In mathematics, the Tsen rank of a field describes conditions under which a system of polynomial equations must have a solution in the field. The concept is named for C. C. Tsen, who introduced their study in 1936.
We consider a system of m polynomial equations in n variables over a field F. Assume that the equations all have constant term zero, so that (0, 0, ... ,0) is a common solution. We say that F is a T<sub>i</sub>-field if every such system, of degrees d<sub>1</sub>, ..., d<sub>m</sub> has a common non-zero solution whenever
The Tsen rank of F is the smallest i such that F is a T<sub>i</sub>-field. We say that the Tsen rank of F is infinite if it is not a T<sub>i</sub>-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=d<sup>i</sup> 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. If E is an extension of F of finite degree n > 1, then the field norm form for E/F is a norm form of level 1. If F admits a norm form of level i then the rational function field F(X) admits a norm form of level i + 1. This allows us to demonstrate the existence of fields of any given Tsen rank.
The Diophantine dimension of a field is the smallest natural number k, if it exists, such that the field of is class C<sub>k</sub>: that is, such that any homogeneous polynomial of degree d in N variables has a non-trivial zero whenever N > d<sup>k</sup>. Algebraically closed fields are of Diophantine dimension 0; quasi-algebraically closed fields of dimension 1.
Clearly if a field is T<sub>i</sub> then it is C<sub>i</sub>, and T<sub>0</sub> and C<sub>0</sub> are equivalent, each being equivalent to being algebraically closed. It is not known whether Tsen rank and Diophantine dimension are equal in general.