{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,20]],"date-time":"2026-08-20T14:33:48Z","timestamp":1787236428093,"version":"build-2736575974"},"reference-count":51,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"1","funder":[{"DOI":"10.13039\/100000001","name":"National Science Foundation","doi-asserted-by":"publisher","award":["DMS-1620014"],"award-info":[{"award-number":["DMS-1620014"]}],"id":[{"id":"10.13039\/100000001","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000001","name":"National Science Foundation","doi-asserted-by":"publisher","award":["CCF-1319632"],"award-info":[{"award-number":["CCF-1319632"]}],"id":[{"id":"10.13039\/100000001","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Appl. Algebra Geometry"],"published-print":{"date-parts":[[2020,1]]},"abstract":"<jats:p>In this paper, we adapt the differential signature construction to the equivalence problem for complex plane algebraic curves under the actions of the projective group and its subgroups. Given an action of a group $G$, a signature map assigns to a plane algebraic curve another plane algebraic curve (a signature curve) in such a way that two generic curves have the same signatures if and only if they are $G$-equivalent. We prove that for any $G$-action, there exists a pair of rational differential invariants, called classifying invariants, that can be used to construct signatures. We derive a formula for the degree of a signature curve in terms of the degree of the original curve, the size of its symmetry group, and some quantities depending on a choice of classifying invariants. We show that all generic curves have signatures of the same degree, and this degree is the sharp upper bound. For the full projective group, as well as for its affine, special affine, and special Euclidean subgroups, we give explicit sets of rational classifying invariants and derive a formula for the degree of the signature curve of a generic curve as a quadratic function of the degree of the original curve.<\/jats:p>","DOI":"10.1137\/19m1242859","type":"journal-article","created":{"date-parts":[[2020,3,10]],"date-time":"2020-03-10T09:22:25Z","timestamp":1583832145000},"page":"185-226","source":"Crossref","is-referenced-by-count":6,"title":["Differential Signatures of Algebraic Curves"],"prefix":"10.1137","volume":"4","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-8212-6296","authenticated-orcid":true,"given":"Irina A.","family":"Kogan","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Michael","family":"Ruddy","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Cynthia","family":"Vinzant","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2020,3,10]]},"reference":[{"key":"atypb1","unstructured":"M. Ackerman and R. Hermann,\n                      Hilbert's Invariant Theory Papers\n                      , Vol. 8, Math Science Press, Brookline, MA, 1978."},{"key":"atypb2","doi-asserted-by":"crossref","unstructured":"D. J. Bates, A. J. Sommese, J. D. Hauenstein, and C. W. Wampler,\n                      Numerically Solving Polynomial Systems with Bertini\n                      , Software Environ. Tools 25, SIAM, Philadelphia, 2013,https:\/\/doi.org\/10.1137\/1.9781611972702.","DOI":"10.1137\/1.9781611972702"},{"key":"atypb3","doi-asserted-by":"publisher","DOI":"10.1006\/S0747-7171(99)90307-3"},{"key":"atypb4","doi-asserted-by":"publisher","DOI":"10.1023\/A:1008139427340"},{"key":"atypb5","first-page":"023","volume":"9","author":"Burdis J. M.","year":"2013","journal-title":"SIGMA Symmetry Integrability Geom. Methods Appl."