Zeros of quaternion polynomials

Serôdio, Rogério; Siu, Lok-shun.

doi:10.1016/s0893-9659(00)00142-7

Cited by 59 publications

(36 citation statements)

References 2 publications

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“…The differentiability of f on the real axis follows from Theorem 2.2 since for any point of the real axis there is a ball in which the function f can be expressed in power series. To prove differentiability outside the real axis consider formula (16) in terms of q = x 0 + ix 1 + jx 2 + kx 3 , namely…”

Section: The Cauchy Integral Formulamentioning

confidence: 99%

A Cauchy kernel for slice regular functions

2009

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In this paper we show how to construct a regular, non commutative Cauchy kernel for slice regular quaternionic functions. We prove an (algebraic) representation formula for such functions, which leads to a new Cauchy formula. We find the expression of the derivatives of a regular function in terms of the powers of the Cauchy kernel, and we present several other consequent results.AMS Classification: 30G35.

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Section: The Cauchy Integral Formulamentioning

confidence: 99%

A Cauchy kernel for slice regular functions

2009

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“…In the associative case, it reduces to known results (cf. [11] for quaternionic regular functions and [22,29] for polynomials).…”

Section: Introductionmentioning

confidence: 99%

Zeros of regular functions of quaternionic and octonionic variable: a division lemma and the camshaft effect

Ghiloni

Perotti

2010

Annali di Matematica

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“…In this case an essentially closed-form solution, requiring a determination of the positive real root of a real cubic equation by Cardano's method, is possible (including a complete enumeration of special cases). Although no closedform solution for the roots of higher-order polynomials is possible, there has been considerable progress in elucidating their fundamental nature, and in developing numerical methods to compute them [1,2,9,10,11,14,16,19,20,21,22,23]. In particular, it has been shown that the set of roots of any polynomial f (X ) ∈ H[X ] is a finite union of singletons and 2-spheres.…”

Section: Introductionmentioning

confidence: 99%

Solution of a quadratic quaternion equation with mixed coefficients

Farouki

Gentili

Giannelli

et al. 2016

Journal of Symbolic Computation

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A comprehensive analysis of the morphology of the solution space for a special type of quadratic quaternion equation is presented. This equation, which arises in a surface construction problem, incorporates linear terms in a quaternion variable and its conjugate with right and left quaternion coefficients, while the quadratic term has a quaternion coefficient placed between the variable and its conjugate. It is proved that, for generic coefficients, the equation has two, one, or no solutions, but in certain special instances the solution set may comprise a circle or a 3-sphere in the quaternion space H. The analysis yields solutions for each case, and intuitive interpretations of them in terms of the four-dimensional geometry of the quaternion space H.

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Zeros of quaternion polynomials

Cited by 59 publications

References 2 publications

A Cauchy kernel for slice regular functions

A Cauchy kernel for slice regular functions

Zeros of regular functions of quaternionic and octonionic variable: a division lemma and the camshaft effect

Solution of a quadratic quaternion equation with mixed coefficients

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