Fred, You need some other hypothesis. For example the elliptic curve y^2 = x^3 - x has it's real locus two disconnected ovals (one of them passes through infinity so looks like an open oval). Since they're disconnected, you can surround one by a circle not encroaching on the other. Victor On Thu, May 29, 2014 at 1:01 PM, Fred Lunnon <fred.lunnon@gmail.com> wrote:
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 --- a situation by no means previously unknown). But I don't know a reference (or a counter-example) --- anybody?
A straightforward way to locate the critical points seems to be to compute the discriminant g of f with respect to (say) x , then find the roots of g(y) = 0 . Is there a more respectable alternative?
WFL
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