Analytical complexity: Development of the topic
Valerii Konstantinovich Beloshapka · Russian Journal of Mathematical Physics · 2012
The complexity of analytic functions of two variables is studied in terms of the order of complexity suggested in [1]. This paper continues [1]. An estimate for the complexity of a polynomial using its degree is given. Examples of homogeneous and harmonic functions are treated. An estimate for the complexity of a power series in terms of the geometry of the support of the series is given. Differential equations defining classes of complexity are considered. For polynomial mappings of complexity one, the Jacobian conjecture is proved. In this connection, the complexity of mappings of the plane is discussed. The research was supported by the Russian Foundation for Basic Research under grants no. 11 - 01 - 00495 a and no. 11 - 01 - 12033_ofi_m_2011.