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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Waring's problem for polynomials in two variables


Authors: Arnaud Bodin and Mireille Car
Journal: Proc. Amer. Math. Soc. 141 (2013), 1577-1589
MSC (2010): Primary 11P05; Secondary 13B25, 11T55
Published electronically: November 7, 2012
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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 \le \deg P + k^3$ and $ s$ that depends on $ k$ and $ \ln (\deg P)$.


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Additional Information

Arnaud Bodin
Affiliation: Laboratoire Paul Painlevé, UFR Mathématiques, Université Lille 1, 59655 Villeneuve d’Ascq Cedex, France
Email: Arnaud.Bodin@math.univ-lille1.fr

Mireille Car
Affiliation: Université Paul Cézanne, Faculté de Saint-Jérôme, 13397 Marseille Cedex, France
Email: Mireille.Car@univ-cezanne.fr

DOI: http://dx.doi.org/10.1090/S0002-9939-2012-11503-5
PII: S 0002-9939(2012)11503-5
Keywords: Several variables polynomials, sum of powers, approximate roots, Vandermonde determinant
Received by editor(s): June 16, 2011
Received by editor(s) in revised form: September 7, 2011
Published electronically: November 7, 2012
Communicated by: Matthew A. Papanikolas
Article copyright: © Copyright 2012 American Mathematical Society