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

A. M. Kytmanov ; A. Sadullaev

Sbornik. Mathematics, Tome 212 (2021) no. 11, pp. 1608-1614

Résumé

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.

Détail

Keywords: Weierstrass's preparation theorem, analytic set, regular point, volume of an analytic set, Wirtinger's theorem.