{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2022,4,2]],"date-time":"2022-04-02T17:14:01Z","timestamp":1648919641011},"reference-count":36,"publisher":"World Scientific Pub Co Pte Lt","issue":"08","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Int. J. Algebra Comput."],"published-print":{"date-parts":[[2019,12]]},"abstract":"<jats:p> For [Formula: see text] denote by [Formula: see text] the free group on [Formula: see text] generators and let [Formula: see text]. For [Formula: see text] and elements [Formula: see text], we study orientable quadratic equations of the form [Formula: see text] with unknowns [Formula: see text] and provide explicit solutions for them for the minimal possible number [Formula: see text]. <\/jats:p><jats:p> In the particular case when [Formula: see text], [Formula: see text] for [Formula: see text] and [Formula: see text] the minimal number which satisfies [Formula: see text], we provide two types of solutions depending on the image of the subgroup [Formula: see text] generated by the solution under the natural homomorphism [Formula: see text]: the first solution, which is called a primitive solution, satisfies [Formula: see text], the second solution satisfies [Formula: see text]. <\/jats:p><jats:p> We also provide an explicit solution of the equation [Formula: see text] for [Formula: see text] in [Formula: see text], and prove that if [Formula: see text], then every solution of this equation is primitive. <\/jats:p><jats:p> As a geometrical consequence, for every solution, we obtain a map [Formula: see text] from the orientable surface [Formula: see text] of genus [Formula: see text] to the torus [Formula: see text] which has the minimal number of roots among all maps from the homotopy class of [Formula: see text]. Depending on the number [Formula: see text], such maps have fundamentally different geometric properties: in some cases, they satisfy the Wecken property and in other cases not. <\/jats:p>","DOI":"10.1142\/s0218196719500589","type":"journal-article","created":{"date-parts":[[2019,7,30]],"date-time":"2019-07-30T02:42:20Z","timestamp":1564454540000},"page":"1451-1466","source":"Crossref","is-referenced-by-count":0,"title":["Explicit solutions of certain orientable quadratic equations in free groups"],"prefix":"10.1142","volume":"29","author":[{"given":"D.","family":"Gon\u00e7alves","sequence":"first","affiliation":[{"name":"Department of Mathematics-IME, University of S\u00e3o Paulo, 05508-090, Rua do Mat\u00e3o 1010, Butanta-S\u00e3o Paulo-SP, Brazil"}]},{"given":"T.","family":"Nasybullov","sequence":"additional","affiliation":[{"name":"Department of Mathematics, KU Leuven Kulak, 8500, Etienne Sabbelaan 53, Kortrijk, Belgium"}]}],"member":"219","published-online":{"date-parts":[[2019,10,24]]},"reference":[{"key":"S0218196719500589BIB002","doi-asserted-by":"publisher","DOI":"10.2478\/BF02476007"},{"key":"S0218196719500589BIB003","doi-asserted-by":"publisher","DOI":"10.1023\/B:MATN.0000015017.47636.4b"},{"key":"S0218196719500589BIB004","first-page":"297","volume":"179","author":"Bogatyi S.","year":"2004","journal-title":"Physics and Chemistry"},{"key":"S0218196719500589BIB005","doi-asserted-by":"publisher","DOI":"10.1007\/s002090100203"},{"key":"S0218196719500589BIB006","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9939-1975-0375287-X"},{"key":"S0218196719500589BIB007","doi-asserted-by":"publisher","DOI":"10.1017\/S0013091500019283"},{"key":"S0218196719500589BIB008","doi-asserted-by":"publisher","DOI":"10.1017\/S0013091500020939"},{"key":"S0218196719500589BIB009","doi-asserted-by":"publisher","DOI":"10.1515\/9783110848397-025"},{"key":"S0218196719500589BIB010","doi-asserted-by":"publisher","DOI":"10.1016\/0040-9383(81)90033-1"},{"issue":"1","key":"S0218196719500589BIB011","first-page":"49","volume":"42","author":"Fenn R.","year":"1996","journal-title":"Enseign. 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