{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,21]],"date-time":"2026-04-21T18:44:37Z","timestamp":1776797077852,"version":"3.51.2"},"reference-count":45,"publisher":"American Mathematical Society (AMS)","issue":"288","license":[{"start":{"date-parts":[[2015,3,13]],"date-time":"2015-03-13T00:00:00Z","timestamp":1426204800000},"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                    We present an algorithm for the computation of the topological type of a real compact Riemann surface associated to an algebraic curve, i.e., its genus and the properties of the set of fixed points of the anti-holomorphic involution\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"tau\">\n                        <mml:semantics>\n                          <mml:mi>\n                            \u03c4\n                            \n                          <\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">\\tau<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    , namely, the number of its connected components, and whether this set divides the surface into one or two connected components. This is achieved by transforming an arbitrary canonical homology basis to a homology basis where the\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"script upper A\">\n                        <mml:semantics>\n                          <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                            <mml:mi class=\"MJX-tex-caligraphic\" mathvariant=\"script\">A<\/mml:mi>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">\\mathcal {A}<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    -cycles are invariant under the anti-holomorphic involution\u00a0\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"tau\">\n                        <mml:semantics>\n                          <mml:mi>\n                            \u03c4\n                            \n                          <\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">\\tau<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    .\n                  <\/p>","DOI":"10.1090\/s0025-5718-2014-02817-2","type":"journal-article","created":{"date-parts":[[2014,3,13]],"date-time":"2014-03-13T09:44:31Z","timestamp":1394703871000},"page":"1823-1846","source":"Crossref","is-referenced-by-count":5,"title":["Computation of the topological type of a real Riemann surface"],"prefix":"10.1090","volume":"83","author":[{"given":"C.","family":"Kalla","sequence":"first","affiliation":[]},{"given":"C.","family":"Klein","sequence":"additional","affiliation":[]}],"member":"14","published-online":{"date-parts":[[2014,3,13]]},"reference":[{"issue":"3","key":"1","first-page":"1","article-title":"The situation of ovals of real plane algebraic curves, the involutions of four-dimensional smooth manifolds, and the arithmetic of integral quadratic forms","volume":"5","author":"Arnol\u2032d, V. I.","year":"1971","journal-title":"Funkcional. Anal. i Prilo\\v{z}en.","ISSN":"https:\/\/id.crossref.org\/issn\/0374-1990","issn-type":"print"},{"issue":"1-2","key":"2","doi-asserted-by":"publisher","first-page":"213","DOI":"10.1016\/S0747-7171(88)80013-0","article-title":"A polynomial-time algorithm for the topological type of a real algebraic curve","volume":"5","author":"Arnon, Dennis S.","year":"1988","journal-title":"J. Symbolic Comput.","ISSN":"https:\/\/id.crossref.org\/issn\/0747-7171","issn-type":"print"},{"key":"3","unstructured":"E. Belokolos, A. Bobenko, V. Enolskii, A. Its, V. Matveev, Algebro-geometric approach to nonlinear integrable equations, Springer Series in nonlinear dynamics (1994)."},{"key":"4","series-title":"Lecture Notes in Mathematics","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-642-17413-1","volume-title":"Computational approach to Riemann surfaces","volume":"2013","year":"2011","ISBN":"https:\/\/id.crossref.org\/isbn\/9783642174124"},{"issue":"43","key":"5","doi-asserted-by":"publisher","first-page":"434009","DOI":"10.1088\/1751-8113\/43\/43\/434009","article-title":"Klein\u2019s curve","volume":"43","author":"Braden, H. W.","year":"2010","journal-title":"J. Phys. A","ISSN":"https:\/\/id.crossref.org\/issn\/1751-8113","issn-type":"print"},{"key":"6","unstructured":"H. Braden, V. Enolskii, T. Northover, Maple packages extcurves and CyclePainter, available at gitorious.org."