Oops --- sequence should have read 0 0 0 2 21 124 605 2736 11963 51416 218709 923680 3879755 On 10/23/06, Fred lunnon <fred.lunnon@gmail.com> wrote:
The plane curve defined as the locus of the (homogeneous) point (X,Y,Z) satisfying the equation X^n + Y^n + Z^n = 0 has a dual form as the envelope of (tangent) line A x + B y + C z = 0, where the (A,B,C) satisfy a polynomial equation of degree m = n(n-1) --- more specifically, a polynomial of degree n-1 in A^n, B^n, C^n.
Example --- n = 4, point poly = X^4 + Y^4 + Z^4, dual poly = A^12 + 3*A^8*B^4 + 3*A^8*C^4 + 3*A^4*B^8 - 21*A^4*B^4*C^4 + 3*A^4*C^8 + B^12 + 3*B^8*C^4 + 3*B^4*C^8 + C^12
Question --- why does m = n(n-1) ?
The coefficients (1 and 3 above) of A^(k+1)n B^(n-k) appear to be binomial coeffcients; but instead of the multinomial 6 expected, we find 21 above.
So define S_n = - coefficient of A^n(n-3) B^n C^n in the dual polynomial. Magma (1000 times faster than Maple by the way!) finds for n = 0..12
S_n = 0 0 0 0 2 21 605 2736 11963 51416 218709 923680 3879755 ...
Question --- what is this sequence? (OEIS doesn't recognise it)
Fred Lunnon