Trott curve
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The Trott curve is an example curve created by Michael Trott of Wolfram Research which is a nonsingular plane algebraic curve of degree four and genus three with twenty-eight real bitangents. It has the equation
- 144(x4 + y4) − 225(x2 + y2) + 350x2y2 + 81 = 0.
The curve has four separated ovals, the maximum number for a curve of degree four, and hence is an M-curve.
Since the curve has twenty-eight (the maximum number) of real bitangents, the dual curve (pictured below) has twenty-eight real ordinary double points.
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 Galois group the Weyl group (Coxeter group) E7 of degree 2903040, generated by reflections.
[edit] References
Michael Trott, Applying GroebnerBasis to Three Problems in Geometry, Mathematica in Education and Research, 6, pp 15-28, 1997
