FOM: How basic is Faltings's theorem?

[JSHIPMAN@bloomberg.net]

Steve, I think you go a little too far in your last posting:
the question "how many solutions do equations have" is
sufficiently foundational that the man on the street (who if he
has taken 9th grade math will have seen polynomial equations)
will take an interest in a general answer to this question.  If
Faltings had handled all curves rather than just the ones in a
hard-to-define-to-the-man-on-the-street subclass, and said "I
found a way to tell how many solutions any equation (in x and y)
has", the man on the street would be impressed.  Similarly,
Matiyasevich could impress Mr. MOTS by saying "I showed no
computer can infallibly tell which equations have solutions."-JS