{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,7,20]],"date-time":"2025-07-20T03:46:27Z","timestamp":1752983187097,"version":"3.40.5"},"reference-count":20,"publisher":"World Scientific Pub Co Pte Ltd","issue":"04","funder":[{"name":"the Austrian Science Fund","award":["P29355-N35"],"award-info":[{"award-number":["P29355-N35"]}]},{"name":"the Spanish Agencia Estatal de Investigaci\u00f3n","award":["MTM2017-82740-P","PID2021-126851NB-100"],"award-info":[{"award-number":["MTM2017-82740-P","PID2021-126851NB-100"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Int. J. Algebra Comput."],"published-print":{"date-parts":[[2024,6]]},"abstract":"<jats:p> Given a finitely generated subgroup H of a free group F, we present an algorithm which computes [Formula: see text], such that the set of elements [Formula: see text], for which there exists a non-trivial H-equation having g as a solution is precisely the disjoint union of the double cosets [Formula: see text]. Moreover, we present an algorithm which, given a finitely generated subgroup [Formula: see text] and an element [Formula: see text], computes a finite set of elements from [Formula: see text] (of the minimum possible cardinality) generating, as a normal subgroup, the \u201cideal\u201d [Formula: see text] of all \u201cpolynomials\u201d [Formula: see text], such that [Formula: see text]. The algorithms, as well as the proofs, are based on the graph-theoretic techniques introduced by Stallings and on the more classical combinatorial techniques of Nielsen transformations. The key notion here is that of dependence of an element [Formula: see text] on a subgroup H. We also study the corresponding notions of dependence sequence and dependence closure of a subgroup. <\/jats:p>","DOI":"10.1142\/s0218196724500176","type":"journal-article","created":{"date-parts":[[2024,4,5]],"date-time":"2024-04-05T15:07:17Z","timestamp":1712329637000},"page":"439-470","source":"Crossref","is-referenced-by-count":1,"title":["Dependence over subgroups of free groups"],"prefix":"10.1142","volume":"34","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-0255-0885","authenticated-orcid":false,"given":"Amnon","family":"Rosenmann","sequence":"first","affiliation":[{"name":"Institut f\u00fcr Diskrete Mathematik, Technische Universit\u00e4t Graz, Austria"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-3519-4135","authenticated-orcid":false,"given":"Enric","family":"Ventura","sequence":"additional","affiliation":[{"name":"Departament de Matem\u00e0tiques, Universitat Polit\u00e8cnica de Catalunya, Spain"},{"name":"Institut de Matem\u00e0tiques de la UPC-Barcelona Tech, Catalonia, Spain"}]}],"member":"219","published-online":{"date-parts":[[2024,5,15]]},"reference":[{"key":"S0218196724500176BIB001","doi-asserted-by":"publisher","DOI":"10.1112\/S0010437X22007709"},{"key":"S0218196724500176BIB003","doi-asserted-by":"publisher","DOI":"10.1016\/j.jalgebra.2023.08.018"},{"key":"S0218196724500176BIB004","doi-asserted-by":"publisher","DOI":"10.1006\/jabr.1999.7881"},{"key":"S0218196724500176BIB005","doi-asserted-by":"publisher","DOI":"10.1515\/JGT.2008.080"},{"issue":"3","key":"S0218196724500176BIB006","first-page":"181","volume":"11","author":"Delgado J.","year":"2022","journal-title":"Trans. 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