Yesterday I proposed the intrinsic criterion for real 2-space zero-detection below. Nobody has yet shot it down --- which may well of course be because nobody has actually read it --- so I shall press on, making the optimistic assumption that it involves no grossly obvious errors. << Lunnon: Theorem: Given a continuously differentiable function f(x, y) , and a compact region R of the plane ( |R^2 ) with f nonzero on the boundary of R : f has a zero within R if and only if there is some (intrinsic critical) point (x, y) in R where *** df/dx = f = 0 ***. Proof (offered tentatively): By the implicit function theorem, the constraint f(x, y) = 0 defines a function x(y) single-valued in any interval of y where df/dx <> 0 . Such an interval meeting R is finite since R is compact, and its endpoints lie in the interior of R since f <> 0 on the boundary; therefore df/dx = 0 at those endpoints. The converse is trivial. QED. WDS: I think this is valid.