},{"key":"atypb6","doi-asserted-by":"publisher","DOI":"10.1023\/A:1007992709392"},{"key":"atypb7","unstructured":"E. Cartan,\n                      La m\u00e9thode du rep\u00e8re mobile, la th\u00e9orie des groupes continus, et les espaces g\u00e9n\u00e9ralis\u00e9s\n                      , Expos\u00e9s de G\u00e9om\u00e9trie 5, Hermann, Paris, 1935."},{"key":"atypb8","unstructured":"E. Cartan,\n                      Les probl\u00e8mes d'\u00e9quivalence\n                      , in Oeuvres Completes, Part II, Volume 2, Gauthier-Villars, Paris, 1953, pp. 1311-1334."},{"key":"atypb9","doi-asserted-by":"crossref","unstructured":"H. Derksen and G. Kemper,\n                      Computational Invariant Theory: Invariant Theory and Algebraic Transformation Groups\n                      , VIII, 2nd enlarged ed., Encyclopaedia of Mathematical Sciences 130, Springer, Heidelberg, 2015,https:\/\/doi.org\/10.1007\/978-3-662-04958-7.","DOI":"10.1007\/978-3-662-04958-7"},{"key":"atypb10","doi-asserted-by":"crossref","unstructured":"O. Faugeras,\n                      Cartan's moving frame method and its application to the geometry and evolution of curves in the Euclidean, affine and projective planes\n                      , in Application of Invariance in Computer Vision, J. L. Mundy, A. Zisserman, and D. Forsyth, eds., Lecture Notes in Comput. Sci. 825, Springer-Verlag, Berlin, 1994, pp. 11-46,https:\/\/doi.org\/10.1007\/3-540-58240-1_2.","DOI":"10.1007\/3-540-58240-1_2"},{"key":"atypb11","doi-asserted-by":"crossref","unstructured":"O. Faugeras and Q. T. Luong,\n                      The Geometry of Multiple Images. The Laws That Govern the Formation of Multiple Images of a Scene and Some of Their Applications\n                      , MIT Press, Cambridge, MA, 2001,https:\/\/doi.org\/10.1108\/ir.2002.29.3.287.2.","DOI":"10.7551\/mitpress\/3259.001.0001"},{"key":"atypb12","doi-asserted-by":"publisher","DOI":"10.1023\/A:1006195823000"},{"key":"atypb13","doi-asserted-by":"crossref","unstructured":"G. Fischer,\n                      Plane Algebraic Curves\n                      , Student Math. Library 15, AMS, Providence, RI, 2001; translated from the 1994 German original by Leslie Kay,https:\/\/doi.org\/10.1090\/stml\/015.","DOI":"10.1090\/stml\/015"},{"key":"atypb14","unstructured":"W. Fulton,\n                      Algebraic Curves. An Introduction to Algebraic Geometry\n                      , Advanced Book Classics, Addison-Wesley, Redwood City, CA, 1989."},{"key":"atypb15","unstructured":"J. H. Grace and A. Young,\n                      The Algebra of Invariants\n                      , Cambridge University Press, Cambridge, UK, 1903."},{"key":"atypb16","first-page":"171","author":"Grim A.","year":"2017","journal-title":"Cham"},{"key":"atypb17","doi-asserted-by":"publisher","DOI":"10.1142\/S0219467816500091"},{"key":"atypb18","doi-asserted-by":"publisher","DOI":"10.4171\/PM\/2032"},{"key":"atypb19","unstructured":"G. Gurevich,\n                      Foundations of the Theory of Algebraic Invariants\n                      , Noordhoff, Groningen, The Netherlands, 1964."},{"key":"atypb20","doi-asserted-by":"crossref","unstructured":"J. Harris,\n                      Algebraic Geometry. A First Course\n                      , corrected reprint of the 1992 original, Grad. Texts in Math. 133, Springer-Verlag, New York, 1995,https:\/\/doi.org\/10.1007\/978-1-4757-2189-8.","DOI":"10.1007\/978-1-4757-2189-8_11"},{"key":"atypb21","doi-asserted-by":"crossref","unstructured":"J. Harris and I. Morrison,\n                      Moduli of Curves\n                      , Springer-Verlag, New York, 1998,https:\/\/doi.org\/10.1007\/b98867.","DOI":"10.1007\/b98867"},{"key":"atypb22","doi-asserted-by":"crossref","unstructured":"R. Hartley and A. 