Tsen rank

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_i-field if every such system, of degrees d₁, ..., d_m has a common non-zero solution whenever

${\displaystyle n>d_{1}^{i}+\cdots +d_{m}^{i}.\,}$

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