Tsen rank

We say that the Tsen rank of F is infinite if it is not a Ti-field for any i (for example, if it is formally real).

We define a norm form of level i on a field F to be a homogeneous polynomial of degree d in n=di variables with only the trivial zero over F (we exclude the case n=d=1).

The existence of a norm form on level i on F implies that F is of Tsen rank at least i − 1.

The Diophantine dimension of a field is the smallest natural number k, if it exists, such that the field of is class Ck: that is, such that any homogeneous polynomial of degree d in N variables has a non-trivial zero whenever N >  dk.

It is not known whether Tsen rank and Diophantine dimension are equal in general.