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 $\deg Q_i^k \le \deg P + k^3$ and $s$ that depends on $k$ and $\ln (\deg P)$.
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Submitted on : Wednesday, May 4, 2016 - 10:47:28 AM
Last modification on : Monday, March 4, 2019 - 2:04:24 PM

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Arnaud Bodin, Mireille Car. Waring's problem for polynomials in two variables. Proceedings of the American Mathematical Society, American Mathematical Society, 2013, 141 (5), pp.1577-1589. ⟨10.1090/S0002-9939-2012-11503-5⟩. ⟨hal-01311352⟩

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