{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T01:41:22Z","timestamp":1760060482509,"version":"build-2065373602"},"reference-count":26,"publisher":"MDPI AG","issue":"3","license":[{"start":{"date-parts":[[2025,9,4]],"date-time":"2025-09-04T00:00:00Z","timestamp":1756944000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["AppliedMath"],"abstract":"<jats:p>We study equations with real polynomials of arbitrary degree, such that each coefficient has a small, individual error; this may originate, for example, from imperfect measuring. In particular, we study the influence of the errors on the roots of the polynomials. The errors are modeled by imprecisions of Sorites type: they are supposed to be stable to small shifts. We argue that such imprecisions are appropriately reflected by (scalar) neutrices, which are convex subgroups of the nonstandard real line; examples are the set of infinitesimals, or the set of numbers of order \u03b5, where \u03b5 is a fixed infinitesimal. The Main Theorem states that the imprecisions of the roots are neutrices, and determines their shape.<\/jats:p>","DOI":"10.3390\/appliedmath5030120","type":"journal-article","created":{"date-parts":[[2025,9,4]],"date-time":"2025-09-04T10:55:37Z","timestamp":1756983337000},"page":"120","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["A General Approach to Error Analysis for Roots of Polynomial Equations"],"prefix":"10.3390","volume":"5","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-8485-3091","authenticated-orcid":false,"given":"Imme van den","family":"Berg","sequence":"first","affiliation":[{"name":"Centro de Investiga\u00e7\u00e3o em Matem\u00e1tica e Aplica\u00e7\u00f5es (CIMA), Universidade de \u00c9vora, Rua Rom\u00e3o Ramalho 59, 7000-671 \u00c9vora, Portugal"}]},{"ORCID":"https:\/\/orcid.org\/0009-0007-6077-9115","authenticated-orcid":false,"given":"Jo\u00e3o Carlos Lopes","family":"Horta","sequence":"additional","affiliation":[{"name":"Faculdade de Ci\u00eancias e Tecnologia, Universidade de Cabo Verde, Campus de Palmarejo Grande, Zona K 379 C, Praia 7943-010, Cape Verde"}]}],"member":"1968","published-online":{"date-parts":[[2025,9,4]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"517","DOI":"10.1007\/s10670-023-00709-z","article-title":"A theory of marginal and large difference","volume":"90","author":"Dinis","year":"2025","journal-title":"Erkenntnis"},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"Koudjeti, F., and Van den Berg, I. (1995). Neutrices, external numbers, and external calculus. Nonstandard Analysis in Practice, Springer.","DOI":"10.1007\/978-3-642-57758-1_7"},{"key":"ref_3","unstructured":"Taylor, J. (2025). Introduction to Error Analysis. The Study of Uncertainties in Physical Measurements, University Science Books. [3rd ed.]."},{"key":"ref_4","doi-asserted-by":"crossref","unstructured":"Dinis, B., and Van den Berg, I. (2019). Neutrices and External Numbers: A Flexible Number System, Chapman and Hall\/CRC.","DOI":"10.1201\/9780429291456"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"281","DOI":"10.1007\/BF02787689","article-title":"Introduction to the neutrix calculus","volume":"7","year":"1959","journal-title":"J. d\u2019Analyse Math."},{"key":"ref_6","doi-asserted-by":"crossref","unstructured":"Murdock, J.A. (1999). Perturbations: Theory and Methods, Society for Industrial and Applied Mathematics.","DOI":"10.1137\/1.9781611971095"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"28542","DOI":"10.3934\/math.20241385","article-title":"On accurate asymptotic approximations of roots for polynomial equations containing a small, but fixed parameter","volume":"9","author":"Saptaingyas","year":"2024","journal-title":"AIMS Math."},{"key":"ref_8","doi-asserted-by":"crossref","unstructured":"Lutz, R., and Goze, M. (1982). Nonstandard Analysis: A Practical Guide with Applications, Springer. Lecture Notes in Mathematics.","DOI":"10.1007\/BFb0093397"},{"key":"ref_9","unstructured":"Remm, E. (2022). Perturbations of polynomials and applications. arXiv."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"2509","DOI":"10.1080\/00927872.2023.2301540","article-title":"Continuity of the roots of a polynomial","volume":"52","author":"Nathanson","year":"2024","journal-title":"Commun. 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