{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T00:55:14Z","timestamp":1760144114646,"version":"build-2065373602"},"reference-count":27,"publisher":"MDPI AG","issue":"4","license":[{"start":{"date-parts":[[2024,3,25]],"date-time":"2024-03-25T00:00:00Z","timestamp":1711324800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>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 Lk(V), k\u22651, i.e., Der (Mk(V),Mk(V)) and proposed a conjecture over its dimension \u03b4k(V) for k\u22650. Later, they proved it true for k up to k=1,2,3,4,5. 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 Lk(V) for k=6 and then compute the upper estimate conjecture of fewnomial isolated singularities. Additionally, we also justify the inequality conjecture \u03b4k+1(V)&lt;\u03b4k(V) for k=6. Our calculated results are innovative and serve as a new addition to the study of singularity theory.<\/jats:p>","DOI":"10.3390\/axioms13040216","type":"journal-article","created":{"date-parts":[[2024,3,25]],"date-time":"2024-03-25T12:32:36Z","timestamp":1711369956000},"page":"216","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["On the Conjecture over Dimensions of Associated Lie Algebra to the Isolated Singularities"],"prefix":"10.3390","volume":"13","author":[{"given":"Naveed","family":"Hussain","sequence":"first","affiliation":[{"name":"Department of Mathematics and Statistics, University of Agriculture, Faisalabad 38000, Pakistan"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-5559-5951","authenticated-orcid":false,"given":"Ahmad N.","family":"Al-Kenani","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah 21589, Saudi Arabia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-8669-942X","authenticated-orcid":false,"given":"Muhammad","family":"Asif","sequence":"additional","affiliation":[{"name":"Department of Mathematics, COMSATS University Islamabad, Lahore Campus, Lahore 54000, Pakistan"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2024,3,25]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"253","DOI":"10.1093\/petrology\/11.2.253","article-title":"The biotite isograd and the lower greenschist facies in the Dalradian rocks of Scotland","volume":"11","author":"Mather","year":"1970","journal-title":"J. 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