The discriminant of quasi m-boundary singularities

Fawaz Alharbi; Eman Al-hudhali

doi:10.1515/dema-2025-0171

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The discriminant of quasi m-boundary singularities

Fawaz Alharbi and Eman Al-hudhali

Published/Copyright: October 4, 2025

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From the journal Demonstratio Mathematica Volume 58 Issue 1

Abstract

We describe the discriminant of deformations of simple quasi m-boundary equivalence classes for m ≥ 2 . All quasi simple m -boundary classes are right equivalent to Arnold’s singularities (ADE). Consequently, their respective discriminants are cylinders over the standard ones, together with some new descriptions involving either submanifolds or generalized Whitney umbrella. Our main results are as follows:

Theorem 3.5 on the bifurcation diagram and caustic of quasi A k singularities.
Theorem 3.9 on the bifurcation diagram and caustic of quasi D 4 + singularity.
Theorem 3.10 on the bifurcation diagram and caustic of quasi D 4 − singularity.
Theorem 3.11 on the bifurcation diagram and caustic of quasi H p , k singularities.

The subsequent section provides a concise overview of the primary results.

Keywords: singularity; submanifold; quasi equivalence; semiborder; m-boundary; Lagrangian projection; bifurcation diagram; caustics; generalized swallowtail; generalized Whitney umbrella; Morin mapping

MSC 2010: 58K05; 58K40; 53A15

1 Introduction

Numerous important equivalence relations in the space of function and map germs have been studied and used in various contexts and applications. New nonstandard equivalence relations, called pseudo – and “quasi” equivalence relations (which are not group actions), were introduced in several recent publications [1–5]. In particular, two function germs f ( z ) and g ( z ) defined on a coordinate space R n in local coordinates z = ( x 1 , … , x m , y 1 , … , y n − m ) , equipped with an algebraic variety Γ (which is called semiborder) are equivalent with respect to the pseudo semi border if there is a diffeomorphism θ , called admissible, maps one germ to another and fulfills the subsequent condition: if c is a critical point of f that lies on Γ then its image or preimage under θ also lies on Γ . This restriction means that Γ is not required to be preserved outside of the critical locus. To ensure that the equivalence relation works properly, additional conditions are imposed on the vector fields that generate the family of admissible diffeomorphisms. The relation that is produced as a result is referred to as quasi-equivalence. The most recent work in this direction was introduced in [1], where the semiborder is regarded as a submanifold of codimension m ≥ 2 . In this case, it is customary to define Γ as the set { x 1 = x 2 = … = x m = 0 } and refer to it as the m -boundary. As a consequence, a classification of simple quasi m -boundary classes of function germs was achieved, where m is greater than or equal to 2. The obtained results can be viewed as a logical continuation of the research conducted in [4], where the set Γ was defined as { x 1 = 0 } and the corresponding relation was referred to as “quasi boundary equivalence.”

The quasi m -boundary classes have fundamental applications in symplectic geometry. In particular, the simple classes are used to classify stable and simple Lagrangian manifolds equipped with a smooth m -dimensional submanifold in it [1].

The study of the bifurcation diagram of a deformation f u ( z ) ( u serves as parameters) of function germ f 0 ( z ) is a fundamental work in singularity theory where one is interested in the set of parameter values u for which f u has a degenerate critical point with critical values equal 0 [6,7]. Caustics are another fundamental concept closely related to the discriminant and is defined as the Lagrangian map critical value set of the Lagrangian manifold [6]. Such studies are known as the catastrophe theory, and it was first introduced by the French mathematician René Thom in the 1960s and gained significant popularity as a consequence of Christopher Zeeman’s efforts in the 1970s. The study is significant for an extensive range of scientific and mathematical areas [8]. Later, many authors have continued further studies on catastrophe theory, including Vladimir Arnold, who gave the catastrophes the ADE classification [9]. Further studies and applications on standard bifurcation diagrams and caustics of singularities can be found in various articles [6,10–15].

