Let $F_0,\ldots,F_n$ be $n+1$ homogeneous polynomials in $n+1$ variables $x_0,\ldots,x_n$ over a commutative ring $K$. The resultant $R(F_0,\ldots,F_n)$ is a certain polynomial in the coefficients of $F_0,\ldots,F_n$; when $K$ is an algebraically closed field, $R(F_0,\ldots,F_n)$ vanishes if and only if $F_0,\ldots,F_n$ have a common nontrivial root. For the general theory, see one of the following: Using Algebraic Geometry, by David A. Cox, John Little, Donal O'shea; Discriminants, Resultants, and Multidimensional Determinants, by Israel M. Gelfand, Mikhail M. Kapranov and Andrei V. Zelevinsky.
i1 : ZZ[t,u][x,y,z] o1 = ZZ[t..u][x..z] o1 : PolynomialRing |
i2 : F = {x^2+3*t*y*z-u*z^2,(t+3*u-1)*x-y,-t*x*y^3+t*x^2*y*z+u*z^4} 2 2 3 2 4 o2 = {x + 3t*y*z - u*z , (t + 3u - 1)x - y, - t*x*y + t*x y*z + u*z } o2 : List |
i3 : time resultant F -- used 0.0512634 seconds 12 11 2 10 3 9 4 8 5 7 6 6 7 5 8 11 10 2 o3 = - 81t u - 1701t u - 15309t u - 76545t u - 229635t u - 413343t u - 413343t u - 177147t u + 567t u + 10206t u + ---------------------------------------------------------------------------------------------------------------------------- 9 3 8 4 7 5 6 6 5 7 4 8 3 9 2 10 10 9 2 76545t u + 306181t u + 688923t u + 826821t u + 413883t u + 1215t u + 1458t u + 729t u - 1701t u - 25515t u - ---------------------------------------------------------------------------------------------------------------------------- 8 3 7 4 6 5 5 6 4 7 3 8 2 9 9 8 2 7 3 153093t u - 459321t u - 689265t u - 414693t u - 2835t u - 3159t u - 1458t u + 2835t u + 33984t u + 152565t u + ---------------------------------------------------------------------------------------------------------------------------- 6 4 5 5 4 6 3 7 2 8 8 7 2 6 3 5 4 4 5 303135t u + 220905t u - 12150t u - 5913t u + 1215t u - 2862t u - 25659t u - 75873t u - 70031t u + 16263t u + ---------------------------------------------------------------------------------------------------------------------------- 3 6 2 7 7 6 2 5 3 4 4 3 5 2 6 6 5 2 4 3 13230t u - 540t u + 1809t u + 10818t u + 15015t u - 6609t u - 9360t u + 135t u - 729t u - 2313t u + 684t u + ---------------------------------------------------------------------------------------------------------------------------- 3 4 2 5 6 5 4 2 3 3 2 4 5 4 3 2 2 3 4 2 2 3 3165t u - 81t u - 54t*u + 189t u + 135t u - 583t u - 38t u + 54t*u - 27t u + 54t u + 47t u - 18t*u - 9t u + 2t*u ---------------------------------------------------------------------------------------------------------------------------- 2 + u o3 : ZZ[t..u] |
Below we compute the general expression of the resultant of a linear, a quadratic and a cubic form on $\mathbb{P}^2$.
i4 : F = genericPolynomials({1,2,3},ZZ) 2 2 2 3 2 2 3 2 o4 = {a x + a x + a x , b x + b x x + b x + b x x + b x x + b x , c x + c x x + c x x + c x + c x x + c x x x + 0 0 1 1 2 2 0 0 1 0 1 3 1 2 0 2 4 1 2 5 2 0 0 1 0 1 3 0 1 6 1 2 0 2 4 0 1 2 ---------------------------------------------------------------------------------------------------------------------------- 2 2 2 3 c x x + c x x + c x x + c x } 7 1 2 5 0 2 8 1 2 9 2 o4 : List |
i5 : time resultant F -- used 9.45588 seconds 6 3 2 5 2 2 2 4 2 2 3 3 3 2 2 4 2 2 3 3 2 4 2 2 2 4 2 2 2 5 2 2 o5 = a b c - 3a a b b c + 3a a b b c - a a b c + 3a a b b c - 6a a b b b c + 3a a b b c + 3a a b b c - 3a a b b c + 2 3 0 1 2 3 4 0 1 2 3 4 0 1 2 4 0 1 2 3 5 0 1 2 3 4 5 0 1 2 4 5 0 1 2 3 5 0 1 2 4 5 0 ---------------------------------------------------------------------------------------------------------------------------- 6 3 2 6 2 5 2 5 2 4 5 2 2 4 2 3 3 2 a b c - a b b c c + a a b b c c + 2a a b b b c c - 2a a b b b c c + a a b b c c - a a b b c c + a a b b c c - 1 5 0 2 1 3 0 1 1 2 2 3 0 1 1 2 1 3 4 0 1 1 2 2 3 4 0 1 0 2 3 4 0 1 1 2 1 4 0 1 1 2 2 4 0 1 ---------------------------------------------------------------------------------------------------------------------------- 