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: $P(x_1,...,x_n)=Q_1(x_1,...,x_n{)}^k+...+Q_s(x_1,...,x_n{)}^k$, 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 $°Q_i^k\le °P+k^3$ and $s$ that depends on $k$ and $\ln (°P)$.