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 doi: 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: • Accepted/Published Online: • Final Version: 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, . 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 .
