The Gram determinant for plane curves

AbstractWe discuss a problem of Arnold, whether every function is stably equivalent to one which is non-degenerate for its Newton diagram. We argue that the answer is negative. We describe a method to make functions non-degenerate after stabilisation and give examples of singularities where this method does not work. We conjecture that they are in fact stably degenerate, that is not stably equivalent to non-degenerate functions.We review the various non-degeneracy concepts in the literature. For finite characteristic, we conjecture that there are no wild vanishing cycles for non-degenerate singularities. This implies that the simplest example of singularities with finite Milnor number, $$x^p+x^q$$ x p + x q in characteristic p, is not stably equivalent to a non-degenerate function. We argue that irreducible plane curves with an arbitrary number of Puiseux pairs (in characteristic zero) are stably non-degenerate. As the stabilisation involves many variables, it becomes very difficult to determine the Newton diagram in general, but the form of the equations indicates that the defining functions are non-degenerate.

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On an area-preserving inverse curvature flow of convex closed plane curves

Journal of Functional Analysis ◽

10.1016/j.jfa.2021.108931 ◽

2021 ◽

Vol 280 (8) ◽

pp. 108931

Author(s):

Laiyuan Gao ◽

Shengliang Pan ◽

Dong-Ho Tsai

Keyword(s):

Curvature Flow ◽

Plane Curves ◽

Area Preserving ◽

Closed Plane

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Cubic spline interpolation for graphs and plane curves

ACM SIGAPL APL Quote Quad ◽

10.1145/586647.586650 ◽

1981 ◽

Vol 11 (4) ◽

pp. 8-12 ◽

Cited By ~ 1

Author(s):

H. R. Krol

Keyword(s):

Spline Interpolation ◽

Cubic Spline ◽

Plane Curves ◽

Cubic Spline Interpolation

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On 3-syzygy and unexpected plane curves

Geometriae Dedicata ◽

10.1007/s10711-021-00602-5 ◽

2021 ◽

Author(s):

Grzegorz Malara ◽

Piotr Pokora ◽

Halszka Tutaj-Gasińska

Keyword(s):

Projective Plane ◽

Plane Curves ◽

Complex Projective Plane ◽

Geometric Properties ◽

Complex Projective

AbstractIn this note we study curves (arrangements) in the complex projective plane which can be considered as generalizations of free curves. We construct families of arrangements which are nearly free and possess interesting geometric properties. More generally, we study 3-syzygy curve arrangements and we present examples that admit unexpected curves.

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