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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
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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
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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.
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I don't see why the Ax-Grothendieck theorem implies Garden-of-Eden theorem for general cellular automata. The reference given is only about
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But for any article about a mathematical object like a theorem or a definition, there needs to be a rigorous definition of the object
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We are also never told over which fields this theorem has been proved for. The theorem needs to be stated clearly and fully.
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The only statement of the Ax-Grothendieck theorem in the article is as follows:
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Garden-of-Eden for any cellular automata on any amenable group (
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