},{"issue":"2","key":"7","first-page":"307","article-title":"Application of Hoare\u2019s theorem to symmetries of Riemann surfaces","volume":"18","author":"Bujalance, E.","year":"1993","journal-title":"Ann. Acad. Sci. Fenn. Ser. A I Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0066-1953","issn-type":"print"},{"key":"8","series-title":"Lecture Notes in Mathematics","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-642-14828-6","volume-title":"Symmetries of compact Riemann surfaces","volume":"2007","author":"Bujalance, Emilio","year":"2010","ISBN":"https:\/\/id.crossref.org\/isbn\/9783642148279"},{"issue":"2","key":"9","doi-asserted-by":"publisher","first-page":"113","DOI":"10.1016\/0022-4049(94)00065-4","article-title":"Symmetries of Riemann surfaces on which \ud835\udc43\ud835\udc46\ud835\udc3f(2,\ud835\udc5e) acts as a Hurwitz automorphism group","volume":"106","author":"Broughton, S. A.","year":"1996","journal-title":"J. Pure Appl. Algebra","ISSN":"https:\/\/id.crossref.org\/issn\/0022-4049","issn-type":"print"},{"issue":"2","key":"10","first-page":"167","article-title":"Basis of homology adapted to the trigonal automorphism of a Riemann surface","volume":"101","author":"Campos, Helena B.","year":"2007","journal-title":"RACSAM. Rev. R. Acad. Cienc. Exactas F\\'{\\i}s. Nat. Ser. A Mat.","ISSN":"https:\/\/id.crossref.org\/issn\/1578-7303","issn-type":"print"},{"key":"11","doi-asserted-by":"crossref","unstructured":"A. Comessati, Sulla connessione delle superficie algebriche reali, Annali di Mat. (3) 23, 215\u2013283 (1915).","DOI":"10.1007\/BF02419577"},{"issue":"1-2","key":"12","doi-asserted-by":"publisher","first-page":"121","DOI":"10.1016\/S0747-7171(88)80008-7","article-title":"Thom\u2019s lemma, the coding of real algebraic numbers and the computation of the topology of semi-algebraic sets","volume":"5","author":"Coste, M.","year":"1988","journal-title":"J. Symbolic Comput.","ISSN":"https:\/\/id.crossref.org\/issn\/0747-7171","issn-type":"print"},{"key":"13","doi-asserted-by":"publisher","first-page":"28","DOI":"10.1016\/S0167-2789(01)00156-7","article-title":"Computing Riemann matrices of algebraic curves","volume":"152\/153","author":"Deconinck, Bernard","year":"2001","journal-title":"Phys. D","ISSN":"https:\/\/id.crossref.org\/issn\/0167-2789","issn-type":"print"},{"issue":"247","key":"14","doi-asserted-by":"publisher","first-page":"1417","DOI":"10.1090\/S0025-5718-03-01609-0","article-title":"Computing Riemann theta functions","volume":"73","author":"Deconinck, Bernard","year":"2004","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"key":"15","isbn-type":"print","doi-asserted-by":"publisher","first-page":"67","DOI":"10.1007\/978-3-642-17413-1_2","article-title":"Computing with plane algebraic curves and Riemann surfaces: the algorithms of the Maple package \u201calgcurves\u201d","author":"Deconinck, Bernard","year":"2011","ISBN":"https:\/\/id.crossref.org\/isbn\/9783642174124"},{"key":"16","doi-asserted-by":"crossref","unstructured":"B.A. Dubrovin, Matrix finite-zone operators, Revs. Sci. Tech. 23, 20\u201350 (1983).","DOI":"10.1007\/BF02104895"},{"issue":"2","key":"17","doi-asserted-by":"publisher","first-page":"267","DOI":"10.1070\/IM1989v032n02ABEH000759","article-title":"Real theta-function solutions of the Kadomtsev-Petviashvili equation","volume":"52","author":"Dubrovin, B. A.","year":"1988","journal-title":"Izv. Akad. Nauk SSSR Ser. Mat.","ISSN":"https:\/\/id.crossref.org\/issn\/0373-2436","issn-type":"print"},{"key":"18","unstructured":"H. Feng, Decomposition and Computation of the Topology of Plane Real Algebraic Curves, Ph.D. thesis, The Royal Institute of Technology, Stockholm (1992)."},{"issue":"1","key":"19","doi-asserted-by":"publisher","first-page":"193","DOI":"10.1016\/j.cam.2003.10.003","article-title":"Hyperelliptic theta-functions and spectral methods","volume":"167","author":"Frauendiener, J.","year":"2004","journal-title":"J. Comput. Appl. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0377-0427","issn-type":"print"},{"issue":"2-3","key":"20","doi-asserted-by":"publisher","first-page":"249","DOI":"10.1007\/s11005-006-0068-4","article-title":"Hyperelliptic theta-functions and spectral methods: KdV and KP solutions","volume":"76","author":"Frauendiener, J.","year":"2006","journal-title":"Lett. Math. Phys.","ISSN":"https:\/\/id.crossref.org\/issn\/0377-9017","issn-type":"print"},{"key":"21","isbn-type":"print","doi-asserted-by":"publisher","first-page":"125","DOI":"10.1007\/978-3-642-17413-1_3","article-title":"Algebraic curves and Riemann surfaces in Matlab","author":"Frauendiener, J\u00f6rg","year":"2011","ISBN":"https:\/\/id.crossref.org\/isbn\/9783642174124"},{"key":"22","unstructured":"A. Gabard, Sur la topologie et la g\u00e9om\u00e9trie des courbes alg\u00e9briques r\u00e9elles, Ph.D. Thesis (2004)."},{"issue":"9","key":"23","doi-asserted-by":"publisher","first-page":"719","DOI":"10.1016\/S0167-8396(02)00167-X","article-title":"Efficient topology determination of implicitly defined algebraic plane curves","volume":"19","author":"Gonzalez-Vega, Laureano","year":"2002","journal-title":"Comput. Aided Geom. Design","ISSN":"https:\/\/id.crossref.org\/issn\/0167-8396","issn-type":"print"},{"key":"24","first-page":"118","article-title":"Complete topological classification of the disposition of ovals of a sixth order curve in the projective plane","volume":"87","author":"Gudkov, D. A.","year":"1969","journal-title":"Gor\\cprime kov. Gos. Univ. U\\v{c}en. Zap."},{"issue":"2","key":"25","doi-asserted-by":"publisher","first-page":"189","DOI":"10.1007\/BF01442458","article-title":"Ueber die Vieltheiligkeit der ebenen algebraischen Curven","volume":"10","author":"Harnack, Axel","year":"1876","journal-title":"Math. Ann.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5831","issn-type":"print"},{"issue":"6","key":"26","doi-asserted-by":"publisher","first-page":"1068","DOI":"10.1137\/0220067","article-title":"Asymptotically fast triangularization of matrices over rings","volume":"20","author":"Hafner, James L.","year":"1991","journal-title":"SIAM J. Comput.","ISSN":"https:\/\/id.crossref.org\/issn\/0097-5397","issn-type":"print"},{"issue":"1","key":"27","doi-asserted-by":"publisher","first-page":"115","DOI":"10.1007\/BF01212696","article-title":"Ueber die reellen Z\u00fcge algebraischer Curven","volume":"38","author":"Hilbert, David","year":"1891","journal-title":"Math. Ann.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5831","issn-type":"print"},{"key":"28","unstructured":"D. Hilbert, Mathematische Probleme, Arch. Math. Phys., 1:43\u201363, (German) (1901)."},{"key":"29","isbn-type":"print","doi-asserted-by":"publisher","first-page":"221","DOI":"10.1017\/CBO9780511661884.014","article-title":"The orientability of subgroups of plane groups","author":"Hoare, A. H. M.","year":"1982","ISBN":"https:\/\/id.crossref.org\/isbn\/0521289742"},{"issue":"4-6","key":"30","doi-asserted-by":"publisher","first-page":"571","DOI":"10.1016\/S0378-4754(96)00034-1","article-title":"An efficient method for analyzing the topology of plane real algebraic curves","volume":"42","author":"Hong, Hoon","year":"1996","journal-title":"Math. Comput. Simulation","ISSN":"https:\/\/id.crossref.org\/issn\/0378-4754","issn-type":"print"},{"issue":"18","key":"31","doi-asserted-by":"publisher","first-page":"4170","DOI":"10.1093\/imrn\/rns175","article-title":"New degeneration of Fay\u2019s identity and its application to integrable systems","author":"Kalla, Caroline","year":"2013","journal-title":"Int. Math. Res. Not. IMRN","ISSN":"https:\/\/id.crossref.org\/issn\/1073-7928","issn-type":"print"},{"issue":"3","key":"32","doi-asserted-by":"publisher","first-page":"569","DOI":"10.1088\/0951-7715\/25\/3\/569","article-title":"On the numerical evaluation of algebro-geometric solutions to integrable equations","volume":"25","author":"Kalla, C.","year":"2012","journal-title":"Nonlinearity","ISSN":"https:\/\/id.crossref.org\/issn\/0951-7715","issn-type":"print"},{"key":"33","volume-title":"On Riemann's theory of algebraic functions and their integrals. A supplement to the usual treatises","author":"Klein, Felix","year":"1963"},{"key":"34","series-title":"Graduate Texts in Mathematics","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4613-0041-0","volume-title":"Algebra","volume":"211","author":"Lang, Serge","year":"2002","ISBN":"https:\/\/id.crossref.org\/isbn\/038795385X","edition":"3"},{"issue":"1","key":"35","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1007\/BF02430624","article-title":"Finite-gap solutions of the Davey-Stewartson equations","volume":"4","author":"Malanyuk, T. M.","year":"1994","journal-title":"J. Nonlinear Sci.","ISSN":"https:\/\/id.crossref.org\/issn\/0938-8974","issn-type":"print"},{"issue":"2","key":"36","doi-asserted-by":"publisher","first-page":"391","DOI":"10.1007\/s00209-002-0480-0","article-title":"Geometry and algebra of real forms of complex curves","volume":"243","author":"Natanzon, S. M.","year":"2003","journal-title":"Math. Z.