The existence of balanced neighborly polynomials

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.