  Inspired by the definition of balanced neighborly spheres, we define balanced
neighborly polynomials and study the existence of these polynomials. The goal
of this article is to construct balanced neighborly polynomials of type
$(k,k,k,k)$ over any field $K$ for all $k \neq 2$, and show that a balanced
neighborly polynomial of type $(2,2,2,2)$ exists if and only if ${\rm char}(K)
\neq 2$. Besides, we also discuss a relation between balanced neighborly
polynomials and balanced neighborly simplicial spheres.
