{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,26]],"date-time":"2026-06-26T13:00:13Z","timestamp":1782478813115,"version":"3.54.5"},"reference-count":11,"publisher":"Springer Science and Business Media LLC","issue":"6","license":[{"start":{"date-parts":[[2022,8,20]],"date-time":"2022-08-20T00:00:00Z","timestamp":1660953600000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2022,8,20]],"date-time":"2022-08-20T00:00:00Z","timestamp":1660953600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["AAECC"],"published-print":{"date-parts":[[2022,12]]},"abstract":"Abstract<\/jats:title>We introduce the concept of subalgebra spectrum, Sp<\/jats:italic>(A<\/jats:italic>), for a subalgebra A<\/jats:italic> of finite codimension in $$\\mathbb {K}[x]$$<\/jats:tex-math>\n \n K<\/mml:mi>\n [<\/mml:mo>\n x<\/mml:mi>\n ]<\/mml:mo>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>. The spectrum is a finite subset of the underlying field. We also introduce a tool, the characteristic polynomial of A<\/jats:italic>, which has the spectrum as its set of zeroes. The characteristic polynomial can be computed from the generators of A<\/jats:italic>, thus allowing us to find the spectrum of an algebra given by generators. We proceed by using the spectrum to get descriptions of subalgebras of finite codimension. More precisely we show that A<\/jats:italic> can be described by a set of conditions that each is either of the type $$f(\\alpha )=f(\\beta )$$<\/jats:tex-math>\n \n f<\/mml:mi>\n (<\/mml:mo>\n \u03b1<\/mml:mi>\n )<\/mml:mo>\n =<\/mml:mo>\n f<\/mml:mi>\n (<\/mml:mo>\n \u03b2<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> for $$\\alpha ,\\beta$$<\/jats:tex-math>\n \n \u03b1<\/mml:mi>\n ,<\/mml:mo>\n \u03b2<\/mml:mi>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> in Sp<\/jats:italic>(A<\/jats:italic>) or of the type stating that some linear combination of derivatives of different orders evaluated in elements of Sp<\/jats:italic>(A<\/jats:italic>) equals zero. We use these types of conditions to, by an inductive process, find explicit descriptions of subalgebras of codimension up to three. These descriptions also include SAGBI bases for each family of subalgebras.<\/jats:p>","DOI":"10.1007\/s00200-022-00573-4","type":"journal-article","created":{"date-parts":[[2022,8,20]],"date-time":"2022-08-20T07:03:00Z","timestamp":1660978980000},"page":"751-789","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":5,"title":["Subalgebras in K[x] of small codimension"],"prefix":"10.1007","volume":"33","author":[{"given":"Rode","family":"Gr\u00f6nkvist","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Erik","family":"Leffler","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Anna","family":"Torstensson","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Victor","family":"Ufnarovski","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"297","published-online":{"date-parts":[[2022,8,20]]},"reference":[{"key":"573_CR1","doi-asserted-by":"crossref","unstructured":"Torstensson, A., Ufnarovski, V., \u00d6fverbeck, H.: On SAGBI bases and resultants. Commutative algebra, singularities and computer algebra (Sinaia, 2002), NATO Sci. Ser. II Math. Phys. Chem., vol. 115, pp. 241\u2013254. Kluwer Academic Publishers, Dordrecht (2003)","DOI":"10.1007\/978-94-007-1092-4_15"},{"key":"573_CR2","unstructured":"Kreuzer, M., Robbiano, L.: Computational Commutative Algebra, vol. 2. Springer, Berlin (2005). ISBN: 978-3-540-25527-7; 3-540-25527-3"},{"key":"573_CR3","first-page":"61","volume-title":"Subalgebra Bases, Commutative Algebra (Salvador, 1988)","author":"L Robbiano","year":"1990","unstructured":"Robbiano, L., Sweedler, M.: Subalgebra Bases, Commutative Algebra (Salvador, 1988), pp. 61\u201387. Springer, Berlin (1990)"},{"key":"573_CR4","doi-asserted-by":"publisher","first-page":"649","DOI":"10.1007\/BF01119685","volume":"6","author":"EA Gorin","year":"1969","unstructured":"Gorin, E.A.: Subalgebras of finite codimension. Math. Notes Acad. Sci. USSR 6, 649\u2013652 (1969). https:\/\/doi.org\/10.1007\/BF01119685","journal-title":"Math. Notes Acad. Sci. USSR"},{"key":"573_CR5","unstructured":"Cox, D.A., Little, J., O\u2019Shea, D.: Using Algebraic Geometry. Graduate Texts in Mathematics. Springer, New York (2005). ISBN: 0-387-20706-6"},{"key":"573_CR6","volume-title":"Polynomials","author":"VV Prasolov","year":"2010","unstructured":"Prasolov, V.V.: Polynomials. Springer, Berlin (2010)"},{"key":"573_CR7","doi-asserted-by":"crossref","unstructured":"Villard, G.: On computing the resultant of generic bivariate polynomials. In: ISSAC\u201918\u2014Proceedings of the 2018 ACM International Symposium on Symbolic and Algebraic Computation, pp. 391\u2013398. New York (2018)","DOI":"10.1145\/3208976.3209020"},{"key":"573_CR8","unstructured":"Bourbaki, N.: Algebra II: Chapters 4\u20137. Elements of Mathematics. Springer (1990). English paperback edition"},{"key":"573_CR9","unstructured":"Lefler, E.: Derivations in univariate polynomial subalgbras of finite codimension. Bachelor\u2019s thesis, Lund Institute of Technology (2021): K39, ISSN 1654-6229, LUFTMA-4007-2021"},{"key":"573_CR10","unstructured":"Gr\u00f6nkvist, R., Leffler, E., Torstensson, A., Ufnarovski, V.: Describing subalgebras of $${\\mathbb{K}}[x]$$ using derivations (2021). arXiv:2017.11916 [math.RA]"},{"key":"573_CR11","doi-asserted-by":"publisher","first-page":"496","DOI":"10.1080\/00029890.1974.11993596","volume":"81","author":"DJ Newman","year":"1974","unstructured":"Newman, D.J.: Point separating algebras of polynomials. Am. Math. Mon. 81, 496\u2013498 (1974)","journal-title":"Am. Math. Mon."}],"container-title":["Applicable Algebra in Engineering, Communication and Computing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s00200-022-00573-4.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s00200-022-00573-4\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s00200-022-00573-4.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2022,11,17]],"date-time":"2022-11-17T16:07:17Z","timestamp":1668701237000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s00200-022-00573-4"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,8,20]]},"references-count":11,"journal-issue":{"issue":"6","published-print":{"date-parts":[[2022,12]]}},"alternative-id":["573"],"URL":"https:\/\/doi.org\/10.1007\/s00200-022-00573-4","relation":{},"ISSN":["0938-1279","1432-0622"],"issn-type":[{"value":"0938-1279","type":"print"},{"value":"1432-0622","type":"electronic"}],"subject":[],"published":{"date-parts":[[2022,8,20]]},"assertion":[{"value":"21 October 2021","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"29 June 2022","order":2,"name":"revised","label":"Revised","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"15 July 2022","order":3,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"20 August 2022","order":4,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}}]}}