# Quasi Semi-Border Singularities

^{*}

## Abstract

**:**

## 1. Introduction

**Theorem**

**1.**

Notation | Normal Form | Restrictions | Codimension |

${\mathbb{A}}_{k}$ | ${x}_{2}^{2}\pm {x}_{1}^{k+1}+{\mathbf{x}}^{2}$ | $k\ge 1$ | $m+k$ |

${\mathbb{D}}_{k}$ | ${x}_{1}^{2}{x}_{2}\pm {x}_{2}^{k-1}+{\mathbf{x}}^{2}$ | $k\ge 4$ | $m+k$ |

${\mathbb{E}}_{6}$ | ${x}_{1}^{3}\pm {x}_{2}^{4}+{\mathbf{x}}^{2}$ | – | $m+6$ |

${\mathbb{E}}_{7}$ | ${x}_{1}^{3}+{x}_{2}{x}_{2}^{3}+{\mathbf{x}}^{2}$ | – | $m+7$ |

${\mathbb{E}}_{8}$ | ${x}_{1}^{3}+{x}_{2}^{5}+{\mathbf{x}}^{2}$ | – | $m+8$ |

${\mathbb{H}}_{\mathbf{p},k}$ | $\sum _{i=1}^{m}}\pm {({x}_{i}\pm {y}_{1}^{{p}_{i}})}^{2}\pm {y}_{1}^{k$ | $k>{p}_{m}\ge \cdots \ge {p}_{1}\ge 2$ | $\sum _{i=1}^{m}}{p}_{i}+k-1$ |

$\mathbf{p}=({p}_{1},{p}_{2},\cdots ,{p}_{m})$ |

**Theorem**

**2.**

- 1.
- A pair $(L,\Gamma )$, consisting of a Lagrangian submanifold and a semi-border, is stable if and only if its arbitrary generating family is versal with respect to the quasi semi-border equivalence and up to the addition of functions in parameters.
- 2.
- Let $(L,\Gamma )$ be a pair of a stable and simple Lagrangian submanifold with an m-boundary. Then, it is symplectically equivalent to the projection determined by a generating family that is a quasi m-boundary versal deformation of one of the classes from the above theorem with respect to the quasi m-border equivalence and up to the addition of functions in parameters.

## 2. Pseudo and Quasi Semi-Border Equivalence Relations

**Definition**

**1.**

**Remark**

**1.**

- 1.
- The general statements below are valid for certain types of varieties. In particular, the variety Γ should be a stratified set, satisfying the Whitney Condition A. In addition, it is assumed in Definition 1 that if a critical point c is contained in a certain stratum of Γ, then its image $\theta \left(c\right)$ must be contained in the same stratum. This means that the diffeomorphism does not need to preserve the variety, but it needs to shift the critical points along the components of the variety.
- 2.
- The pseudo semi-border equivalence will be just called the pseudo border if the variety is a hypersurface.
- 3.
- In contrast to the standard equivalence relation, the pseudo semi-border equivalence is not a group action due to the constraints on the set of admissible diffeomorphisms.

**Definition**

**2.**

**Remark**

**2.**

**Lemma**

**1.**

**Remark**

**3.**

- 1.
- The details of the proof of Lemma 1 can be found in [6].
- 2.
- If for a given function $\frac{\partial {f}_{t}}{\partial t}$ in Equation (1), we can find a decomposition in the right side form, then the vector field ${v}_{t}$ in the $(w,t)$-space can be integrated to generate the phase flow ${\theta}_{t}$. Of course, we need to be sure that the germs of diffeomorphisms are defined on some neighborhood of the base point. This is usually achieved if the vector field vanishes at the base point. This method is called Moser’s homotopy method.
- 3.
- 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 $Rad\left\{{I}_{t}\right\}$ by the ideal ${I}_{t}$ in the equivalence definition. The new relation, described explicitly in the following definition, will be called the quasi semi-border equivalence.

