In answer to Andy: it shouldn't matter which compactification is used --- so long as we don't forget about the necessity. "Or some other ..." --- trick question? There aren't any others! In answer to Victor: [see point (2) in my list]. If we reason about z = f(x,y) in 3-space, there is a critical point at (x,y,z) = (-0.98,-0.980,-45) approx, which projects into the disc near the origin: so my (3-space) algorithm reports that it cannot confirm that there is no zero of f within R . Note that the criterion is one-way: it can only establish that there is NO zero in the region! I'll post an example shortly ... WFL On 5/30/14, Andy Latto <andy.latto@pobox.com> wrote:
On Fri, May 30, 2014 at 10:52 AM, Fred Lunnon <fred.lunnon@gmail.com> wrote:
(5) Andy's counter-example --- a straight line of zeros within an infinite strip R --- I did foresee, but ignored on the grounds that it can be fixed by compactification. Adjoin a complex point or projective line at infinity: the boundary of R then includes points at infinity where the line meets it.
If you know the conjecture you're making is false, and have a patched-up version of the conjecture that's actually what you're asking about, I'd find it more useful if you actually stated the patched-up conjecture, rather than having us each guess what patching up you intend.
So the question you were asking about R2, you are actually meaning to ask about some compactification of R2. Would that be the one-point compactification? or RP2, the real projective plane? Or some other compactification?
Andy
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