PIMS MSS Colloquium: Detlev Hoffmann

Consider two finitely generated field extensions K and L of the same transcendence degree over some base field F. Suppose these extensions become isomorphic over F after adjoining some variables. Does this imply that K isalready isomorphic to L over F? This is commonly known as the Zariski problem. It is known that in general the answer is negative. The quadratic Zariski problem QZP (a terminology coined by Jack Ohm) now asks this question in the case where K and L are function fields of projective quadrics of the same dimension over F. QZP has a positive answer for quadrics of small dimension or for quadrics defined by quadratic forms of certain types, but no counterexamples are known. We extend the list of quadratic forms and for base fields F for which QZP has a positive answer. For example, we show that QZP has a positive answer for number fields with at most one real place, or for the field of Laurent series over the reals.