Waring's problem for polynomials in two variables

Bodin, Arnaud; Car, Mireille

Document type :

Compte-rendu et recension critique d'ouvrage

DOI :

10.1090/S0002-9939-2012-11503-5

Title :

Waring's problem for polynomials in two variables

Author(s) :

Bodin, Arnaud [Auteur]

Laboratoire Paul Painlevé - UMR 8524 [LPP]
Car, Mireille [Auteur]
Laboratoire d'Analyse, Topologie, Probabilités [LATP]

Journal title :

Proceedings of the American Mathematical Society

Pages :

1577-1589

Publisher :

American Mathematical Society

Publication date :

2013-05

ISSN :

0002-9939

English keyword(s) :

several variables polynomials
sum of powers
approximate roots
Vandermonde determinant

HAL domain(s) :

Mathématiques [math]/Théorie des nombres [math.NT]

English abstract : [en]

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 ...
Show more >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 $\ln (\deg P)$.Show less >

Language :

Anglais

Popular science :

Non

Collections :

Laboratoire Paul Painlevé - UMR 8524

Source :

Harvested from HAL

Files

1104.0472
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