On the Conjecture over Dimensions of Associated Lie Algebra to the Isolated Singularities

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On the Conjecture over Dimensions of Associated Lie Algebra to the Isolated Singularities

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06 February 2024

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06 February 2024

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Abstract

Lie algebra plays an important role in the study of singularity theory and other field of sciences. Finding numerous invariants linked with isolated singularities is always a main interest in the classification theory of isolated singularities. Any Lie algebra that characterizes simple singularity produces a natural question. The study of properties such as to find the dimensions of newly defined algebra is a remarkable work. Hussain, Yau and Zuo \cite{HYZ10} have been found a new class of Lie algebra \texorpdfstring{ $ \mathcal{L}_ k (V)$}{LG}, \texorpdfstring{$k\geq 1$}{LG} i.e., Der ($M_{k}(V), M_{k}(V)$) and purposed a conjecture over its dimension $\delta_{k}(V)$ for $k\geq0$. Later they proved it true for $k$ up to $k=1,2,3,4,5$. In this work, the main concern is whether it's true for a higher value of $k$. According to this, we calculate first, the dimension of Lie algebra \texorpdfstring{$\mathcal{L}_ k (V)$}{LG} for $k=6$ and then compute the upper estimate conjecture of fewnomial isolated singularities. Along with, we also justify the inequality conjecture: \texorpdfstring{$\delta_{k+1}(V) < \delta_{k}(V)$}{LG} for $k=6$. Our calculated results are innovative and a new addition to the study of singularity theory.

Keywords:

Subject: Physical Sciences - Mathematical Physics

1. Introduction

A mathematical structure known as a Lie algebra is made up of vector spaces and a particular binary operation known as the Lie bracket which used to measure the "twist" between two infinitesimal transformations geometrically. Lie algebras are essential to many branches of mathematics and science. In quantum mechanics and particle physics, they are widely used, and in physical phenomena, they arise as symmetry groups. In this regard, Brieskorn established the relationship between Lie algebras and simple singularities. Recent studies have investigated relations between finite-dimensional solvable Lie algebras and isolated hypersurface singularities in complex analysis. In different areas of science and mathematics, singularities have arisen naturally. That’s why singularity theory has become the interlinking path between the various applications of mathematics with its conclusive parts. As an example, it coordinates regular polyhedra theory and simple Lie algebra with the optimal caustics investigations, and relates knot theory to wave fronts of hyperbolic PDE, while it also connects commutative algebra to the theory of solids shape. In most problems of singularity theory, the core aim is to determine the dependence of various physical phenomena while deal with the geometric objects.

The germs of holomorphic functions are widely accepted to be at the origin of

C^{n}

and

O_{n}

. Naturally, the

O_{n}

can be used to identify the algebra of n indeterminate power series. Yau takes into account the Lie algebras derived from the moduli algebra

A (V) : = O_{n} / (g, \frac{\partial g}{\partial y_{1}}, \dots, \frac{\partial g}{\partial y_{n}})

, where

L (V)

define as

Der (A (V), A (V))

and V denote the isolated hypersurface singularity. Lie algebra

L (V)

is a famous solvable finite-dimensional algebra ([1,2,3]). Yau algebra of V is used in singularity theory to distinguish

L (V)

from the other types of Lie algebra ([4,5]). The complex analytical set of isolated hypersurface singularities and the finite set of solvable dimensional Lie algebras (nilpotent) have many new natural connections that Hussain, Yau, Zuo, and their research associates have discovered in recent years ([6,7,8,9,10,11,12]). They have presented three distinct methods for connecting isolated hypersurface singularities to Lie algebra.

These associations are helpful to understand the solvable Lie algebra (nilpotent), from a geometric point of view [8]. A lot of work has been done by Yau and his research collaborates, from 1980s [8,12,13,14,15,16,17,18,19,20,21,22,23].

Let a holomorphic function

g : (C^{n}, 0) \to (C, 0)

associated to the isolated hypersurface singularity

(V, 0)

and its multiplicity denoted as

m u l t (g)

. The

L_{k} (V)

[24] defined as:

Let ideal

J_{k} (g)

generated by

< \frac{\partial^{k} g}{\partial y_{i_{1}} \dots \partial y_{i_{k}}} | 1 \leq i_{1}, \dots, i_{k} \leq n >

and

m u l t (g) = m

1 \leq k \leq m

. Then

M_{k} (V) : = O_{n} / (g + J_{1} (g) + \dots + J_{k} (g))

define the k-th local algebra and

L_{k} (V)

denoted the derivations Lie algebras and its dimension denoted as

δ_{k} (V)

. It is further note that the

L_{k} (V)

is the generalization of Yau algebra. Further detail can be check from ([12,24]).

In [24], the sharp upper estimate and inequality conjectures introduced in following pattern:

Conjecture 1.

[24] Let

δ_{k} ({y_{1}^{α_{1}} + \dots + y_{n}^{α_{n}} = 0}) = h_{k} (α_{1}, \dots, α_{n})

0 \leq k \leq n

and

(V, 0) = {(y_{1}, y_{2}, \dots, y_{n}) \in C^{n} : g (y_{1}, y_{2}, \dots, y_{n}) = 0}, (n \geq 2)

be an isolated singularity with weight type

(w_{1}, w_{2}, \dots, w_{n}; 1)

. Then

δ_{k} (V) \leq h_{k} (1 / w_{1}, \dots, 1 / w_{n})

Conjecture 2.

[24] using the above notations, suppose

(V, 0)

defined by

g \in O_{n}, n \geq 2

. Then

δ_{(k + 1)} (V) < δ_{k} (V), k \geq 1 .

The Conjecture 1 have been proved for binomial and trinomial singularities when

k = 1, 2, 3, 4

([11,17,20,24,25]) and the Conjecture 2 also have been proved for

k = 1, 2, 3

([24,25]).

The main results of this paper is to prove the Conjecture 1 and 2) for binomial and trinomial singularities for particular value of k. The key findings of this paper following as:

Theorem 3.

For

(V, 0) = {(y_{1}, y_{2}, \dots, y_{n}) \in C^{n} : y_{1}^{α_{1}} + \dots + y_{n}^{α_{n}} = 0}, (n \geq 2; α_{l} \geq 7, 1 \leq j \leq n)

δ_{6} (V) = h_{6} (α_{1}, \dots, α_{n}) = \sum_{j = 1}^{n} \frac{α_{j} - 7}{α_{j} - 6} \prod_{l = 1}^{n} (α_{l} - 6) .

Theorem 4.

For binomial singularity

(V, 0)

defined by

g (y_{1}, y_{2})

a weighted homogeneous polynomial with

m u l t (g) \geq 8

and weight type

(w_{1}, w_{2}; 1)

δ_{6} (V) \leq h_{6} (\frac{1}{w_{1}}, \frac{1}{w_{2}}) = \sum_{l = 1}^{2} \frac{\frac{1}{w_{l}} - 7}{\frac{1}{w_{l}} - 6} \prod_{j = 1}^{2} (\frac{1}{w_{j}} - 6) .

Theorem 5.

For binomial singularity

(V, 0)

defined by

g (y_{1}, y_{2})

a weighted homogeneous polynomial with

m u l t (g) \geq 8

and weight type

(w_{1}, w_{2}; 1)

δ_{6} (V) < δ_{5} (V) .

Theorem 6.

For feunomial singularity

(V, 0)

defined by

g (y_{1}, y_{2}, y_{3})

a weighted homogeneous polynomial with

m u l t (g) \geq 8

and weight type

(w_{1}, w_{2}, w_{3}; 1)

, Then

δ_{6} (V) \leq h_{6} (\frac{1}{w_{1}}, \frac{1}{w_{2}}, \frac{1}{w_{3}}) = \sum_{l = 1}^{3} \frac{\frac{1}{w_{l}} - 7}{\frac{1}{w_{l}} - 6} \prod_{j = 1}^{3} (\frac{1}{w_{j}} - 6) .

Theorem 7.

For trinomial singularity

(V, 0)

defined by

g (y_{1}, y_{2}, y_{3})

a weighted homogeneous polynomial with

m u l t (g) \geq 8

and weight type

(w_{1}, w_{2}, w_{3}; 1)

. Then

δ_{6} (V) < δ_{5} (V) .

