Waring’s problem for polynomials in two variables

Abstract:

We prove that all polynomials in several variables can be decomposed as the sums of $k$th powers: $P(x_1,\ldots ,x_n) = Q_1(x_1,\ldots ,x_n)^k+\cdots + Q_s(x_1,\ldots ,x_n)^k$, 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-variable polynomials of large degree to get a decomposition $P(x,y) = Q_1(x,y)^k+\cdots + Q_s(x,y)^k$ with $\deg Q_i^k \leqslant \deg P + k^3$ and $s$ that depends on $k$ and $\ln (\deg P)$.