On partial differential equations of Waring's-problem form in several complex variables

Abstract

In this paper, we first consider the pseudoprimeness of meromorphic solutions $u$ to a family of partial differential equations (PDEs) $H(u_{z_1},u_{z_2},\dots ,u_{z_n})=P(u)$ of Waring's-problem form, where $H(z_1,z_2,\dots ,z_n)$ is a nontrivial homogenous polynomial of degree $\ell$ in ${\mathbf{C}}^n$ and $P(w)$ is a polynomial of degree $\hslash$ in $\mathbf{C}$ with all zeros distinct. Then, we study when these PDEs can admit entire solutions in ${\mathbf{C}}^n$ and further find these solutions for important cases including particularly $u_{z_1}^{\ell }+u_{z_2}^{\ell }+\cdots +u_{z_n}^{\ell }=u^{\hslash }$, which are (often said to be) PDEs of super-Fermat form if $\hslash =0,\ell$ and an eikonal equation if $\ell =2$ and $\hslash =0$.