Talk:Ax–Grothendieck theorem

The theorem is not precisely stated! — Preceding unsigned comment added by 157.92.4.75 (talk) 21:12, 10 February 2012 (UTC)

This article needs a rigorous statement of its subject

The only statement of the Ax-Grothendieck theorem in the article is as follows:

"The theorem is often given as this special case: If P is an injective 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 Cⁿ"

But for any article about a mathematical object like a theorem or a definition, there needs to be a rigorous definition of the object somewhere in the article. But this has none.

(In the informal statement of the Ax-Grothendieck theorem that appears in the current article, the "polynomial mapping" P is not defined as a polynomial mapping over the same field as the vector space. This may be implicit, but it should be explicit.

We are also never told over which fields this theorem has been proved for. The theorem needs to be stated clearly and fully.2600:1700:E1C0:F340:E11E:B0DE:F074:B85C (talk) 22:36, 28 May 2018 (UTC)

Link with Garden of Eden

I don't see why the Ax-Grothendieck theorem implies Garden-of-Eden theorem for general cellular automata. The reference given is only about algebraic cellular automata. Moreover the statement of Garden-of-Eden theorem is "pre-injectivity is equivalent to surjectivity" which implies "injectivity implies surjectivity" but is stronger. In particular, the surjunctivity holds on the free group while Garden-of-Eden doesn't. The classical proof of Garden-of-Eden for any cellular automata on any amenable group (cellular automata and groups) doesn't use Ax-Grothendieck. — Preceding unsigned comment added by 147.94.77.12 (talk) 12:09, 6 November 2018 (UTC)