The initial elucidation of the quasi bifurcation diagrams and caustics associated with the deformation of the simple quasi function germ was carried out in [16]. Following [16], the bifurcation diagrams of the deformation F ( z , λ ) = F ˜ ( z , λ ) + λ 0 depending on parameters λ = ( λ 0 , λ 1 … , λ μ ) of a quasi function germ f ( z ) = F ˜ ( z , 0 ) defined on Γ , typically comprise two distinct components denoted as W 0 and W 1 . The initial part W 0 is contained in R λ μ + 1 and given by the equations F = 0 and ∂ F ∂ z = 0 . The stratum W 0 corresponds to the standard discriminant. The second part W 2 is contained in W 1 and fulfills supplemental equations that construct the semiborder. The caustics of the deformation F ( w , λ ) of a function germ f ( z ) = F ( z , 0 ) also consists of two parts, Σ 0 and Σ 1 within the truncated deformation base λ ˜ = ( λ 1 , … , λ μ ) (forgotten λ 0 ). The first part Σ 0 is the image of the singular set W 0 under the projection π : λ → λ ˜ , and the second one is the set π ( W 1 ) . Discriminant will refer to bifurcation diagrams and caustics in the present article. Such a setting appears in different situations; for example, in [18,19], the geometry of cuspidal edge with boundary was considered, in which case the configuration corresponds to the bifurcation diagrams of the quasi simple class B 3 .

The primary objective of the present study is to provide a comprehensive description of the discriminant associated with deformations of simple quasi m -boundary classes, where m ≥ 2 . The significance of bifurcation diagrams of quasi classes extends to the area of physics and variational problems. In particular, the quasi equivalence relation describes the character of the behavior of critical points of a function on a specific domain endowed with a submanifold of codimension m ≥ 2 . The further research which goes beyond the present work might be related to study the differential geometry, topology, and concrete applications of the discriminants of the quasi simple classes.

This article has been organized as follows. In Section 2, we recall the basic results of the quasi m -boundary classification from [1] and other definitions and results mainly from [17]. In Section 3, we introduce new definitions of the bifurcation diagrams and caustics of the quasi function deformation and describe in detail the discriminant of simple quasi m -boundary classes.

2 Preliminary

In this section, we review from [1] the classification of simple function germs with respect to the quasi m -boundary equivalence relation for m ≥ 2 . We also recall the standard definitions and results from singularity theory, which will be used in the following section.

Let R n be the n -dimensional Euclidean space equipped with submanifold Γ S m of codimension m in it. Let z = ( x 1 , … , x m , y 1 , … , y n − m ) be the local coordinates of R n , where 1 ≤ m ≤ n . Then, we may assume that Γ S m = { x 1 = x 2 = … = x m = 0 } .

Let C z represents the ring that consists of all the C ∞ germs of functions f : ( R n , 0 ) → R , in local coordinates z as mentioned earlier. Let ℳ z denotes the maximal ideal within C z .

The description of simple quasi m -boundary singularities, with m ≥ 2 , is given in the following.

Theorem 2.1

[1] Let f : ( R n , 0 ) → R , be a simple germ with respect to the quasi m-boundary equivalence relation. Then, f is stably quasi m-boundary equivalent to one of the following simple classes:

Notation	Normal form	Restrictions	Codimension
A k	x 2 2 ± x 1 k + 1 + x 2	k ≥ 1	m + k
D k	x 1 2 x 2 ± x 2 k − 1 + x 2	k ≥ 4	m + k
E 6	x 1 3 ± x 2 4 + x 2	—	m + 6
E 7	x 1 3 + x 2 x 2 3 + x 2	—	m + 7
E 8	x 1 3 + x 2 5 + x 2	—	m + 8
H p , k p = ( p 1 , p 2 , … , p m )	∑ i = 1 m ± ( x i ± y 1 p i ) 2 ± y 1 k	k > p m ≥ … ≥ p 1 ≥ 2	∑ i = 1 m p i + k − 1

Remarks 2.2

The results are presented up to the rearrangements of the coordinates x i .
In Theorem 2.1, x 2 = ∑ i = 3 m ± x i 2 .
The A k classes may be expressed in other ways, i.e., ∑ i = 1 m ± ( x i ± y 1 ) 2 ± y 1 k + 1 . Consequently, these classes can be involved into the series H p , k + 1 as H 1 m , k + 1 , where 1 m = ( 1 , 1 , … , 1 ) .
The tangent space to the quasi m -boundary equivalence singularity of f at f is
T Q ( Γ S m ) f = ∑ i = 1 m ∂ f ∂ x i ∑ j = 1 m x j A i j + ∑ l = 1 n ∂ f ∂ w l B i l + ∑ j = 1 n − m ∂ f ∂ y j E j , A i j , B i l , E i ∈ C z .