4 2 2 3 3 2 4 3 3 4 2 3 3 4 2 2a a a b b c c + a a a b c c - 2a a b b b c c + 2a a b b b c c - 2a a a b b c c + 2a a b b b c c - 2a a b b b c c + 0 1 2 3 4 0 1 0 1 2 4 0 1 1 2 1 3 5 0 1 1 2 2 3 5 0 1 0 1 2 3 5 0 1 1 2 1 4 5 0 1 1 2 2 4 5 0 1 ---------------------------------------------------------------------------------------------------------------------------- 2 3 3 2 2 4 2 2 5 2 3 2 2 4 2 5 3 6a a a b b b c c - 4a a a b b c c - a a b b c c + a a b b c c - 4a a a b b c c + 5a a a b b c c - 2a a b c c + 0 1 2 3 4 5 0 1 0 1 2 4 5 0 1 1 2 1 5 0 1 1 2 2 5 0 1 0 1 2 3 5 0 1 0 1 2 4 5 0 1 0 1 5 0 1 ---------------------------------------------------------------------------------------------------------------------------- 6 2 2 5 2 2 5 2 4 2 2 4 2 2 2 3 2 2 2 4 2 2 3 2 a b b c - a a b b c - 2a a b b b c + 2a a a b b b c + a a b b c - a a a b b c + 2a a b b b c - 2a a a b b b c + 2 0 3 1 0 2 2 3 1 1 2 0 3 4 1 0 1 2 2 3 4 1 1 2 0 4 1 0 1 2 2 4 1 1 2 0 3 5 1 0 1 2 2 3 5 1 ---------------------------------------------------------------------------------------------------------------------------- 2 4 2 2 3 3 2 3 2 2 2 3 2 2 2 2 2 2 4 2 2 2 4 2 2 2 2 2 2 2 a a b b c - 2a a b b b c + 2a a a b b b c - 2a a a b b b c + a a a b b c + a a b b c - a a a b b c + 2a a a b b c - 0 2 3 5 1 1 2 0 4 5 1 0 1 2 2 4 5 1 0 1 2 3 4 5 1 0 1 2 4 5 1 1 2 0 5 1 0 1 2 2 5 1 0 1 2 3 5 1 ---------------------------------------------------------------------------------------------------------------------------- 2 3 2 2 2 4 3 2 5 2 2 4 2 5 3 2 4 3 3 4 2 2a a a b b c + a a b c + a a b b c c - a a b b c c - 2a a b c c - 2a a b b b c c + 2a a b b b c c + 5a a a b b c c + 0 1 2 4 5 1 0 1 5 1 1 2 1 3 0 2 1 2 2 3 0 2 0 2 3 0 2 1 2 1 3 4 0 2 1 2 2 3 4 0 2 0 1 2 3 4 0 2 ---------------------------------------------------------------------------------------------------------------------------- 3 3 2 4 2 2 2 3 2 3 2 3 3 3 4 2 2 3 2 a a b b c c - a a b b c c - 4a a a b b c c + a a a b c c + 2a a b b b c c - 2a a b b b c c - 4a a a b b c c - 1 2 1 4 0 2 1 2 2 4 0 2 0 1 2 3 4 0 2 0 1 2 4 0 2 1 2 1 3 5 0 2 1 2 2 3 5 0 2 0 1 2 3 5 0 2 ---------------------------------------------------------------------------------------------------------------------------- 4 2 5 3 2 4 2 5 2 6 2 4 2 2a a b b b c c + 2a a b b b c c + 6a a a b b b c c - 2a a a b b c c + a a b b c c - a b b c c - 2a a a b b c c + 1 2 1 4 5 0 2 1 2 2 4 5 0 2 0 1 2 3 4 5 0 2 0 1 2 4 5 0 2 1 2 1 5 0 2 1 2 5 0 2 0 1 2 3 5 0 2 ---------------------------------------------------------------------------------------------------------------------------- 5 2 5 2 5 2 4 2 2 4 4 2 3 a a b b c c - 2a a b b c c + a a b b c c + a a a b b c c + 4a a b b b c c - 2a a a b b b c c - 2a a a b b b c c - 0 1 4 5 0 2 1 2 0 3 1 2 0 2 1 3 1 2 0 1 2 2 3 1 2 1 2 0 3 4 1 2 0 1 2 1 3 4 1 2 0 1 2 2 3 4 1 2 ---------------------------------------------------------------------------------------------------------------------------- 2 4 2 3 3 2 2 3 2 3 2 2 2 3 2 2 2 2 3 3 3 a a b b c c - 2a a b b c c + a a a b b c c + a a a b b c c + 2a a a b b c c - a a a b c c - 4a a b b b c c + 0 2 3 4 1 2 1 2 0 4 1 2 0 1 2 1 4 1 2 0 1 2 2 4 1 2 0 1 2 3 4 1 2 0 1 2 4 1 2 1 2 0 3 5 1 2 ---------------------------------------------------------------------------------------------------------------------------- 2 3 3 2 4 2 3 2 4 2 2 2 2a a a b b b c c + 2a a a b b b c c + 4a a b b b c c - 2a a a b b b c c - 2a a a b b b c c - 2a a a b b b c c + 0 1 2 1 3 5 1 2 0 1 2 2 3 5 1 2 1 2 0 4 5 1 2 0 1 2 1 4 5 1 2 0 1 2 2 4 5 1 2 0 1 2 3 4 5 1 2 ---------------------------------------------------------------------------------------------------------------------------- 2 3 2 5 2 4 2 5 2 2 4 2 2 4 2 2 4 2 2 2 4 3 2 2a a a b b c c - 2a a b b c c + a a a b b c c + a a b b c c - a a b b c c + a a b b c - a a a b b c + a