{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,20]],"date-time":"2026-04-20T23:40:45Z","timestamp":1776728445053,"version":"3.51.2"},"reference-count":25,"publisher":"American Mathematical Society (AMS)","issue":"240","license":[{"start":{"date-parts":[[2002,12,4]],"date-time":"2002-12-04T00:00:00Z","timestamp":1038960000000},"content-version":"am","delay-in-days":365,"URL":"https:\/\/www.ams.org\/publications\/copyright-and-permissions"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Math. Comp."],"abstract":"<p>\n                    In this paper, we consider some practical applications of the symbolic Hamburger-Noether expressions for plane curves, which are introduced as a symbolic version of the so-called Hamburger-Noether expansions. More precisely, we give and develop in symbolic terms algorithms to compute the resolution tree of a plane curve (and the adjunction divisor, in particular), rational parametrizations for the branches of such a curve, special adjoints with assigned conditions (connected with different problems, like the so-called Brill-Noether algorithm), and the Weierstrass semigroup at\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper P\">\n                        <mml:semantics>\n                          <mml:mi>P<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">P<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    together with functions for each value in this semigroup, provided\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper P\">\n                        <mml:semantics>\n                          <mml:mi>P<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">P<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    is a rational branch of a singular plane model for the curve. Some other computational problems related to algebraic curves over perfect fields can be treated symbolically by means of such expressions, but we deal just with those connected with the effective construction and decoding of algebraic geometry codes.\n                  <\/p>","DOI":"10.1090\/s0025-5718-01-01390-4","type":"journal-article","created":{"date-parts":[[2002,9,20]],"date-time":"2002-09-20T14:12:48Z","timestamp":1032531168000},"page":"1759-1780","source":"Crossref","is-referenced-by-count":14,"title":["Symbolic Hamburger-Noether expressions of plane curves and applications to AG codes"],"prefix":"10.1090","volume":"71","author":[{"given":"A.","family":"Campillo","sequence":"first","affiliation":[]},{"given":"J.","family":"Farr\u00e1n","sequence":"additional","affiliation":[]}],"member":"14","published-online":{"date-parts":[[2001,12,4]]},"reference":[{"key":"1","series-title":"Lecture Notes in Mathematics","isbn-type":"print","volume-title":"Algebroid curves in positive characteristic","volume":"813","author":"Campillo, Antonio","year":"1980","ISBN":"https:\/\/id.crossref.org\/isbn\/3504100229"},{"issue":"1","key":"2","doi-asserted-by":"publisher","first-page":"71","DOI":"10.1006\/ffta.1999.0266","article-title":"Computing Weierstrass semigroups and the Feng-Rao distance from singular plane models","volume":"6","author":"Campillo, A.","year":"2000","journal-title":"Finite Fields Appl.","ISSN":"https:\/\/id.crossref.org\/issn\/1071-5797","issn-type":"print"},{"issue":"3","key":"3","doi-asserted-by":"publisher","first-page":"429","DOI":"10.1007\/BF01446904","article-title":"Infinitely near imposed singularities and singularities of polar curves","volume":"287","author":"Casas-Alvero, E.","year":"1990","journal-title":"Math. Ann.