Mathematics desk
< April 29	<< Mar \| April \| May >>	May 1 >

Welcome to the Wikipedia Mathematics Reference Desk Archives
The page you are currently viewing is an archive page. While you can leave answers for any questions shown below, please ask new questions on one of the current reference desk pages.

April 30[edit]

Bivariate polynomial curves[edit]

In each of these decision problems, do we know whether an answer can be obtained in polynomial time?

(1) Given a bivariate polynomial equation of degree n with numerical coefficients, is the corresponding curve closed?

(2) Given a bivariate polynomial equation of degree n with numerical coefficients, with corresponding curve known to be closed, does the curve enclose a convex region?

(3) Given a bivariate polynomial equation of degree n with numerical coefficients, is the corresponding curve self-intersecting?

Thanks in advance! Loraof (talk) 19:51, 30 April 2017 (UTC)[reply]

The usual question to ask for (3) is "is a given algebraic curve a nonsingular variety?" in which case it is also a smooth manifold. --JBL (talk) 01:51, 1 May 2017 (UTC)[reply]

Curves of constant width[edit]

Does a curve of constant width necessarily enclose a convex region? Our article describes it as convex in the first sentence, implying that this is either a necessary property or a definitional feature, but is it really necessary? (Certainly the width is defined for non-convex shapes, as shown in the diagram in the lead of Mean width.) Thanks. Loraof (talk) 19:59, 30 April 2017 (UTC)[reply]

A non-convex region is constant width if and only if its convex hull is, so there is no reason to consider the nonconvex case. --JBL (talk) 01:44, 1 May 2017 (UTC)[reply]

Okay. Now the convex hull of a non-convex curve will have a line segment over each concave part. Is it possible for a curve of constant width to have any linear portions? (If not, that would preclude non-convex curves from having constant width.) Loraof (talk) 14:57, 1 May 2017 (UTC)[reply]

I think your second sentence is probably false -- surely there are fractal curves whose convex hull is the unit circle, for example? --JBL (talk) 18:41, 1 May 2017 (UTC)[reply]