{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,21]],"date-time":"2026-02-21T20:44:07Z","timestamp":1771706647245,"version":"3.50.1"},"reference-count":47,"publisher":"World Scientific Pub Co Pte Ltd","issue":"07","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Int. J. Algebra Comput."],"published-print":{"date-parts":[[2024,11]]},"abstract":"<jats:p> For the purposes of our analysis, we shall work with the Steenrod algebra [Formula: see text] over the field of two elements, [Formula: see text]. Our paper focuses on expanding the outcomes from our preceding study, which serves as Part I. Precisely, we will initially compute the dimension of the \u201ccohit\u201d space [Formula: see text], where [Formula: see text] is defined as the graded polynomial ring [Formula: see text], for the specific case of [Formula: see text]. This computation will be carried out explicitly for the generic degree [Formula: see text], where t is a positive integer. We have developed a novel algorithm implemented in SageMath to compute the case where [Formula: see text]. We subsequently study the dimension of [Formula: see text] in degrees [Formula: see text] for [Formula: see text]. One of our findings is the correction of some erroneous results in the previous work by Moetele and Mothebe. We, additionally, give an explicit formula for the dimension of [Formula: see text] in degree fourteen for all [Formula: see text] and in degree fifteen for all [Formula: see text]. We also describe a computational method implemented in SageMath to explicitly account for the number of spikes of degree [Formula: see text] in the [Formula: see text]-module [Formula: see text], applicable for all positive integers [Formula: see text]. In application, we examine W. Singer\u2019s conjecture on the algebraic transfer of rank 5 in degrees [Formula: see text] for all [Formula: see text] and of ranks [Formula: see text] in internal degrees d ranging from 13 to 15. Considering higher homological degrees, we analyze the behavior of the algebraic transfer of rank 7 in the generic degrees [Formula: see text] for [Formula: see text] and [Formula: see text] for [Formula: see text]. We then demonstrate that the (in)decomposable elements [Formula: see text],\u00a0[Formula: see text], [Formula: see text], [Formula: see text] and [Formula: see text] are not in the image of the Singer transfer. <\/jats:p>","DOI":"10.1142\/s0218196724500401","type":"journal-article","created":{"date-parts":[[2024,7,5]],"date-time":"2024-07-05T06:05:46Z","timestamp":1720159546000},"page":"1001-1057","source":"Crossref","is-referenced-by-count":2,"title":["On the dimensions of the graded space \ud835\udd3d2 \u2297\ud835\udc9c\ud835\udd3d2[x1,x2,\u2026,xs] at degrees s + 5 and its relation to algebraic transfers"],"prefix":"10.1142","volume":"34","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-6885-3996","authenticated-orcid":false,"given":"\u0110\u1eb7ng","family":"V\u00f5 Ph\u00fac","sequence":"first","affiliation":[{"name":"Department of Information Technology, FPT University, Quy Nhon AI Campus, An Phu Thinh New Urban Area, Quy Nhon City Binh Dinh, Vietnam"},{"name":"Faculty of Mathematics and Statistics, Quy Nhon University, 170 An Duong Vuong, Quy Nhon City, Binh Dinh, Vietnam"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"219","published-online":{"date-parts":[[2024,8,31]]},"reference":[{"key":"S0218196724500401BIB001","doi-asserted-by":"publisher","DOI":"10.1007\/BF02564578"},{"key":"S0218196724500401BIB002","doi-asserted-by":"publisher","DOI":"10.2307\/1970147"},{"key":"S0218196724500401BIB003","doi-asserted-by":"publisher","DOI":"10.1073\/pnas.38.8.720"},{"key":"S0218196724500401BIB004","doi-asserted-by":"publisher","DOI":"10.1090\/conm\/146\/01215"},{"key":"S0218196724500401BIB005","doi-asserted-by":"publisher","DOI":"10.1006\/jsco.1996.0125"},{"key":"S0218196724500401BIB006","doi-asserted-by":"publisher","DOI":"10.2140\/gtm.2007.11.435"},{"key":"S0218196724500401BIB009","doi-asserted-by":"publisher","DOI":"10.1016\/j.topol.2011.01.002"},{"key":"S0218196724500401BIB012","doi-asserted-by":"publisher","DOI":"10.1016\/j.crma.2010.11.008"},{"key":"S0218196724500401BIB013","doi-asserted-by":"publisher","DOI":"10.1007\/s00229-011-0487-0"},{"key":"S0218196724500401BIB014","doi-asserted-by":"publisher","DOI":"10.1016\/j.topol.2014.10.013"},{"key":"S0218196724500401BIB015","doi-asserted-by":"publisher","DOI":"10.2140\/pjm.1988.134.27"},{"key":"S0218196724500401BIB016","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-0348-9018-2_9"},{"key":"S0218196724500401BIB017","first-page":"101","volume":"11","author":"H\u00e0 L. 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