Abstract
In the present paper we introduce the class of slice-polynomial functions: slice regular functions defined over the quaternions, outside the real axis, whose restriction to any complex half-plane is a polynomial. These functions naturally emerge in the twistor interpretation of slice regularity introduced in Gentili et al. (J Eur Math Soc 16(11):2323–2353, 2014) and developed in Altavilla (J Geom Phys 123:184–208, 2018). To any slice-polynomial function P we associate its companion\(P^\vee \) and its extension to the real axis \(P_{\mathbb {R}}\), that are quaternionic functions naturally related to P. Then, using the theory of twistor spaces, we are able to show that for any quaternion q the cardinality of simultaneous pre-images of q via P, \(P^\vee \) and \(P_{\mathbb {R}}\) is generically constant, giving a notion of degree. With the brand new tool of slice-polynomial functions, we compute the twistor discriminant locus of a cubic scroll \(\mathcal {C}\) in \(\mathbb {CP}^3\) and we conclude by giving some qualitative results on the complex structures induced by \(\mathcal {C}\) via the twistor projection.
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Notes
The name “wing” is due to Ghiloni and Perotti.
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A. Altavilla and G. Sarfatti partially supported by GNSAGA of INdAM, FIRB 2012 Geometria differenziale e teoria geometrica delle funzioni and by SIR 2014 AnHyC—Analytic aspects in complex and hypercomplex geometry n. RBSI14DYEB. A. Altavilla partially supported by SIR Grant NEWHOLITE—New methods in holomorphic iteration n. RBSI14CFME. G. Sarfatti partially supported by Finanziamento Premiale FOE 2014 SUNRISE—Splines for accurate numerics: adaptive models for simulation environments. The authors would like to thank Simon Salamon for his useful suggestions on the first draft of this paper. G. Sarfatti wishes to thank the Institut Montpelliérain Alexander Grothendieck where part of this project was carried out.
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Altavilla, A., Sarfatti, G. Slice-polynomial functions and twistor geometry of ruled surfaces in \(\mathbb {CP}^3\). Math. Z. 291, 1059–1092 (2019). https://doi.org/10.1007/s00209-018-2225-8
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DOI: https://doi.org/10.1007/s00209-018-2225-8
Keywords
- Twistor spaces
- Slice regular functions
- Functions of hypercomplex variables
- Rational and ruled surfaces
- Slice-polynomial functions