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April 30

Bivariate polynomial curves

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

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]