Function fields
Function fields are defined for geometric objects as curves or surfaces. A quadratic form defines itself such a geometric object - a quadric. A central problem in the theory of quadratic forms is to determine which anisotropic forms become isotropic when extended to the function field of the projective quadric associated to a given form.
Function fields of Pfister quadratics
- Becher, O'Shea. Isotropy over function fields of conics, work in progress.
- O'Shea. Isotropy over function fields of Pfister forms, submitted.
Function fields and Differential forms
- Dolphin, Hoffmann. Differential forms and bilinear forms under field extensions, submitted.
Computing invariants of function fields
The determination of field invariants for function fields (even for curves)
over a given field k is only understood for the case where k is real closed,
algebraically closed or finite.
Using the new method of Field Patching due to Hartmann, Harbater,
and Krashen, we computed the pythagoras number and the u-invariant for
function fields of curves over the field of Laurent series with real coefficients.
- Becher, Grimm, Van Geel. Quadratic forms over funtion fields in one variable over a complete valued field, work in progress.
- Grimm. Sums of Squares in Function Fields of Conics, work in progress.
- Grimm. Function fields in one variable with Pythagoras number two, submitted.
- Becher. The u-invariant of a real function field, Mathematische Annalen 346 (2010): 245-249.
- Becher, Van Geel. Sums of squares in function fields of hyperelliptic curves, Mathematische Zeitschrift 261 (2009): 829-844.