Defining equations for real analytic real hypersurfaces in Cⁿ

A defining function for a real analytic real hypersurface can be uniquely written as 2 Re(H) + E, where H is holomorphic and E contains no pure terms. We study how H and E change when we perform a local biholo- morphic change of coordinates, or multiply by a unit. One of the main results is necesary and sufficient conditions on the first nonvanishing homogeneous part of E (expanded in terms of H) beyond Eqq that serve as obstructions to writing a defining equation as 2 Re(h) 4+e, where e is independent of h. We also find necessary pluriharmonic obstructions to doing this, which arise from the easier case of attempting to straighten the hypersurface.