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

$n > d_1^i + \cdots + d_m^i. \,$

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).

Properties

• A field has Tsen rank zero if and only if it is algebraically closed.
• A finite field has Tsen rank 1: this is the Chevalley–Warning theorem.
• If F is algebraically closed then rational function field F(X) has Tsen rank 1.
• If F has Tsen rank i, then the rational function field F(X) has Tsen rank at most i + 1.
• If F has Tsen rank i, then an algebraic extension of F has Tsen rank at most i.
• If F has Tsen rank i, then an extension of F of transcendence degree k has Tsen rank at most i + k.
• There exist fields of Tsen rank i for every integer i ≥ 0.

Norm form

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. 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.

Diophantine dimension

The Diophantine dimension of a field is the smallest natural number k, if it exists, such that the field of is class C_k: that is, such that any homogeneous polynomial of degree d in N variables has a non-trivial zero whenever N >  d^k. Algebraically closed fields are of Diophantine dimension 0; quasi-algebraically closed fields of dimension 1.^[1]

Clearly if a field is T_i then it is C_i, and T₀ and C₀ are equivalent, each being equivalent to being algebraically closed. It is not known whether Tsen rank and Diophantine dimension are equal in general.

See also

• Tsen's theorem

References

1. Neukirch, Jürgen; Schmidt, Alexander; Wingberg, Kay (2008). Cohomology of Number Fields. Grundlehren der Mathematischen Wissenschaften 323 (2nd ed.). Springer-Verlag. p. 361. ISBN 3-540-37888-X.

• Tsen, C. (1936). "Zur Stufentheorie der Quasi-algebraisch-Abgeschlossenheit kommutativer Körper". J. Chinese Math. Soc. 171: 81–92. Zbl 0015.38803.
• Lorenz, Falko (2008). Algebra. Volume II: Fields with Structure, Algebras and Advanced Topics. Springer. ISBN 978-0-387-72487-4.