Sums of squares in algebraic function fields over a complete discretely valued field

Abstract

A recently found local-global principle for quadratic forms over function fields of curves over a complete discretely valued field is applied to the study of quadratic forms, sums of squares, and related field invariants.

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

“…p k((y))(x) = sup p (x) /k a finite field extension and u k((y))(x) = 2 sup p (x) /k a finite field extension have been obtained recently by Becher, Grimm and Van Geel [3], using a local-global principle proved by Colliot-Thélène, Parimala and Suresh [8] and some valuationtheoretic arguments. These imply the "≤" part of Theorem 1.1.…”

“…We can choose a closed point on E whose residue field is 2 and blow up X at that point. Then we get a regular scheme X (3) which is birational to X and which contains a divisor isomorphic to P n−1 2 . Repeating this procedure sufficiently many times produces a regular scheme birational to X which contains P n−1 as a divisor.…”

“…It is conjectured in [3,Conjecture 4.16] that the inequality in assertion (i) of Theorem 3.1 is actually an equality or, equivalently, that p( (x)) ≤ p(k(x)) for all finite extensions /k.…”

“…Moreover, if [3,Conjecture 4.15] were to hold for all real fields, then we could replace sup{p( (x 1 , . .…”

Sums of squares in function fields over Henselian local fields

“…One has s(L) ≤ Completing A at η D yields an embedding K ⊂ κ(η D )((t)). Since a is a sum of squares in K hence also in κ(η D )((t)), either the t-adic valuation of a ∈ κ(η D )((t)) is even, or κ(η D ) is not formally real, by [4,Proposition 4.2]. In the first case, e is even and res D (α) = 0.…”