Estimates for the volume of the zeros of a holomorphic function depending on a complex parameter

Given a holomorphic function $f(\sigma,z)$, $\sigma\in\mathbb{C}^{m}$, $z\in\mathbb{C}^{n}$, an estimate for the volume of the zero set $\{z\colon f(\sigma,z)=0\}$ is presented which holds uniformly in $\sigma $. Such estimates are quite useful in investigations of oscillatory integrals of the form
$$ J(\lambda,\sigma)=\int_{\mathbb{R}^{n} }a(\sigma, x)e^{i\lambda \Phi (\sigma, x)}\,dx $$
as $\lambda \to \infty $. Here $a(\sigma, x)\in C_{0}^{\infty } (\mathbb{R}^{n} \times\mathbb{R}^{m})$ is a so-called amplitude function and $\Phi (\sigma, x)$ is a phase function.
Bibliography: 9 titles.