**Definition**

**3.**

- 1.
- ${\theta}_{0}=id$,
- 2.
- ${f}_{t}\circ {\theta}_{t}={f}_{0}$, and
- 3.
- the components of the vector field ${v}_{t}$ generating ${\theta}_{t}$ on each segment of smoothness have the forms:$${\dot{X}}_{i}\left(t\right)={\dot{x}}_{i}+{I}_{t},\phantom{\rule{2.em}{0ex}}{\dot{Y}}_{j}\left(t\right)={\dot{y}}_{i}\in {\mathbf{C}}_{w}.$$

**Remark**

**4.**

- 1.
- Each one of the vector fields ${v}_{t}$ and their phase flow ${\theta}_{t}$ will be called admissible with respect to the family ${f}_{t}$.
- 2.
- The tangent space to the quasi semi-border equivalence class of f, denoted by $TQ{\Gamma}_{f}$, has the following description:$$TQ{\Gamma}_{f}=\left(\right)open="\{"\; close="\}">\sum _{i=1}^{m}\frac{\partial f}{\partial {x}_{i}}\left(\right)open="("\; close=")">{\dot{x}}_{i}+\sum _{k=1}^{n}\frac{\partial f}{\partial {w}_{k}}{A}_{i,k}.$$

**Property**

**1.**

**Property**

**2.**

**Definition**

**4.**

#### 2.1. Basic Techniques of the Classification

**Lemma**

**2.**

- 1.
- There is an admissible vector field $\dot{w}={\displaystyle \sum _{i=1}^{m}}{\dot{X}}_{i}\frac{\partial}{\partial {x}_{i}}+\sum _{j=1}^{n-m}{\dot{Y}}_{j}\frac{\partial}{\partial {y}_{j}}$ for ${f}_{0}$, that is,$${\dot{X}}_{i}={\dot{x}}_{i}+\sum _{k=1}^{n}\frac{\partial f}{\partial {w}_{k}}{A}_{i,k}\phantom{\rule{1.em}{0ex}}and\phantom{\rule{1.em}{0ex}}{\dot{Y}}_{j}={\dot{y}}_{i}$$$$\phi =\sum _{i=1}^{m}\frac{\partial {f}_{0}}{\partial {x}_{i}}{\dot{X}}_{i}+\sum _{j=1}^{n-m}\frac{\partial {f}_{0}}{\partial {y}_{i}}{\dot{Y}}_{i}+\widehat{\phi}+\sum _{i=1}^{s}{c}_{i}{e}_{i}\left(w\right),$$
- 2.
- Moreover, for any $\delta ,$$N<\delta <\gamma ,$ and any $\psi \in {S}_{\delta}$, the expression:$$E(\psi ,\phi )=\sum _{i=1}^{m}\frac{\partial \psi}{\partial {x}_{i}}\left(\right)open="["\; close="]">{\dot{X}}_{i}+\sum _{k=1}^{n}{A}_{i,k}\frac{\partial \psi}{\partial {w}_{k}}+\sum _{j=1}^{n-m}\frac{\partial \psi}{\partial {y}_{j}}{\dot{Y}}_{j}$$Then, any germ $f={f}_{0}+{f}_{*}$ is quasi semi-border equivalent to a germ ${f}_{0}+{\displaystyle \sum _{i=1}^{s}}{a}_{i}{e}_{i},$ where ${a}_{i}\in \mathbb{R}$.

**Remark**

**5.**

#### 2.2. Prenormal Forms of Quasi Classes

**Lemma**

**3.**

**Proof.**

**Lemma**

**4.**

**Lemma**

**5.**

**Lemma**

**6.**

**Remark**

**6.**

**Lemma**

**7.**

$\mathit{\kappa}$ | Notation | Normal Form | Restrictions |

0 | ${F}_{00}$ | $\sum _{i=1}^{m}\pm {x}_{i}^{2}$ | |

1 | ${F}_{10}$ | ${\tilde{f}}_{2}\left(x\right)+{\tilde{f}}_{3}\left(x\right)$ | $rank\left(\right)open="("\; close=")">{d}_{0}^{2}{\tilde{f}}_{2}$, ${\tilde{f}}_{3}\in {\mathcal{M}}_{x}^{3}$ |

