Warings problem for fields

TL;DR

This paper investigates conditions under which sums of kth powers in a field can be uniformly bounded, extending results from squares to higher powers, with constructive proofs but large bounds.

Contribution

It establishes that finiteness of bounds for sums of squares implies finiteness for higher powers under certain density conditions, using a non-archimedean approach.

Findings

Finiteness of w(K, 2) implies finiteness of w(K, k) for k > 2.

Constructive proofs with explicit but large upper bounds.

Method does not rely on deep arithmetical properties of the field.

Abstract

Denote by P(K, k) the members of the field K which are sums of kth powers of field elements, by P+(K, k) the set of members of K which are sums of kth powers of totally positive elements of K. We are interested in deciding whether or not there exist integers w(K, k) and g(K, k) such that: a in P(K, k) implies that a is the sum of at most w(K, k) kth powers; a in P+(K, k) implies that a is the sum of at most g(K, k) totally positive kth powers. We will show that if w(K, 2) is finite and provided that the kth powers are dense (in a sense described explicitly in theorem 3) in K, then w(K, k) is also finite for k > 2. The proofs are constructive, but the implied upper bounds for w(K, k) are large. This is to be expected since the method of proof does not use any deep arithmetical or algebraic properties of the field K.