FOM: Simpson writes JSHIPMAN at
Wed Oct 22 09:30:46 EDT 1997

>Could you please remind me what Faltings' theorem says?
If the soltution set of a polynomial p(x,y)=0 has more than one
hole, then there are only finitely many rational solutions.
This isn't satisfactory as stated, it is in no way clear to the
uninitiated what it means for a solution set to have holes.
I'd be willing to concede the point that Faltings' theorem is
as foundational as Matiyasevich's if "solution sets have >1
hole" could be replaced by a condition on p(x,y) whose meaning
was intuitively clear (even if it was a less general condition).
-- Joe Shipman

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