2. Preliminaries

Proposition 1.

Analytically, the series Type A., Type B., and Type C. are the homogeneous fewnomial Isolated singularity g having mult

(g) \geq 3

Type A.

y_{1}^{α_{1}} + y_{2}^{α_{2}} + \dots + y_{n - 1}^{α_{n - 1}} + y_{n}^{α_{n}}

n \geq 1

Type B.

y_{1}^{α_{1}} y_{2} + y_{2}^{α_{2}} y_{3} + \dots + y_{n - 1}^{α_{n - 1}} y_{n} + y_{n}^{α_{n}}

n \geq 2

Type C.

y_{1}^{α_{1}} y_{2} + y_{2}^{α_{2}} y_{3} + \dots + y_{n - 1}^{α_{n - 1}} y_{n} + y_{n}^{α_{n}} y_{1}

n \geq 2

Corollary 1.

Analytically, the series A, B, and C are the Homogeneous fewnomial Isolated binomial singularities g having mult

(g) \geq 3

y_{1}^{α_{1}} + y_{2}^{α_{2}}

y_{1}^{α_{1}} y_{2} + y_{2}^{α_{2}}

y_{1}^{α_{1}} y_{2} + y_{2}^{α_{2}} y_{1}

Proposition 2. ([28]) Analytically

g (y_{1}, y_{2}, y_{3})

, mult

(g) \geq 3

is,

Type 1.

y_{1}^{α_{1}} + y_{2}^{α_{2}} + y_{3}^{α_{3}}

Type 2.

y_{1}^{α_{1}} y_{2} + y_{2}^{α_{2}} y_{3} + y_{3}^{α_{3}}

Type 3.

y_{1}^{α_{1}} y_{2} + y_{2}^{α_{2}} y_{3} + y_{3}^{α_{3}} y_{1}

Type 4.

y_{1}^{α_{1}} + y_{2}^{α_{2}} + y_{3}^{α_{3}} y_{1}

Type 5.

y_{1}^{α_{1}} y_{2} + y_{2}^{α_{2}} y_{1} + y_{3}^{α_{3}} .

3. Proof of theorems

Before going to the proof of main theorems, first we will prove some propositions.

Proposition 3.

For

(V, 0)

defined by

g = y_{1}^{α_{1}} + y_{2}^{α_{2}} + \dots + y_{n}^{α_{n}}

(

α_{l} \geq 8, l = 1, 2, \dots, n

) with weight type

(\frac{1}{α_{1}}, \frac{1}{α_{2}}, \dots, \frac{1}{α_{n}}; 1)

\begin{matrix} δ_{6} (V) = \sum_{j = 1}^{n} \frac{α_{j} - 7}{α_{j} - 6} \prod_{l = 1}^{n} (α_{l} - 6) . \end{matrix}

Proof.

\prod_{l = 1}^{n} (α_{l} - 6)

be the dimension of moduli algebra

M_{6} (V)

and monomial basis are of the form

\begin{matrix} {y_{1}^{l_{1}} y_{2}^{l_{2}} \dots y_{n}^{l_{n}}, 0 \leq l_{1} \leq α_{1} - 7, 0 \leq l_{2} \leq α_{2} - 7, \dots, 0 \leq l_{n} \leq α_{n} - 7}, \end{matrix}

with following relations:

\begin{matrix} y_{1}^{α_{1} - 6} = 0, y_{2}^{α_{2} - 6} = 0, y_{3}^{α_{3} - 6} = 0, \dots, y_{n}^{α_{n} - 6} = 0 . \end{matrix}

(1)

This implies

D y_{j} = \sum_{l_{1} = 0}^{α_{1} - 7} \sum_{l_{2} = 0}^{α_{2} - 7} \dots \sum_{l_{n} = 0}^{α_{n} - 7} α_{l_{1}, l_{2}, \dots, l_{n}}^{i} y_{1}^{l_{1}} y_{2}^{l_{2}} \dots y_{n}^{l_{n}}, j = 1, 2, \dots, n .

Using 1 we may determine a derivation of

M_{6} (V)

\begin{matrix} α_{0, l_{2}, l_{3},, \dots, l_{n}}^{1} = 0; 0 \leq l_{2} \leq α_{2} - 7, 0 \leq l_{3} \leq α_{3} - 7, \dots, 0 \leq l_{n} \leq α_{n} - 7; \\ α_{l_{1}, 0, l_{3},, \dots, l_{n}}^{2} = 0; 0 \leq l_{1} \leq α_{1} - 7, 0 \leq l_{3} \leq α_{3} - 7, \dots, 0 \leq l_{n} \leq α_{n} - 7; \\ α_{l_{1}, l_{2}, 0, \dots, l_{n}}^{3} = 0; 0 \leq l_{1} \leq α_{1} - 7, 0 \leq l_{2} \leq α_{2} - 7, \dots, 0 \leq l_{n} \leq α_{n} - 7; \\ ⋮ \\ α_{l_{1}, l_{2}, l_{3}, \dots, l_{n - 1}, 0}^{n} = 0; 0 \leq l_{1} \leq α_{1} - 7, 0 \leq l_{2} \leq α_{2} - 7, \dots, 0 \leq l_{n - 1} \leq α_{n - 1} - 7 . \end{matrix}

The derivation basis are of the form

\begin{matrix} y_{1}^{l_{1}} y_{2}^{l_{2}} \dots y_{n}^{l_{n}} \partial_{1}, 1 \leq l_{1} \leq α_{1} - 7, 0 \leq l_{2} \leq α_{2} - 7, 0 \leq l_{3} \leq α_{3} - 7, \dots, 0 \leq l_{n} \leq α_{n} - 7; \\ y_{1}^{l_{1}} y_{2}^{l_{2}} \dots y_{n}^{l_{n}} \partial_{2}, 0 \leq l_{1} \leq α_{1} - 7, 1 \leq l_{2} \leq α_{2} - 7, 0 \leq l_{3} \leq α_{3} - 7, \dots, 0 \leq l_{n} \leq α_{n} - 7; \\ y_{1}^{l_{1}} y_{2}^{l_{2}} \dots y_{n}^{l_{n}} \partial_{3}, 0 \leq l_{1} \leq α_{1} - 7, 0 \leq l_{2} \leq α_{2} - 7, 1 \leq l_{3} \leq α_{3} - 7, 0 \leq l_{4} \leq α_{4} - 7, \\ 0 \leq l_{5} \leq α_{5} - 7, 0 \leq l_{6} \leq α_{6} - 7, \dots, 0 \leq l_{n} \leq α_{n} - 7; \\ ⋮ \\ y_{1}^{l_{1}} y_{2}^{l_{2}} \dots y_{n}^{l_{n}} \partial_{n}, 0 \leq l_{1} \leq α_{1} - 7, 0 \leq l_{2} \leq α_{2} - 7, 0 \leq l_{3} \leq α_{3} - 7, \dots, 1 \leq l_{n} \leq α_{n} - 7 . \end{matrix}

This implies

\begin{matrix} δ_{6} (V) = \sum_{j = 1}^{n} \frac{α_{j} - 7}{α_{j} - 6} \prod_{l = 1}^{n} (α_{l} - 6) . \end{matrix}

□

Remark 1.

For weighted isolated homogeneous fewnomial singularity

(V, 0)

of type A defined by

g = y_{1}^{α_{1}} + y_{2}^{α_{2}}

(

α_{1} \geq 8, α_{2} \geq 8

) with weight type

(\frac{1}{α_{1}}, \frac{1}{α_{2}}; 1)

the Proposition 3 implies

δ_{6} (V) = 2 α_{1} α_{2} - 13 (α_{1} + α_{2}) + 84 .

Proposition 4.

For isolated binomial singularity

(V, 0)

of type B defined by

g = y_{1}^{α_{1}} y_{2} + y_{2}^{α_{2}}

(

α_{1} \geq 7, α_{2} \geq 8

) with weight type

(\frac{α_{2} - 1}{α_{1} α_{2}}, \frac{1}{α_{2}}; 1)

δ_{6} (V) = 2 α_{1} α_{2} - 13 (α_{1} + α_{2}) + 87 .

