Minimal Balanced Neighborly Polynomials

In this paper we introduce minimal balanced neighborly polynomials and show some methods to construct such polynomials. In particular, using this notion, we prove the existence of balanced neighborly polynomials of the following types: (i) type \((p,\dots ,p)\) $(p,\cdots ,p)$ for most prime numbers p, (ii) types \((d-1,d,d,d)\) $(d-1,d,d,d)$ , \((d-1,d-1,d,d)\) $(d-1,d-1,d,d)$ and \((d-1,d-1,d-1,d)\) $(d-1,d-1,d-1,d)$ when d is odd or is divisible by 4. We also construct balanced neighborly simplicial spheres of type \((2,4k-1,4k-1,4k-1)\) $(2,4k-1,4k-1,4k-1)$ .