An automorphic classification of real cubic curves. (English) Zbl 1452.14021

Summary: The action of ring automorphisms of \(\mathbb{R}[x, y]\) on real plane curves is considered. The orbits containing degree-three polynomials are computed, with one representative per orbit being selected.


14H10 Families, moduli of curves (algebraic)
14P05 Real algebraic sets
14R10 Affine spaces (automorphisms, embeddings, exotic structures, cancellation problem)
14H37 Automorphisms of curves
Full Text: DOI arXiv Euclid


[1] S. S. Abhyankar and T. T. Moh, “Embeddings of the line in the plane”, J. Reine Angew. Math. 276 (1975), 148-166. Mathematical Reviews (MathSciNet): MR379502
Zentralblatt MATH: 0332.14004
· Zbl 0332.14004
[2] S. S. Abhyankar, W. J. Heinzer, and A. Sathaye, “Translates of polynomials”, pp. 51-124 in A tribute to C. S. Seshadri, edited by V. Lakshmibai et al., Hindustan Book Agency, Gurgaon, India, 2003. Mathematical Reviews (MathSciNet): MR2017580
Zentralblatt MATH: 1053.14011
· Zbl 1053.14011
[3] M. Bly, Classifications of real conic and cubic curves, Ph.D. Thesis, University of Tennessee, 2018, https://trace.tennessee.edu/utk_graddiss/5021/. URL: Link to item
[4] R. S. Burington, An invariant classification of plane cubic curves under the affine group, Ph.D. Thesis, Ohio State University, 1931. Zentralblatt MATH: 57.0831.04
· JFM 57.0831.04
[5] A. Cayley, “On the classification of cubic curves”, pp. 81-128 in Transactions of the Cambridge Philosophical Society, vol. 11, 1866. · JFM 01.0168.01
[6] A. B. Korchagin, “Newtonian and affine classifications of irreducible cubics”, Algebra i Analiz 24:5 (2012), 94-123. Mathematical Reviews (MathSciNet): MR3087822
Zentralblatt MATH: 1308.51022
· Zbl 1308.51022
[7] M. Nadjafikah and A.-R. Forough, “Classification of cubics up to affine transformations”, Differ. Geom. Dyn. Syst. 8 (2006), 184-195. Mathematical Reviews (MathSciNet): MR2220723
Zentralblatt MATH: 1161.53301
· Zbl 1161.53301
[8] I. Newton, Enumeratio linearum tertii ordinis, 1704. Zentralblatt MATH: 0079.24203
· Zbl 0079.24203
[9] J. Plücker, System der analytishen geometrie, Dunker und Humlot, Berlin, 1835.
[10] D. · Zbl 0668.14021
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