On reducibility of zero sets of entire functions of several variables (Q1903875)

Let \(\Omega\) be a domain in \(\mathbb{C}^n\), \(n \geq 2\) and let \(m\) be a fixed natural number. The author defines, by an integral representation, the class of plurisubharmonic functions in \(\Omega\) which are \(m\)-logarithmic potentials. Based on this class and on the properties of the Radon transform the author further describes a class of finite functions possessing a representation in the form of infinite product with irreducible polynomials of degree at most \(m\) and exponentials. In this connection the Theorem 3 establishes a necessary and sufficient condition for the zero set of an entire function to be representable as \(\bigcup^\infty_{k = 1} \{z : P_k(z) = 0\}\), where \(P_k\) are holomorphic polynomials with \(\text{deg } P_k \leq m\), \(k = 1,2,\dots\).