a b c - 0 1 2 4 5 1 2 1 2 0 5 1 2 0 1 2 1 5 1 2 0 1 2 5 1 2 0 1 4 5 1 2 1 2 0 3 2 0 1 2 1 3 2 0 2 3 2 ---------------------------------------------------------------------------------------------------------------------------- 3 3 2 2 3 2 2 3 2 2 4 2 2 2 3 2 2 2 2 2 2 2 2 4 2 2 3 2 2 2a a b b b c + 2a a a b b b c - 2a a a b b c + a a b b c - a a a b b c + a a a b b c + 2a a b b b c - 2a a a b b b c 1 2 0 3 4 2 0 1 2 1 3 4 2 0 1 2 3 4 2 1 2 0 4 2 0 1 2 1 4 2 0 1 2 3 4 2 1 2 0 3 5 2 0 1 2 1 3 5 2 ---------------------------------------------------------------------------------------------------------------------------- 2 2 2 2 2 5 2 4 2 2 3 2 6 2 2 5 2 2 2 4 2 2 6 2 + 2a a a b b c - 2a a b b b c + 2a a a b b b c - 2a a a b b b c + a b b c - a a b b c + a a b b c + a b b c c - 0 1 2 3 5 2 1 2 0 4 5 2 0 1 2 1 4 5 2 0 1 2 3 4 5 2 1 0 5 2 0 1 1 5 2 0 1 3 5 2 2 1 3 0 3 ---------------------------------------------------------------------------------------------------------------------------- 5 2 4 2 6 2 5 2 5 2 2 4 3 3 2 2a a b b b c c + a a b b c c - 2a b b c c + 2a a b b c c - a a b b c c + 2a a b b b c c - a a b b c c + 1 2 1 2 3 0 3 1 2 2 3 0 3 2 0 3 0 3 0 2 2 3 0 3 1 2 1 4 0 3 1 2 1 2 4 0 3 1 2 2 4 0 3 ---------------------------------------------------------------------------------------------------------------------------- 5 5 4 2 4 2 4 2 2 4 2 2 3 3 4a a b b b c c - 2a a b b b c c - 2a a a b b b c c - 2a a b b c c + 2a a a b b c c + a a b b c c - a a a b c c + 1 2 0 3 4 0 3 0 2 1 3 4 0 3 0 1 2 2 3 4 0 3 1 2 0 4 0 3 0 1 2 1 4 0 3 0 2 3 4 0 3 0 1 2 4 0 3 ---------------------------------------------------------------------------------------------------------------------------- 2 4 2 3 3 4 2 2 2 4 4 2 4 2 3 3 a a b b c c - 2a a b b b c c + a a b b c c - 4a a b b b c c + 4a a a b b b c c - 2a a b b c c + 4a a b b b c c - 1 2 1 5 0 3 1 2 1 2 5 0 3 1 2 2 5 0 3 1 2 0 3 5 0 3 0 1 2 1 3 5 0 3 0 2 3 5 0 3 1 2 0 4 5 0 3 ---------------------------------------------------------------------------------------------------------------------------- 2 3 3 2 2 2 2 2 4 2 2 3 2 2 4 2 6a a a b b b c c + 2a a a b b b c c + 3a a a b b c c - 2a a b b c c + 4a a a b b c c - 2a a a b b c c - 0 1 2 1 4 5 0 3 0 1 2 2 4 5 0 3 0 1 2 4 5 0 3 1 2 0 5 0 3 0 1 2 1 5 0 3 0 1 2 2 5 0 3 ---------------------------------------------------------------------------------------------------------------------------- 2 3 2 2 4 3 6 5 5 4 2 5 4a a a b b c c + 2a a b c c - a b b b c c + a a b b b c c + a a b b b c c - a a a b b c c + a a b b b c c - 0 1 2 4 5 0 3 0 1 5 0 3 2 0 1 3 1 3 1 2 0 2 3 1 3 0 2 1 2 3 1 3 0 1 2 2 3 1 3 1 2 0 1 4 1 3 ---------------------------------------------------------------------------------------------------------------------------- 2 4 4 2 3 2 5 2 4 4 2 2 3 2 a a b b b c c - a a a b b b c c + a a a b b c c + a a b b b c c - a a b b b c c - a a a b b c c + a a a b b c c - 1 2 0 2 4 1 3 0 1 2 1 2 4 1 3 0 1 2 2 4 1 3 0 2 0 3 4 1 3 0 2 2 3 4 1 3 0 1 2 0 4 1 3 0 1 2 2 4 1 3 ---------------------------------------------------------------------------------------------------------------------------- 2 4 3 3 2 3 3 2 2 4 2 4 2 3 a a b b b c c + a a b b b c c + a a a b b b c c - a a a b b c c - 2a a a b b b c c - a a b b b c c + 3a a a b b b c c 1 2 0 1 5 1 3 1 2 0 2 5 1 3 0 1 2 1 2 5 1 3 0 1 2 2 5 1 3 0 1 2 0 3 5 1 3 0 2 1 3 5 1 3 0 1 2 2 3 5 1 3 ---------------------------------------------------------------------------------------------------------------------------- 2 3 2 3 2 2 2 3 3 3 2 2 3 2 2 + 3a a a b b b c c + a a a b b b c c - 4a a a b b b c c + a a b b b c c - a a a b b c c - 2a a a b b c c - 0 1 2 0 4 5 1 3 0 1 2 1 4 5 1 3 0 1 2 2 4 5 1 3 0 2 3 4 5 1 3 0 1 2 4 5 1 3 0 1 2 0 5 1 3 ---------------------------------------------------------------------------------------------------------------------------- 2 2 2 2 2 3 2 3 2 