Uniform bounds in Waring's problem over some diagonal forms

TL;DR

This paper studies how large positive integers can be expressed as sums of k-th powers within certain diagonal forms, aiming to find uniform bounds on the number of variables needed across a family of forms.

Contribution

It introduces uniform bounds for the number of variables required in Waring's problem over specific diagonal forms, extending classical results to a broader family.

Findings

Established uniform bounds for variable counts across a family of diagonal forms

Extended classical Waring's problem results to new classes of forms

Provided conditions under which all large integers can be represented

Abstract

We investigate the existence of representations of every large positive integer as a sum of $k$-th powers of integers represented as certain diagonal forms. In particular, we consider a family of diagonal forms and discuss the problem of giving a uniform upper bound over the family for the number of variables needed to have such representations.