Page 160 - Textos de Matemática Vol. 44

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H. NAKAZATO, A. KOVACˇEC, N. BEBIANO AND J. DA PROVIDEˆNCIA
Next we have
{aω+gj (a)ω2 + c : 0 ≤ a ≤ 1, j = 1, 2}∪
∪ {aω + gj (a)ω2 + c : 0 ≤ a ≤ 1/9, j = 3, 4} ={1(2eiθ +e−2iθ):0≤θ≤2π}
3
= { 1 ( 2 cos θ + cos(2θ) + i(2 sin θ − sin(2θ)) ) : 0 ≤ θ ≤ 2π}. 3
This curve is known as a deltoid. To see the validity of the transformation to
trigonometric functions given in the last equation, use c = 1 − a − b and check
that aω+bω2 +c is equal to (1/2)(2−3a−3b)+(1/2)√3(a−b)i. Let u,v be
the real and imaginary part of this expression. The map (a,b) → (u,v) is an
affine bijection of R2 to itself and its inverse is given by a = (1/3)(1−u+√3v),
√
and b = (1/3)(1 − u −
above, we get a polynomial relation in u, v, namely
3v). Plugging this into the polynomial relation xpr==0
g(u, v) = −(1/27) + (2u2)/9 − (8u3)/27 + u4/9
+ (2v2)/9 + (8uv2)/9 + (2u2v2)/9 + v4/9
= 0,
which is an affine image of the closed curve above. One finally checks that
g( 1 (2 cos t + cos(2t)), 1 (2 sin t − sin(2t)) = 0. Since the pair of trigonometric 33
functions here can be seen to be injective on [0, 2π[ it is a simple closed curve and for topological reasons must coincide with the union of sets above. An alternative way to show this would be to establish irreducibility of g in C[u, v]. This is what we wished to show.
Now for a matrix A = A(a,b,c) ∈ TU(3) as above we consider a product as in(2.3)withλj =ωj:
33
( aljλj) l=1 j=1
Hence
= (aω+bω2 +c)(cω+aω2 +b)(bω+cω2 +a)
2
= ωj(aω+bω2+c)
j=0
= (aω+bω2 +c)3
{(aω + bω2 + c)3 : A(a, b, c) ∈ T U (3)} = { 1 (2eiθ + e−2iθ )3 : 0 ≤ θ ≤ 2π} 27

⊆ W3 (D0),