In mathematics, the Ax–Grothendieck theorem is a result about
injectivity
In mathematics, an injective function (also known as injection, or one-to-one function) is a function that maps distinct elements of its domain to distinct elements; that is, implies . (Equivalently, implies in the equivalent contrapositi ...
and
surjectivity
In mathematics, a surjective function (also known as surjection, or onto function) is a function that every element can be mapped from element so that . In other words, every element of the function's codomain is the image of one element of i ...
of
polynomials that was proved independently by
James Ax
James Burton Ax (10 January 1937 – 11 June 2006) was an American mathematician who made groundbreaking contributions in algebra and number theory using model theory. He shared, with Simon B. Kochen, the seventh Frank Nelson Cole Prize in ...
and
Alexander Grothendieck.
The theorem is often given as this special case: If ''P'' is an
injective
In mathematics, an injective function (also known as injection, or one-to-one function) is a function that maps distinct elements of its domain to distinct elements; that is, implies . (Equivalently, implies in the equivalent contrapositiv ...
polynomial function from an ''n''-dimensional
complex vector space to itself then ''P'' is
bijective. That is, if ''P'' always maps distinct arguments to distinct values, then the values of ''P'' cover all of C
''n''.
The full theorem generalizes to any
algebraic variety over an
algebraically closed field
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 ...
.
Proof via finite fields
Grothendieck's proof of the theorem
is based on proving the analogous theorem for
finite fields and their
algebraic closures. That is, for any field ''F'' that is itself finite or that is the closure of a finite field, if a polynomial ''P'' from ''F
n'' to itself is injective then it is bijective.
If ''F'' is a finite field, then ''F
n'' is finite. In this case the theorem is true for trivial reasons having nothing to do with the representation of the function as a polynomial: any injection of a finite set to itself is a bijection. When ''F'' is the algebraic closure of a finite field, the result follows from
Hilbert's Nullstellensatz. The Ax–Grothendieck theorem for complex numbers can therefore be proven by showing that a counterexample over C would translate into a counterexample in some algebraic extension of a finite field.
This method of proof is noteworthy in that it is an example of the idea that finitistic algebraic relations in fields of
characteristic 0 translate into algebraic relations over finite fields with large characteristic.
Thus, one can use the arithmetic of finite fields to prove a statement about C even though there is no
homomorphism from any finite field to C. The proof thus uses
model-theoretic principles such as the
compactness theorem to prove an elementary statement about polynomials. The proof for the general case uses a similar method.
Other proofs
There are other proofs of the theorem.
Armand Borel
Armand Borel (21 May 1923 – 11 August 2003) was a Swiss mathematician, born in La Chaux-de-Fonds, and was a permanent professor at the Institute for Advanced Study in Princeton, New Jersey, United States from 1957 to 1993. He worked in alg ...
gave a proof using topology.
The case of ''n'' = 1 and field C follows since C is algebraically closed and can also be thought of as a special case of the result that for any
analytic function ''f'' on C, injectivity of ''f'' implies surjectivity of ''f''. This is a corollary of
Picard's theorem
In complex analysis, Picard's great theorem and Picard's little theorem are related theorems about the range of an analytic function. They are named after Émile Picard.
The theorems
Little Picard Theorem: If a function f: \mathbb \to\mathbb is ...
.
Related results
Another example of reducing theorems about
morphisms of finite type to finite fields can be found in
EGA IV: There, it is proved that a
radicial ''S''-endomorphism of a scheme ''X'' of finite type over ''S'' is bijective (10.4.11), and that if ''X''/''S'' is of finite presentation, and the endomorphism is a monomorphism, then it is an automorphism (17.9.6). Therefore, a scheme of finite presentation over a base ''S'' is a cohopfian object in the category of ''S''-schemes.
The Ax–Grothendieck theorem may also be used to prove the
Garden of Eden theorem
In a cellular automaton, a Garden of Eden is a configuration that has no predecessor. It can be the initial configuration of the automaton but cannot arise in any other way.
John Tukey named these configurations after the Garden of Eden in Abr ...
, a result that like the Ax–Grothendieck theorem relates injectivity with surjectivity but in
cellular automata rather than in algebraic fields. Although direct proofs of this theorem are known, the proof via the Ax–Grothendieck theorem extends more broadly, to automata acting on
amenable groups.
Some partial converses to the Ax-Grothendieck Theorem:
*A generically surjective polynomial map of ''n''-dimensional affine space over a finitely generated extension of Z or Z/''p''Z
't''is bijective with a polynomial inverse rational over the same ring (and therefore bijective on affine space of the algebraic closure).
*A generically surjective rational map of ''n''-dimensional affine space over a Hilbertian field is generically bijective with a rational inverse defined over the same field. ("Hilbertian field" being defined here as a field for which Hilbert's Irreducibility Theorem holds, such as the rational numbers and function fields.)
[.]
References
External links
*.
{{DEFAULTSORT:Ax-Grothendieck theorem
Theorems in algebra
Model theory