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Texts in Math. 52, Springer-Verlag, New York, Heidelberg, 1977,https:\/\/doi.org\/10.1007\/978-1-4757-3849-0.","DOI":"10.1007\/978-1-4757-3849-0"},{"key":"atypb24","doi-asserted-by":"publisher","DOI":"10.1007\/s10851-011-0303-1"},{"key":"atypb25","doi-asserted-by":"publisher","DOI":"10.1007\/s10851-012-0358-7"},{"key":"atypb26","doi-asserted-by":"publisher","DOI":"10.1007\/s10851-013-0454-3"},{"key":"atypb27","doi-asserted-by":"publisher","DOI":"10.1016\/j.jsc.2006.03.005"},{"key":"atypb28","doi-asserted-by":"publisher","DOI":"10.1007\/s10208-006-0219-0"},{"key":"atypb29","doi-asserted-by":"crossref","unstructured":"T. W. Hungerford,\n                      Algbera\n                      , Grad. Texts in Math. 73, Springer-Verlag, New York, Berlin, 1980,https:\/\/doi.org\/10.1007\/978-1-4612-6101-8.","DOI":"10.1007\/978-1-4612-6101-8"},{"key":"atypb30","unstructured":"E. L. Ince,\n                      Ordinary Differential Equations\n                      , Dover Publications, New York, 1944."},{"key":"atypb31","unstructured":"I. A. Kogan,\n                      Inductive Approach to Cartan's Moving Frames Method with Applications to Classical Invariant Theory\n                      , preprint,https:\/\/arxiv.org\/abs\/1909.02055, 2019."},{"key":"atypb32","doi-asserted-by":"publisher","DOI":"10.4153\/CJM-2003-013-2"},{"key":"atypb33","unstructured":"I. A. Kogan, M. G. Ruddy, and C. Vinzant,\n                      Supplementary materials for \u201cDifferential Signature of Algebraic Curves\u201d paper\n                      ,https:\/\/mgruddy.wixsite.com\/home\/dsag-supplementarymaterials, 2019."},{"key":"atypb34","doi-asserted-by":"publisher","DOI":"10.1007\/s00029-015-0220-z"},{"key":"atypb35","doi-asserted-by":"crossref","unstructured":"S. Lie,\n                      Vorlesungen \u00fcber continuierliche Gruppen mit geometrischen und anderen Anwendungen\n                      , Chelsea Publishing, New York, NY, 1971; Bearbeitet und herausgegeben von Georg Scheffers, Nachdruck der Auflage des Jahres, 1893.","DOI":"10.5962\/bhl.title.18549"},{"key":"atypb36","doi-asserted-by":"publisher","DOI":"10.1007\/s10851-009-0155-0"},{"key":"atypb37","doi-asserted-by":"crossref","unstructured":"P. J. Olver,\n                      Applications of Lie Groups to Differential Equations\n                      , 2nd ed., Springer, New York, 1993,https:\/\/doi.org\/10.1007\/978-1-4684-0274-2.","DOI":"10.1007\/978-1-4612-4350-2"},{"key":"atypb38","doi-asserted-by":"crossref","unstructured":"P. J. 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Wears,\n                      Signature Varieties of Polynomial Functions\n                      , Ph.D. thesis, North Carolina State University, Raleigh, NC, 2011,http:\/\/www.lib.ncsu.edu\/resolver\/1840.16\/7336."}],"container-title":["SIAM Journal on Applied Algebra and Geometry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/19M1242859","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,20]],"date-time":"2026-08-20T13:26:31Z","timestamp":1787232391000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/19M1242859"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,1]]},"references-count":51,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2020,1]]}},"alternative-id":["10.1137\/19M1242859"],"URL":"https:\/\/doi.org\/10.1137\/19m1242859","relation":{},"ISSN":["2470-6566"],"issn-type":[{"value":"2470-6566","type":"electronic"}],"subject":[],"published":{"date-parts":[[2020,1]]}}}