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Niveau: Supérieur, Doctorat, Bac+8

WARING'S PROBLEM FOR POLYNOMIALS IN TWO VARIABLES ARNAUD BODIN AND MIREILLE CAR Abstract. We prove that all polynomials in several variables can be de- composed as the sums of kth powers: P (x1, . . . , xn) = Q1(x1, . . . , xn)k+ · · · +Qs(x1, . . . , xn)k, provided that elements of the base field are them- selves sums of kth powers. We also give bounds for the number of terms s and the degree of the Qki . We then improve these bounds in the case of two variables polynomials of large degree to get a decomposition P (x, y) = Q1(x, y)k + · · · + Qs(x, y)k with degQki 6 degP + k 3 and s that depends on k and ln(degP ). 1. Introduction For any domain A and any integer k > 2, let W (A, k) denote the subset of A formed by all finite sums of kth powers ak with a ? A. Let wA(k) denote the least integer s, if it exists, such that for every element a ?W (A, k), the equation a = ak1 + · · ·+ a k s admits solutions (a1, .

WARING'S PROBLEM FOR POLYNOMIALS IN TWO VARIABLES ARNAUD BODIN AND MIREILLE CAR Abstract. We prove that all polynomials in several variables can be de- composed as the sums of kth powers: P (x1, . . . , xn) = Q1(x1, . . . , xn)k+ · · · +Qs(x1, . . . , xn)k, provided that elements of the base field are them- selves sums of kth powers. We also give bounds for the number of terms s and the degree of the Qki . We then improve these bounds in the case of two variables polynomials of large degree to get a decomposition P (x, y) = Q1(x, y)k + · · · + Qs(x, y)k with degQki 6 degP + k 3 and s that depends on k and ln(degP ). 1. Introduction For any domain A and any integer k > 2, let W (A, k) denote the subset of A formed by all finite sums of kth powers ak with a ? A. Let wA(k) denote the least integer s, if it exists, such that for every element a ?W (A, k), the equation a = ak1 + · · ·+ a k s admits solutions (a1, .

- polynomial ring
- then
- large degree
- k2 avec k2
- has prime characteristic
- every polynomial
- let wa
- strict waring problem
- ring z

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Published by | mijec |

Reads | 13 |

Language | English |

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