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The literature on quaternionic polynomials and, in particular, on methods for finding and classifying their zero sets, is fast developing and reveals a growing interest in this subject. In contrast, polynomials defined over the algebra of coquaternions have received very little attention from researchers. One of the few exceptions is the very recent paper by Janovska and Opfer [Electron Trans Numer Anal. 2017;46:55-70], where, among other results, we can find a first attempt to prove that a unilateral coquaternionic polynomial of degree n has, at most, zeros. In this paper we present a full proof of this result, using a totally different and, from our point of view, much simpler approach. Also, we give a complete characterization of the zero sets of such polynomials and present a new result giving conditions which guarantee the existence of a special type of zeros. An algorithm to compute and classify all the zeros of a coquaternionic polynomial is proposed and several numerical examples are carefully constructed., Research at CMAT was financed by Portuguese Funds through FCT -Fundacao para a Ciencia e a Tecnologia, within the [project number UID/MAT/00013/2013]. Research at NIPE was carried out within the funding with COMPETE reference number POCI-01-0145-FEDER-006683 [project number UID/ECO/03182/2013], with the FCT/MEC's (Fundacao para a Ciencia e a Tecnologia, I.P.) financial support through national funding and by the ERDF through the Operational Programme on 'Competitiveness and Internationalization - COMPETE 2020' under the PT2020 Partnership Agreement.

Fundamentals of a function theory in co-dimension one for Clifford algebra valued functions over ℝn+1 are considered. Special attention is given to their origins in analytic properties of holomorphic functions of one and, by some duality reasons, also of several complex variables. Due to algebraic peculiarities caused by non-commutativity of the Clifford product, generalized holomorphic functions are characterized by two different but equivalent properties: on one side by local derivability (existence of a well defined derivative related to co-dimension one) and on the other side by differentiability (existence of a local approximation by linear mappings related to dimension one). As important applications, sequences of harmonic Appell polynomials are considered whose definition and explicit analytic representations rely essentially on both dual approaches., The work of the first, second and fourth authors was supported by Portuguese funds through the CIDMA - Center for Research and Development in Mathematics and Applications, and the Portuguese Foundation for Science and Technology (“FCT-Fundação para a Ciência e Tecnologia”), within project PEst-OE/MAT/UI4106/2013. The work of the second author was supported by Portuguese funds through the CMAT - Centre of Mathematics and FCT within the Project UID/MAT/00013/2013.

Unfortunately, the reference Pogoruy [1] has been published with the incorrect Journal’s name.

In this paper we give a complete characterization of the nth roots of a coquaternion q. In particular, we show that the number and type of roots-isolated and/or hyperboloidal-depend on the nature of q, on the parity of n and (eventually) on the sign of the real part of q. We also show how the coquaternionic formalism can be used to obtain, in a simple manner, explicit expressions for the real nth roots of any 2 x 2 real matrix., Research at CMAT was financed by Portuguese Funds through FCT - Fundacao para a Ciencia e a Tecnologia, within the Project UID/MAT/00013/2013. Research at NIPE was carried out within the funding with COMPETE reference number POCI-01-0145-FEDER-006683 (UID/ECO/03182/2013), with the FCT/MEC's (Fundacao para a Ciencia e a Tecnologia, I.P.) financial support through national funding and by the ERDF through the Operational Programme on "Competitiveness and Internationalization-COMPETE 2020" under the PT2020 Partnership Agreement., info:eu-repo/semantics/publishedVersion

Quaternions, introduced by Hamilton in 1843 as a generalization of complex numbers, have found, in more recent years, a wealth of applications in a number of different areas that motivated the design of efficient methods for numerically approximating the zeros of quaternionic polynomials. In fact, one can find in the literature recent contributions to this subject based on the use of complex techniques, but numerical methods relying on quaternion arithmetic remain scarce. In this paper, we propose a Weierstrass-like method for finding simultaneously all the zeros of unilateral quaternionic polynomials. The convergence analysis and several numerical examples illustrating the performance of the method are also presented., Research at CMAT was financed by Portuguese Funds through FCT, within the Project UID/MAT/00013/2013. Research at NIPE was carried out within the funding with COMPETE reference number POCI-01-0145-FEDER-006683 (UID/ECO/03182/2013), with the FCT/MEC’s (Funda¸c˜ao para a Ciˆencia e a Tecnologia, I.P.) financial support through national funding and by the ERDF through the Operational Programme on “Competitiveness and Internationalization - COMPETE 2020” under the PT2020 Partnership Agreement., info:eu-repo/semantics/publishedVersion

In this paper we consider the extension, to the algebra of coquaternions, of a complex quadratic map with a real super-attractive 8-cycle. We establish that, in addition to the real cycle, this new map has sets of non-isolated periodic points of period 8, forming four attractive 8-cycles. Here , the cycles are to be interpreted as cycles of sets and an appropriate notion of attractivity is used. Some characteristics of the basins of attraction of the five attracting 8-cycles are discussed and plots revealing the intertwined nature of these basins are shown., Research at CMAT was financed by Portuguese Funds through FCT - Funda¸c˜ao para a Ciˆencia e a Tecnologia, within the Project UID/MAT/00013/2013. Research at NIPE has been carried out within the funding with COMPETE reference number POCI-01- 0145-FEDER-006683, with the FCT/MEC’s (Funda¸c˜ao para a Ciˆencia e a Tecnologia, I.P.) financial support through national funding and by the ERDF through the Operational Programme on “Competitiveness and Internationalization – COMPETE 2020” under the PT2020 Partnership Agreement., info:eu-repo/semantics/publishedVersion

Series title : Lecture notes in computer science, vol. 7971, ISSN 0302-9743, The aim of our contribution is to call attention to the relationship between totally regular variables, introduced by R. Delanghe in 1970, and Appell sequences with respect to the hypercomplex derivative. Under some natural normalization condition the set of all paravector valued totally regular variables defined in the three dimensional Euclidean space will be completely characterized. Together with their integer powers they constitute automatically Appell sequences, since they are isomorphic to the complex variables., Fundação para a Ciência e a Tecnologia (FCT)