Abstract: | Let $A_1, A_2\in\mathbb{C}\backslash\{0\}$ and $n, m\in\mathbb{N}\backslash\{0\}.$ Using algebraic methods, we prove that there exist three analytic functions $\varphi:\mathbb{C}^m\rightarrow\mathbb{C}$ and
$g_1, g_2:\mathbb{C}^n\rightarrow\mathbb{C}$ such that $v$ is convex and strictly plurisubharmonic on $\mathbb{C}^n\times\mathbb{C}^m$ if and only if $m=1,$ $n\in\{1, 2\},$ there exists $c\in\mathbb{C}$ such that $\mid\varphi+c\mid^2$ is convex and strictly subharmonic on $\mathbb{C}$ and the functions $g_1$ and $g_2$ have fundamental representations over $\mathbb{C}^n.$ $v(z,w)=\mid A_1\varphi(w)-overline{g_1}(z)\mid^2+\mid A_2\varphi(w)-\overline{g_2}(z)\mid^2,$ for
$(z,w)\in\mathbb{C}^n\times\mathbb{C}^m.$
At the end, we prove an additional theorem by analytic and algebraic methods. |
