Homogeneous quadratic polynomials $f$ in $n$ complex variables are investigated and various necessary and sufficient conditions are given for $f$ to be nonzero in the set ${\Gamma }^{\left(n\right)}=\left\{z\in C^{\left(n\right)}:\text{R}ez>0\right\}$. Conclusions for the theory of multivariable positive real functions are formulated with applications in multivariable electrical network theory.