Let G ( z , u ) be deformation of f : ( R n , 0 ) → ( R , 0 ) , where u ∈ R l represents that parameters. We shall set G u ( z ) = G ( z , u ) ; consequently, G 0 = f .

The initial speeds of G are defined by

G ˙ i = ∂ G ∂ u i ( z , 0 ) , ∀ i ∈ { 1,2 , … , l } .

The following result is a version of Theorem 3 in [20].

Proposition 2.3

A deformation G of a germ of a function f is versal with respect to the quasi m-boundary equivalence relation if and only if

T Q ( Γ S m ) f + R { 1 , G ˙ 1 , … G ˙ l } = C z .

Assume that the elements h 0 , … , h l − 1 ∈ C z form a basis of the quotient space C z ⁄ T Q ( Γ S m ) f . Then, Proposition 2.3 implies that a miniversal deformation of a function germ f may take the form:

(1) G ( z , u ) = f ( z ) + ∑ i = 0 l − 1 u i h i ( z ) ,

Suppose h 0 = 1 and h i ∈ ℳ x , y . Then, the space R l − 1 in the local coordinates u 1 , … , u l − 1 , is referred to as the base of a truncated quasi m -boundary miniversal deformation of f .

Proposition 2.4

The miniversal deformations of simple quasi m-boundary singularities are described in the following table:

Class	Miniversal deformation	Constraints
A k	± x 1 k + 1 ± x 2 2 + x 2 + λ 0 + ∑ i = 1 k λ i x 1 i + ∑ j = 2 m μ j x j	k ≥ 1
D k	x 1 2 x 2 ± x 2 k − 1 + x 2 + λ 0 + λ 1 x 1 2 + λ 2 x 2 x 1 + ∑ i = 2 k − 2 λ i + 1 x 2 i + ∑ i = 1 m μ j x j	k ≥ 4
E 6	x 1 3 ± x 2 4 + x 2 + λ 0 + ∑ i = 1 m λ i x i + μ 0 x 1 2 + x 1 ( ∑ j = 1 2 μ j x 2 j ) + ∑ j = 3 4 μ j x 2 j − 1	—
E 7	x 1 3 + x 1 x 2 3 + x 2 + λ 0 + ∑ i = 1 m λ i x i + ∑ j = 1 2 μ j x 1 j x 2 2 − j + ∑ j = 3 5 μ j x 2 j − 1	—
E 8	x 1 3 + x 2 5 + x 2 + λ 0 + ∑ i = 1 m λ i x i + μ 0 x 1 2 + x 1 ( ∑ j = 1 3 μ j x 2 j ) + ∑ j = 4 6 μ j x 2 j − 2	—
H p , k	∑ i = 1 m ± ( x i ± y 1 p i ) 2 ± y 1 k + λ 0 + ∑ i = 1 m x i ( ∑ j = 0 p i − 1 λ i , j y 1 j ) + ∑ l = 1 k − 2 μ l y l l	k > p m ≥ … ≥ p 1 ≥ 2

Recall from [17] that a smooth map-germ H : ( R n , 0 ) → ( R m , 0 ) is said to be of corank 1 if the rank of the matrix of Jacobian evaluated at the origin is equal to min ( n , m ) − 1 .

Theorem 2.5

(Morin) [17] A stable germ H : ( R n , 0 ) → ( R m , 0 ) ; of corank 1 is right-left equivalent to a suspension of the germ of generalized Whitney umbrella:

Φ : z ∈ R λ ↦ u 1 = z k + 0 + λ 1,2 z k − 2 + … + λ 1 , k − 1 z u 2 = λ 2,1 z k − 1 + λ 2,2 z k − 2 + … + λ 2 , k − 1 z ⋮ u t = λ t , 1 z k − 1 + λ t , 2 z k − 2 + … + λ t , k − 1 z v = λ ,

where t ( k − 1 ) ≤ n , n ≤ m , rank ≔ r = n − 1 , t = m − r = m − n + 1 .

