{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,18]],"date-time":"2026-03-18T02:26:38Z","timestamp":1773800798530,"version":"3.50.1"},"reference-count":15,"publisher":"Association for Computing Machinery (ACM)","issue":"3\/4","license":[{"start":{"date-parts":[[2015,2,5]],"date-time":"2015-02-05T00:00:00Z","timestamp":1423094400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.acm.org\/publications\/policies\/copyright_policy#Background"}],"content-domain":{"domain":["dl.acm.org"],"crossmark-restriction":true},"short-container-title":["ACM Commun. Comput. Algebra"],"published-print":{"date-parts":[[2015,2,5]]},"abstract":"<jats:p>We consider linear ordinary differential or difference systems of the form L(y) = 0 where L is an operator with matrix coefficients, the unknown vector y has m components y1, . . . , ym, m &gt; 1. The matrix coefficients are of size m x m, their entries belong to a differential or difference field K of characteristic 0. For any such a system the solution space VL is considered, and the components of each solution are in a fixed appropriate differential or difference extension of K (e.g., in the universal Picard-Vessiot extension). We prove that dim VLM = dim VL+dim VM for arbitrary operators L and M of the considered form, and discuss some algorithms based on this property of operators. In particular, we propose an algorithm to compute dim VL, as well as a new algorithm having a low complexity for recognizing unimodular operators and constructing the inverse of a unimodular operator.<\/jats:p>","DOI":"10.1145\/2733693.2733719","type":"journal-article","created":{"date-parts":[[2015,2,10]],"date-time":"2015-02-10T13:19:47Z","timestamp":1423574387000},"page":"155-165","update-policy":"https:\/\/doi.org\/10.1145\/crossmark-policy","source":"Crossref","is-referenced-by-count":3,"title":["On Solution Spaces of Products of Linear Differential or Difference Operators"],"prefix":"10.1145","volume":"48","author":[{"given":"Sergei A.","family":"Abramov","sequence":"first","affiliation":[{"name":"Computing Centre of the Russian Academy of Sciences, Moscow, Russia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Moulay A.","family":"Barkatou","sequence":"additional","affiliation":[{"name":"Institut XLIM, DMI, Universit\u00e9 de Limoges; CNRS, Limoges, France"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"320","published-online":{"date-parts":[[2015,2,5]]},"reference":[{"key":"e_1_2_1_1_1","doi-asserted-by":"publisher","DOI":"10.1080\/10236199908808199"},{"key":"e_1_2_1_2_1","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-319-02297-0_1"},{"key":"e_1_2_1_3_1","unstructured":"S.A. Abramov M.A. Barkatou D.E. Khmelnov. On full-rank differential systems with power series coefficients. J. of Symbolic Computation accepted.  S.A. Abramov M.A. Barkatou D.E. Khmelnov. On full-rank differential systems with power series coefficients. J. of Symbolic Computation accepted."},{"key":"e_1_2_1_4_1","doi-asserted-by":"publisher","DOI":"10.1145\/384101.384102"},{"key":"e_1_2_1_5_1","doi-asserted-by":"publisher","DOI":"10.1134\/S0361768813020023"},{"key":"e_1_2_1_6_1","doi-asserted-by":"publisher","DOI":"10.1016\/j.jsc.2011.12.016"},{"key":"e_1_2_1_7_1","doi-asserted-by":"publisher","DOI":"10.1016\/S0377-0427(96)00120-3"},{"key":"e_1_2_1_8_1","doi-asserted-by":"publisher","DOI":"10.1016\/j.jsc.2005.10.002"},{"key":"e_1_2_1_9_1","volume-title":"Free Rings and their Relations","author":"Cohn P.M.","year":"1971","unstructured":"P.M. Cohn . Free Rings and their Relations . Academic Press , London & New York, 1971 . P.M. Cohn. Free Rings and their Relations. 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