For

m u l t (g) \geq 8

2 α_{1} α_{2} - 13 (α_{1} + α_{2}) + 87 \leq \frac{2 α_{1} α_{2}^{2}}{α_{2} - 1} - 13 (\frac{α_{1} α_{2}}{α_{2} - 1} + α_{2}) + 84 .

Proof.

α_{1} α_{2} - 6 (α_{1} + α_{2}) + 37

be the dimension of

\begin{matrix} M_{6} (V) & = C {y_{1}, y_{2}} / (g_{y_{1} y_{1} y_{1} y_{1} y_{1}}, g_{y_{2} y_{2} y_{2} y_{2} y_{2}}, g_{y_{1} y_{2} y_{2} y_{2} y_{2}}, g_{y_{1} y_{1} y_{2} y_{2} y_{2}}, g_{y_{1} y_{1} y_{1} y_{2} y_{2}}, g_{y_{1} y_{1} y_{1} y_{1} y_{2}}) \end{matrix}

and monomial basis are of the form

\begin{matrix} {y_{1}^{l_{1}} y_{2}^{l_{2}}, 0 \leq l_{1} \leq α_{1} - 7; 0 \leq l_{2} \leq α_{2} - 7; y_{1}^{α_{1} - 6}} . \end{matrix}

(2)

This implies,

D y_{j} = \sum_{l_{1} = 0}^{α_{1} - 7} \sum_{l_{2} = 0}^{α_{2} - 7} α_{l_{1}, l_{2}}^{i} y_{1}^{l_{1}} y_{2}^{l_{2}} + α_{α_{1} - 6, 0}^{i} y_{1}^{α_{1} - 6}, i = 1, 2 .

The basis of

L_{6} (V)

are

y_{1}^{l_{1}} y_{2}^{l_{2}} \partial_{1}, 1 \leq l_{1} \leq α_{1} - 7, 0 \leq l_{2} \leq α_{2} - 7; y_{1}^{l_{1}} y_{2}^{l_{2}} \partial_{2}, 0 \leq l_{1} \leq α_{1} - 7, 1 \leq l_{2} \leq α_{2} - 7;

y_{2}^{α_{2} - 7} \partial_{1}; y_{1}^{α_{1} - 6} \partial_{1}; y_{1}^{α_{1} - 6} \partial_{2} .

We get following formula

δ_{6} (V) = 2 α_{1} α_{2} - 13 (α_{1} + α_{2}) + 87 .

Finally we need to show that

2 α_{1} α_{2} - 13 (α_{1} + α_{2}) + 87 \leq \frac{2 α_{1} α_{2}^{2}}{α_{2} - 1} - 13 (\frac{α_{1} α_{2}}{α_{2} - 1} + α_{2}) + 84 .

(3)

After solving 3 we have

α_{1} (α_{2} - 10) + α_{2} (α_{1} - 6) + 6 \geq 0

. □

Proposition 5.

For isolated binomial singularity

(V, 0)

of type C defined by

g = y_{1}^{α_{1}} y_{2} + y_{2}^{α_{2}} y_{1}

(

α_{1} \geq 7, α_{2} \geq 7

) with weight type

(\frac{α_{2} - 1}{α_{1} α_{2} - 1}, \frac{α_{1} - 1}{α_{1} α_{2} - 1}; 1)

δ_{6} (V) = \{\begin{matrix} 2 α_{1} α_{2} - 13 (α_{1} + α_{2}) + 90; & α_{1} \geq 8, α_{2} \geq 8 \\ α_{2} - 3; & α_{1} = 7, α_{2} \geq 7 . \end{matrix}

For

m u l t (g) \geq 8

2 α_{1} α_{2} - 13 (α_{1} + α_{2}) + 90 \leq \frac{2 {(α_{1} α_{2} - 1)}^{2}}{(α_{1} - 1) (α_{2} - 1)} - 13 (α_{1} α_{2} - 1) (\frac{α_{1} + α_{2} - 2}{(α_{1} - 1) (α_{2} - 1)}) + 84 .

Proof.

The Moduli algebra has dimenssion

α_{1} α_{2} - 6 (α_{1} + α_{1}) + 38 .

The monomial basis of

\begin{matrix} M_{5} (V) & = C {y_{1}, y_{2}} / (g_{y_{1} y_{1} y_{1} y_{1} y_{1}}, g_{y_{2} y_{2} y_{2} y_{2} y_{2}}, g_{y_{1} y_{2} y_{2} y_{2} y_{2}}, g_{y_{1} y_{1} y_{2} y_{2} y_{2}}, g_{y_{1} y_{1} y_{1} y_{2} y_{2}}, g_{y_{1} y_{1} y_{1} y_{1} y_{2}}) \end{matrix}

are of the form

\begin{matrix} {y_{1}^{l_{1}} y_{2}^{l_{2}}, 0 \leq l_{1} \leq α_{1} - 7; 0 \leq l_{2} \leq α_{2} - 7; y_{1}^{α_{1} - 6}; y_{2}^{α_{2} - 6}} . \end{matrix}

(4)

This implies,

D y_{j} = \sum_{l_{1} = 0}^{α_{1} - 7} \sum_{l_{2} = 0}^{α_{2} - 7} α_{l_{1}, l_{2}}^{i} y_{1}^{l_{1}} y_{2}^{l_{2}} + α_{α_{1} - 6, 0}^{i} y_{1}^{α_{1} - 6} + α_{0, α_{2} - 6}^{i} y_{2}^{α_{2} - 6}, i = 1, 2 .

The derivation basis are of the form

y_{1}^{l_{1}} y_{2}^{l_{2}} \partial_{1}, 1 \leq l_{1} \leq α_{1} - 7, 0 \leq l_{2} \leq α_{2} - 7; y_{1}^{l_{1}} y_{2}^{l_{2}} \partial_{2}, 0 \leq l_{1} \leq α_{1} - 7, 1 \leq l_{2} \leq α_{2} - 7;

y_{2}^{α_{2} - 7} \partial_{1}; y_{2}^{α_{2} - 6} \partial_{1}; y_{1}^{α_{1} - 6} \partial_{1}; y_{2}^{α_{2} - 6} \partial_{2}; y_{1}^{α_{1} - 7} \partial_{2}; y_{1}^{α_{1} - 6} \partial_{2} .

This implies,

δ_{6} (V) = 2 α_{1} α_{2} - 13 (α_{1} + α_{2}) + 90 .

For

α_{1} = 7, α_{2} \geq 7

, we get

L_{6} (V)

y_{2}^{l_{2}} \partial_{2}, 1 \leq l_{2} \leq α_{2} - 6; y_{2}^{α_{2} - 6} \partial_{1}; y_{1} \partial_{1}; y_{1} \partial_{2} .

Next we will prove that

2 α_{1} α_{2} - 13 (α_{1} + α_{2}) + 90 \leq \frac{2 {(α_{1} α_{2} - 1)}^{2}}{(α_{1} - 1) (α_{2} - 1)} - 13 (α_{1} α_{2} - 1) (\frac{α_{1} + α_{2} - 2}{(α_{1} - 1) (α_{2} - 1)}) + 84 .

(5)

After solving 5 we have

α_{1} α_{2}^{2} [(α_{2} - 5) (α_{1} - 5) - α_{1} (α_{2} - 8)] + α_{2}^{3} + 4 α_{1}^{2} α_{2} + 10 α_{2}^{2} (α_{1} - 6) + 6 α_{1} α_{2} (α_{1} - 6)

+ 3 α_{1}^{2} (α_{2} - 6) + α_{1} α_{2} (α_{1} - 6) + 15 α_{1} + 2 (α_{2} - 6) \geq 0 .

Similarly, the conjecture 1 hold true for

α_{1} = 7, α_{2} \geq 7 .

□

Remark 2.

For fewnomial surface isolated singularity

(V, 0)

of the type 1 defined by

g = y_{1}^{α_{1}} + y_{2}^{α_{2}} + y_{3}^{α_{3}}

(

α_{1} \geq 8, α_{2} \geq 8, α_{3} \geq 8

) with weight type

(\frac{1}{α_{1}}, \frac{1}{α_{2}}, \frac{1}{α_{3}}; 1)

. From Proposition 3, we get

δ_{6} (V) = 3 α_{1} α_{2} α_{3} + 120 (α_{1} + α_{2} + α_{3}) - 19 (α_{1} α_{2} + α_{1} α_{3} + α_{2} α_{3}) - 756 .