2 3 2 2 3 3 3 5 5 2 a a a b b c c + 3a a a b b c c - 2a a a b b c c + 3a a a b b c c - 2a a b c c + a a b b b c c - a a b b c c - 0 1 2 1 5 1 3 0 1 2 2 5 1 3 0 1 2 3 5 1 3 0 1 2 4 5 1 3 0 1 5 1 3 1 2 0 1 3 2 3 0 2 1 3 2 3 ---------------------------------------------------------------------------------------------------------------------------- 2 4 4 5 2 2 4 2 2 4 4 2 3 3 a a b b b c c + a a a b b b c c + 2a a b b c c - 2a a b b c c - a a b b b c c + a a a b b c c + a a b b b c c - 1 2 0 2 3 2 3 0 1 2 1 2 3 2 3 0 2 0 3 2 3 0 2 2 3 2 3 1 2 0 1 4 2 3 0 1 2 1 4 2 3 1 2 0 2 4 2 3 ---------------------------------------------------------------------------------------------------------------------------- 2 3 4 2 4 2 3 2 3 2 2 3 2 a a a b b b c c - 5a a a b b b c c + 2a a b b b c c + 3a a a b b b c c + 3a a a b b c c - 2a a a b b c c - 0 1 2 1 2 4 2 3 0 1 2 0 3 4 2 3 0 2 1 3 4 2 3 0 1 2 2 3 4 2 3 0 1 2 0 4 2 3 0 1 2 1 4 2 3 ---------------------------------------------------------------------------------------------------------------------------- 2 2 2 2 3 3 2 3 2 3 3 3 2 3 2 4 2 3 2 a a a b b c c - a a b b c c + a a a b c c + a a b b b c c - a a a b b c c - a a b b b c c + a a a b b b c c + 0 1 2 2 4 2 3 0 2 3 4 2 3 0 1 2 4 2 3 1 2 0 1 5 2 3 0 1 2 1 5 2 3 1 2 0 2 5 2 3 0 1 2 1 2 5 2 3 ---------------------------------------------------------------------------------------------------------------------------- 2 3 2 3 2 2 2 3 3 2 3 2 2 2 2 6a a a b b b c c - 3a a a b b b c c - 3a a a b b b c c + 2a a b b c c - 7a a a b b b c c + 5a a a b b b c c + 0 1 2 0 3 5 2 3 0 1 2 1 3 5 2 3 0 1 2 2 3 5 2 3 0 2 3 5 2 3 0 1 2 0 4 5 2 3 0 1 2 1 4 5 2 3 ---------------------------------------------------------------------------------------------------------------------------- 2 3 3 2 3 2 2 4 2 2 3 2 2 4 2 3 2 2 2a a a b b b c c - a a a b b b c c - 2a a a b b c c + 4a a a b b c c - 3a a a b b c c - a a b b c c + 2a a a b b c c 0 1 2 2 4 5 2 3 0 1 2 3 4 5 2 3 0 1 2 4 5 2 3 0 1 2 0 5 2 3 0 1 2 1 5 2 3 0 1 2 5 2 3 0 1 2 3 5 2 3 ---------------------------------------------------------------------------------------------------------------------------- 3 3 2 6 2 2 5 2 2 4 2 2 5 2 2 4 2 2 3 2 2 2 4 2 2 + a a b b c c + a b b c - 2a a b b b c + a a b b c - a a b b c + 2a a a b b b c - a a a b b c + a a b b c - 0 1 4 5 2 3 2 0 3 3 0 2 0 2 3 3 0 2 2 3 3 1 2 0 4 3 0 1 2 0 2 4 3 0 1 2 2 4 3 1 2 0 5 3 ---------------------------------------------------------------------------------------------------------------------------- 2 3 2 2 2 2 2 2 2 4 2 3 3 2 2 3 2 3 2 2 2 2 2 2 2 2a a a b b b c + a a a b b c + 2a a b b b c - 2a a b b b c - 2a a a b b b c + 2a a a b b b c + 2a a a b b c - 0 1 2 0 2 5 3 0 1 2 2 5 3 0 2 0 3 5 3 0 2 2 3 5 3 0 1 2 0 4 5 3 0 1 2 2 4 5 3 0 1 2 0 5 3 ---------------------------------------------------------------------------------------------------------------------------- 3 2 2 2 4 2 2 2 4 2 2 4 2 3 2 5 2 2 4 3 3 2 5 2 2a a a b b c + a a b b c - a a a b b c + a a b c - a a b b c c + 2a a b b b c c - a a b b c c + 2a a b b c c + 0 1 2 2 5 3 0 2 3 5 3 0 1 2 4 5 3 0 1 5 3 1 2 1 3 0 4 1 2 1 2 3 0 4 1 2 2 3 0 4 1 2 0 3 0 4 ---------------------------------------------------------------------------------------------------------------------------- 5 2 4 2 2 4 2 3 3 4 2 2 2 4 2 3 a a b b c c - 3a a a b b c c + a a b b c c - 2a a b b b c c + a a b b c c - 4a a b b b c c + 4a a a b b b c c - 0 2 1 3 0 4 0 1 2 2 3 0 4 1 2 1 4 0 4 1 2 1 2 4 0 4 1 2 2 4 0 4 1 2 0 3 4 0 4 0 1 2 2 3 4 0 4 ---------------------------------------------------------------------------------------------------------------------------- 2 4 2 3 3 2 2 3 2 3 2 2 2 3 2 3 3 2 4 2 a a b b c c + 2a a b b c c - a a a b b c c - a a a b b c c + a a a b b c c - a a b b c c + 2a a b b b c c - 0 2 3 4 0 4 1 2 0 4 0 4 0 1 2 1 4 0 4 0 1 2 2 4 0 4 0 1 2 3 4 0 4 1 2 1 5 0 4 1 2 