mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, the Tsen rank of a

field Field may refer to: Expanses of open ground * Field (agriculture), an area of land used for agricultural purposes * Airfield, an aerodrome that lacks the infrastructure of an airport * Battlefield * Lawn, an area of mowed grass * Meadow, a grass ...

describes conditions under which a system of

polynomial equations In mathematics, an algebraic equation or polynomial equation is an equation of the form :P = 0 where ''P'' is a polynomial with coefficients in some field, often the field of the rational numbers. For many authors, the term ''algebraic equation'' ...

must have a solution in the

. 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 ''d''₁, ..., ''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 In mathematics, a field is algebraically closed if every non-constant polynomial in (the univariate polynomial ring with coefficients in ) has a root in . Examples As an example, the field of real numbers is not algebraically closed, because ...

. * A finite field has Tsen rank 1: this is the

Chevalley–Warning theorem In number theory, the Chevalley–Warning theorem implies that certain polynomial, polynomial equations in sufficiently many variables over a finite field have solutions. It was proved by and a slightly weaker form of the theorem, known as Cheval ...

. * 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 In abstract algebra, the transcendence degree of a field extension ''L'' / ''K'' is a certain rather coarse measure of the "size" of the extension. Specifically, it is defined as the largest cardinality of an algebraically independent subset of ...

''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''=''d''^''i'' 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 In mathematics, a norm form is a homogeneous form in ''n'' variables constructed from the field norm of a field extension ''L''/''K'' of degree ''n''. That is, writing ''N'' for the norm mapping to ''K'', and selecting a basis ''e''1, ..., ''e ...

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 field In mathematics, a field ''F'' is called quasi-algebraically closed (or C1) if every non-constant homogeneous polynomial ''P'' over ''F'' has a non-trivial zero provided the number of its variables is more than its degree. The idea of quasi-algebra ...

s of dimension 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.

References

* * {{cite book , first=Falko , last=Lorenz , title=Algebra. Volume II: Fields with Structure, Algebras and Advanced Topics , year=2008 , publisher=Springer , isbn=978-0-387-72487-4 Field (mathematics) Diophantine geometry

Properties

Norm form

Diophantine dimension

See also

References