Proposition 6.

For fewnomial surface isolated singularity

(V, 0)

of the type 2 defined by

g = y_{1}^{α_{1}} y_{2} + y_{2}^{α_{2}} y_{3} + y_{3}^{α_{3}}

(

α_{1} \geq 7, α_{2} \geq 7, α_{3} \geq 8

) with weight type

(\frac{1 - α_{3} + α_{2} α_{3}}{α_{1} α_{2} α_{3}}, \frac{α_{3} - 1}{α_{2} α_{3}}, \frac{1}{α_{3}}; 1)

δ_{6} (V) = \{\begin{matrix} 3 α_{1} α_{2} α_{3} - 19 (α_{1} α_{2} + α_{1} α_{3} + α_{2} α_{3}) + 124 (α_{1} + α_{3}) \\ + 120 α_{2} - 807; & α_{1} \geq 7, α_{2} \geq 8, α_{3} \geq 8 \\ 2 α_{1} α_{3} - 9 α_{1} - 11 α_{3} + 47; & α_{1} \geq 7, α_{2} = 7, α_{3} \geq 8 . \end{matrix}

For

α_{1} \geq 7, α_{2} \geq 8, α_{3} \geq 8

, we need to prove following inequality:

\begin{matrix} 3 α_{1} α_{2} α_{3} - 19 (α_{1} α_{2} + α_{1} α_{3} + α_{2} α_{3}) + 124 (α_{1} + α_{3}) + 120 α_{2} - 807 \leq \frac{3 α_{1} α_{2}^{2} α_{3}^{3}}{(1 - α_{3} + α_{2} α_{3}) (α_{3} - 1)} \\ - 19 (\frac{α_{1} α_{2}^{2} α_{3}^{2}}{(1 - α_{3} + α_{2} α_{3}) (α_{3} - 1)} + \frac{α_{1} α_{2} α_{3}^{2}}{1 - α_{3} + α_{2} α_{3}} + \frac{α_{2} α_{3}^{2}}{α_{3} - 1}) + 120 (\frac{α_{1} α_{2} α_{3}}{1 - α_{3} + α_{2} α_{3}} \\ + \frac{α_{2} α_{3}}{α_{3} - 1} + α_{3}) - 756 . \end{matrix}

Proof.

The Moduli algebra has dimension

α_{1} α_{2} α_{3} - 6 (α_{1} α_{2} + α_{1} α_{3} + α_{2} α_{3}) + 37 (α_{1} + α_{3}) + 36 α_{2} - 228 .

The monomial basis of

M_{6} (V)

are of the form:

\begin{matrix} {y_{1}^{l_{1}} y_{2}^{l_{2}} y_{3}^{l_{3}}, 0 \leq l_{1} \leq α_{1} - 7; 0 \leq l_{2} \leq α_{2} - 7; 0 \leq l_{3} \leq α_{3} - 7; y_{1}^{α_{1} - 6} y_{3}^{l_{3}}, 0 \leq l_{3} \leq α_{3} - 7; \\ y_{1}^{l_{1}} y_{3}^{α_{3} - 6}, 0 \leq l_{1} \leq α_{1} - 7} . \end{matrix}

This implies,

\begin{matrix} D y_{j} = & \sum_{l_{1} = 0}^{α_{1} - 7} \sum_{l_{2} = 0}^{α_{2} - 7} \sum_{l_{3} = 0}^{α_{3} - 7} α_{l_{1}, l_{2}, l_{3}}^{i} y_{1}^{l_{1}} y_{2}^{l_{2}} y_{3}^{l_{3}} + \sum_{l_{1} = 0}^{α_{1} - 7} α_{l_{1}, 0, α_{3} - 6}^{i} y_{1}^{l_{1}} y_{3}^{α_{3} - 6} + \sum_{l_{3} = 0}^{α_{3} - 7} α_{α_{1} - 6, 0, l_{3}}^{i} y_{3}^{l_{3}} y_{1}^{α_{1} - 6}, i = 1, 2, 3 . \end{matrix}

The derivation basis are of the form

\begin{matrix} y_{1}^{l_{1}} y_{2}^{l_{2}} y_{3}^{l_{3}} \partial_{1}, 1 \leq l_{1} \leq α_{1} - 7, 0 \leq l_{2} \leq α_{2} - 7, 0 \leq l_{3} \leq α_{3} - 7; y_{1}^{α_{1} - 6} y_{3}^{l_{3}} \partial_{1}, 0 \leq l_{3} \leq α_{3} - 7, \\ y_{2}^{α_{2} - 7} y_{3}^{l_{3}} \partial_{1}, 1 \leq l_{3} \leq α_{3} - 7; y_{1}^{l_{1}} y_{2}^{α_{2} - 6} \partial_{1}, 0 \leq l_{1} \leq α_{1} - 7, \\ y_{1}^{l_{1}} y_{2}^{l_{2}} y_{3}^{l_{3}} \partial_{2}, 0 \leq l_{1} \leq α_{1} - 7, 1 \leq l_{2} \leq α_{2} - 7, 0 \leq l_{3} \leq α_{3} - 7; y_{1}^{α_{1} - 6} y_{3}^{l_{3}} \partial_{2}, 0 \leq l_{3} \leq α_{3} - 7, \\ y_{1}^{l_{1}} y_{2}^{α_{2} - 6} \partial_{2}, 0 \leq l_{1} \leq α_{1} - 7; y_{1}^{l_{1}} y_{3}^{α_{3} - 7} \partial_{2}, 1 \leq l_{1} \leq α_{1} - 7, \\ y_{1}^{l_{1}} y_{2}^{l_{2}} y_{3}^{l_{3}} \partial_{3}, 0 \leq l_{1} \leq α_{1} - 7, 0 \leq l_{2} \leq α_{2} - 7, 1 \leq l_{3} \leq α_{3} - 7, y_{1}^{l_{1}} y_{2}^{α_{2} - 6} \partial_{3}, 0 \leq l_{1} \leq α_{1} - 7, \\ y_{1}^{α_{1} - 6} y_{3}^{l_{3}} \partial_{3}, 1 \leq l_{3} \leq α_{3} - 7 . \end{matrix}

We get

δ_{6} (V) = 3 α_{1} α_{2} α_{3} - 19 (α_{1} α_{2} + α_{1} α_{3} + α_{2} α_{3}) + 124 (α_{1} + α_{3}) + 120 α_{2} - 807 .

For

α_{1} \geq 7, α_{2} = 7, α_{3} \geq 8,

we get the following basis:

\begin{matrix} y_{1}^{l_{1}} y_{3}^{l_{3}} \partial_{1}, 1 \leq l_{1} \leq α_{1} - 6, 0 \leq l_{3} \leq α_{3} - 7; y_{1}^{l_{1}} y_{2} \partial_{1}, 0 \leq l_{1} \leq α_{1} - 7, \\ y_{1}^{l_{1}} y_{2} \partial_{2}, 0 \leq l_{1} \leq α_{1} - 7; y_{1}^{l_{1}} y_{3}^{α_{3} - 7} \partial_{2}, 1 \leq l_{1} \leq α_{1} - 6, \\ y_{1}^{l_{1}} y_{3}^{l_{3}} \partial_{3}, 0 \leq l_{1} \leq α_{1} - 6, 1 \leq l_{3} \leq α_{3} - 7; y_{1}^{l_{1}} y_{2} \partial_{3}, 0 \leq l_{1} \leq α_{1} - 7 . \end{matrix}

We get

δ_{6} (V) = 2 α_{1} α_{3} - 9 α_{1} - 11 α_{3} + 47 .