1 2 5 0 4 ---------------------------------------------------------------------------------------------------------------------------- 5 2 3 3 2 3 3 2 2 3 2 4 2 a a b b c c + 4a a b b b c c - 2a a a b b b c c - 2a a a b b b c c + 4a a a b b c c - 4a a b b b c c + 1 2 2 5 0 4 1 2 0 3 5 0 4 0 1 2 1 3 5 0 4 0 1 2 2 3 5 0 4 0 1 2 3 5 0 4 1 2 0 4 5 0 4 ---------------------------------------------------------------------------------------------------------------------------- 3 2 2 2 2 2 3 2 5 2 4 2 5 2 2 3 2 4a a a b b b c c - 6a a a b b b c c + a a a b b c c + 2a a b b c c - 3a a a b b c c + a a b b c c + 4a a a b b c c - 0 1 2 1 4 5 0 4 0 1 2 3 4 5 0 4 0 1 2 4 5 0 4 1 2 0 5 0 4 0 1 2 1 5 0 4 0 1 2 5 0 4 0 1 2 3 5 0 4 ---------------------------------------------------------------------------------------------------------------------------- 2 4 2 5 2 4 4 2 3 2 5 2 2 4 2 a a b b c c + a a b b b c c - a a b b b c c - a a a b b b c c + a a a b b c c - 2a a b b c c + 2a a b b c c - 0 1 4 5 0 4 1 2 0 1 3 1 4 1 2 0 2 3 1 4 0 1 2 1 2 3 1 4 0 1 2 2 3 1 4 0 2 0 3 1 4 0 2 2 3 1 4 ---------------------------------------------------------------------------------------------------------------------------- 2 4 3 3 2 3 3 2 2 4 2 3 2 3 2 a a b b b c c + a a b b b c c + a a a b b b c c - a a a b b c c + 3a a a b b b c c - 3a a a b b b c c - a a a b b c c 1 2 0 1 4 1 4 1 2 0 2 4 1 4 0 1 2 1 2 4 1 4 0 1 2 2 4 1 4 0 1 2 0 3 4 1 4 0 1 2 2 3 4 1 4 0 1 2 0 4 1 4 ---------------------------------------------------------------------------------------------------------------------------- 2 2 2 2 3 3 4 2 3 2 4 2 2 3 + a a a b b c c + a a b b b c c - a a b b b c c - a a a b b b c c + a a a b b c c - 2a a a b b b c c + 0 1 2 2 4 1 4 1 2 0 1 5 1 4 1 2 0 2 5 1 4 0 1 2 1 2 5 1 4 0 1 2 2 5 1 4 0 1 2 0 3 5 1 4 ---------------------------------------------------------------------------------------------------------------------------- 2 3 2 2 2 3 3 2 3 2 2 2 2 3 2 a a a b b b c c + a a a b b b c c - 2a a b b c c + a a a b b b c c - a a a b b b c c + 3a a a b b b c c - 0 1 2 1 3 5 1 4 0 1 2 2 3 5 1 4 0 2 3 5 1 4 0 1 2 0 4 5 1 4 0 1 2 1 4 5 1 4 0 1 2 3 4 5 1 4 ---------------------------------------------------------------------------------------------------------------------------- 3 2 2 2 3 2 2 4 2 3 2 2 3 3 2 2 4 4 2 a a a b b c c + a a a b b c c - a a b b c c - 2a a a b b c c + a a b b c c - a a b b b c c + a a a b b c c + 0 1 2 4 5 1 4 0 1 2 1 5 1 4 0 1 2 5 1 4 0 1 2 3 5 1 4 0 1 4 5 1 4 1 2 0 1 3 2 4 0 1 2 1 3 2 4 ---------------------------------------------------------------------------------------------------------------------------- 3 3 2 3 2 4 2 2 3 2 3 3 2 3 2 4 2 a a b b b c c - a a a b b b c c - a a b b c c + a a a b b c c + a a b b b c c - a a a b b c c - a a b b b c c + 1 2 0 2 3 2 4 0 1 2 1 2 3 2 4 0 2 1 3 2 4 0 1 2 2 3 2 4 1 2 0 1 4 2 4 0 1 2 1 4 2 4 1 2 0 2 4 2 4 ---------------------------------------------------------------------------------------------------------------------------- 3 2 2 3 2 2 2 3 3 2 3 2 2 2 2 2 2 3 2 2 a a a b b b c c + a a a b b b c c - a a a b b b c c + a a b b c c - a a a b b c c + a a a b b c c - a a a b b c c - 0 1 2 1 2 4 2 4 0 1 2 0 3 4 2 4 0 1 2 2 3 4 2 4 0 2 3 4 2 4 0 1 2 0 4 2 4 0 1 2 1 4 2 4 0 1 2 3 4 2 4 ---------------------------------------------------------------------------------------------------------------------------- 4 2 3 2 2 5 4 3 2 2 2 2 a a b b b c c + a a a b b c c + a a b b b c c - a a a b b b c c - 2a a a b b b c c + a a a b b b c c + 1 2 0 1 5 2 4 0 1 2 1 5 2 4 1 2 0 2 5 2 4 0 1 2 1 2 5 2 4 0 1 2 0 3 5 2 4 0 1 2 1 3 5 2 4 ---------------------------------------------------------------------------------------------------------------------------- 2 3 3 2 2 4 2 3 3 2 5 2 a a a b b b c c - 2a a a b b c c + 3a a a b b b c c - 3a a a b b b c c + 3a a a b b b c c - 2a a b b c c + 0 1 