${F}_{11}$ | $\sum _{i=1}^{m}\pm {x}_{i}^{2}+{\tilde{f}}_{3}(x,{y}_{1})$ | ${\tilde{f}}_{3}\in {\mathcal{M}}_{x,{y}_{1}}^{3}$, ${y}_{1}\in \mathbb{R}$ | |

$\ge 2$ | ${F}_{\kappa 0}$ | ${\tilde{f}}_{2}\left(x\right)+{\tilde{f}}_{3}$ | $rank\left(\right)open="("\; close=")">{d}_{0}^{2}{\tilde{f}}_{2}$, ${\tilde{f}}_{3}\in {\mathcal{M}}_{x}^{3}$ |

${F}_{\kappa (\mu -1)}$ | ${\tilde{f}}_{2}\left(x\right)+{\tilde{f}}_{3}(x,\tilde{y})$ | $rank\left(\right)open="("\; close=")">{d}_{0}^{2}{\tilde{f}}_{2}$, ${\tilde{f}}_{3}\in {\mathcal{M}}_{x,\tilde{y}}^{3}$, | |

$\tilde{y}=({y}_{1},{y}_{2},\cdots ,{y}_{\mu -1})$ | |||

${F}_{\kappa \kappa}$ | $\sum _{i=1}^{m}\pm {x}_{i}^{2}+{\tilde{f}}_{3}(x,\tilde{y})$ | ${\tilde{f}}_{3}\in {\mathcal{M}}_{x,\tilde{y}}^{3}$, $\tilde{y}=({y}_{1},{y}_{2},\cdots ,{y}_{\kappa})$ |

## 3. Classifications of Simple Functions

**Remark**

**7.**

- 1.
- The case when the semi-border is a smooth hypersurface ($\Gamma {S}_{1}$) was discussed in [1].
- 2.
- The quasi semi-border equivalence will be also called the quasi m-boundary for the respective type of $\Gamma {S}_{m}$.
- 3.
- All results are given for permutations of ${x}_{1},{x}_{2},\cdots $ and ${x}_{m}$.

#### 3.1. Simple Quasi Boundary

**Theorem**

**3.**

Notation | Normal Form | Restrictions | Codimension |

${B}_{k}$ | $\pm {y}_{1}^{2}\pm {x}_{1}^{k}$ | $k\ge 2$ | k |

${F}_{p,k}$ | $\pm {({x}_{1}\pm {y}_{1}^{p})}^{2}\pm {y}_{1}^{k}$ | $k>p\ge 2$ | $p+k-1$ |

**Remark**

**8.**

- 1.
- The classes ${B}_{k}$ can be written in the form $\pm {({x}_{1}\pm {y}_{1})}^{2}\pm {y}_{1}^{k}$, and hence, it can be included in the series ${F}_{p,k}$ as ${F}_{1,k}.$
- 2.
- Corank one germs are either simple, hence they are quasi boundary equivalent to one of the germs in the above theorem, or they belong to a subset of infinite codimension in the space of all germs.
- 3.
- All germs with the quadratic part of corank greater than one are non-simple.
- 4.
- The uni-modal class ${\tilde{S}}_{5}:{y}_{1}^{3}+{x}_{1}^{3}+a{x}_{1}^{2}{y}_{1},$$a\in \mathbb{R}\backslash \left\{\frac{-3}{\sqrt[3]{4}}\right\}$ is the only fencing singularity that is adjacent to ${F}_{2,3}$.
- 5.
- The graph of low codimension adjacencies is as follows:$$\begin{array}{ccccccccccc}{B}_{2}& \leftarrow & {B}_{3}& \leftarrow & {B}_{4}& \leftarrow & {B}_{5}& \leftarrow & {B}_{6}& \leftarrow & \dots \\ & & \uparrow & & \uparrow & & \uparrow & & \uparrow & & \\ & & {F}_{2,3}& \leftarrow & {F}_{2,4}& \leftarrow & {F}_{2,5}& \leftarrow & {F}_{2,6}& \leftarrow & \dots \\ & & \uparrow & & \uparrow & & \uparrow & & \uparrow & & \\ & & {\tilde{S}}_{5}& & {F}_{3,4}& \leftarrow & {F}_{3,5}& \leftarrow & {F}_{3,6}& \leftarrow & \dots \\ & & & & & & \uparrow & & \uparrow & & \\ & & & & & & \dots & & \dots & \dots & \end{array}$$

