Quasi Cusp Singularities

Quasi Semi-Border Singularities

“…In the present paper, we study an even more general setting and consider an arbitrary algebraic variety of any dimension, which will be called a semi-border, and the respective relation will be named quasi semi-border equivalence. In the definition of the quasi border equivalence relation in [3], it was assumed that the border was stratified set and the stratification satisfied the Whitney Condition A. Therefore, in many cases, we will be able to establish similar results to that in [1][2][3] for semi-borders that satisfy such conditions.…”

“…In the definition of the quasi border equivalence relation in [3], it was assumed that the border was stratified set and the stratification satisfied the Whitney Condition A. Therefore, in many cases, we will be able to establish similar results to that in [1][2][3] for semi-borders that satisfy such conditions. In particular, two function germs are pseudo semi-border equivalent, if there is a diffeomorphism acting as a change of variables, taking one germ to the other and satisfying the following condition: if one of these functions has a critical point at the semi-borders, then its image (or respectively the inverse image) also belongs to the semi-borders.…”

“…Definition 1. [3] Two functions f 0 , f 1 : R n → R are called pseudo semi-border equivalent if there exists a diffeomorphism θ : R n → R n such that f 1 • θ = f 0 , and if a critical point c of the function f 0 belongs to the border Γ, then θ(c) also belongs to Γ, and vice versa, if c is a critical point of f 1 and belongs to Γ, then θ −1 (c) also belongs to Γ.…”

“…The aim of the relation is to preserve only the positions of critical points of functions defined on a space along a smooth hypersurface. Using Zakalyukin's idea, a similar quasi equivalence relation was introduced in which case the smooth hypersurface was replaced by an arbitrary one, called a border, in particular, when the border is a cylinder over a corner (a union of two transversal intersecting smooth hypersurfaces) or a cuspidal edge, as considered in [2,3], respectively. In the present paper, we study an even more general setting and consider an arbitrary algebraic variety of any dimension, which will be called a semi-border, and the respective relation will be named quasi semi-border equivalence.…”

“…Due to the bad behavior of the radical of an ideal depending on a parameter (see [3]), we will put extra constraints on the pseudo equivalence relation. In particular, we exchange Rad{I t } by the ideal I t in the equivalence definition.…”