OK, now to complete the proof of the 3D version of the theorem; I will use tools (i) (ii)... (ix) from my previous post: CLAIM: 1. If f(x,y,z) is a real polynomial, then its zero set {x,y,z} such that f(x,y,z)=0 in general describe algebraic surfaces (although curves and isolated points can be possible) in 3-space; call this zero-set C. 2. Let R be a compact simply-connected region with connected interior in xyz space. 3. Then C avoids R provided: (A) C avoids boundary(R). (B) R does not contain a critical point of f, meaning a point where grad f = 0. PROOF:
From tools listed plus simple-connected compact assumption, R must be topologically a ball.
The only way C can fail to avoid R is (i) C hits boundary(R). But that case is already handled in the theorem statement. (ii) A connected component CC of C lies entirely within R's interior. So from now on we only consider case (ii) since that is all we need to consider. Note that any surface-surface or curve-surface self-crossing points, or isolated points, curve end-points, curves, surface-boundary points automatically are critical points of f. (Actually I do not think some of those are possible, but their possibility or impossibility will not matter.) Hence wlog we need only consider the case where CC has none of those things. Since CC is by assumption entirely contained in a compact region with no endpoints and no crossing points (and it is not just an isolated point) it necessarily is a boundaryless manifold. Furthermore, we claim it must be compact and closed. Why? Because any kind of infinite-spiraling-surface confined within a compact region of 3-space, could not arise from a finite degree polynomial. [Really there should be some lemma here, that any connected component of an algebraic surface, in any dimension D, which happens to lie entirely within a compact subregion of D-space, automatically has finite measure and finitely bounded geodesic distance from anyplace on it to anyplace else.] So CC is a closed compact boundaryless 2-manifold embeddable in 3-space, hence by Mobius is an h-holed torus for h=0,1,2,3,...m where m is finitely bounded by some function of the degree of polynomial f. And CC has an interior. Now since f=0 on CC and wlog f>0 for all points (x,y,) interior to CC and within distance epsilon from CC (for all epsilon with 0<epsilon<eps0, for some finite positive eps0) and since interior(CC) U CC is compact, we see f must have a max which must be positive and must be located strictly interior to CC. This max is a critical point. Now given that CC is wholy contained inside R, and R is ball-shaped, it follows that interior(CC) also is wholy contained inside R ("interior" meaning the finite measure side -- exterior has infinite measure). Hence the critical point lies in R. Q.E.D. ================== Now one may ask -- can this result be generalized to any dimension, not just 2D and 3D? I'm not sure. I think my proof will prove it in any dimension D where the necessary ingredient-tool-lemmas are available in that dimension. I think we need these tools: 1. finite number of possible topologies for any surface curve compact connected component CC of finite algebraic degree. 2. measure and geodesic diameter automatically bounded for such a CC. 3. R automatically must be homeomorphic to ball. 4. Any kind of surface self-crossing, lower dimensionality than D-1, etc automatically yields a critical point so we can wlog assume CC is a non-wild boundaryless compact (D-1)-manifold. 5. CC "interior" and "exterior" then automatically always exist. We can wlog demand f<0 in exterior and f>0 in interior, at least if we stay within distance epsilon of CC for all small-enough epsilon. 6. If ball-shaped R contains CC then R automatically also contains interior(CC). If those tools are valid in some dimension then I think the theorem follows in that dimension. The only ones I don't feel entirely confident about are #3 and #5 (I feel confident about them in 2D and 3D, but in dimensions higher than 4 they are not so obvious... although likely the answers are known...).