#### 3.2. Simple Quasi m-Boundary Classes

**Theorem**

**4.**

Notation | Normal Form | Restrictions | Codimension |

${\mathbb{A}}_{k}$ | ${x}_{2}^{2}\pm {x}_{1}^{k+1}+{\mathbf{x}}^{2}$ | $k\ge 1$ | $m+k$ |

${\mathbb{D}}_{k}$ | ${x}_{1}^{2}{x}_{2}\pm {x}_{2}^{k-1}+{\mathbf{x}}^{2}$ | $k\ge 4$ | $m+k$ |

${\mathbb{E}}_{6}$ | ${x}_{1}^{3}\pm {x}_{2}^{4}+{\mathbf{x}}^{2}$ | – | $m+6$ |

${\mathbb{E}}_{7}$ | ${x}_{1}^{3}+{x}_{2}{x}_{2}^{3}+{\mathbf{x}}^{2}$ | – | $m+7$ |

${\mathbb{E}}_{8}$ | ${x}_{1}^{3}+{x}_{2}^{5}+{\mathbf{x}}^{2}$ | – | $m+8$ |

${\mathbb{H}}_{\mathbf{p},k}$ | $\sum _{i=1}^{m}}\pm {({x}_{i}\pm {y}_{1}^{{p}_{i}})}^{2}\pm {y}_{1}^{k$ | $k>{p}_{m}\ge \cdots \ge {p}_{1}\ge 2$ | $\sum _{i=1}^{m}}{p}_{i}+k-1$ |

$\mathbf{p}=({p}_{1},{p}_{2},\cdots ,{p}_{m})$ |

**Remark**

**9.**

- 1.
- In Theorem 4, ${\mathbf{x}}^{2}={\displaystyle \sum _{i=3}^{m}}\pm {x}_{i}^{2}$.
- 2.
- The classes ${\mathbb{A}}_{k}$ can be written equivalently in the forms $\sum _{i=1}^{m}}\pm {({x}_{i}\pm {y}_{1})}^{2}\pm {y}_{1}^{k+1$, and hence, they can be included in the series ${\mathbb{H}}_{\mathbf{p},k+1}$ as ${\mathbb{H}}_{{1}_{m,k+1}}$, where ${1}_{m}=(1,1,\cdots ,1)$.
- 3.
- The graph of adjacencies of simple quasi m-boundary classes in low codimension is as follows:$$\begin{array}{ccccccccccc}{\mathbb{A}}_{1}& \leftarrow & {\mathbb{A}}_{2}& \leftarrow & {\mathbb{A}}_{3}& \leftarrow & {\mathbb{A}}_{4}& \leftarrow & {\mathbb{A}}_{5}& \leftarrow & \dots \\ & & & & \uparrow & & & & & & \\ & & & & {\mathbb{D}}_{4}& \leftarrow & {\mathbb{D}}_{5}& \leftarrow & {\mathbb{D}}_{4}& \leftarrow & \dots \\ & & & & \uparrow & & & & & & \\ & & & & {\mathbb{E}}_{6}& & & & & & \\ & & & & \uparrow & & & & & & \\ & & & & {\mathbb{E}}_{7}& & & & & & \\ & & & & \uparrow & & & & & & \\ & & & & {\mathbb{E}}_{8}& & & & & & \end{array}$$$$\begin{array}{ccccc}{\mathbb{A}}_{k}& \leftarrow & {\mathbb{H}}_{\mathbf{p},k+1}& \leftarrow & {\mathbb{H}}_{\mathbf{p},k+2}\\ & & \uparrow & & \\ & & {\mathbb{H}}_{\mathbf{p}+1,k+1}& & \end{array}$$$$\begin{array}{ccc}{\mathbb{D}}_{4}& & \\ \uparrow & & \\ {\mathbb{E}}_{6}& \leftarrow & {\mathbb{P}}_{8}\\ \uparrow & & \\ {\mathbb{E}}_{7}& \leftarrow & {\mathbb{X}}_{9}\\ \uparrow & & \\ {\mathbb{E}}_{8}& \leftarrow & {\mathbb{J}}_{10}\end{array}$$
- 4.
- The fencing class with respect to the quasi two-boundary equivalence relation is a stabilization of the following:
**Notation****Class****Restrictions****Codimension**${\mathbb{P}}_{8}$ ${x}_{1}^{3}+{x}_{2}^{3}+{y}_{1}^{3}+a{x}_{1}{x}_{2}{y}_{1}$ ${a}^{3}+27\ne 0$ 10 ${\mathbb{X}}_{9}$ ${x}_{1}^{4}+{x}_{2}^{4}+a{x}_{1}^{2}{x}_{2}^{2}$ ${a}^{2}\ne 4$ 11 ${\mathbb{J}}_{10}$ ${x}_{1}^{3}+{x}_{2}^{6}+a{x}_{1}{x}_{2}$ $4{a}^{3}+27\ne 0$ 12

