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 Warings problem for polynomials in two variables


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We prove that all polynomials in several variables can be decomposed as the sums of $k$th powers: $Px 1,

.,x n = Q 1x 1,

.,x n^k+

.+ Q sx 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 $Px,y = Q 1x,y^k+

.+ Q sx,y^k$ with $\deg Q i^k \le \deg P + k^3$ and $s$ that depends on $k$ and $\ln \deg P$.



Author: Arnaud Bodin; Mireille Car

Source: https://archive.org/







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