For

α_{1} \geq 7, α_{2} \geq 8, α_{3} \geq 8

, we need to prove following inequality:

\begin{matrix} 3 α_{1} α_{2} α_{3} - 19 (α_{1} α_{2} + α_{1} α_{3} + α_{2} α_{3}) + 124 (α_{1} + α_{3}) + 120 α_{2} - 807 \leq \frac{3 α_{1} α_{2}^{2} α_{3}^{3}}{(1 - α_{3} + α_{2} α_{3}) (α_{3} - 1)} \\ - 19 (\frac{α_{1} α_{2}^{2} α_{3}^{2}}{(1 - α_{3} + α_{2} α_{3}) (α_{3} - 1)} + \frac{α_{1} α_{2} α_{3}^{2}}{1 - α_{3} + α_{2} α_{3}} + \frac{α_{2} α_{3}^{2}}{α_{3} - 1}) + 120 (\frac{α_{1} α_{2} α_{3}}{1 - α_{3} + α_{2} α_{3}} \\ + \frac{α_{2} α_{3}}{α_{3} - 1} + α_{3}) - 756 . \end{matrix}

After simplification we get

{(α_{1} - 5)}^{3} (α_{2} - 7) α_{3} + (α_{2} - 6) α_{1} α_{3} ((α_{3} - 5) (α_{1} - 7) + (α_{2} - 5) (α_{3} - 5)) + α_{2} (3 α_{3} - 6) (α_{1} - 5) + α_{2} (α_{1} - 4) + 6 \geq 0 .

Similarly, one can prove that for

α_{1} \geq 7, α_{2} = 7, α_{3} \geq 8

the conjecture 1 hold true. □

Proposition 7.

For isolated fewnomial surface singularity

(V, 0)

of type 3 defined by

g = y_{1}^{α_{1}} y_{2} + y_{2}^{α_{2}} y_{3} + y_{3}^{α_{3}} y_{1}

(

α_{1} \geq 7, α_{2} \geq 7, α_{3} \geq 7

) with weight type

(\frac{1 - α_{3} + α_{2} α_{3}}{1 + α_{1} α_{2} α_{3}}, \frac{1 - α_{1} + α_{1} α_{3}}{1 + α_{1} α_{2} α_{3}}, \frac{1 - α_{2} + α_{1} α_{2}}{1 + α_{1} α_{2} α_{3}}; 1)

δ_{6} (V) = \{\begin{matrix} 3 α_{1} α_{2} α_{3} + 124 (α_{1} + α_{2} + α_{3}) - 19 (α_{1} α_{2} + α_{1} α_{3} + α_{2} α_{3}) \\ - 831; & α_{1} \geq 8, α_{2} \geq 8, α_{3} \geq 8 \\ 2 α_{2} α_{3} - 11 α_{2} - 9 α_{3} + 51; & α_{1} = 7, α_{2} \geq 8, α_{3} \geq 7 \\ 2 α_{1} α_{3} - 9 α_{1} - 11 α_{3} + 51; & α_{1} \geq 7, α_{2} = 7, α_{3} \geq 7 \\ 2 α_{1} α_{2} - 11 α_{1} - 9 α_{2} + 51; & α_{1} \geq 8, α_{2} \geq 8, α_{3} = 7 \end{matrix}

Assuming that

α_{1} \geq 8, α_{2} \geq 8, α_{3} \geq 8

, then we need to prove following inequality:

3 α_{1} α_{2} α_{3} + 124 (α_{1} + α_{2} + α_{3}) - 19 (α_{1} α_{2} + α_{1} α_{3} + α_{2} α_{3}) - 831 \leq \frac{3 {(1 + α_{1} α_{2} α_{3})}^{3}}{(1 - α_{3} + α_{2} α_{3}) (1 - α_{1} + α_{1} α_{3}) (1 - α_{2} + α_{1} α_{2})} + 120 (\frac{1 + α_{1} α_{2} α_{3}}{1 - α_{3} + α_{2} α_{3}} + \frac{1 + α_{1} α_{2} α_{3}}{1 - α_{1} + α_{1} α_{3}} + \frac{1 + α_{1} α_{2} α_{3}}{1 - α_{2} + α_{1} α_{2}}) - 19 (\frac{{(1 + α_{1} α_{2} α_{3})}^{2}}{(1 - α_{3} + α_{2} α_{3}) (1 - α_{1} + α_{1} α_{3})} + \frac{{(1 + α_{1} α_{2} α_{3})}^{2}}{(1 - α_{1} + α_{1} α_{3}) (1 - α_{2} + α_{1} α_{2})} + \frac{{(1 + α_{1} α_{2} α_{3})}^{2}}{(1 - α_{3} + α_{2} α_{3}) (1 - α_{2} + α_{1} α_{2})}) - 756 .

Proof.

(α_{1} α_{2} α_{3} - 6 (α_{1} α_{2} + α_{1} α_{3} + α_{2} α_{3}) + 37 (α_{1} + α_{2} + α_{3}) - 234)

be the dimension of

M_{5} (V)

and monomial basis are of the form

\begin{matrix} {y_{1}^{l_{1}} y_{2}^{l_{2}} y_{3}^{l_{3}}, 0 \leq l_{1} \leq α_{1} - 7; 0 \leq l_{2} \leq α_{2} - 7; 0 \leq l_{3} \leq α_{3} - 7; y_{1}^{α_{1} - 6} y_{3}^{l_{3}}, 0 \leq l_{3} \leq α_{3} - 7; \\ y_{2}^{l_{2}} y_{3}^{α_{3} - 6}, 0 \leq l_{2} \leq α_{2} - 7; y_{1}^{l_{1}} y_{2}^{α_{2} - 6}, 0 \leq l_{1} \leq α_{1} - 7} . \end{matrix}

This implies,

\begin{matrix} D y_{j} = & \sum_{l_{1} = 0}^{α_{1} - 7} \sum_{l_{2} = 0}^{α_{2} - 7} \sum_{l_{3} = 0}^{α_{3} - 7} α_{l_{1}, l_{2}, l_{3}}^{i} y_{1}^{l_{1}} y_{2}^{l_{2}} y_{3}^{l_{3}} + \sum_{l_{1} = 0}^{α_{1} - 7} α_{l_{1}, α_{2} - 6, 0}^{i} y_{1}^{l_{1}} y_{2}^{α_{2} - 6} + \sum_{l_{3} = 0}^{α_{3} - 7} α_{α_{1} - 6, 0, l_{3}}^{i} y_{1}^{α_{1} - 6} y_{3}^{l_{3}} \\ + \sum_{l_{2} = 0}^{α_{2} - 7} α_{0, l_{2}, α_{3} - 6}^{i} y_{2}^{l_{2}} y_{3}^{α_{3} - 6}, i = 1, 2, 3 . \end{matrix}

The derivation basis are of the form

\begin{matrix} y_{1}^{l_{1}} y_{2}^{l_{2}} y_{3}^{l_{3}} \partial_{1}, 1 \leq l_{1} \leq α_{1} - 7, 0 \leq l_{2} \leq α_{2} - 7, 0 \leq l_{3} \leq α_{3} - 7; y_{2}^{l_{2}} y_{3}^{α_{3} - 6} \partial_{1}, 0 \leq l_{2} \leq α_{2} - 7, \\ y_{2}^{α_{2} - 7} y_{3}^{l_{3}} \partial_{1}, 1 \leq l_{3} \leq α_{3} - 7; y_{1}^{l_{1}} y_{2}^{α_{2} - 6} \partial_{1}, 0 \leq l_{1} \leq α_{1} - 7; y_{1}^{α_{1} - 6} y_{3}^{l_{3}} \partial_{1}, 0 \leq l_{3} \leq α_{3} - 7, \\ y_{1}^{l_{1}} y_{2}^{l_{2}} y_{3}^{l_{3}} \partial_{2}, 0 \leq l_{1} \leq α_{1} - 7, 1 \leq l_{2} \leq α_{2} - 7, 0 \leq l_{3} \leq α_{3} - 7; y_{1}^{α_{1} - 6} y_{3}^{l_{3}} \partial_{2}, 0 \leq l_{3} \leq α_{3} - 7, \\ y_{1}^{l_{1}} y_{2}^{α_{2} - 6} \partial_{2}, 0 \leq l_{1} \leq α_{1} - 7; y_{1}^{l_{1}} y_{3}^{α_{3} - 7} \partial_{2}, 1 \leq l_{1} \leq α_{1} - 7; y_{2}^{l_{2}} y_{3}^{α_{3} - 6} \partial_{2}, 0 \leq l_{2} \leq α_{2} - 7, \\ y_{1}^{l_{1}} y_{2}^{l_{2}} y_{3}^{l_{3}} \partial_{3}, 0 \leq l_{1} \leq α_{1} - 7, 0 \leq l_{2} \leq α_{2} - 7, 1 \leq l_{3} \leq α_{3} - 7; y_{1}^{l_{1}} y_{2}^{α_{2} - 6} \partial_{3}, 0 \leq l_{1} \leq α_{1} - 7, \\ y_{1}^{α_{1} - 7} y_{2}^{l_{2}} \partial_{3}, 1 \leq l_{2} \leq α_{2} - 7; y_{2}^{l_{2}} y_{3}^{α_{3} - 6} \partial_{3}, 0 \leq l_{2} \leq α_{2} - 7; y_{1}^{α_{1} - 4} y_{3}^{l_{3}} \partial_{3}, 0 \leq l_{3} \leq α_{3} - 7 . \end{matrix}

Therefore we have

δ_{6} (V) = 3 α_{1} α_{2} α_{3} + 124 (α_{1} + α_{2} + α_{3}) - 19 (α_{1} α_{2} + α_{1} α_{3} + α_{2} α_{3}) - 831 .

