The Hilbert-Waring theorem
Waring’s problem for exponent k is to prove that the set of nonnegative integers is a basis of finite order, that is, to prove that every nonnegative integer can be written as the sum of a bounded number of kth powers. We denote by g(k) the smallest number s such that every nonnegative integer is the sum of exactly s kth powers of nonnegative integers. Waring’s problem is to show that g(k) is finite; Hilbert proved this in 1909. The goal of this chapter is to prove the Hilbert-Waring theorem: the kth powers are a basis of finite order for every positive integer k.
KeywordsNonnegative Integer Hermite Polynomial Finite Order Polynomial Identity Fourth Power
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