**Property**

**3.**

**Proof.**

**Corollary**

**1.**

**Lemma**

**8.**

**Proof.**

**Lemma**

**9.**

**Proof.**

**Lemma**

**10.**

**Proof.**

#### Proof of Theorem 4

- If f is a function germ in x only, then Lemma 8 implies that f is quasi m-boundary equivalent to $g({x}_{m},{x}_{m-1})+\mathbf{x}$, where $\mathbf{x}=\sum _{i=1}^{m-2}\pm {x}_{i}^{2}$, and g belongs to one of Arnold’s simple classes $\left(\mathbb{A}\mathbb{D}\mathbb{E}\right)$.
- Let $n>m$. Then, Lemma 7 and Lemma 10 imply that f is simple if ${f}_{2}$ is the non-degenerate quadratic form and $\kappa =1$, in which case, by Lemma 9, f is stably quasi m-boundary equivalent to $\tilde{f}(x,{y}_{1})=\sum _{i=1}^{m}\left(\right)open="["\; close="]">\pm {x}_{i}^{2}+{x}_{i}{\varphi}_{i}\left({y}_{1}\right)+\varphi \left({y}_{1}\right),$ where ${\varphi}_{i}\in {\mathcal{M}}_{{y}_{1}}^{2}$ and $\varphi \in {\mathcal{M}}_{{y}_{1}}^{3}$. Considering the lowest non-zero terms in ${\varphi}_{i}$ and $\varphi $, and using Lemma 2, one can easily show that $\tilde{f}$ is reduced to one of the following forms:
- $\sum _{i=1}^{m}\pm {x}_{i}^{2}\pm {y}_{1}^{k}$, $k\ge 3,$ which can be written equivalently as:$${\mathbb{H}}_{\mathbf{p},k}:\sum _{i=1}^{m}\pm {({x}_{i}\pm {y}_{1}^{k-1})}^{2}\pm {y}_{1}^{k},\phantom{\rule{3.33333pt}{0ex}}\mathrm{where}\phantom{\rule{3.33333pt}{0ex}}\mathbf{p}=({p}_{1},{p}_{2},\cdots ,{p}_{m}),{p}_{i}=k-1,\phantom{\rule{3.33333pt}{0ex}}\mathrm{for}\text{}\mathrm{all}\phantom{\rule{3.33333pt}{0ex}}i.$$
- $\sum _{i=1}^{m}\left(\right)open="("\; close=")">\pm {x}_{i}^{2}\pm {x}_{i}{y}_{1}^{k}$, $k\ge 2,$ which is also quasi m-boundary equivalent to:$${\mathbb{H}}_{\mathbf{p},2k}:\sum _{i=1}^{m}\pm {({x}_{i}\pm {y}_{1}^{k})}^{2}\pm {y}_{1}^{2k},\phantom{\rule{3.33333pt}{0ex}}\mathrm{where}\phantom{\rule{3.33333pt}{0ex}}\mathbf{p}=({p}_{1},{p}_{2},\cdots ,{p}_{m}),{p}_{i}=k,\phantom{\rule{3.33333pt}{0ex}}\mathrm{for}\text{}\mathrm{all}\phantom{\rule{3.33333pt}{0ex}}i.$$
- $\sum _{i=1}^{m}\pm {x}_{i}^{2}+\sum _{i=1}^{s}\pm {x}_{i}{y}_{1}^{k}+\sum _{i=s+1}^{m}\pm {x}_{i}{y}_{1}^{{p}_{i}}$, $2\le k<{p}_{s+1}\le {p}_{s+2}\le \cdots \le {p}_{m}<2k$, which can be written equivalently as:$${\mathbb{H}}_{\mathbf{p},2k}:\sum _{i=1}^{m}\pm {({x}_{i}\pm {y}_{1}^{{p}_{i}})}^{2}\pm {y}_{1}^{2k},\phantom{\rule{3.33333pt}{0ex}}\mathrm{where}\phantom{\rule{3.33333pt}{0ex}}\mathbf{p}=({p}_{1},{p}_{2},\cdots ,{p}_{m}),{p}_{i}=k,\phantom{\rule{3.33333pt}{0ex}}i=1,2,\cdots ,s.$$
- $\sum _{i=1}^{m}\left(\right)open="("\; close=")">\pm {x}_{i}^{2}\pm {x}_{i}{y}_{1}^{{p}_{i}}\pm {y}_{1}^{k}$, k-odd with $k\ge 5$ and $\frac{k-1}{2}<{p}_{1}\le {p}_{2}\le \cdots \le {p}_{m}<k-1$ or equivalently to:$${\mathbb{H}}_{\mathbf{p},k}:\sum _{i=1}^{m}\pm {({x}_{i}\pm {y}_{1}^{{p}_{i}})}^{2}\pm {y}_{1}^{k},\phantom{\rule{3.33333pt}{0ex}}\mathrm{where}\phantom{\rule{3.33333pt}{0ex}}\mathbf{p}=({p}_{1},{p}_{2},\cdots ,{p}_{m}).$$A general formula for all simple quasi m-boundary classes contained in $\tilde{f}$ is:$${\mathbb{H}}_{\mathbf{p},k}:\sum _{i=1}^{m}\pm {({x}_{i}\pm {y}_{1}^{{p}_{i}})}^{2}\pm {y}_{1}^{k},$$

