An algebraic surface with affine equation
where
is a Chebyshev polynomial of the first kind and 
is a polynomial defined by
where the matrices have dimensions
. These represent surfaces in 
with only ordinary double points as singularities. The first few surfaces are given by
The dth order such surface has
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
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
(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)
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