On the Conjecture over Dimensions of Associated Lie Algebra to the Isolated Singularities - Naveed Hussain; Ahmad N. Al-Kenani; Muhammad Asif (Axioms, 2024)
Lie algebra plays an important role in the study of singularity theory and other fields of sciences. Finding numerous invariants linked with isolated singularities has always been a primary interest in the field of 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 have found a new class of Lie algebra <semantics>Lk(V)</semantics>, <semantics>k≥1</semantics>, i.e., Der (<semantics>Mk(V),Mk(V)</semantics>) and proposed a conjecture over its dimension <semantics>δk(V)</semantics> for <semantics>k≥0</semantics>. Later, they proved it true for k up to <semantics>k=1,2,3,4,5</semantics>. In this work, the main concern is whether it is true for a higher value of k. According to this, we first calculate the dimension of Lie algebra <semantics>Lk(V)</semantics> for <semantics>k=6</semantics> and then compute the upper estimate conjecture of fewnomial isolated singularities. Additionally, we also justify the inequality conjecture <semantics>δk+1(V)<δk(V)</semantics> for <semantics>k=6</semantics>. Our calculated results are innovative and serve as a new addition to the study of singularity theory.