## 4. Application to Lagrangian Semi-Border Singularities

#### 4.1. Versal Deformations of m-Boundary Classes

**Property**

**4.**

Singularity | Miniversal Deformation | Restrictions |

${\mathbb{A}}_{k}$ | $\pm {x}_{1}^{k+1}\pm {x}_{2}^{2}+{\mathbf{x}}^{2}+{\lambda}_{0}+{\displaystyle \sum _{i=1}^{m}}{\lambda}_{i}{x}_{i}+{\displaystyle \sum _{j=2}^{k}}{\mu}_{j}{x}_{1}^{j}$ | $k\ge 1$ |

${\mathbb{D}}_{k}$ | ${x}_{1}^{2}{x}_{2}\pm {x}_{2}^{k-1}+{\mathbf{x}}^{2}+{\lambda}_{0}+{\displaystyle \sum _{i=1}^{m}}{\lambda}_{i}{x}_{i}+{\mu}_{1}{x}_{1}^{2}+{\mu}_{2}{x}_{2}{x}_{1}+{\displaystyle \sum _{j=2}^{k-2}}{\mu}_{j}{x}_{2}^{j}$ | $k\ge 4$ |

${\mathbb{E}}_{6}$ | ${x}_{1}^{3}\pm {x}_{2}^{4}+{\mathbf{x}}^{2}+{\lambda}_{0}+{\displaystyle \sum _{i=1}^{m}}{\lambda}_{i}{x}_{i}+{\mu}_{0}{x}_{1}^{2}+{x}_{1}\left(\right)open="("\; close=")">{\displaystyle \sum _{j=1}^{2}}{\mu}_{j}{x}_{2}^{j}$ | – |