In case of

α_{1} = 7, α_{2} \geq 8, α_{3} \geq 7,

we obtain basis as

\begin{matrix} y_{2}^{α_{2} - 7} y_{3}^{l_{3}} \partial_{1}, 1 \leq l_{3} \leq α_{3} - 6; y_{1} y_{3}^{l_{3}} \partial_{1}, 0 \leq l_{3} \leq α_{3} - 7; y_{2}^{α_{2} - 6} \partial_{1}; y_{3}^{α_{3} - 6} \partial_{2}, \\ y_{1} y_{3}^{l_{3}} \partial_{2}, 0 \leq l_{3} \leq α_{3} - 7; y_{2}^{α_{2} - 6} \partial_{2}; y_{2}^{l_{2}} y_{3}^{l_{3}} \partial_{2}, 1 \leq l_{2} \leq α_{2} - 7, 0 \leq l_{3} \leq α_{3} - 6, \\ y_{2}^{l_{2}} y_{3}^{l_{3}} \partial_{3}, 0 \leq l_{2} \leq α_{2} - 7, 1 \leq l_{3} \leq α_{3} - 6, y_{1} y_{3}^{l_{3}} \partial_{3}, 0 \leq l_{3} \leq α_{3} - 7; y_{2}^{α_{2} - 6} \partial_{3} . \end{matrix}

Therefore we have

δ_{6} (V) = 2 α_{2} α_{3} - 11 α_{2} - 9 α_{3} + 51 .

Similarly, we can get bases for

α_{1} \geq 8, α_{2} \geq 8, α_{3} = 7

and

α_{1} \geq 7, α_{2} = 7, α_{3} \geq 7

For

α_{1} \geq 8, α_{2} \geq 8, α_{3} \geq 8,

we need to prove following inequality:

3 α_{1} α_{2} α_{3} + 124 (α_{1} + α_{2} + α_{3}) - 19 (α_{1} α_{2} + α_{1} α_{3} + α_{2} α_{3}) - 851 \leq \frac{3 {(1 + α_{1} α_{2} α_{3})}^{3}}{(1 - α_{3} + α_{2} α_{3}) (1 - α_{1} + α_{1} α_{3}) (1 - α_{2} + α_{1} α_{2})} + 120 (\frac{1 + α_{1} α_{2} α_{3}}{1 - α_{3} + α_{2} α_{3}} + \frac{1 + α_{1} α_{2} α_{3}}{1 - α_{1} + α_{1} α_{3}} + \frac{1 + α_{1} α_{2} α_{3}}{1 - α_{2} + α_{1} α_{2}}) - 19 (\frac{{(1 + α_{1} α_{2} α_{3})}^{2}}{(1 - α_{3} + α_{2} α_{3}) (1 - α_{1} + α_{1} α_{3})} + \frac{{(1 + α_{1} α_{2} α_{3})}^{2}}{(1 - α_{1} + α_{1} α_{3}) (1 - α_{2} + α_{1} α_{2})} + \frac{{(1 + α_{1} α_{2} α_{3})}^{2}}{(1 - α_{3} + α_{2} α_{3}) (1 - α_{2} + α_{1} α_{2})}) - 756 .

After simplification we get

4 (α_{1} α_{2} + α_{2} α_{3} + α_{1} α_{3}) + α_{1} (α_{2} - 7) + α_{2} (α_{3} - 7) + α_{3} (α_{1} - 7) + 4 α_{1}^{2} [α_{2} (α_{3} - 7) + α_{3} (α_{2} - 7)] + 3 α_{2}^{2} [α_{1} (α_{3} - 6) + α_{3} (α_{1} - 7)] + 5 α_{3}^{2} [α_{1} (α_{2} - 7) + α_{2} (α_{1} - 6)] + 2 (α_{1}^{2} + α_{2}^{2} + α_{3}^{2}) + 3 (α_{1}^{3} α_{2} + α_{2}^{3} α_{3} + α_{3}^{3} α_{1}) + 2 α_{1}^{2} α_{2}^{2} α_{3}^{2} + 5 (α_{1} α_{2}^{2} α_{3} + α_{1} α_{2} α_{3}^{2}) + 2 α_{1}^{2} α_{2} α_{3} + α_{1} α_{2} α_{3} [2 α_{1} - 11] + α_{1}^{3} α_{2} α_{3}^{2} (α_{3} - 7) (α_{2} - 7) + α_{1}^{2} α_{3}^{2} (α_{3} - 7) (α_{1} α_{2} - 7) + α_{1}^{2} α_{2} α_{3}^{2} (α_{3} + α_{2} - 8) + 3 α_{1} α_{2} α_{3}^{3} (α_{1} - 7) + α_{1}^{2} α_{2}^{3} α_{3} (α_{3} - 7) (α_{1} - 6) + α_{1}^{2} α_{2}^{2} (α_{1} - 7) (α_{2} a_{3} - 6) + α_{1}^{3} α_{2} α_{3} (α_{2} - 7) + α_{1}^{2} α_{2}^{2} α_{3} (α_{1} - 6 + (α_{3} - 7)) + α_{1} α_{2}^{2} α_{3}^{3} (α_{2} - 7) (α_{1} - 6) + α_{2}^{2} α_{3}^{2} (α_{2} - 7) (α_{1} α_{3} - 7) + 12 \geq 0 .

Similarly, conjecture 1 we can proved for

α_{1}, α_{3} \geq 7, α_{2} = 7; α_{1} \geq 8, α_{2} \geq 8, α_{3} = 7

and

α_{1} = 7, α_{2} \geq 8, α_{3} \geq 7 .

□

Proposition 8.

For isolated fewnomial surface singularity

(V, 0)

of type 4 defined by

g = y_{1}^{α_{1}} + y_{2}^{α_{2}} + y_{3}^{α_{3}} y_{2}

(

α_{1} \geq 8, α_{2} \geq 8, α_{3} \geq 7

) with weight type

(\frac{1}{α_{1}}, \frac{1}{α_{2}}, \frac{α_{2} - 1}{α_{2} α_{3}}; 1)

δ_{6} (V) = 3 α_{1} α_{2} α_{3} + 124 α_{1} + 120 (α_{2} + α_{3}) - 19 (α_{1} α_{2} + α_{1} α_{3} + α_{2} α_{3}) - 781 .

Assuming that

m u l t (g) \geq 8

, we need to prove following inequality

3 α_{1} α_{2} α_{3} + 124 α_{1} + 120 (α_{2} + α_{3}) - 19 (α_{1} α_{2} + α_{1} α_{3} + α_{2} α_{3}) - 781 \leq \frac{3 α_{2}^{2} α_{1} α_{3}}{α_{2} - 1} + 120 (α_{1} + α_{2} + \frac{α_{2} α_{3}}{α_{2} - 1}) - 19 (α_{1} α_{2} + \frac{α_{1} α_{2} α_{3}}{α_{2} - 1} + \frac{α_{2}^{2} α_{3}}{α_{2} - 1}) - 756 .

Proof.

