Pythagoras numbers of function fields of hyperelliptic curves with good reduction (Q818804)

The Pythagoras number of a field \(k\) is the least number \(n\) such that every sum of squares is a sum of at most \(n\) squares. The authors determine the Pythagoras number of function fields of hyperelliptic curves defined over the field \(\mathbb{R}((t))\) of formal power series in one variable. If the hyperelliptic curve \(C\) has good reduction then the Pythagoras number is 2 or 3, according as the field \(\mathbb{R}((t))(C)\) is real or is not real. The main step of the proof is to determine the central simple \(\mathbb{R}((t))(C)\)-algebras that are unramified and whose class in the Brauer group has order 2. The authors comment that the results are true if \(\mathbb{R}\) is replaced by any real closed field and if the formal power series field is replaced by any Henselian discretely valued field whose residue field is real closed.