Function fields, valuations and sums of squares

Karim Johannes Becher (University of Antwerp)

Frank Adams 1,

The study of sums of squares in rings and fields has a long history. In certain fields, for example global fields, the property of being a sum of \(n\) squares can be verified via a local-global principle.
For example, the Hasse-Minkowski Theorem provides such a local-global principle for quadratic forms over a global field, where the localisations are given by completions with respect to absolute values.

Recently, a similar local-global principle was obtained over function fields of curves over a complete discretely valued field, for example over \(\mathbb{R}((t))\), the field of Laurent series with real coefficients.
In a joint work with David Grimm and Jan Van Geel we study the consequences for sums of squares.
In particular, in a function field \(F\) over \(\mathbb{R}((t))\), the defect of \(F\) from having that all sums of squares are actually sums of two squares is measured by a special set of valuations, which turns out to be finite in any case. We show that the number of these valuations is bounded by the arithmetic genus of \(F\).
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