Chmutov曲面[Chmutov Surface] - 达人(其实～我是卧底，来自外星)

2005/01/20 | Chmutov曲面[Chmutov Surface]
类别(∑〖数学〗) | 评论(0) | 阅读(32) | 发表于 12:34: An algebraic surface with affine equation
(1)

where is a Chebyshev polynomial of the first kind and is a polynomial defined by

(2)

where the matrices have dimensions . These represent surfaces in with only ordinary double points as singularities. The first few surfaces are given by
(3)

(4)

(5)

The dth order such surface has
(6)

singular points (Chmutov 1992), giving the sequence 0, 1, 3, 14, 28, 57, 93, 154, 216, 321, 425, 576, 732, 949, 1155, ... (Sloane's A057870) for d = 1, 2, .... For a number of orders d, Chmutov surfaces have more ordinary double points than any other known equations of the same degree.

Based on Chmutov's equations, Banchoff (1991) defined the simpler set of surfaces
(7)

where n is even and is again a Chebyshev polynomial of the first kind. For example, the surfaces illustrated above have orders 2, 4, and 6 and are given by the equations
(8)

(9)

(10); 0