scholarly journals Counterexamples to the local-global principle for non-singular plane curves and a cubic analogue of Ankeny-Artin-Chowla-Mordell conjecture

2020 ◽  
pp. 1
Author(s):  
Yoshinosuke Hirakawa ◽  
Yosuke Shimizu
Keyword(s):  
Plane Curves ◽  
Mordell Conjecture ◽  
Global Principle

scholarly journals The local–global principle for symmetric determinantal representations of smooth plane curves

2016 ◽  
Vol 43 (1) ◽  
pp. 141-162 ◽  
Author(s):  
Yasuhiro Ishitsuka ◽  
Tetsushi Ito
Keyword(s):  
Plane Curves ◽  
Smooth Plane ◽  
Global Principle

scholarly journals The local–global principle for symmetric determinantal representations of smooth plane curves in characteristic two

2017 ◽  
Vol 221 (6) ◽  
pp. 1316-1321 ◽  
Author(s):  
Yasuhiro Ishitsuka ◽  
Tetsushi Ito
Keyword(s):  
Plane Curves ◽  
Smooth Plane ◽  
Characteristic Two ◽  
Global Principle

scholarly journals On a method of dealing with the intersections of plane curves

1904 ◽  
Vol 5 (4) ◽  
pp. 385-385 ◽  
Author(s):  
F. S. Macaulay
Keyword(s):  
Plane Curves

scholarly journals Conjectures on Stably Newton Degenerate Singularities

2021 ◽  
Author(s):  
Jan Stevens
Keyword(s):  
Characteristic Zero ◽  
Arbitrary Number ◽  
Milnor Number ◽  
Plane Curves ◽  
Newton Diagram ◽  
Finite Characteristic ◽  
Vanishing Cycles

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.

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

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

Cubic spline interpolation for graphs and plane curves

1981 ◽  
Vol 11 (4) ◽  
pp. 8-12 ◽  
Author(s):  
H. R. Krol
Keyword(s):  
Spline Interpolation ◽  
Cubic Spline ◽  
Plane Curves ◽  
Cubic Spline Interpolation

scholarly journals On 3-syzygy and unexpected plane curves

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.

scholarly journals Newton transformations and motivic invariants at infinity of plane curves

2021 ◽  
Author(s):  
Pierrette Cassou-Noguès ◽  
Michel Raibaut
Keyword(s):  
Plane Curves ◽  
Newton Transformations
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