Oscillatory integrals and Weierstrass polynomials

The well-known Weierstrass theorem states that if $f\left( z,w \right)$ is holomorphic at a point $\left( {{z}^{0}},{{w}^{0}} \right)\in \mathbb{C}_{z}^{n}\times {{\mathbb{C}}_{w}}$ and $f\left( {{z}^{0}},{{w}^{0}} \right)=0,$ but $f\left( {{z}^{0}},w \right) \not \equiv 0,$ then in some neighborhood $U=V\times W$of this point $f$ is represented as
\begin{equation} f\left( z,w \right)=\left[ {{\left( w-{{w}^{0}} \right)}^{m}}+{{c}_{m-1}}\left( z \right){{\left( w-{{w}^{0}} \right)}^{m-1}}+...+{{c}_{0}}\left( z \right) \right]\varphi \left( z,w \right),\,\,\,\,(1) \end{equation}
where ${c}_{k}\left( z \right) $ are holomorphic in $V$ and $\varphi(z,w)$ is holomorphic in $U,$ $\varphi(z,w)\neq 0,\, (z,w) \in U$.
In recent years, the Weierstrass representation (1) has found a number of applications in the theory of oscillatory integrals. Using a version of Weierstrass representation the first author (see [ikr]) obtained a solution of famous Sogge-Stein problem (see [SS]). He obtained also close to a sharp bound for maximal operators associated to analytic hypersurfaces.
In the obtained results the phase function is an analytic function at a fixed critical point without requiring the condition $f\left( {{z}^{0}},w \right)\not\equiv 0.$ It is natural to expect the validity of Weierstrass theorem without requiring a condition $f\left( {{z}^{0}},w \right)\not\equiv 0,$ in form
\begin{equation} f\left( z,w \right)=\left[ {{c}_{m}}\left( z \right){{\left( w-{{w}^{0}} \right)}^{m}}+{{c}_{m-1}}\left( z \right){{\left( w-{{w}^{0}} \right)}^{m-1}}+...+{{c}_{0}}\left( z \right) \right]\varphi \left( z,w \right). (2) \end{equation}
Such kind of results may be useful to studying of the oscillatory integrals and in estimates for maximal operators on a Lebesgue spaces. However, the well-known Osgood counterexample [O], p.90 (see also [F], p. 68) shows that when $n>1$ it is not always possible.
In the talk we will discuss, that there is a global option (see [S1], [S2]), also a global multidimensional (in $w$) analogue of (2) is true without requiring the condution $f\left( {{z}^{0}},w \right)\not\equiv 0$. In addition, for an arbitrary germ of a holomorphic function, we will prove one representation, that is useful in the study of oscilatory integrals.
This is a joint work with I. Ikramov (Samarkand State University, Samarkand, Uzbekistan).