| Résume |
Let F |R denote a (formally) real function field, H its Real Holomorphy Ring, E its group of
units and E+ = E ∩ ∑ F 2 its subgroup of totally positive units.
Theorem 1 1. E+ is the largest subset W of F ∗ with the Waring Property, i.e.
for every exponent n there exists a bound w such that every element of W is a sum of at
most w terms of n-th powers of elements in W ,
2. in the case of E+ let wn denote the lowest bound w for the representation of its elements
as sums of n-th powers and p the Pythagoras number of F then we have
wn ≤
(2n + w2
2n
)
and w2 ≤ p ,
3. if F = R(X, Y ) then 3 ≤ w2 ≤ p = 4 .
Let M denote the compact space of the real places of F . Throughout the proofs the following
two representations are used
H → C(M, R), Sk(H) → C(M, Sk) where Sk = Sk(R).
In the proof of statement 1 and the first inequality in statement 2, the representation of H will
be applied as well as a valuation criterion for sums of 2n-th powers and a famous identity of
Hilbert introduced by him in his celebrated solution of the original Waring Problem for N in
1909.
The proof of the second inequality makes use of the representation of Sk(H) and the descrip-
tion of the real holomorphy ring and the space of real places M of F by the family of smooth
affine models of F with compact real locus. This description allows invoking differential topology
and a result of W. Kucharz on the ring of continuous rational functions on compact smooth real
algebraic sets. Let d be the transcendence degree of F . We get:
if k ≥ d + 1 then E+ ∩
k∑
1
F 2 =
k∑
1
E2
+ .
Using the inequality p ≥ d + 1 the claim follows.
To prove the third statement the element (1 + X4 + Y 4)/(1 + X2 + Y 2)2 gets analyzed.
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