Example 2.6

[17] Let n = 2 and m = 3 . Then, we obtain the ordinary Whitney umbrella: u 1 = z 2 , u 2 = λ z , v = λ (Figure 1). Here, r = 1 , t = 2 , and k = 2 .

Figure 1

Whitney umbrella.

3 Bifurcation diagrams and Caustics of simple quasi m -boundary classes

In this section, we begin by establishing the definitions of bifurcation diagrams and caustics of quasi singularities. We then describe the bifurcations diagram and caustics of simple quasi m -boundary singularities, providing detailed proofs.

Let G u be a quasi m -boundary miniversal deformation of a function germ f , where u ∈ R l . Let V = { G u = 0 } ⊆ R l .

Definition 3.1

The bifurcation diagram of f is the set of points u such that V is singular or a singularity of V lies on Γ S m .

Remark 3.2

The Definition 3.1 indicates that the bifurcation diagram is made up of two parts: W 0 and W m , in which case W m ⊆ W 0 , where dim ( W j ) = τ − ( j + 1 ) and j ∈ { 0 , m } . More precisely, the initial part W 0 is the projection to the base space R l of the subset χ 0 , which is located in the total space ( x , y , u ) , and specified by the equations: G u = 0 and ∂ G u ∂ x = ∂ G u ∂ y = 0 . The second part W m is contained in W 0 , and it fulfills the additional equations x 1 = x 2 = … = x m = 0 .

Remark 3.3

Bifurcation diagrams and caustics will be referred as discriminant.

Definition 3.4

Let π : R τ → R τ − 1 be the natural projection map that ignoring μ 0 . The quasi m-boundary caustic of a function g is a hypersurface in the base R τ − 1 that is formed by the union of two sets: the π -image Σ 0 of the singular points of the set W 0 ⊂ R τ , and the set Σ m = π ( W m ) .

3.1 Bifurcation diagrams and caustics of A k singularity

Theorem 3.5

The first part of the bifurcation diagram of the A k classes is the product of the space R m and a generalized swallowtail. The second part is a ( k − 1 )-dimensional submanifold that smoothly passes through the vertex of the first component.
The caustic of the A k series consists of a cylindrical generalized swallowtail (with m generators) and a smooth submanifold intersecting with the first component in ( k − 2 ) -dimensional space.

Proof

Consider the quasi versal deformation

F ( x , λ , μ ) = ± x 1 k + 1 ± x 2 2 + x 2 + λ 0 + ∑ i = 1 k λ i x 1 i + ∑ j = 1 m − 1 μ j x j + 1

of A k singularity. The first strata of the bifurcation diagram are determined by solving the equations F = 0 and ∂ F ∂ x i = 0 for λ and μ . Thus, we obtain the surface, which is parameterized as follows:

(2) Φ : x λ 2 ⋮ λ k μ ↦ λ 0 = ± k x 1 k + 1 ± x 2 2 + x + ∑ i = 2 k ( i − 1 ) λ i x 1 i λ 1 = ∓ ( k + 1 ) x 1 k − ∑ i = 2 k i λ i x 1 i − 1 λ 2 ⋮ λ k μ 1 = ∓ 2 x 2 μ 2 = ∓ 2 x 3 ⋮ μ m − 1 = ∓ 2 x m ,

where x = ∑ i = 3 m ± x i 2 . This mapping is exactly a formula of generalized swallowtail times R m . On the other hand, if we set x 1 = x 2 = … = x m = 0 in (2), we obtain the R k − 1 -space in local coordinates λ 2 , … , λ k , which defines the second component of the bifurcation diagram.

The singular set Σ F (the vertex) is determined by the equation

det ∂ F ∂ x i ∂ x j = 0 ,

which gives that

λ 2 = ∓ k ( k + 1 ) x 1 k − 1 − ∑ i = 3 k i ( i − 1 ) 2 λ i x 1 i − 2 .

Clearly, Σ F intersects the second strata in the space R k − 2 in local coordinates μ 3 , μ 4 , … , μ k .

The second result follows from projecting the singular set together with the second component along the λ 0 -axis.□

Now we discuss the bifurcation diagrams and caustics for m = 2 and specific values of k in low dimensions.

Remark 3.6

The case when m = 1 is discussed in [16].

