On a quaternionic Picard theorem

Bisi, Cinzia; Winkelmann, Jochen

doi:10.1090/bproc/54

Cited by 13 publications

(6 citation statements)

References 12 publications

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“…We measure the size of a function by using the sup-norm. From the euclidean scalar product on C ≃ R 2 and on R 3 ≃ R 8 we deduce a scalar product on R 3 ⊗ C ≃ R 16 . We call the norm induced by this scalar product by .…”

Section: Runge's Theoremmentioning

confidence: 97%

See 1 more Smart Citation

On a Runge Theorem over $\mathbb{R}_3$

Bisi¹,

Martino²,

Winkelmann³

2021

Preprint

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Section: Runge's Theoremmentioning

confidence: 97%

“…The theory of slice regular functions has given already many fruitful results, both on the analytic and the geometric side, see for example [3,4,6,8,9,10,11,12,13,14,16].…”

Section: Preliminaries and Notationsmentioning

confidence: 99%

On a Runge Theorem over $\mathbb{R}_3$

Bisi¹,

Martino²,

Winkelmann³

2021

Preprint

Self Cite

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show abstract

“…The theory of slice regular functions has given already many fruitful results, both on the analytic and the geometric side, see for example [1,2,3,5,6,7,8,9,10,11,13]. Now we will see some basic notions about the real Clifford algebra R 3 and its quadratic cone Q R 3 , introduced in the papers [21,22].…”

Section: Preliminary Materialsmentioning

confidence: 99%

On the Quadratic Cone of $\mathbb{R}_3.$

Bisi,

De Martino

2021

Preprint

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In this paper we study the following type of functions f : Q R3 → R 3 , where Q R3 is the quadratic cone of the algebra R 3 . From the fact that it is possible to write the algebra R 3 as a direct sum of quaternions, we get the observation that it is possible to find a clever representation for Q R3 . By using this result a slice regular theory was introduced and a Cauchy formula is discussed. Moreover, a detailed study of the zeros is performed. Finally, we find a formula for the determinant of a matrix with entries in Q R3 .

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“…The theory of slice regular functions has given already many fruitful results, both on the analytic and the geometric side, see for example [ANB,BG,BW3,CS]. Moreover, slice hyperholomorphic functions have several applications in operator theory and in Mathematical Physics [GMP].…”

Section: Let Now D ⊆ C Be An Open Set and Zmentioning

confidence: 99%

On Brolin's theorem over the quaternions

Bisi¹,

Martino²

2020

Preprint

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In this paper we investigate the Brolin's theorem over H, the skew field of quaternions. Moreover, considering a quaternionic polynomial p with real coefficients, we focus on the properties of its equilibrium measure, among the others, the mixing property and the Lyapunov exponents of the measure. We prove a central limit theorem and we compute the topological entropy and measurable entropy with respect to the quaternionic equilibrium measure. We prove that they are equal considering both a quaternionic polynomial with real coefficients and a polynomial with coefficients in a slice but not all real. Brolin's theorems for the one slice preserving polynomials and for generic polynomials are also proved.

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On a quaternionic Picard theorem

Abstract: The classical theorem of Picard states that a non-constant holomorphic function f : C → C can avoid at most one value. We investigate how many values a non-constant slice regular function of a quaternionic variable f : H → H may avoid.

Cited by 13 publications

References 12 publications

On a Runge Theorem over $\mathbb{R}_3$

On a Runge Theorem over $\mathbb{R}_3$

On the Quadratic Cone of $\mathbb{R}_3.$

On Brolin's theorem over the quaternions

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