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Root multiplicities are adaptively estimated within the iterative process, avoiding additional function evaluations beyond those required for parallel updates. The robustness and stability of the proposed methods are assessed using both random and distant initial guesses and validated on benchmark polynomials as well as nonlinear models from biomedical engineering. The numerical results show notable improvements in residual error, iteration count, CPU time, memory usage, and overall convergence rate compared with established classical techniques. These findings demonstrate that the proposed schemes provide reliable, high-order, and computationally efficient tools for solving challenging nonlinear problems in science and engineering.<\/jats:p>","DOI":"10.3390\/a19010021","type":"journal-article","created":{"date-parts":[[2025,12,24]],"date-time":"2025-12-24T14:27:51Z","timestamp":1766586471000},"page":"21","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Adaptive Parallel Methods for Polynomial Equations with Unknown Multiplicity"],"prefix":"10.3390","volume":"19","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-2980-5801","authenticated-orcid":false,"given":"Mudassir","family":"Shams","sequence":"first","affiliation":[{"name":"Department of Mathematics, Faculty of Arts and Science, Balikesir University, 10145 Bal\u0131kesir, Turkey"},{"name":"Faculty of Engineering, Free University of Bozen-Bolzano, 39100 Bolzano, Italy"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-0516-0033","authenticated-orcid":false,"given":"Bruno","family":"Carpentieri","sequence":"additional","affiliation":[{"name":"Faculty of Engineering, Free University of Bozen-Bolzano, 39100 Bolzano, Italy"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2025,12,24]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"034001","DOI":"10.1115\/1.4001333","article-title":"Nonlinear dynamics and its applications in micro-and nanoresonators","volume":"132","author":"Rhoads","year":"2010","journal-title":"J. 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