The Pythagoras number and the u-invariant of Laurent series fields in several variables

Abstract

We show that every sum of squares in the three-variable Laurent series field R( (x, y, z) ) is a sum of 4 squares, as was conjectured in a paper of Choi, Dai, Lam and Reznick in the 1980's. We obtain this result by proving that every sum of squares in a finite extension of R( (x, y) ) is a sum of 3 squares. It was already shown in Choi, Dai, Lam and Reznick's paper that every sum of squares in R( (x, y) ) itself is a sum of two squares. We give a generalization of this result where R is replaced by an arbitrary real field. Our methods yield similar results about the u-invariant of fields of the same type.

A Cohomological Hasse Principle Over Two-dimensional Local Rings

“…Together with earlier results in [CDLR82], Theorem 1.4 suggests that at least for n ≤ 3, upper bound estimates of the Pythagoras number of a Laurent series field in n variables can be reduced to the case of a rational function field in n − 1 variables. This is consistent in philosophy with Conjecture 5.4 of [Hu15]. (That conjecture combined with [BGVG14, Conjecture 4.15] will imply p(k((x, y, z))) = p(k(x, y)) in the general case.…”

“…, t n ))) ≤ 2 n−1 is still true when n ≥ 3 ([CDLR82, p.80, Problem 9]). The n = 3 case is recently proved in [Hu15]. For n ≥ 4, the best upper bound until now is 2 n .…”

Refinements to Patching and Applications to Field Invariants

“…Then q is isotropic over F Q ′ for every Q ′ ∈ π −1 (Q), by the special case just shown. By Proposition 4.4(a), q is isotropic over (b) Corollary 4.8(b) is related to Theorem 4.9 in [Hu15]. The proof in [Hu15] used that the function field there was assumed to be purely transcendental.…”

Representation of positive semidefinite elements as sum of squares in 2-dimensional local rings

“…In addition, τ (κ) = τ (κ((x))), see [36,Lem.5.13]. The Pythagoras number p(κ [[x, y]]) has been also studied by Hu in [28,Sect. 3] where he showed that…”