Corollary 3.7

The first part of the bifurcation diagram of A 1 singularity is an elliptic paraboloid, and the second one is point in it (Figure 2).

$Figure 2 The bifurcation diagram of A 1 {{\mathbb{A}}}_{1} singularity. The origin is the second component.$

Figure 2

The bifurcation diagram of A 1 singularity. The origin is the second component.

Corollary 3.8

The bifurcation diagram of A 2 singularity is a cuspidal edge times R and a line in it (Figure 3).
The A 2 caustic is a union of a parabolic cylinder and a line tangent to it (Figure 4).

$Figure 3 Cross-section of the bifurcation diagram of A 2 {{\mathbb{A}}}_{2} singularity. The origin is the second component.$

Figure 3

Cross-section of the bifurcation diagram of A 2 singularity. The origin is the second component.

$Figure 4 The A 2 {{\mathbb{A}}}_{2} caustic. The red axis is the second component.$

Figure 4

The A 2 caustic. The red axis is the second component.

3.2 Caustic of D 4 ± singularity

Theorem 3.9

The caustic of D 4 + singularity is a union of hyperbolic umbilic times R m and a two-dimensional submanifold (Figure 5).

$Figure 5 The hyperbolic umbilic: cross-section of the caustics of the quasi D 4 + {{\mathbb{D}}}_{4}^{+} singularity.$

Figure 5

The hyperbolic umbilic: cross-section of the caustics of the quasi D 4 + singularity.

Proof

Consider the D 4 + singularity : f = x 1 2 x 2 + x 2 3 + x , which can be written equivalently as f = x 1 3 + x 2 3 via the diffeomorphism

ϕ : ( x 1 , x 2 , x 3 , … , x m ) ↦ ( x 1 , x 2 − x 1 , x 3 , … , x m )

followed by rescaling the coefficients. Its quasi versal deformation may take the form

(3) F = x 1 3 + x 2 3 + x + λ 0 − λ 1 x 1 − λ 2 x 2 + λ 3 x 1 x 2 + λ 4 x 1 2 + λ 5 x 2 2 − ∑ i = 3 m μ j x j .

Note here that, the signs of λ 1 , λ 2 , and μ i are changed to make the calculations easier.

The first component of the caustic of the D 4 + singularity is determined by the following relations:

(4) ∂ F ∂ x 1 = 3 x 1 2 − λ 1 + λ 3 x 2 + 2 λ 4 x 1 = 0 ,

(5) ∂ F ∂ x 2 = 3 x 2 2 − λ 2 + λ 3 x 1 + 2 λ 5 x 2 = 0 ,

(6) ∂ F ∂ x i = ± 2 x i − μ i = 0 , i = 3 , 4 , … , m ;

and the singular set

Σ F = 6 x 1 + 2 λ 4 λ 3 0 0 … 0 λ 3 6 x 2 + 2 λ 5 0 0 … 0 0 0 ± 2 0 0 0 0 0 0 ± 2 0 0 ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ 0 0 0 0 0 ± 2 = 0 ,

i.e.,

(7) Σ F = ( 6 x 1 + 2 λ 4 ) ( 6 x 2 + 2 λ 2 ) − λ 3 2 = 0 .

If we apply the diffeomorphism:

Φ : ( x 1 , x 2 , x 3 , … , x m ) ↦ x 1 − 1 3 λ 4 , x 2 − 1 3 λ 5 , x 3 , … , x m ,

followed by a change in the parameters only via

ψ : ( λ 1 , λ 2 , λ 3 , λ 3 , λ 5 , μ ) ↦ λ 1 − λ 4 2 3 − λ 3 λ 5 3 , λ 2 − λ 5 2 3 − λ 3 λ 4 3 , λ 3 , λ 4 . λ 5 , μ ,

then (4)–(7) become

(8) λ 1 = 3 x 1 2 + λ 3 x 2 ,

(9) λ 2 = 3 x 2 2 + λ 3 x 1 ,

(10) μ i = ± 2 x i , i = 3 , 4 , … , m ,

and

(11) λ 3 2 = 36 x 1 x 2 ,

respectively, which define a cylinder (with m generators) over the standard hyperbolic umbilic in the parameter space with local coordinates in λ 1 , λ 2 , λ 3 , λ 4 , λ 5 and μ .

