{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T01:17:17Z","timestamp":1760145437838,"version":"build-2065373602"},"reference-count":5,"publisher":"MDPI AG","issue":"7","license":[{"start":{"date-parts":[[2024,7,18]],"date-time":"2024-07-18T00:00:00Z","timestamp":1721260800000},"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>Let P(x) be a system of polynomials in s variables, where x\u2208Cs. If z0 is an isolated zero of P, then the multiplicity and its structure at z0 can be revealed by the normal set of the quotient ring R(&lt;P&gt;) or its dual space R* or by certain numerical methods. In his book titled \u201cNumerical Polynomial Algebra\u201d, Stetter described the so-called distinguished points, which are embedded in a zero manifold of P, and the author defined their multiplicities. In this note, we will generalize the definition of distinguished points and give a more appropriate definition for their multiplicity, as well as show how to calculate the multiplicity of these points.<\/jats:p>","DOI":"10.3390\/axioms13070479","type":"journal-article","created":{"date-parts":[[2024,7,18]],"date-time":"2024-07-18T08:39:12Z","timestamp":1721291952000},"page":"479","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["A Note on the Multiplicity of the Distinguished Points"],"prefix":"10.3390","volume":"13","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-2866-7179","authenticated-orcid":false,"given":"Weiping","family":"Li","sequence":"first","affiliation":[{"name":"Department of Mathematics, Oklahoma State University, Stillwater, OK 74078, USA"}]},{"given":"Xiaoshen","family":"Wang","sequence":"additional","affiliation":[{"name":"Department of Mathematics and Statistics, University of Arkansas at Little Rock, Little Rock, AR 72204, USA"}]}],"member":"1968","published-online":{"date-parts":[[2024,7,18]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Stetter, H. (2004). Numerical Polynomial Algebra, Society for Industrial and Applied Mathematics (SIAM).","DOI":"10.1137\/1.9780898717976"},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"Dayton, B., and Zeng, Z. (2005, January 24\u201327). Computing the multiplicity structure in solving polynomial systems. Proceedings of the 2005 International Symposium on Symbolic and Algebraic Computation, Beijing, China.","DOI":"10.1145\/1073884.1073902"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"183","DOI":"10.1007\/BF01075595","article-title":"The number of roots of a system of equations","volume":"9","author":"Bernshtein","year":"1975","journal-title":"Funct. Anal. Appl."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"1541","DOI":"10.1090\/S0025-5718-1995-1297471-4","article-title":"A Polyhedral Method for Solving Sparse Polynomial Systems","volume":"64","author":"Huber","year":"1995","journal-title":"Math. Comput."},{"key":"ref_5","doi-asserted-by":"crossref","unstructured":"Cox, D., Little, J., O\u2019shea, D., and Sweedler, M. (1991). Ideals, Varieties, and Algorithms, Springer.","DOI":"10.1007\/978-1-4757-2181-2"}],"container-title":["Axioms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2075-1680\/13\/7\/479\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T15:18:45Z","timestamp":1760109525000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2075-1680\/13\/7\/479"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,7,18]]},"references-count":5,"journal-issue":{"issue":"7","published-online":{"date-parts":[[2024,7]]}},"alternative-id":["axioms13070479"],"URL":"https:\/\/doi.org\/10.3390\/axioms13070479","relation":{},"ISSN":["2075-1680"],"issn-type":[{"type":"electronic","value":"2075-1680"}],"subject":[],"published":{"date-parts":[[2024,7,18]]}}}