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Hermitean Clifford analysis is a higher dimensional function theory centeredaround the simultaneous null solutions, called Hermitean monogenic functions,of two Hermitean conjugate complex Dirac operators. As an essential steptowards the construction of an orthogonal basis of Hermitean monogenicpolynomials, in this paper a Cauchy-Kovalevskaya extension theorem isestablished for such polynomials. The minimal number of initial polynomialsneeded to obtain a unique Hermitean monogenic extension is determined, alongwith the compatibility conditions they have to satisfy. The Cauchy-Kovalevskayaextension principle then allows for a dimensional analysis of the spaces ofspherical Hermitean monogenics, i.e. homogeneous Hermitean monogenicpolynomials. A version of this extension theorem for specific real-analyticfunctions is also obtained.

The Cauchy-Kovalevskaya Extension Theorem in Hermitean Clifford Analysis