Abstract
Let G i be a closed Lie subgroup of U(n), Ω i be a bounded G i -invariant domain in Cn which contains 0, and \(O{\left( {{\mathbb{C}^n}} \right)^{{G_i}}} = \mathbb{C}\), for i = 1; 2. If f: Ω1 → Ω2 is a biholomorphism, and f(0) = 0, then f is a polynomial mapping (see Ning et al. (2017)). In this paper, we provide an upper bound for the degree of such polynomial mappings. It is a natural generalization of the well-known Cartan’s theorem.
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Acknowledgements
This work was supported by National Natural Science Foundation of China (Grant Nos. 11501058 and 11431013), the Fundamental Research Funds for the Central Universities (Grant No. 0208005202035) and Key Research Program of Frontier Sciences, Chinese Academy of Sciences (Grant No. QYZDY-SSW-SYS001).
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In memory of Professor LU QiKeng (1927–2015)
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Ning, J., Zhou, X. The degree of biholomorphic mappings between special domains in ℂn preserving 0. Sci. China Math. 60, 1077–1082 (2017). https://doi.org/10.1007/s11425-017-9048-y
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DOI: https://doi.org/10.1007/s11425-017-9048-y