mathbb R\) numbers and hyperbolic complex numbers are found. The quadratic equation with hyperbolic fourth-\(\mathbb R\) coefficients and variables is solved. The Cauchy-Riemann system for holomorphicity of fourth-\(\mathbb R\) functions is recalled. Holomorphic homogeneous polynomials of fourth-\(\mathbb R\) variables are listed.

MSC:

30C10 Polynomials and rational functions of one complex variable
32A30 Other generalizations of function theory of one complex variable
30G35 Functions of hypercomplex variables and generalized variables
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