The existence of balanced neighborly polynomials

TL;DR

This paper introduces balanced neighborly polynomials, constructs them for most types over any field, and explores their relation to balanced neighborly simplicial spheres, revealing characteristic-dependent existence conditions.

Contribution

It constructs balanced neighborly polynomials of type (k,k,k,k) over any field for all k ≠ 2 and characterizes the existence of type (2,2,2,2) based on the field's characteristic.

Findings

Balanced neighborly polynomials of type (k,k,k,k) exist over any field for all k ≠ 2.

A balanced neighborly polynomial of type (2,2,2,2) exists iff the field's characteristic is not 2.

Established a relation between balanced neighborly polynomials and balanced neighborly simplicial spheres.

Abstract

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.