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5831","issn-type":"print"},{"key":"4","doi-asserted-by":"crossref","unstructured":"E. Casas-Alvero, \u201cSingularities of plane curves\", London Math. Soc. Lecture Notes Series  276, Cambridge University Press (2000).","DOI":"10.1017\/CBO9780511569326"},{"issue":"2","key":"5","first-page":"119","article-title":"Rational Puiseux expansions","volume":"70","author":"Duval, Dominique","year":"1989","journal-title":"Compositio Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0010-437X","issn-type":"print"},{"key":"6","doi-asserted-by":"crossref","unstructured":"F. Enriques and O. Chisini, \u201cTeoria geometrica delle equazioni e delle funzioni algebriche\", Bologna (1918).","DOI":"10.2307\/2974229"},{"key":"7","unstructured":"J.I. Farr\u00e1n and Ch. Lossen, \u201cbrnoeth.lib\", A Singular 2.0 library for the Brill-Noether algorithm, Weierstrass semigroups and AG codes (2001). Available via \\url{http:\/\/www.singular.uni-kl.de\/}."},{"issue":"1","key":"8","doi-asserted-by":"publisher","first-page":"37","DOI":"10.1109\/18.179340","article-title":"Decoding algebraic-geometric codes up to the designed minimum distance","volume":"39","author":"Feng, Gui Liang","year":"1993","journal-title":"IEEE Trans. Inform. Theory","ISSN":"https:\/\/id.crossref.org\/issn\/0018-9448","issn-type":"print"},{"issue":"6","key":"9","first-page":"1289","article-title":"Codes on algebraic curves","volume":"259","author":"Goppa, V. D.","year":"1981","journal-title":"Dokl. Akad. Nauk SSSR","ISSN":"https:\/\/id.crossref.org\/issn\/0002-3264","issn-type":"print"},{"issue":"4","key":"10","first-page":"762","article-title":"Algebraic-geometric codes","volume":"46","author":"Goppa, V. D.","year":"1982","journal-title":"Izv. Akad. Nauk SSSR Ser. Mat.","ISSN":"https:\/\/id.crossref.org\/issn\/0373-2436","issn-type":"print"},{"key":"11","first-page":"285","article-title":"Sur les inverses des \u00e9l\u00e9ments d\u00e9rivables dans un anneau abstrait","volume":"209","author":"Hebroni, P.","year":"1939","journal-title":"C. R. Acad. Sci. Paris","ISSN":"https:\/\/id.crossref.org\/issn\/0001-4036","issn-type":"print"},{"key":"12","unstructured":"G.-M. Greuel, G. Pfister and H. Sch\u00f6nemann, \u201cSingular 2.0\", A computer algebra system for polynomial computations, Centre for Computer Algebra, University of Kaiserslautern (2001). Available via \\url{http:\/\/www.singular.uni-kl.de\/}."},{"key":"13","unstructured":"G. Hach\u00e9, \u201cConstruction effective des codes g\u00e9om\u00e9triques\", Ph.D. thesis, Univ. Paris 6 (1996)."},{"key":"14","isbn-type":"print","doi-asserted-by":"publisher","first-page":"262","DOI":"10.1007\/3-540-60114-7_19","article-title":"Computation in algebraic function fields for effective construction of algebraic-geometric codes","author":"Hach\u00e9, Ga\u00e9tan","year":"1995","ISBN":"https:\/\/id.crossref.org\/isbn\/3540601147"},{"issue":"6","key":"15","doi-asserted-by":"publisher","first-page":"1615","DOI":"10.1109\/18.476233","article-title":"Effective construction of algebraic geometry codes","volume":"41","author":"Hach\u00e9, Ga\u00e9tan","year":"1995","journal-title":"IEEE Trans. Inform. Theory","ISSN":"https:\/\/id.crossref.org\/issn\/0018-9448","issn-type":"print"},{"issue":"1","key":"16","doi-asserted-by":"publisher","first-page":"67","DOI":"10.1007\/BF01810849","article-title":"Group codes on certain algebraic curves with many rational points","volume":"1","author":"Hansen, Johan P.","year":"1990","journal-title":"Appl. Algebra Engrg. Comm. Comput.","ISSN":"https:\/\/id.crossref.org\/issn\/0938-1279","issn-type":"print"},{"key":"17","series-title":"Graduate