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5874","issn-type":"print"},{"issue":"1","key":"37","doi-asserted-by":"publisher","first-page":"189","DOI":"10.2307\/1968723","article-title":"On the topology of real plane algebraic curves","volume":"39","author":"Petrowsky, I.","year":"1938","journal-title":"Ann. of Math. (2)","ISSN":"https:\/\/id.crossref.org\/issn\/0003-486X","issn-type":"print"},{"issue":"2","key":"38","doi-asserted-by":"publisher","first-page":"177","DOI":"10.1007\/BF01456711","article-title":"Die Maximalzahl und Anordnung der Ovale bei der ebenen Kurve 6. Ordnung und bei der Fl\u00e4che 4. Ordnung","volume":"73","author":"Rohn, K.","year":"1913","journal-title":"Math. Ann.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5831","issn-type":"print"},{"issue":"1","key":"39","doi-asserted-by":"publisher","first-page":"37","DOI":"10.1017\/S0004972700028756","article-title":"The topological configuration of a real algebraic curve","volume":"43","author":"Sakkalis, Takis","year":"1991","journal-title":"Bull. Austral. Math. Soc.","ISSN":"https:\/\/id.crossref.org\/issn\/0004-9727","issn-type":"print"},{"key":"40","isbn-type":"print","doi-asserted-by":"publisher","first-page":"107","DOI":"10.1145\/1064092.1064111","article-title":"On the exact computation of the topology of real algebraic curves","author":"Seidel, Raimund","year":"2005","ISBN":"https:\/\/id.crossref.org\/isbn\/1581139918"},{"issue":"2","key":"41","doi-asserted-by":"publisher","first-page":"151","DOI":"10.1007\/BF01160673","article-title":"Moduli spaces for real algebraic curves and real abelian varieties","volume":"201","author":"Sepp\u00e4l\u00e4, M.","year":"1989","journal-title":"Math. Z.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5874","issn-type":"print"},{"key":"42","isbn-type":"print","doi-asserted-by":"publisher","first-page":"467","DOI":"10.1090\/conm\/033\/767125","article-title":"Combinatorial group theory, Riemann surfaces and differential equations","author":"Tretkoff, C. L.","year":"1984","ISBN":"https:\/\/id.crossref.org\/isbn\/0821850350"},{"key":"43","unstructured":"M. Trott, Applying Groebner basis to three problems in geometry, Mathematica in Education and Research 6 (1): 15\u201328 (1997)."},{"issue":"3","key":"44","doi-asserted-by":"publisher","first-page":"453","DOI":"10.1007\/BF01445115","article-title":"Selfadjoint determinantal representations of real plane curves","volume":"296","author":"Vinnikov, Victor","year":"1993","journal-title":"Math. Ann.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5831","issn-type":"print"},{"key":"45","isbn-type":"print","doi-asserted-by":"publisher","first-page":"187","DOI":"10.1007\/BFb0099934","article-title":"Gluing of plane real algebraic curves and constructions of curves of degrees 6 and 7","author":"Viro, O. Ya.","year":"1984","ISBN":"https:\/\/id.crossref.org\/isbn\/3540133372"}],"container-title":["Mathematics of Computation"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/www.ams.org\/mcom\/2014-83-288\/S0025-5718-2014-02817-2\/S0025-5718-2014-02817-2.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"},{"URL":"https:\/\/www.ams.org\/mcom\/2014-83-288\/S0025-5718-2014-02817-2\/S0025-5718-2014-02817-2.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,4,21]],"date-time":"2026-04-21T17:58:05Z","timestamp":1776794285000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.ams.org\/mcom\/2014-83-288\/S0025-5718-2014-02817-2\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2014,3,13]]},"references-count":45,"journal-issue":{"issue":"288","published-print":{"date-parts":[[2014,7]]}},"alternative-id":["S0025-5718-2014-02817-2"],"URL":"https:\/\/doi.org\/10.1090\/s0025-5718-2014-02817-2","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":[[2014,3,13]]}}}