The Sharp Upper Estimate Conjecture for the Dimension δ k ( V ) of New Derivation Lie Algebra

work. Abstract: Hussain, Yau, and Zuo introduced the Lie algebra L k ( V ) from the derivation of the local algebra M k ( V ) : = O n / ( g + J 1 ( g ) + · · · + J k ( g )) . To ﬁnd the dimension of a newly deﬁned algebra is an important task in order to study its properties. In this regard, we compute the dimension of Lie algebra L 5 ( V ) and justify the sharp upper estimate conjecture for fewnomial isolated singularities. We also verify the inequality conjecture: δ 5 ( V ) < δ 4 ( V ) for a general class of singularities. Our ﬁndings are novel and an addition to the study of Lie algebra.


Introduction
It is commonly known that at the origin of C n , O n are the germs of holomorphic functions. Naturally, the algebra of n indeterminate power series may be identified by the O n . Yau considered the Lie algebras of the derivation of moduli algebra A(V) := O n /(g, ∂g ∂x 1 , · · · , ∂g ∂x n ), where L(V) = Der(A(V), A(V)), and V denotes the isolated hypersurface singularity. L(V) is well recognized as solvable finite dimensional Lie algebra ([1-3]). L(V) distinguished from the other types of Lie algebra present in singularity theory ( [4,5]) is known as the Yau algebra of V [6]. Several new natural connections have been developed in recent years by Hussain, Yau, Zuo, and their research fellows ( [7][8][9][10][11][12]) between the finite set of solvable dimensional Lie algebras (nilpotent) and the complex analytical set of isolated hypersurface singularities. Three different ways have been introduced to associate isolated hypersurface singularities with Lie algebra. From a geometric point of view, these associations support understanding the solvable Lie algebra (nilpotent), [9]. Since the 1980s, Yau and their research fellows have provided much work on singularities [9,[13][14][15][16][17][18][19][20][21][22].
Let a holomorphic function g : (C n , 0) → (C, 0) be defined by the isolated hypersurface singularity (V, 0), with its multiplicity mult(g). mult(g) in the power series expansion is the order of the nonvanishing lowest term of g at o. In [23], the new derivation Lie algebras are defined in the following way: Let J k (g) =< ∂ k g ∂x i 1 ···∂x i k | 1 ≤ i 1 , · · · , i k ≤ n > be an ideal. For mult(g) = m and 1 ≤ k ≤ m, M k (V) := O n /(g + J 1 (g) + · · · + J k (g)) are the new k-th local algebra and L k (V) its new Lie algebras of derivations with dimension δ k (V), which is a new numerical analytic invariant. L k (V) is the generalization of Yau algebra. More details can be found in ( [23]).
The main goal of this study is to confirm Conjecture 1 (resp. Conjecture 2) for binomial and trinomial singularities when k = 5 (resp. k = 4). The following are our key findings.

Preliminaries
Proposition 1.2 of [25] states: Let finite dimension associative algebras A and B have units for the tensor product, (1) The following result is used in this work.
Theorem 7 ([17]). For ideal J in R = C{x 1 , · · · , x n }, The linear endomorphism D of commutative associative algebra A with D(ab) = D(a)b + aD(b) is called a derivation of A. Proposition 1. Analytically, a weighted homogeneous fewnomial singularity g with mult(g) ≥ 3 is equivalent to a linear combination of the series: Corollary 1. Analytically, each binomial isolated singularity is equivalent to one of the three series:

Proof of Theorems
The following propositions will be used to prove the main results of this paper.
Proposition 3. Let (V(g), 0) be an isolated singularity and g = x b 1 1 + x b 2 2 + · · · + x b n n (b j ≥ 7, j = 1, 2, · · · , n) be a weighted homogeneous polynomial with weight type ( 1 b 1 , 1 b 2 , · · · , 1 b n ; 1). Then, Proof. After simple calculation, the moduli algebra M 5 (V) has a monomial basis of the form with the following relations: Without loss of generality, one can write derivation D in terms of the monomial basis in the following way: The sufficient and necessary conditions may be found using the relations (2) to define a derivation of M 5 (V) in following way: . . .
The Lie algebra L 5 (V) has the following basis: This implies ; 1). Then, For mult(g) ≥ 7, we conclude that Proof. After simple calculation, the moduli algebra M 5 (V) defined as has a monomial basis of the form Without loss of generality, one can write derivation D in terms of the monomial basis in the following way: The Lie algebra L 5 (V) has the following basis: We obtain the following formula Finally, we need to show that After For mult(g) ≥ 7, we conclude that ) + 60.
Proof. After simple calculation, the following moduli algebra has a monomial basis of the form Without loss of generality, one can write derivation D in terms of the monomial basis in the following way: The Lie algebra L 5 (V) has the following basis: Therefore, we obtain For b 1 = 6, b 2 ≥ 6, we obtain the following bases of Lie algebra L 5 (V): We also need to show that ) + 60. (6) After solving 6, we have Similarly, we can check that Conjecture 1 holds true for b 1 = 6, b 2 ≥ 6.

Remark 2.
Let (V, 0) be a trinomial singularity of type 1 defined by g = ; 1). Then, from Proposition 3, we obtain Proposition 6. Let (V, 0) be a trinomial singularity of type 2 defined by g = ; 1). Then, For b 1 ≥ 6, b 2 ≥ 7, b 3 ≥ 7, we conclude that: Proof. After simple calculation, the moduli algebra M 5 (V) has the following basis: Without loss of generality, one can write derivation D in terms of the monomial basis in the following way: The Lie algebra L 5 (V) has following basis: We obtain For b 1 ≥ 6, b 2 = 6, b 3 ≥ 7, we obtain the following basis: We obtain For b 1 ≥ 6, b 2 ≥ 7, b 3 ≥ 7, we need to prove following inequality: After solving the above inequality, we obtain Similarly, one can prove that for b 1 ≥ 6, b 2 = 6, b 3 ≥ 7 Conjecture 1 holds true. Then, Proof. The moduli algebra M 5 (V) has the following monomial basis Without loss of generality, one can write derivation D in terms of the monomial basis in the following way: The Lie algebras L 5 (V) have the following bases: Therefore, we have In case of b 1 = 6, b 2 ≥ 7, b 3 ≥ 6, we obtain the following basis: Therefore, we have Similarly, we can obtain bases for b 1 ≥ 7, b 2 ≥ 7, b 3 = 6 and b 1 ≥ 6, b 2 = 6, b 3 ≥ 6. For b 1 ≥ 7, b 2 ≥ 7, b 3 ≥ 7, we need to prove following inequality: After solving the above inequality, we obtain Similarly, we can check that Conjecture 1 holds true for 1) ; 1). Then, For mult(g) ≥ 7, we conclude that: Proof. The moduli algebra M 5 (V) has the following monomial basis Without loss of generality, one can write derivation D in terms of the monomial basis in the following way: The Lie algebras L 5 (V) have the following bases: Therefore, we have Next, we also need to show that when b 1 ≥ 7, b 2 ≥ 7, b 3 ≥ 6, From the above inequality, we obtain Proposition 9. Let (V, 0) be a trinomial singularity of type 5 defined by g = x b 1 1 ; 1). Then, Proof. The moduli algebra M 5 (V) has the following monomial basis Without loss of generality, one can write derivation D in terms of the monomial basis in the following way: The Lie algebras L 5 (V) have the following bases: Therefore, we have For b 1 = 6, b 2 ≥ 6, b 3 ≥ 7, we obtain the following basis: We have δ 5 (V) = 2b 2 b 3 − 11b 2 − 6b 3 + 34.