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Generalization of the Gauss–Lucas theorem for bicomplex polynomials

The aim of this paper is to extend the domain of the Gauss–Lucas theorem from the set of complex numbers to the set of bicomplex numbers. We also discuss a bicomplex version of another compact generalization of the Gauss–Lucas theorem. | Turk J Math (2017) 41: 1618 – 1627 ¨ ITAK ˙ c TUB ⃝ Turkish Journal of Mathematics http://journals.tubitak.gov.tr/math/ doi:10.3906/mat-1512-19 Research Article Generalization of the Gauss–Lucas theorem for bicomplex polynomials Mahmood BIDKHAM∗, Sara AHMADI Faculty of Mathematics, Statistics, and Computer Science, Semnan University, Semnan, Iran Received: 04.12.2015 • Accepted/Published Online: 08.02.2017 • Final Version: 23.11.2017 Abstract: The aim of this paper is to extend the domain of the Gauss–Lucas theorem from the set of complex numbers to the set of bicomplex numbers. We also discuss a bicomplex version of another compact generalization of the Gauss–Lucas theorem. Key words: Bicomplex polynomial, Gauss–Lucas theorem 1. Introduction Corrado Segre published a paper [13] in 1892, in which he studied an infinite set of algebra whose elements he called bicomplex numbers. The work of Segre remained unnoticed for almost a century, but recently mathematicians have started taking interest in the subject and a new theory of special functions has started coming up[6, 9]. In this paper, we introduce the mathematical tools necessary to investigate the Gauss–Lucas theorem for bicomplex polynomials. We also discuss a bicomplex version of another compact generalization of the Gauss–Lucas theorem proved by Aziz and Rather [1] for complex polynomials. Let BC denote the set of bicomplex numbers, i.e. BC = {x1 + ix2 + j(x3 + ix4 ) : x1 , x2 , x3 , x4 ∈ R}, with i2 = −1, j 2 = −1 and ij = ji, and then we can write bicomplex number Z = x1 + ix2 + j(x3 + ix4 ) as z1 + jz2 where z1 , z2 ∈ C. The addition and the multiplication of two bicomplex numbers are defined in the usual way. If we denote e1 = 1+ij 2 , e2 = 1−ij 2 , then the bicomplex number Z = z1 + jz2 , z1 , z2 ∈ C , is uniquely represented as (z1 − iz2 )e1 + (z1 + iz2 )e2 . It can be easily verified that for every two bicomplex numbers Z1 = α1 e1 + β1 e2 , Z2 = α2 e1 + β2 e2 , we can write the .