The continuity of reduction of hypersurfaces in \(\mathbb{C}^ 2\) to a normal form (Q1337921)

Let \(M\) be a real analytic hypersurface in \(\mathbb{C}^ 2\) given by an equation \(v=F(z,u,y)\) with complex coordinates \(z=x+iy\), \(w=u+iv\). Suppose that the Levi form of \(M\) is nondegenerate at some point. After a plane normalization given in the author's previous paper [Math. Notes 52, No. 1, 687-694 (1992); translation from Mat. Zametki 52, No. 1, 76-86 (1992; Zbl 0769.32007)], we obtain a new equation for \(M:F(x,u,y) = 2y + \sum_{k \geq 4} F_ k (x,u)y^ k\), where coefficients \(F_ 4(x,u)\) and \(F_ 5(x,u)\) satisfy some additional conditions. Let \(\{M_ \alpha\}\) be a family of hypersurfaces \(M_ \alpha\) depending continuously on a real parameter \(\alpha\). The author considers the reduction of \(M_ \alpha\) to a plane normal form and obtains the conditions under which the normal equations and normalizing mappings depend on \(\alpha\) continuously.