{"title":"Centralizers of Jacobian derivations","authors":"D. Efimov, A. Petravchuk, M. Sydorov","doi":"10.12958/adm2186","DOIUrl":null,"url":null,"abstract":"Let K be an algebraically closed field of characte-ristic zero, K[x,y] the polynomial ring in variables x, y and let W2(K) be the Lie algebra of all K-derivations on K[x,y]. A derivation D∈W2(K) is called a Jacobian derivation if there exists f∈K[x,y] such that D(h)=det J(f,h) for any h∈K[x,y] (hereJ(f,h) is the Jacobian matrix for f and h). Such a derivation is denoted by Df. The kernel of Df in K[x,y] is a subalgebra K[p] where p=p(x,y) is a polynomial of smallest degree such that f(x,y)=φ(p(x,y) for some φ(t)∈K[t]. Let C=CW2(K)(Df) be the centralizer of Df in W2(K). We prove that C is the free K[p]-module of rank 1 or 2 over K[p] and point out a criterion of being a module of rank 2. These results are used to obtain a classof integrable autonomous systems of differential equations.","PeriodicalId":364397,"journal":{"name":"Algebra and Discrete Mathematics","volume":null,"pages":null},"PeriodicalIF":0.0000,"publicationDate":"2023-11-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Algebra and Discrete Mathematics","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.12958/adm2186","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 0
Abstract
Let K be an algebraically closed field of characte-ristic zero, K[x,y] the polynomial ring in variables x, y and let W2(K) be the Lie algebra of all K-derivations on K[x,y]. A derivation D∈W2(K) is called a Jacobian derivation if there exists f∈K[x,y] such that D(h)=det J(f,h) for any h∈K[x,y] (hereJ(f,h) is the Jacobian matrix for f and h). Such a derivation is denoted by Df. The kernel of Df in K[x,y] is a subalgebra K[p] where p=p(x,y) is a polynomial of smallest degree such that f(x,y)=φ(p(x,y) for some φ(t)∈K[t]. Let C=CW2(K)(Df) be the centralizer of Df in W2(K). We prove that C is the free K[p]-module of rank 1 or 2 over K[p] and point out a criterion of being a module of rank 2. These results are used to obtain a classof integrable autonomous systems of differential equations.