On the Waring problem for polynomial rings

TL;DR

This paper explores an analog of the classical Waring problem for polynomial rings, demonstrating that general homogeneous polynomials can be expressed as sums of a bounded number of k-th powers, matching naive dimension count predictions.

Contribution

It establishes an upper bound for representing general homogeneous polynomials as sums of k-th powers, aligning with naive dimension count estimates.

Findings

General homogeneous polynomials of degree divisible by k can be expressed as sums of at most k^n k-th powers.

The bound matches the naive dimension count prediction.

The result extends the classical Waring problem to polynomial rings.

Abstract

In this note we discuss an analog of the classical Waring problem for C[x_0, x_1,...,x_n]. Namely, we show that a general homogeneous polynomial p \in C[x_0,x_1,...,x_n] of degree divisible by k\ge 2 can be represented as a sum of at most k^n k-th powers of homogeneous polynomials in C[x_0, x_1,...,x_n]. Noticeably, k^n coincides with the number obtained by naive dimension count.