Automorphisms of certain forms of higher degree over ordered fields

We are interested in forms of even degree d 4 over ordered fields, which are linear combinations with positive coefficients of dth powers of linear forms. Usually such a representation is not unique. However certain forms have so-called unique good representation. Linear preservers of such forms are studied. Since −idV is a linear preserver of any form of even degree, orthogonal groups of considered forms have a central involution. We show that any finite group with a central involution is isomorphic with the orthogonal group of such forms of even degree.