{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2022,4,2]],"date-time":"2022-04-02T13:33:22Z","timestamp":1648906402216},"reference-count":8,"publisher":"World Scientific Pub Co Pte Lt","issue":"03","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Int. J. Algebra Comput."],"published-print":{"date-parts":[[2008,5]]},"abstract":"<jats:p>Lyndon initiated a study of equations over cyclic groups [8] and his approach led him to consider a class of equations of the form a<jats:sup>h<\/jats:sup>= (t<jats:sup>p<\/jats:sup>at<jats:sup>-p<\/jats:sup>)(t<jats:sup>q<\/jats:sup>at<jats:sup>-q<\/jats:sup>)(t<jats:sup>p<\/jats:sup>at<jats:sup>-p<\/jats:sup>)<jats:sup>-1<\/jats:sup>(t<jats:sup>q<\/jats:sup>at<jats:sup>-q<\/jats:sup>)<jats:sup>-1<\/jats:sup>where t is the variable and the cyclic group is C<jats:sub>n<\/jats:sub>= \u3008a|a<jats:sup>n<\/jats:sup>\u3009. Work by Ferlini, Goldstein, and Salpukis [3] concentrated, more generally, on equations with the exponent sum of the variable being zero over C<jats:sub>2<\/jats:sub>. This paper continues along the path started by Lyndon and shows that a large class of the equations above do have solutions over C<jats:sub>n<\/jats:sub>where n \u2265 4. Our approach involves the use of pictures. We also include a result that the equations with h = 1, 2, p = 1, and q = 2 do not have a solution over C<jats:sub>3<\/jats:sub>.<\/jats:p>","DOI":"10.1142\/s0218196708004494","type":"journal-article","created":{"date-parts":[[2008,6,2]],"date-time":"2008-06-02T10:26:32Z","timestamp":1212402392000},"page":"423-441","source":"Crossref","is-referenced-by-count":0,"title":["SOLUTIONS TO ZERO-SUM EXPONENT EQUATIONS OVER FINITE CYCLIC GROUPS OF EXPONENT GREATER THAN TWO"],"prefix":"10.1142","volume":"18","author":[{"given":"VINCENT","family":"FERLINI","sequence":"first","affiliation":[{"name":"Mathematics Department, Keene State College, Keene, NH 03435, USA"}]}],"member":"219","published-online":{"date-parts":[[2011,11,20]]},"reference":[{"key":"rf1","doi-asserted-by":"publisher","DOI":"10.1112\/jlms\/53.2.271"},{"key":"rf3","doi-asserted-by":"crossref","first-page":"709","DOI":"10.1142\/S0218196700000339","volume":"10","author":"Ferlini V.","journal-title":"Int. J. Algebra Comput."},{"key":"rf4","doi-asserted-by":"publisher","DOI":"10.1073\/pnas.48.9.1531"},{"key":"rf5","doi-asserted-by":"publisher","DOI":"10.1017\/S0013091500028108"},{"key":"rf6","first-page":"507","volume":"59","author":"Howie J.","journal-title":"Proc. London Math. Soc."},{"key":"rf7","doi-asserted-by":"publisher","DOI":"10.1007\/BF02584882"},{"key":"rf9","doi-asserted-by":"crossref","unstructured":"R. C.\u00a0Lyndon, Combinatorial Group Theory and Topology, Annals of Mathematical Studies\u00a0111, eds. S.\u00a0Gersten and J.\u00a0Stallings (Princeton University Press, 1987)\u00a0pp. 3\u201333.","DOI":"10.1515\/9781400882083-002"},{"key":"rf10","unstructured":"S.\u00a0Pride, Group Theory From a Geometric Viewpoint (World Scientific, 1991)\u00a0pp. 687\u2013717."}],"container-title":["International Journal of Algebra and Computation"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.worldscientific.com\/doi\/pdf\/10.1142\/S0218196708004494","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,5,6]],"date-time":"2020-05-06T02:57:21Z","timestamp":1588733841000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.worldscientific.com\/doi\/abs\/10.1142\/S0218196708004494"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2008,5]]},"references-count":8,"journal-issue":{"issue":"03","published-online":{"date-parts":[[2011,11,20]]},"published-print":{"date-parts":[[2008,5]]}},"alternative-id":["10.1142\/S0218196708004494"],"URL":"https:\/\/doi.org\/10.1142\/s0218196708004494","relation":{},"ISSN":["0218-1967","1793-6500"],"issn-type":[{"value":"0218-1967","type":"print"},{"value":"1793-6500","type":"electronic"}],"subject":[],"published":{"date-parts":[[2008,5]]}}}