Trott curve

In real algebraic geometry, the Trott curve is the set of points ("x","y") satisfying the degree four polynomial equation:$displaystyle\; 144(x^4+y^4)-225(x^2+y^2)+350x^2y^2+81=0.$These points form a nonsingular quartic plane curve that has genus three and that has twenty-eight real bitangents.

This curve was described in 1997 by Michael Trott of Wolfram Research. An explicit quartic with twenty-eight real bitangents and a very similar geometry to Trott's curve was already given by Plücker (1839); see e.g. Gray (1982). Tetsuji Shioda (1995) gave a different construction of a quartic with twenty-eight bitangents, formed by projecting a cubic surface; twenty-seven of the bitangents to Shioda's curve are real while the twenty-eighth is the line at infinity.

The Trott curve has four separated ovals, the maximum number for a curve of degree four, and hence is an M-curve. The four ovals can be grouped into six different pairs of ovals; for each pair of ovals there are four bitangents touching both ovals in the pair, two that separate the two ovals, and two that do not. Additionally, each oval bounds a nonconvex region of the plane and has one bitangent spanning the nonconvex portion of its boundary. The twenty-eight real bitangents of the Trott curve are the most possible for any degree four curve.

The dual curve to the Trott curve (pictured below) has twenty-eight real ordinary double points, dual to the twenty-eight bitangents of the primal Trott curve.

By adding a small fourth-degree polynomial with rational coefficients to the equation of the Trott curve, a process which has been called a "classical small perturbation" of the curve, we can almost always obtain another nonsingular curve with twenty-eight real bitangents, all of which have finite slope, and none of which have zero slope. In almost all cases, the slopes of these bitangents are algebraically conjugate values for an irreducible polynomial of degree twenty-eight over the rationals with all real roots, which has as its Galois group the Weyl group (Coxeter group) E₇ of degree 2903040, generated by reflections.

References

*cite journal
author = Gray, Jeremy
title = From the history of a simple group
journal = The Mathematical Intelligencer
volume = 4
issue = 2
year = 1982
id = MathSciNet | id = 0672918
pages = 59–67 [http://www.msri.org/communications/books/Book35/files/gray.pdf Reprinted] in cite book
author = Levy, Silvio, ed.
title = The Eightfold Way
series = MSRI Publications, 35
publisher = Cambridge University Press
year = 1999
pages = 115–131
id = MathSciNet | id = 1722415, ISBN 0521660661

*cite book
author = Plücker, J.
authorlink = Julius Plücker
title = Theorie der algebraischen Curven: gegrundet auf eine neue Behandlungsweise der analytischen Geometrie
year = 1839
publisher = Adolph Marcus
location = Berlin

*cite journal
author = Shioda, Tetsuji
title = Weierstrass transformations and cubic surfaces
journal = Commentarii Mathematici Universitatis Sancti Pauli
volume = 44
year = 1995
issue = 1
pages = 109–128
id = MathSciNet | id = 1336422
url = http://www.rkmath.rikkyo.ac.jp/math/shioda/papers/wtcs.PDF

*cite journal
author = Trott, Michael
title = Applying GroebnerBasis to Three Problems in Geometry
journal = Mathematica in Education and Research
volume = 6
issue = 1
pages = 15–28
year = 1997