2 2 3 5 2 4 0 1 2 3 5 2 4 0 1 2 0 4 5 2 4 0 1 2 1 4 5 2 4 0 1 2 3 4 5 2 4 0 1 0 5 2 4 ---------------------------------------------------------------------------------------------------------------------------- 2 4 2 3 3 2 5 2 5 4 2 4 2 3 2 2a a b b c c - 2a a b b c c - 2a a b b c c + a a b b b c c + 3a a a b b b c c - a a b b b c c - a a a b b c c + 0 1 1 5 2 4 0 1 3 5 2 4 1 2 0 3 3 4 0 2 0 1 3 3 4 0 1 2 0 2 3 3 4 0 2 1 2 3 3 4 0 1 2 2 3 3 4 ---------------------------------------------------------------------------------------------------------------------------- 2 4 2 4 2 3 2 3 2 2 2 2 2 4 3 3 2a a b b c c - a a a b b b c c - 3a a a b b b c c + a a a b b b c c + a a a b b c c - a a b b b c c + a a b b b c c + 1 2 0 4 3 4 0 1 2 0 1 4 3 4 0 1 2 0 2 4 3 4 0 1 2 1 2 4 3 4 0 1 2 2 4 3 4 0 2 0 3 4 3 4 0 2 2 3 4 3 4 ---------------------------------------------------------------------------------------------------------------------------- 2 3 2 3 2 2 3 3 2 2 3 3 2 2 2 2 2 3 2 a a a b b c c - a a a b b c c - 2a a b b c c + a a a b b b c c + 3a a a b b b c c - a a a b b b c c - a a a b b c c - 0 1 2 0 4 3 4 0 1 2 2 4 3 4 1 2 0 5 3 4 0 1 2 0 1 5 3 4 0 1 2 0 2 5 3 4 0 1 2 1 2 5 3 4 0 1 2 2 5 3 4 ---------------------------------------------------------------------------------------------------------------------------- 2 3 3 3 3 2 2 2 2 3 2 4 2 2a a a b b b c c + a a b b b c c + a a a b b b c c + a a a b b b c c - a a a b b b c c - a a b b b c c + 0 1 2 0 3 5 3 4 0 2 1 3 5 3 4 0 1 2 2 3 5 3 4 0 1 2 0 4 5 3 4 0 1 2 1 4 5 3 4 0 2 3 4 5 3 4 ---------------------------------------------------------------------------------------------------------------------------- 4 2 2 3 2 3 2 2 3 3 2 4 2 2 2 4 2 2 4 2 a a a b b c c - 2a a a b b c c + a a a b b c c + a a b b c c - a a b b c c + a a b b c - a a a b b b c - 0 1 2 4 5 3 4 0 1 2 0 5 3 4 0 1 2 1 5 3 4 0 1 2 5 3 4 0 1 4 5 3 4 1 2 0 3 4 0 1 2 0 1 3 4 ---------------------------------------------------------------------------------------------------------------------------- 2 3 2 2 3 2 2 4 2 2 3 3 2 2 3 3 2 2 2 3 2 3 2 2 2 2 2 2 a a a b b b c + a a a b b b c + a a b b c - a a b b c - a a b b c + a a a b b b c + a a a b b b c - a a a b b b c - 0 1 2 0 2 3 4 0 1 2 1 2 3 4 0 2 0 3 4 0 2 2 3 4 1 2 0 4 4 0 1 2 0 1 4 4 0 1 2 0 2 4 4 0 1 2 1 2 4 4 ---------------------------------------------------------------------------------------------------------------------------- 2 3 2 3 2 2 4 2 2 2 3 2 2 4 2 2 3 2 2 2 2 2 a a a b b b c + a a a b b b c + a a b b c - a a a b b b c - a a a b b b c + a a a b b b c + 2a a a b b b c - 0 1 2 0 3 4 4 0 1 2 2 3 4 4 1 2 0 5 4 0 1 2 0 1 5 4 0 1 2 0 2 5 4 0 1 2 1 2 5 4 0 1 2 0 3 5 4 ---------------------------------------------------------------------------------------------------------------------------- 3 2 2 3 2 2 4 2 2 2 2 3 2 3 2 2 4 2 2 4 2 2 3 3 2 2 a a a b b b c - a a a b b b c + a a b b c - a a a b b b c + a a a b b b c - a a a b b b c + a a b b c - a a b b c + 0 1 2 1 3 5 4 0 1 2 2 3 5 4 0 2 3 5 4 0 1 2 0 4 5 4 0 1 2 1 4 5 4 0 1 2 3 4 5 4 0 1 0 5 4 0 1 1 5 4 ---------------------------------------------------------------------------------------------------------------------------- 4 2 2 2 2 4 2 3 3 4 2 2 2 4 2 4 2 2 3 2 2 4 3 a a b b c + a a b b c c - 2a a b b b c c + a a b b c c - 2a a b b c c - 2a a a b b c c + 4a a a b b c c + 2a a b c c 0 1 3 5 4 1 2 1 3 0 5 1 2 1 2 3 0 5 1 2 2 3 0 5 1 2 0 3 0 5 0 1 2 1 3 0 5 0 1 2 2 3 0 5 0 2 3 0 5 ---------------------------------------------------------------------------------------------------------------------------- 3 3 2 4 2 5 2 3 3 2 3 3 2 2 3 2 - a a b b c c + 2a a b b b c c - a a b b c c + 4a a b b b c c + 2a a a b b b c c - 6a a a b b b c c - 4a a a b b c c 1 2 1 4 0 5 1 2 1 2 4 0 5 1 2 2 4 0 5 1 2 0 3 4 0 5 0 1 2 1 3 4 0 5 0 1 2 2 3 4 0 5 0 1 2 3 4 0 5 ---------------------------------------------------------------------------------------------------------------------------- 4 2 2 4 2 2 2 2 2 2 3 3 4 2 2 5 6 2 - 2a a b b c c + 2a a a b b c c + 3a a a b b c c - a a a b c c + a a b b c c - 2a a b b b c c + a b b c c - 1 2 0 4 0 5 0 1 2 2 4 0 5 0 1 2 3 4 0 5 0 1 2 4 0 5 1 2 1 5 0 5 1 2 1 2 5 0 5 1 2 5 0 5 ---------------------------------------------------------------------------------------------------------------------------- 4 2 4 5 4 5 2 4 2 6 2 4a a b b b c c + 4a a a b b b c c + 4a a b b b c c - 2a a a b b b c c - 2a a b b b c c + a a b b c c - 2a b b c c + 1 2 0 3 5 0 5 0 1 2 2 3 5 0 5 1 2 0 4 5 0 5 0 1 2 1 4 5 0 5 0 1 2 4 5 0 5 0 1 4 5 0 5 1 0 5 0 5 ---------------------------------------------------------------------------------------------------------------------------- 5 2 2 4 2 2 4 3 3 2 3 3 2 2 4 2 2a a b b c c - 2a a b b c c - a a b b b c c + a a b b b c c + a a a b b b c c - a a a b b c c + 4a a a b b c c - 0 1 1 5 0 5 0 1 3 5 0 5 1 2 0 1 3 1 5 1 2 0 2 3 1 5 0 1 2 1 2 3 1 5 0 1 2 2 3 1 5 0 1 2 0 3 1 5 ---------------------------------------------------------------------------------------------------------------------------- 2 4 2 2 3 2 3 3 4 2 3 2 4 2 2 3 a a b b c c - 3a a a b b c c + a a b b b c c - a a b b b c c - a a a b b b c c + a a a b b c c - 7a a a b b b c c + 0 2 1 3 1 5 0 1 2 2 3 1 5 1 2 0 1 4 1 5 1 2 0 2 4 1 5 0 1 2 1 2 4 1 5 0 1 2 2 4 1 5 0 1 2 0 3 4 1 5 ---------------------------------------------------------------------------------------------------------------------------- 2 3 2 2 2 3 3 2 3 2 2 2 2 2 2 2 3 2 3 2 2 2a a a b b b c c + 5a a a b b b c c + a a b b c c + 3a a a b b c c - a a a b b c c - 2a a a b b c c - 2a a a b b c c 0 1 2 1 3 4 1 5 0 1 2 2 3 4 1 5 0 2 3 4 1 5 0 1 2 0 4 1 5 0 1 2 1 4 1 5 0 1 2 2 4 1 5 0 1 2 3 4 1 5 ---------------------------------------------------------------------------------------------------------------------------- 3 2 3 4 2 5 4 5 2 3 2 2 2 2 + a a a b c c - a a b b b c c + a a b b b c c + a a a b b b c c - a a b b c c + 6a a a b b b c c - 3a a a b b b c c - 0 1 2 4 1 5 1 2 0 1 5 1 5 1 2 0 2 5 1 5 0 1 2 1 2 5 1 5 0 1 2 5 1 5 0 1 2 0 3 5 1 5 0 1 2 1 3 5 1 5 ---------------------------------------------------------------------------------------------------------------------------- 2 3 3 2 2 4 2 3 2 4 3 2 3a a a b b b c c + 2a a a b b c c - 5a a a b b b c c + 3a a a b b b c c + 2a a b b b c c - a a a b b b c c - 0 1 2 2 3 5 1 5 0 1 2 3 5 1 5 0 1 2 0 4 5 1 5 0 1 2 1 4 5 1 5 0 1 2 4 5 1 5 0 1 2 3 4 5 1 5 ---------------------------------------------------------------------------------------------------------------------------- 3 3 2 5 2 2 4 2 3 3 2 3 3 2 3 2 4 2 a a b b c c + 2a a b b c c - 2a a b b c c + 2a a b b c c + a a b b b c c - a a a b b c c - a a b b b c c + 0 1 4 5 1 5 0 1 0 5 1 5 0 1 1 5 1 5 0 1 3 5 1 5 1 2 0 1 3 2 5 0 1 2 1 3 2 5 1 2 0 2 3 2 5 ---------------------------------------------------------------------------------------------------------------------------- 3 2 2 3 2 2 3 2 2 2 2 2 3 3 3 4 2 3 2 2 a a a b b b c c - 2a a a b b c c + 3a a a b b c c - a a a b b c c - 2a a b c c - a a b b b c c + a a a b b c c + 0 1 2 1 2 3 2 5 0 1 2 0 3 2 5 0 1 2 1 3 2 5 0 1 2 2 3 2 5 0 2 3 2 5 1 2 0 1 4 2 5 0 1 2 1 4 2 5 ---------------------------------------------------------------------------------------------------------------------------- 5 4 3 2 2 2 2 2 3 3 2 2 a a b b b c c - a a a b b b c c + 3a a a b b b c c - 4a a a b b b c c + a a a b b b c c + 3a a a b b c c - 1 2 0 2 4 2 5 0 1 2 1 2 4 2 5 0 1 2 0 3 4 2 5 0 1 2 1 3 4 2 5 0 1 2 2 3 4 2 5 0 1 2 3 4 2 5 ---------------------------------------------------------------------------------------------------------------------------- 4 2 2 3 2 3 2 2 5 4 2 6 5 a a a b b c c + a a a b b c c - a a a b b c c + a a b b b c c - a a a b b c c - a b b b c c + a a b b b c c - 0 1 2 0 4 2 5 0 