Texts in Mathematics, No. 52","isbn-type":"print","doi-asserted-by":"crossref","DOI":"10.1007\/978-1-4757-3849-0","volume-title":"Algebraic geometry","author":"Hartshorne, Robin","year":"1977","ISBN":"https:\/\/id.crossref.org\/isbn\/0387902449"},{"issue":"6","key":"18","doi-asserted-by":"publisher","first-page":"1589","DOI":"10.1109\/18.476214","article-title":"On the decoding of algebraic-geometric codes","volume":"41","author":"H\u00f8holdt, Tom","year":"1995","journal-title":"IEEE Trans. Inform. Theory","ISSN":"https:\/\/id.crossref.org\/issn\/0018-9448","issn-type":"print"},{"key":"19","doi-asserted-by":"crossref","unstructured":"M.D. Huang and D. Ierardi, \u201cEfficient algorithms for Riemann-Roch problem and for addition in the jacobian of a curve\", Proceedings 32nd Annual Symposium on Foundations of Computer Sciences, pp. 678-687, IEEE Comput. Soc. Press (1991).","DOI":"10.1109\/SFCS.1991.185435"},{"issue":"6","key":"20","doi-asserted-by":"publisher","first-page":"1720","DOI":"10.1109\/18.476245","article-title":"The minimum distance of codes in an array coming from telescopic semigroups","volume":"41","author":"Kirfel, Christoph","year":"1995","journal-title":"IEEE Trans. Inform. Theory","ISSN":"https:\/\/id.crossref.org\/issn\/0018-9448","issn-type":"print"},{"issue":"2","key":"21","doi-asserted-by":"crossref","first-page":"231","DOI":"10.24033\/bsmf.2096","article-title":"Algorithme de Brill-Noether et codes de Goppa","volume":"116","author":"Le Brigand, D.","year":"1988","journal-title":"Bull. Soc. Math. France","ISSN":"https:\/\/id.crossref.org\/issn\/0037-9484","issn-type":"print"},{"key":"22","isbn-type":"print","first-page":"203","article-title":"On complete ideals in regular local rings","author":"Lipman, Joseph","year":"1988","ISBN":"https:\/\/id.crossref.org\/isbn\/4314100028"},{"key":"23","series-title":"Cambridge Studies in Advanced Mathematics","isbn-type":"print","volume-title":"Commutative ring theory","volume":"8","author":"Matsumura, Hideyuki","year":"1986","ISBN":"https:\/\/id.crossref.org\/isbn\/0521259169"},{"key":"24","unstructured":"M. Rybowicz, \u201cSur le calcul des places et des anneaux d\u2019entiers d\u2019un corps de fonctions alg\u00e9briques\", Ph.D. thesis, Limoges (1990)."},{"key":"25","series-title":"Mathematics and its Applications (Soviet Series)","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1007\/978-94-011-3810-9","volume-title":"Algebraic-geometric codes","volume":"58","author":"Tsfasman, M. A.","year":"1991","ISBN":"https:\/\/id.crossref.org\/isbn\/0792307275"}],"container-title":["Mathematics of Computation"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/www.ams.org\/mcom\/2002-71-240\/S0025-5718-01-01390-4\/S0025-5718-01-01390-4.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"},{"URL":"https:\/\/www.ams.org\/mcom\/2002-71-240\/S0025-5718-01-01390-4\/S0025-5718-01-01390-4.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,4,20]],"date-time":"2026-04-20T23:04:24Z","timestamp":1776726264000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.ams.org\/mcom\/2002-71-240\/S0025-5718-01-01390-4\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2001,12,4]]},"references-count":25,"journal-issue":{"issue":"240","published-print":{"date-parts":[[2002,10]]}},"alternative-id":["S0025-5718-01-01390-4"],"URL":"https:\/\/doi.org\/10.1090\/s0025-5718-01-01390-4","archive":["CLOCKSS","Portico"],"relation":{},"ISSN":["1088-6842","0025-5718"],"issn-type":[{"value":"1088-6842","type":"electronic"},{"value":"0025-5718","type":"print"}],"subject":[],"published":{"date-parts":[[2001,12,4]]}}}