Sums of even powers of k-regulous functions

Abstract

We prove that the Pythagoras number of the ring of 0regulous functions R 0 (R n ) is finite and bounded from above by 2 n . We also show that a k-regulous function which is nonnegative on R n cannot be necessarily written as a sum of squares of k-regulous functions, provided that k > 0 and n > 1. We then obtain lower bounds for p(R k (R n )) of k-regulous functions on R n , which tends to infinity as k tends to infinity for a fixed n ≥ 3.

On Waring numbers of henselian rings

“…Hence, we get the result that S n l (Z) is dense in Z p if and only if l ≥ w n (Z p ) and S n l (Q) is dense in Q p if and only if l ≥ w n (Q p ). In particular, the minimal value of l such that S n l (Q) is dense in Q p agrees with the minimal value of l such that R(S n l (Z)) is dense in Q p for all pairs (p, n) except for (2, 2), (2,4) and (2,8) , where…”

“…In general, it is a very difficult task to determine whether nth Waring number is finite or not. There are also some results in this manner for the rings of k-regulous functions on R n (see [2]).…”

“…Let R be a henselian local ring with the total ring of fractions Q(R) = R, the maximal ideal m and the residue field k. Let n be a positive integer and m be the char(k)-free part of n. Assume that char(k) ∤ n or char(R) = char(k). Then,(2)…”