If K is a field, denote by P(K,k) the a ∈ K which are sums of kth powers of elements of K, by P⁺(K,k) the set of a ∈ K which are sums of kth powers of totally positive elements of K. We give some simple conditions for which there exist integers w(K,k) and g(K,k) such that: a ∈ P(K,k) implies that a is the sum of at most w(K,k) kth powers; a ∈ P⁺(K,k) implies that a is the sum of at most g(K,k) totally positive kth powers. We apply the results to characterise functions that are sums of kth powers in certain function fields K(X).
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-aa159-4-2,
author = {William Ellison},
title = {Waring's problem for fields},
journal = {Acta Arithmetica},
volume = {161},
year = {2013},
pages = {315-330},
zbl = {1310.11101},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-aa159-4-2}
}
William Ellison. Waring's problem for fields. Acta Arithmetica, Tome 161 (2013) pp. 315-330. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-aa159-4-2/