×

On a quaternionic Picard theorem. (English) Zbl 1462.30091

Let \(\mathbb H\) be the field of quaternions. A map \(f: \mathbb H \to \mathbb H\) is an entire slice regular function if it can be expressed as the sum of a globally convergent power series \(f(q) = \sum_{k = 0}^\infty q^k a_k\) with quaternionic coefficients \(a_k\). These functions are quaternionic analogues of the entire holomorphic functions \(f: \mathbb C \to \mathbb C\), being the latter globally convergent power series as well. By the Picard theorem, the image \(f(\mathbb C)\) of a non-constant entire holomorphic function can avoid at most one complex number. Here the authors present a quaternionic version of this, namely they prove that:
(a) the image \(f(\mathbb H)\) of a non-constant entire slice regular function can avoid not more than four quaternions in general position (i.e., is not contained in any three dimensional real affine subspace);
(b) for any prescribed triple \(q_1\), \(q_2\), \(q_3\), there exists an entire slice regular function \(f\) such that \( f(\mathbb H)\) does not include such a triple.
The question on whether for each prescribed quadruple of quaternions in general position there exists an entire slice regular function avoiding such values remains open.

MSC:

30G35 Functions of hypercomplex variables and generalized variables
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Angella, Daniele; Bisi, Cinzia, Slice-quaternionic Hopf surfaces, J. Geom. Anal., 29, 3, 1837-1858 (2019) · Zbl 1435.30137 · doi:10.1007/s12220-018-0064-9
[2] Bisi, Cinzia; Gentili, Graziano, On quaternionic tori and their moduli space, J. Noncommut. Geom., 12, 2, 473-510 (2018) · Zbl 1405.30051 · doi:10.4171/JNCG/284
[3] Bisi, Cinzia; Stoppato, Caterina, Landau’s theorem for slice regular functions on the quaternionic unit ball, Internat. J. Math., 28, 3, 1750017, 21 pp. (2017) · Zbl 1368.30023 · doi:10.1142/S0129167X17500173
[4] Gentili, Graziano; Stoppato, Caterina; Struppa, Daniele C., Regular functions of a quaternionic variable, Springer Monographs in Mathematics, x+185 pp. (2013), Springer, Heidelberg · Zbl 1269.30001 · doi:10.1007/978-3-642-33871-7
[5] Gentili, Graziano; Struppa, Daniele C., A new theory of regular functions of a quaternionic variable, Adv. Math., 216, 1, 279-301 (2007) · Zbl 1124.30015 · doi:10.1016/j.aim.2007.05.010
[6] Ghiloni, R.; Perotti, A., Slice regular functions on real alternative algebras, Adv. Math., 226, 2, 1662-1691 (2011) · Zbl 1217.30044 · doi:10.1016/j.aim.2010.08.015
[7] Mongodi, Samuele, Holomorphicity of slice-regular functions, Complex Anal. Oper. Theory, 14, 3, Paper No. 37, 26 pp. (2020) · Zbl 1436.30044 · doi:10.1007/s11785-020-00996-2
[8] Noguchi, Junjiro, On holomorphic curves in semi-abelian varieties, Math. Z., 228, 4, 713-721 (1998) · Zbl 0949.32011 · doi:10.1007/PL00004640
[9] Noguchi, Junjiro, Lemma on logarithmic derivatives and holomorphic curves in algebraic varieties, Nagoya Math. J., 83, 213-233 (1981) · Zbl 0429.32003
[10] P1 E. Picard, Sur une propri\'et\'e des fonctions enti\`eres, C. R. Acad. Sci. Paris 88 (1879), 1024-1027. · JFM 11.0267.01
[11] P2 E. Picard, Sur les fonctions enti\`eres, C. R. Acad. Sci. Paris 89 (1879), 662-665. · JFM 11.0268.01
[12] Picard, \'{E}mile, M\'{e}moire sur les fonctions enti\`eres, Ann. Sci. \'{E}cole Norm. Sup. (2), 9, 145-166 (1880) · JFM 12.0327.01
[13] Sangwine, Stephen J.; Alfsmann, Daniel, Determination of the biquaternion divisors of zero, including the idempotents and nilpotents, Adv. Appl. Clifford Algebr., 20, 2, 401-410 (2010) · Zbl 1243.16019 · doi:10.1007/s00006-010-0202-3
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.