(1) Dan has admonished me in private, and perfectly correctly, for confusing surfaces and curves with functions. In future I shall try to restrain my wayward imagination: fundamentally, it's functions we're discussing. In particular, my "critical points" --- where some relevant combination of differentials vanishes --- depend essentially on the coordinate frame (x,y) , the particular polynomial function f(x,y) , and whether I'm discussing: z = f(x,y) in 3-space, with CR's at df/dx = df/dy = 0 ; or y(x) or x(y) defined implicitly (and locally) by f = 0 in 2-space, with CR's at f = df/x = 0 , etc. (2) That confusion misled me into proposing a bogus counter-example to my 3-space algorithm, based on a "tilting cigar" surface meeting the z-plane in the boundary of the 2-space region of interest R , but rising to a maximum z at a point P which projects down to outside R . Intuitively (aha!) this is impossible for simply-connected R , since the surface would have to buckle under itself, and would no longer represent a single-valued function. (3) But as Warren's example of a circle of zeros within annular R shows, simple-connectivity is necessary (the surface can be a paraboloid). This means that any proof is going to involve more topological nous than I can currently muster. (4) Furthermore a "pedestrian" 2-space algorithm actually provides more information about what region R can be guaranteed free of zeros. (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. (6) I didn't follow Victor's reasoning concerning elliptic curves, I'm afraid; which may be well a problem of communication, given my own earlier muddle. In particular, at least two points on his interior oval will satisfy f = df/x = 0 , (7) But it's probably more constructive to work through an actual case illustrating my proposal in action: so I shall go prepare an example. Fred Lunnon On 5/30/14, Victor Miller <victorsmiller@gmail.com> wrote:
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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