${\mathbb{E}}_{7}$ | ${x}_{1}^{3}+{x}_{1}{x}_{2}^{3}+{\mathbf{x}}^{2}+{\lambda}_{0}+{\displaystyle \sum _{i=1}^{m}}{\lambda}_{i}{x}_{i}+{\displaystyle \sum _{j=1}^{2}}{\mu}_{j}{x}_{1}^{j}{x}_{2}^{2-j}+{\displaystyle \sum _{j=3}^{5}}{\mu}_{j}{x}_{2}^{j-1}$ | – |

${\mathbb{E}}_{8}$ | ${x}_{1}^{3}+{x}_{2}^{5}+{\mathbf{x}}^{2}+{\lambda}_{0}+{\displaystyle \sum _{i=1}^{m}}{\lambda}_{i}{x}_{i}+{\mu}_{0}{x}_{1}^{2}+{x}_{1}\left(\right)open="("\; close=")">{\displaystyle \sum _{j=1}^{3}}{\mu}_{j}{x}_{2}^{j}$ | – |

${\mathbb{H}}_{\mathbf{p},k}$ | $\sum _{i=1}^{m}}\pm {({x}_{i}\pm {y}_{1}^{{p}_{i}})}^{2}\pm {y}_{1}^{k}+{\lambda}_{0}+{\displaystyle \sum _{i=1}^{m}}{x}_{i}\left({\displaystyle \sum _{j=0}^{{p}_{i}-1}}{\lambda}_{i,j}{y}_{1}^{j}\right)+{\displaystyle \sum _{l=1}^{k-2}}{\mu}_{l}{y}_{l}^{l$ | $k>{p}_{m}\ge \cdots \ge {p}_{1}\ge 2$ |

#### 4.2. Lagrangian Submanifolds with Smooth Varieties

**Definition**

**5.**

**Definition**

**6.**

**Definition**

**7.**

**Property**

**5.**

**quasi semi-border**equivalent, up to the addition of functions that depend on the parameters.

**Theorem**

**5.**

- 1.
- A pair $(L,\Gamma )$, consisting of a Lagrangian submanifold and a semi-border, is stable if and only if its arbitrary generating family is versal with respect to the quasi semi-border equivalence and up to the addition of functions in the parameters.
- 2.
- Let $(L,\Gamma )$ be a pair of stable and simple Lagrangian submanifolds with the m-boundary. Then, it is symplectically equivalent to the projection determined by a generating family that is a quasi m-boundary versal deformation, listed in Proposition 4, and up to the addition of functions in parameters.

**Proof.**

## 5. Conclusions

**example**of a Lagrangian submanifold with a regular boundary or a corner is presented by a set of Hamilton vector field trajectories issued from an initial set being an isotropic submanifold subset determined by some inequalities. This construction is needed for various settings in geometry and physics. For example, given an initial hypersurface H with a boundary ${H}_{1}$ in Euclidean space, the envelope of the family of normals to H forms the ordinary caustics and the union of normals to H at the points of ${H}_{1}$ forms the second component of the caustic of the projection of the respective Lagrange submanifold with a boundary. Other motivations to study the singularities of Lagrange projections with boundaries were mentioned in [7]. More complicated semi-borders appear in various applications in physics. For example, the Lagrangian manifold with a corner is the solution of the Hamilton–Jacobi equation with the initial data embedded into the cotangent bundle of the configuration space as a manifold with a boundary or a corner [4].

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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Alharbi, F.; Alsaeed, S.
Quasi Semi-Border Singularities. *Mathematics* **2019**, *7*, 495.
https://doi.org/10.3390/math7060495

**AMA Style**

Alharbi F, Alsaeed S.
Quasi Semi-Border Singularities. *Mathematics*. 2019; 7(6):495.
https://doi.org/10.3390/math7060495

**Chicago/Turabian Style**

Alharbi, Fawaz, and Suliman Alsaeed.
2019. "Quasi Semi-Border Singularities" *Mathematics* 7, no. 6: 495.
https://doi.org/10.3390/math7060495