On the other hand, the second component is obtained by imposing the conditions x 1 = x 2 = … = x m = 0 into (8)–(11) and the result follows.□

Theorem 3.10

The caustic of D 4 − singularity is a union of elliptic umbilic times R m and a two-dimensional submanifold (Figure 6).

$Figure 6 The elliptic umbilic: cross-section of the caustics of quasi D 4 − {{\mathbb{D}}}_{4}^{-} singularity.$

Figure 6

The elliptic umbilic: cross-section of the caustics of quasi D 4 − singularity.

Proof

Up to the quasi equivalence, the D 4 − singularity can be written as follows:

f = 1 3 x 1 3 − x 1 x 2 2 + x .

Thus, the respective quasi miniversal deformation may take the form

(12) F = 1 3 x 1 3 − x 1 x 2 2 + x + λ 0 − λ 1 x 1 + λ 2 x 2 + λ 3 x 1 x 2 + λ 4 x 1 2 + λ 5 x 2 2 − ∑ i = 3 m μ j x j .

By standard versality theorem with respect to right equivalence relation, (12) can be written equivalently as

(13) F = 1 3 x 1 3 − x 1 x 2 2 + x + λ 0 − λ 1 x 1 + λ 2 x 2 + λ 3 ( x 1 2 + x 2 2 ) − ∑ i = 3 m μ j x j .

Thus, the first component of the caustic is determined by the equations

(14) ∂ F ∂ x 1 = x 1 2 − x 2 2 + 2 λ 3 x 1 − λ 1 = 0 ,

(15) ∂ F ∂ x 2 = − 2 x 1 x 2 + 2 λ 3 x 2 + λ 2 = 0 ,

(16) ∂ F ∂ x i = ± 2 x i − μ i = 0 , i = 3 , 4 , … , m ,

and the singular set

Σ F = 2 x 1 + 2 λ 3 − 2 x 2 0 0 … 0 − 2 x 2 − 2 x 1 + 2 λ 3 0 0 … 0 0 0 ± 2 0 0 0 0 0 0 ± 2 0 0 ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ 0 0 0 0 0 ± 2 = 0 ,

or equivalently

(17) λ 3 = x 1 2 + x 2 2 .

Note, (17) implies that the points ( x 1 , x 2 ) lie on a circle centered at the origin with radius λ 3 . So, if λ 3 is constant, then we may write

(18) x 1 = λ 3 cos ( θ ) , x 2 = λ 3 sin ( θ ) , where 0 ≤ θ ≤ 2 π .

Therefore, when we substitute the values of x 1 and x 2 from (18) into (14) and (15), we obtain

λ 1 = λ 3 2 [ cos ( 2 θ ) + 2 cos ( θ ) ] and λ 2 = λ 3 2 [ sin ( 2 θ ) − 2 sin ( θ ) ] ,

which are exactly the parametrization of the elliptic umbilic. Hence, the first components of the caustics of the quasi D 4 − singularity is a cylinder with m generators over the elliptic umbilic.

Finally, set the conditions x 1 = x 2 = … = x m = 0 into (14)–(17) for the second component; and the result follows.□

3.3 Bifurcation diagrams and caustics of H p , k singularity

Theorem 3.11

The first part of the bifurcation diagram of the H p , k classes is a cylindrical generalized swallowtail of type A k − 2 . The second component, on the other hand, is a smooth variety contained inside this cylinder.
The caustic of H p , k classes is a union of a cylinder over a generalized swallowtail of type A k − 2 and a generalized Whitney umbrella.

Proof

Clearly, the quasi H p , k classes are right equivalent to standard A k − 1 classes. Therefore, the first component of the bifurcation diagram (caustic) is a product of a generalized swallow tail and R r − ( k − 1 ) ( R ( r − 1 ) − ( k − 1 ) ) , where r is the dimension of the base of the quasi versal deformation of the class H p , k . The second part is obtained by imposing the conditions x 1 = … = x m = 0 in the equations that define the first component.

For the second component of the caustics, the quasi versal deformation may take the form

(19) F = ∑ i = 1 m ± ( x i ± y p i ) 2 + ∑ i = 1 m x i ∑ j = 0 p i − 1 λ i , j y j ± y k + ∑ i = 0 k − 2 μ i y i .

