Waring's problem for polynomials in two variables

TL;DR

This paper extends Waring's problem to multivariable polynomials, providing decomposition bounds and improvements for two-variable polynomials of large degree.

Contribution

It proves that multivariable polynomials can be expressed as sums of kth powers under certain conditions and improves bounds specifically for two-variable polynomials.

Findings

Decomposition of polynomials into sums of kth powers is possible under field conditions.

Bounds for the number of terms and degrees in the decomposition are established.

Improved bounds for two-variable polynomials of large degree are provided.

Abstract

We prove that all polynomials in several variables can be decomposed as the sums of $k$th powers: $P(x_1,...,x_n) = Q_1(x_1,...,x_n)^k+...+ Q_s(x_1,...,x_n)^k$, provided that elements of the base field are themselves sums of $k$th powers. We also give bounds for the number of terms $s$ and the degree of the $Q_i^k$. We then improve these bounds in the case of two variables polynomials of large degree to get a decomposition $P(x,y) = Q_1(x,y)^k+...+ Q_s(x,y)^k$ with $\deg Q_i^k \le \deg P + k^3$ and $s$ that depends on $k$ and $\ln (\deg P)$.