(α_{1} α_{2} α_{3} - 6 (α_{1} α_{2} + α_{1} α_{3} + α_{2} α_{3}) + 36 (α_{2} + α_{3}) + 37 α_{1} - 222)

be the dimension of

M_{6} (V)

and monomial basis are of the form

\begin{matrix} {y_{1}^{l_{1}} y_{2}^{l_{2}} y_{3}^{l_{3}}, 0 \leq l_{1} \leq α_{1} - 7; 0 \leq l_{2} \leq α_{2} - 7; 0 \leq l_{3} \leq α_{3} - 7; y_{1}^{l_{1}} y_{3}^{α_{3} - 6}, 0 \leq l_{2} \leq α_{1} - 7} . \end{matrix}

This implies,

\begin{matrix} D y_{j} = & \sum_{l_{1} = 0}^{α_{1} - 7} \sum_{l_{2} = 0}^{α_{2} - 7} \sum_{l_{3} = 0}^{α_{3} - 7} α_{l_{1}, l_{2}, l_{3}}^{i} y_{1}^{l_{1}} y_{2}^{l_{2}} y_{3}^{l_{3}} + \sum_{l_{1} = 0}^{α_{1} - 7} α_{l_{1}, 0, α_{3} - 6}^{i} y_{1}^{l_{1}} y_{3}^{α_{3} - 6}, i = 1, 2, 3 . \end{matrix}

The derivation basis are of the form

\begin{matrix} y_{1}^{l_{1}} y_{2}^{l_{2}} y_{3}^{l_{3}} \partial_{1}, 1 \leq l_{1} \leq α_{1} - 7, 0 \leq l_{2} \leq α_{2} - 7, 0 \leq l_{3} \leq α_{3} - 7; y_{1}^{l_{1}} y_{3}^{α_{3} - 6} \partial_{1}, 1 \leq l_{1} \leq α_{1} - 7, \\ y_{1}^{l_{1}} y_{2}^{l_{2}} y_{3}^{l_{3}} \partial_{2}, 1 \leq l_{1} \leq α_{1} - 7, 1 \leq l_{2} \leq α_{2} - 7, 0 \leq l_{3} \leq α_{3} - 7; y_{1}^{l_{1}} y_{3}^{α_{3} - 6} \partial_{2}, 0 \leq l_{1} \leq α_{1} - 7, \\ y_{2}^{l_{2}} y_{3}^{l_{3}} \partial_{2}, 1 \leq l_{2} \leq α_{2} - 7, 0 \leq l_{3} \leq α_{3} - 7, y_{1}^{l_{1}} y_{2}^{α_{2} - 7} \partial_{3}, 0 \leq l_{1} \leq α_{1} - 7 \\ y_{1}^{l_{1}} y_{2}^{l_{2}} y_{3}^{l_{3}} \partial_{3}, 0 \leq l_{1} \leq α_{1} - 7, 0 \leq l_{2} \leq α_{2} - 7, 1 \leq l_{3} \leq α_{3} - 7, y_{1}^{l_{1}} y_{3}^{α_{3} - 6} \partial_{3}, 0 \leq l_{1} \leq α_{1} - 7 . \end{matrix}

Therefore we have

δ_{6} (V) = 3 α_{1} α_{2} α_{3} + 124 α_{1} + 120 (α_{2} + α_{3}) - 19 (α_{1} α_{2} + α_{1} α_{3} + α_{2} α_{3}) - 781 .

Next, we also need to show that when

α_{1} \geq 8, α_{2} \geq 8, α_{3} \geq 7

, then

3 α_{1} α_{2} α_{3} + 124 α_{1} + 120 (α_{2} + α_{3}) - 19 (α_{1} α_{2} + α_{1} α_{3} + α_{2} α_{3}) - 781 \leq \frac{3 α_{2}^{2} α_{1} α_{3}}{α_{2} - 1} + 120 (α_{1} + α_{2} + \frac{α_{2} α_{3}}{α_{2} - 1}) - 19 (α_{1} α_{2} + \frac{α_{1} α_{2} α_{3}}{α_{2} - 1} + \frac{α_{2}^{2} α_{3}}{α_{2} - 1}) - 756 .

From above inequality, we get

\frac{α_{1} α_{3} (2 α_{2} - 12)}{α_{2} - 7} + α_{2} α_{3} + α_{3} (α_{2} - 5) + \frac{6 α_{3}}{α_{2} - 6} + \frac{α_{1} [α_{2} (α_{3} - 6) + 7]}{α_{2} - 6} \geq 0 .

□

Proposition 9.

For isolated fewnomial surface singularity

(V, 0)

of type 5 defined by

g = y_{1}^{α_{1}} y_{2} + y_{2}^{α_{2}} y_{1} + y_{3}^{α_{3}}

(

α_{1} \geq 7, α_{2} \geq 7, α_{3} \geq 8

) with weight type

(\frac{α_{2} - 1}{α_{1} α_{2} - 1}, \frac{α_{1} - 1}{α_{1} α_{2} - 1}, \frac{1}{α_{3}}; 1)

δ_{6} (V) = \{\begin{matrix} 3 α_{1} α_{2} α_{3} + 120 (α_{1} + α_{2}) + 128 α_{3} - 19 (α_{1} α_{2} + α_{1} α_{3} + α_{2} α_{3}) \\ - 806; & α_{1} \geq 8, α_{2} \geq 8, α_{3} \geq 8 \\ 2 α_{2} α_{3} - 13 α_{2} - 8 α_{3} + 53; & α_{1} = 7, α_{2} \geq 7, α_{3} \geq 8 \end{matrix}

Assuming that

α_{1} \geq 8, α_{2} \geq 8, α_{3} \geq 8

, then we need to prove following inequality:

3 α_{1} α_{2} α_{3} + 120 (α_{1} + α_{2}) + 128 α_{3} - 19 (α_{1} α_{2} + α_{1} α_{3} + α_{2} α_{3}) - 806 \leq \frac{3 α_{3} {(α_{1} α_{2} - 1)}^{2}}{(α_{2} - 1) (α_{1} - 1)} + 120 (\frac{α_{1} α_{2} - 1}{α_{2} - 1} + \frac{α_{1} α_{2} - 1}{α_{1} - 1} + α_{3}) - 19 (\frac{{(α_{1} α_{2} - 1)}^{2}}{(α_{2} - 1) (α_{1} - 1)} + \frac{α_{3} (α_{1} α_{2} - 1)}{α_{1} - 1} + \frac{α_{3} (α_{1} α_{2} - 1)}{α_{2} - 1}) - 756 .

Proof.

α_{1} α_{2} α_{3} - 6 (α_{1} α_{2} + α_{1} α_{3} + α_{2} α_{3}) + 36 (α_{1} + α_{2}) + 38 α_{3} - 228

be the number of dimension of

M_{6} (V)

and monomial basis are of the form

\begin{matrix} {y_{1}^{l_{1}} y_{2}^{l_{2}} y_{3}^{l_{3}}, 0 \leq l_{1} \leq α_{1} - 7; 0 \leq l_{2} \leq α_{2} - 7; 0 \leq l_{3} \leq α_{3} - 7; y_{1}^{α_{1} - 6} y_{3}^{l_{3}}, 0 \leq l_{3} \leq α_{3} - 7; \\ y_{2}^{α_{2} - 6} y_{3}^{l_{3}}, 0 \leq l_{3} \leq α_{3} - 7}, \end{matrix}

This implies,

\begin{matrix} D y_{j} = & \sum_{l_{1} = 0}^{α_{1} - 7} \sum_{l_{2} = 0}^{α_{2} - 7} \sum_{l_{3} = 0}^{α_{3} - 7} α_{l_{1}, l_{2}, l_{3}}^{i} y_{1}^{l_{1}} y_{2}^{l_{2}} y_{3}^{l_{3}} + \sum_{l_{3} = 0}^{α_{3} - 7} α_{α_{1} - 6, 0, l_{3}}^{i} y_{1}^{α_{1} - 6} y_{3}^{l_{3}} + \sum_{l_{3} = 0}^{α_{3} - 7} α_{0, α_{2} - 6, l_{3}}^{i} y_{2}^{α_{2} - 6} y_{3}^{l_{3}}, i = 1, 2, 3 . \end{matrix}