Note that F is quasi equivalent to the deformation

(20) G = ∑ i = 1 m ± ( x i ± y p i ) 2 + 2 ∑ i = 1 m ( x i + y p i − 1 ) ∑ j = 0 p i − 1 λ i , j y j ± y k + ∑ l = 0 k − 2 μ l y l .

Also, note that if we add the portion Ω = ∑ i = 1 m ∑ j = 0 p i − 1 λ i , j y j 2 to (20), then G remains versal due to the versality theorem as ∂ Ω ∂ λ i , j ∣ λ i , j = 0 = 0 . Thus, we may take the alternative form of the versal deformation as follows:

G ˜ = ∑ i = 1 m ± ( x i ± y p i ) 2 + 2 ∑ i = 1 m ( x i + y p i − 1 ) ∑ j = 0 p i − 1 λ i , j y j + ∑ i = 1 m ∑ j = 0 p i − 1 λ i , j y j 2 ± y k + ∑ l = 0 k − 2 μ l y l = ∑ i = 1 m ± x i ± y p i + ∑ j = 0 p i − 1 λ i , j y j 2 ± y k + ∑ l = 0 k − 2 μ l y l

Let A i = ± y p i + ∑ j = 0 p i − 1 λ i , j y j and B = ± y k + ∑ l = 0 k − 2 μ l y l . Then, we have

(21) G ˜ = ∑ i = 1 m ± ( x i + A i ) 2 + B .

The second component of the caustics satisfies the following relations:

∂ G ˜ ∂ x 1 = ± 2 ( x 1 + A 1 ) = 0 ∂ G ˜ ∂ x 2 = ± 2 ( x 2 + A 2 ) = 0 ⋮ ⋮ ∂ G ˜ ∂ x m = ± 2 ( x m + A m ) = 0 ∂ G ˜ ∂ y = ∑ i = 1 m ± 2 ( x i + A i ) ∂ A i ∂ y + ∂ B ∂ y = 0

and

x 1 = x 2 = … = x m = 0 ,

which gives the mapping.

Φ : y ∈ R λ ˜ i , j , μ ˜ ↦ μ 1 = ∓ k y k − 1 − ∑ l = 1 k − 3 ( j + 1 ) μ j + 1 y j λ 1 , 0 = ∓ y p 1 − ∑ j = 1 p 1 − 1 λ 1 , j y j λ 2,0 = ∓ y p 2 − ∑ j = 1 p 2 − 1 λ 2 , j y j ⋮ λ m , 0 = ∓ y p m − ∑ j = 1 p m − 1 λ m , j y j λ ˜ i , j μ ˜ ,

where λ ˜ i , j = λ i , j , for which 1 ≤ i ≤ m , 1 ≤ j ≤ p i , and μ ˜ = ( μ 2 , μ 3 , … , μ k − 2 ) .

By left-right equivalence relation, one can reduce Φ to the stable Morin mapping (generalized Whitney umbrella), which is described in Theorem 2.5, and this finishes the proof.□

Corollary 3.12

The caustics of H 2,3 singularity (i.e., when m = 1 , p 1 = 2 , and k = 3 ) is a union of standard Whitney umbrella and a smooth surface tangent to it in R 3 (Figure 7).

$Figure 7 The caustics of H 2,3 {{\mathbb{H}}}_{\mathrm{2,3}} singularity.$

Figure 7

The caustics of H 2,3 singularity.

Acknowledgments

The authors express their sincere gratitude to the anonymous referees for their careful reading of the manuscript and valuable comments.

Funding information: Authors state no funding involved.
Author contributions: Both authors have accepted responsibility for the entire content of this manuscript and approved its submission.
Conflict of interest: The authors state no conflict of interest.
Data availability statement: Data sharing is not applicable to this article as no datasets were generated or analysed during the current study.

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Received: 2024-03-25

Revised: 2025-03-03

Accepted: 2025-08-19

Published Online: 2025-10-04

This work is licensed under the Creative Commons Attribution 4.0 International License.

Articles in the same Issue

https://doi.org/10.1515/dema-2025-0171

Keywords for this article

singularity; submanifold; quasi equivalence; semiborder; m-boundary; Lagrangian projection; bifurcation diagram; caustics; generalized swallowtail; generalized Whitney umbrella; Morin mapping

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