Waring's problem for pseudo-polynomials

TL;DR

This paper extends Waring's problem to pseudo-polynomials, which are like polynomials but include at least one non-integer exponent, exploring their additive representations and building on prior work for real powers greater than 12.

Contribution

It introduces the concept of pseudo-polynomials into Waring's problem and extends existing results for real powers above 12 to this broader class.

Findings

Extended Waring's problem to pseudo-polynomials.

Established representability results for pseudo-polynomials.

Built on prior work for real powers > 12.

Abstract

Waring's problem has a long history in additive number theory. In its original form it deals with the representability of every positive integer as sum of $k$-th powers with integer $k$. Instead of these powers we deal with pseudo-polynomials in this paper. A pseudo-polynomial is a ``polynomial'' with at least one exponent not being an integer. Our work extends earlier results on the related problem of Waring for arbitrary real powers $k>12$ by Deshouillers and Arkhipov and Zhitkov.