Despite trying hard, I managed to foul up my proposed criterion --- what I actually had in mind was "surface z = f(x, y) has no critical points within R " --- but Warren has apparently discerned what I meant, and skewered it. But maybe things can be patched up by specifying that R is simply connected? [ " Dear Mr ... Thankyou for your proof of Fred's last theorem. The first mistake is on line ... of page ... " ] WFL On 5/29/14, Warren D Smith <warren.wds@gmail.com> wrote:
Lunnon: Given a plane curve C defined by f = 0 with f(x, y) polynomial, and a region R of the plane (possibly extending to infinity), I assert that C avoids R provided C avoids the boundary of R ; and C has no critical points within R .
There must be a well-known theorem to this effect (unless, of course, it's actually false...
WDS: a counterexample: C=unit circle, R = thickened version of unit circle (annulus).
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