Infinite Easier Waring Constants for Commutative Rings

TL;DR

This paper proves that for any fixed number of terms, there is no universal bound on expressing elements of certain subrings in commutative rings as sums of 2^n-th powers with signs, highlighting an infinite complexity in such representations.

Contribution

It establishes the non-existence of a universal finite Waring constant for subrings generated by 2^n-th powers in all commutative rings.

Findings

No finite Waring constant exists for these subrings.

The result applies universally across all commutative rings with identity.

It extends understanding of power representations in ring theory.

Abstract

Suppose n >= 2. We show that there is no integer v >= 1 such that for all commutative rings R with identity, every element of the subring J(2^n,R) of R generated by 2^n-th powers can be written in the form \pm f_1^{2^n} \pm \cdots \pm f_v^{2^n} for some f_1,...,f_v \in R and some choice of signs.