The derivation basis are of the form

\begin{matrix} y_{1}^{l_{1}} y_{2}^{l_{2}} y_{3}^{l_{3}} \partial_{1}, 1 \leq l_{1} \leq α_{1} - 7, 0 \leq l_{2} \leq α_{2} - 7, 0 \leq l_{3} \leq α_{3} - 7; y_{1}^{α_{1} - 6} y_{3}^{l_{3}} \partial_{1}, 0 \leq l_{3} \leq α_{3} - 7, \\ y_{2}^{α_{2} - 6} y_{3}^{l_{3}} \partial_{1}, 0 \leq l_{3} \leq α_{3} - 7; y_{2}^{α_{2} - 7} y_{3}^{l_{3}} \partial_{1}, 0 \leq l_{3} \leq α_{3} - 7, \\ y_{1}^{l_{1}} y_{2}^{l_{2}} y_{3}^{l_{3}} \partial_{2}, 0 \leq l_{1} \leq α_{1} - 7, 1 \leq l_{2} \leq α_{2} - 7, 0 \leq l_{3} \leq α_{3} - 7; y_{1}^{α_{1} - 6} y_{3}^{l_{3}} \partial_{2}, 0 \leq l_{3} \leq α_{3} - 7, \\ y_{2}^{α_{2} - 6} y_{3}^{l_{3}} \partial_{2}, 0 \leq l_{3} \leq α_{3} - 7; y_{1}^{α_{1} - 7} y_{3}^{l_{3}} \partial_{2}, 0 \leq l_{3} \leq α_{3} - 7, \\ y_{1}^{l_{1}} y_{2}^{l_{2}} y_{3}^{l_{3}} \partial_{3}, 0 \leq l_{1} \leq α_{1} - 7, 0 \leq l_{2} \leq α_{2} - 7, 1 \leq l_{3} \leq α_{3} - 7; y_{1}^{α_{1} - 6} y_{3}^{l_{3}} \partial_{3}, 1 \leq l_{3} \leq α_{3} - 7, \\ y_{2}^{α_{2} - 6} y_{3}^{l_{3}} \partial_{3}, 1 \leq l_{3} \leq α_{3} - 7 . \end{matrix}

Therefore we have

δ_{6} (V) = 3 α_{1} α_{2} α_{3} + 120 (α_{1} + α_{2}) + 128 α_{3} - 19 (α_{1} α_{2} + α_{1} α_{3} + α_{2} α_{3}) - 806 .

For

α_{1} = 7, α_{2} \geq 7, α_{3} \geq 8,

we obtain the following basis:

\begin{matrix} y_{2}^{l_{2}} y_{3}^{l_{3}} \partial_{2}, 1 \leq l_{2} \leq α_{2} - 7, 0 \leq l_{3} \leq α_{3} - 7; y_{2}^{α_{2} - 6} y_{3}^{l_{3}} \partial_{1}, 0 \leq l_{3} \leq α_{3} - 7, \\ y_{1} y_{3}^{l_{3}} \partial_{1}, 0 \leq l_{3} \leq α_{3} - 7; y_{2}^{α_{2} - 6} y_{3}^{l_{3}} \partial_{2}, 0 \leq l_{3} \leq α_{3} - 7, \\ y_{2}^{l_{2}} y_{3}^{l_{3}} \partial_{3}, 0 \leq l_{2} \leq α_{2} - 7, 1 \leq l_{3} \leq α_{3} - 7; y_{1} y_{3}^{l_{3}} \partial_{2}, 0 \leq l_{3} \leq α_{3} - 7, \\ y_{1} y_{3}^{l_{3}} \partial_{3}, 1 \leq l_{3} \leq α_{3} - 7 . \end{matrix}

We have

δ_{6} (V) = 2 α_{2} α_{3} - 13 α_{2} - 8 α_{3} + 53 .

Next, we need to show that when

α_{1} \geq 8, α_{2} \geq 8, α_{3} \geq 8

, then

3 α_{1} α_{2} α_{3} + 120 (α_{1} + α_{2}) + 128 α_{3} - 19 (α_{1} α_{2} + α_{1} α_{3} + α_{2} α_{3}) - 806 \leq \frac{3 α_{3} {(α_{1} α_{2} - 1)}^{2}}{(α_{2} - 1) (α_{1} - 1)} + 120 (\frac{α_{1} α_{2} - 1}{α_{2} - 1} + \frac{α_{1} α_{2} - 1}{α_{1} - 1} + α_{3}) - 19 (\frac{{(α_{1} α_{2} - 1)}^{2}}{(α_{2} - 1) (α_{1} - 1)} + \frac{α_{3} (α_{1} α_{2} - 1)}{α_{1} - 1} + \frac{α_{3} (α_{1} α_{2} - 1)}{α_{2} - 1}) - 756 .

After simplification, we get

\begin{matrix} α_{1} (α_{1} - 7) (α_{2} - 6) (α_{3} + (α_{1} - 5) α_{2} (α_{2} - 7) α_{3}) + α_{1}^{2} (α_{3} - 6) (α_{2} - 5) + α_{2}^{2} α_{1} + 4 α_{1} (α_{2} - 6) \\ + 4 α_{2} (α_{1} - 6) + 4 α_{3} (α_{1} - 5) + 12 α_{1} α_{2} + 14 α_{1} α_{3} + 5 α_{2} α_{3} + 22 α_{2} + α_{1} α_{2} (α_{1} - 6) \\ + (α_{1} - 5) α_{2} (α_{2} - 6) (α_{3} - 5) + (α_{1} - 6) (α_{3} - 7) + 22 \geq 0 . \end{matrix}

Similarly, for

α_{1} = 7, α_{2} \geq 7, α_{3} \geq 8

the conjecture 1 also hold true. □

Proof of Theorem 3.

Proof.

The proof of theorem 3 is the consequence of Proposition 3. □

Proof of Theorem 4.

Proof.

The Proposition 4, 5 and Remark 1 implies Theorem 4 as a corollary. □

Proof of Theorem 5.

Proof.

The Propositions 4, 5, 6 and Remark 3 of [24] along with Remark 1 Propositions 4 and 5 implies that the inequality

δ_{6} (V) < δ_{5} (V)

holds true for binomial singularities. □

Proof of Theorem 6.

Proof.

The Theorem 6 is the consequence of the Remark 2 and Propositions 6, 7, 8, 9. □

Proof of Theorem 7.

Proof.

The Propositions 6, 7, 8, 9 and Remark 4 of [24] along with Remark 2 and Propositions 6, 7, 8, 9 implies that the inequality

δ_{6} (V) < δ_{5} (V)

holds true trinomial singularities. □

4. Conclusion

Finding the dimension of an algebra is critical for studying its applications. This work is the partial proof of the conjecture over dimensions

δ_{k} (V)

k -

th Yau Algebra. First of all we determine general formula fo the dimension of fewnomial isolated singularities for

L_{6} (V)

in Prepositions 3. Then on the behave of this result determine formulas for weighted binomial isolated singularities of all the three types listed in Corollary 1 in Preposition 2 4 and 5. Then Using general formula for dimension

δ_{6} (V)

L_{6} (V)

determine formulas for weighted binomial isolated singularities in Preposition 6, 7, 8, and 9 of all the five types listed in Preposition 2. On the behave of all these findings and dimension of

L_{5} (V)

determind in [29], we proved inequality conjecture:

δ_{6} (V) < δ_{5} (V)

for binomial and trinomial isolated singularities in theorem 3 and 3.

Author Contributions

Conceptualization, N.H., M. A.; methodology, N.H., A.N.A.-K., M. A; validation, A.N.A.-K., N.H., M. A.; writing draft, editing, N.H., A.N.A.-K., M. A; review, N.H., A.N.A.-K., M. A; funding , A.N.A.-K; supervision, N.H.;. All authors have read and agreed to the published version of the manuscript.

Data Availability Statement

Acknowledgments

Conflicts of Interest

The authors don’t have any conflicts of interest.

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On the Conjecture over Dimensions of Associated Lie Algebra to the Isolated Singularities

Abstract

1. Introduction

2. Preliminaries

3. Proof of theorems

4. Conclusion

Author Contributions

Data Availability Statement

Acknowledgments

Conflicts of Interest

References

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