On the order of 4-dimensional regular polytope numbers

TL;DR

This paper proves asymptotic formulas for representing large integers as sums of 4-dimensional regular polytope numbers, extending to degree-four polynomials with specific boundary conditions, thus generalizing Waring's problem to higher dimensions.

Contribution

It establishes asymptotic formulas for sums of 4-polytope numbers and generalizes the results to degree-four polynomials with particular boundary conditions.

Findings

Asymptotic formulas for large integer representations using 4-polytope numbers

Extension of results to degree-four polynomials with f(0)=0 and f(1)=1

Confirmation of Kim's conjecture on 4-dimensional regular polytopes

Abstract

In light of Kim's conjecture on regular polytopes of dimension four, which is a generalization of Waring's problem, we establish asymptotic formulas for representing any sufficiently large integer as a sum of numbers in the form of those regular 4-polytopes. Moreover, we are able to obtain a more general result of the asymptotics for any degree-four polynomial $f$ satisfying $f(0)=0$ and $f(1)=1$.