FOM: Faltings' theorem isn't basic

Lou van den Dries vddries at
Wed Oct 22 17:48:51 EDT 1997

Well, what is "basic" for Steve seems to change from one email to the
next. In an earlier email he says:

"Matijasevic's theorem is about objects of a much more basic kind:
polynomial equations with integer coefficients"

and later:

"Matijasevic's theorem may be called foundational while the Lang
conjectures may not".

Now, if the difference between the basicness of Matijasevic and
Faltings (I mean, between the questions addressed by their theorems)
is that Matijasevic considers integer solutions and Faltings considers
rational solutions, then I should have talked about Siegel's theorem
instead, which does consider integer solutions, and gives an effective
necessary and sufficient condition for equations p(x,y) = 0 to
have infinitely many integer solutions, and this condition also
involves the genus, etc.  Sincerely,
                                     Lou van den Dries

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