{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T02:00:53Z","timestamp":1760061653843,"version":"3.41.0"},"reference-count":12,"publisher":"Association for Computing Machinery (ACM)","issue":"3","license":[{"start":{"date-parts":[[2019,2,16]],"date-time":"2019-02-16T00:00:00Z","timestamp":1550275200000},"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":[[2019,2,16]]},"abstract":"<jats:p>\n            In this poster we present the results of [10]. We consider the problem of finding the common roots of a set of polynomial functions defining a zero-dimensional ideal\n            <jats:italic>I<\/jats:italic>\n            in a ring\n            <jats:italic>R<\/jats:italic>\n            of polynomials over C. We propose a general algebraic framework to find the solutions and to compute the structure of the quotient ring\n            <jats:italic>R\/I<\/jats:italic>\n            from the cokernel of a resultant map. This leads to what we call Truncated Normal Forms (TNFs). Algorithms for generic dense and sparse systems follow from the classical resultant constructions. In the presented framework, the concept of a border basis is generalized by relaxing the conditions on the set of basis elements. This allows for algorithms to adapt the choice of basis in order to enhance the numerical stability. We present such an algorithm. The numerical experiments show that the methods allow to compute all zeros of challenging systems (high degree, with a large number of solutions) in small dimensions with high accuracy.\n          <\/jats:p>","DOI":"10.1145\/3313880.3313888","type":"journal-article","created":{"date-parts":[[2019,2,19]],"date-time":"2019-02-19T20:54:15Z","timestamp":1550609655000},"page":"78-81","update-policy":"https:\/\/doi.org\/10.1145\/crossmark-policy","source":"Crossref","is-referenced-by-count":2,"title":["Truncated normal forms for solving polynomial systems"],"prefix":"10.1145","volume":"52","author":[{"given":"Simon","family":"Telen","sequence":"first","affiliation":[]},{"given":"Bernard","family":"Mourrain","sequence":"additional","affiliation":[]},{"given":"Marc","family":"Van Barel","sequence":"additional","affiliation":[]}],"member":"320","published-online":{"date-parts":[[2019,2,16]]},"reference":[{"key":"e_1_2_1_1_1","unstructured":"E. Cattani D. A. Cox G. Ch\u00e8ze A. Dickenstein M. Elkadi I. Z. Emiris A. Galligo A. Kehrein M. Kreuzer and B. Mourrain. Solving polynomial equations: foundations algorithms and applications (Algorithms and Computation in Mathematics). 2005.   E. Cattani D. A. Cox G. Ch\u00e8ze A. Dickenstein M. Elkadi I. Z. Emiris A. Galligo A. Kehrein M. Kreuzer and B. Mourrain. Solving polynomial equations: foundations algorithms and applications (Algorithms and Computation in Mathematics). 2005."},{"key":"e_1_2_1_2_1","doi-asserted-by":"crossref","unstructured":"D. A. Cox J. Little and D. O'Shea. Ideals varieties and algorithms volume 3. Springer 1992.  D. A. Cox J. Little and D. O'Shea. Ideals varieties and algorithms volume 3. Springer 1992.","DOI":"10.1007\/978-1-4757-2181-2"},{"volume-title":"Springer Science & Business Media","year":"2006","author":"Cox D. A.","key":"e_1_2_1_3_1"},{"key":"e_1_2_1_4_1","doi-asserted-by":"crossref","unstructured":"M.\n       \n      Elkadi\n     and \n      \n      \n      B.\n       \n      Mourrain\n      \n  \n  . \n  Introduction \u00e0 la r\u00e9solution des syst\u00e8mes polynomiaux volume \n  59\n   of \n  Math\u00e9matiques et Applications\n  . \n  Springer 2007\n  .  M. Elkadi and B. Mourrain. Introduction \u00e0 la r\u00e9solution des syst\u00e8mes polynomiaux volume 59 of Math\u00e9matiques et Applications. Springer 2007.","DOI":"10.1007\/978-3-540-71647-1"},{"journal-title":"Journal of Symbolic Computation, 28(1--2):3--44","year":"1999","author":"Emiris I. Z.","key":"e_1_2_1_5_1"},{"key":"e_1_2_1_6_1","first-page":"491","volume-title":"21st International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2009), Discrete Math. Theor. Comput. Sci. Proc., AK","author":"Joswig M.","year":"2009"},{"key":"e_1_2_1_7_1","doi-asserted-by":"publisher","DOI":"10.5555\/646031.677375"},{"key":"e_1_2_1_8_1","doi-asserted-by":"publisher","DOI":"10.1145\/242961.242966"},{"volume-title":"Amer. Math. Soc.","year":"2002","author":"Sturmfels B.","key":"e_1_2_1_9_1"},{"key":"e_1_2_1_10_1","unstructured":"S. Telen B. Mourrain and M. Van Barel. Solving polynomial systems via a stabilized representation of quotient algebras. arXiv preprint arXiv:1711.04543 2017.  S. Telen B. Mourrain and M. Van Barel. Solving polynomial systems via a stabilized representation of quotient algebras. arXiv preprint arXiv:1711.04543 2017."},{"key":"e_1_2_1_11_1","doi-asserted-by":"publisher","DOI":"10.1016\/j.cam.2018.04.021"},{"key":"e_1_2_1_12_1","doi-asserted-by":"publisher","DOI":"10.1145\/317275.317286"}],"container-title":["ACM Communications in Computer Algebra"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/dl.acm.org\/doi\/10.1145\/3313880.3313888","content-type":"unspecified","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/dl.acm.org\/doi\/pdf\/10.1145\/3313880.3313888","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,6,17]],"date-time":"2025-06-17T23:23:46Z","timestamp":1750202626000},"score":1,"resource":{"primary":{"URL":"https:\/\/dl.acm.org\/doi\/10.1145\/3313880.3313888"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2019,2,16]]},"references-count":12,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2019,2,16]]}},"alternative-id":["10.1145\/3313880.3313888"],"URL":"https:\/\/doi.org\/10.1145\/3313880.3313888","relation":{},"ISSN":["1932-2240"],"issn-type":[{"type":"print","value":"1932-2240"}],"subject":[],"published":{"date-parts":[[2019,2,16]]},"assertion":[{"value":"2019-02-16","order":2,"name":"published","label":"Published","group":{"name":"publication_history","label":"Publication History"}}]}}