1 2 1 4 2 5 0 1 2 3 4 2 5 1 2 0 1 5 2 5 0 1 2 1 5 2 5 1 0 2 5 2 5 0 1 1 2 5 2 5 ---------------------------------------------------------------------------------------------------------------------------- 4 2 3 2 4 3 2 2 5 2 4 3 3 2a a a b b b c c + 3a a a b b b c c - a a b b b c c - 2a a a b b c c + a a b b b c c - a a b b b c c + a a b b b c c 0 1 2 0 3 5 2 5 0 1 2 1 3 5 2 5 0 1 2 3 5 2 5 0 1 2 3 5 2 5 0 1 0 4 5 2 5 0 1 1 4 5 2 5 0 1 3 4 5 2 5 ---------------------------------------------------------------------------------------------------------------------------- 2 4 2 4 2 4 2 2 3 2 2 2 2 2 4 2 3 3 2 + 2a a b b c c - 2a a a b b b c c + a a b b c c - 2a a a b b b c c + a a a b b c c - 2a a b b c c + 2a a b b c c - 1 2 0 3 3 5 0 1 2 0 1 3 3 5 0 2 1 3 3 5 0 1 2 0 2 3 3 5 0 1 2 2 3 3 5 0 2 0 3 3 5 0 2 2 3 3 5 ---------------------------------------------------------------------------------------------------------------------------- 3 3 2 2 3 2 3 2 3 2 2 3 2 2 3 2a a b b c c + 2a a a b b b c c - a a a b b c c + 2a a a b b b c c - a a a b b c c + 6a a a b b b c c - 1 2 0 4 3 5 0 1 2 0 1 4 3 5 0 1 2 1 4 3 5 0 1 2 0 2 4 3 5 0 1 2 2 4 3 5 0 1 2 0 3 4 3 5 ---------------------------------------------------------------------------------------------------------------------------- 3 3 3 2 2 2 2 2 3 2 2 3 2 2 4 2 2 4 3 2a a b b b c c - 4a a a b b b c c - 4a a a b b c c + 2a a a b b c c + 2a a a b b c c + a a b b c c - a a a b c c + 0 2 1 3 4 3 5 0 1 2 2 3 4 3 5 0 1 2 0 4 3 5 0 1 2 1 4 3 5 0 1 2 2 4 3 5 0 2 3 4 3 5 0 1 2 4 3 5 ---------------------------------------------------------------------------------------------------------------------------- 4 2 2 3 2 2 2 2 2 4 2 4 2 2 2 2 2a a b b c c - 2a a a b b b c c + a a a b b c c - 2a a a b b b c c + a a b b c c - 4a a a b b b c c + 1 2 0 5 3 5 0 1 2 0 1 5 3 5 0 1 2 1 5 3 5 0 1 2 0 2 5 3 5 0 1 2 5 3 5 0 1 2 0 3 5 3 5 ---------------------------------------------------------------------------------------------------------------------------- 3 2 3 2 4 2 2 2 3 3 2 3 3 2a a a b b b c c + 2a a a b b b c c - 2a a b b c c + 6a a a b b b c c - 4a a a b b b c c - 2a a b b b c c + 0 1 2 1 3 5 3 5 0 1 2 2 3 5 3 5 0 2 3 5 3 5 0 1 2 0 4 5 3 5 0 1 2 1 4 5 3 5 0 1 2 4 5 3 5 ---------------------------------------------------------------------------------------------------------------------------- 4 4 2 2 2 4 2 3 3 2 4 2 2 3 3 2 2 3 2a a a b b b c c + a a b b c c - 2a a b b c c + 2a a b b c c - 2a a b b c c - 2a a b b c c + 3a a a b b b c c - 0 1 2 3 4 5 3 5 0 1 4 5 3 5 0 1 0 5 3 5 0 1 1 5 3 5 0 1 3 5 3 5 1 2 0 3 4 5 0 1 2 0 1 3 4 5 ---------------------------------------------------------------------------------------------------------------------------- 2 3 2 3 2 2 2 2 2 3 2 3 3 2 3 2 2 4 2 2 a a a b b c c + a a a b b b c c - a a a b b b c c - 2a a a b b c c + a a b b c c + a a a b b c c + 2a a b b c c - 0 1 2 1 3 4 5 0 1 2 0 2 3 4 5 0 1 2 1 2 3 4 5 0 1 2 0 3 4 5 0 2 1 3 4 5 0 1 2 2 3 4 5 1 2 0 4 4 5 ---------------------------------------------------------------------------------------------------------------------------- 3 2 2 2 2 2 4 2 3 2 2 2 3 2 3a a a b b b c c + a a a b b c c - a a a b b b c c + a a a b b b c c + a a a b b b c c - a a a b b b c c - 0 1 2 0 1 4 4 5 0 1 2 1 4 4 5 0 1 2 0 2 4 4 5 0 1 2 1 2 4 4 5 0 1 2 0 3 4 4 5 0 1 2 2 3 4 4 5 ---------------------------------------------------------------------------------------------------------------------------- 4 2 2 2 3 2 3 2 2 4 2 5 2 4 2 3 2 a a b b c c + a a a b b c c - a a a b b c c + a a a b b c c - 2a a b b c c + 3a a a b b b c c - a a a b b c c + 0 2 3 4 4 5 0 1 2 0 4 4 5 0 1 2 1 4 4 5 0 1 2 3 4 4 5 1 2 0 5 4 5 0 1 2 0 1 5 4 5 0 1 2 1 5 4 5 ---------------------------------------------------------------------------------------------------------------------------- 5 2 4 2 3 3 2 3 3 2 4 3 3 a a b b b c c - a a b b b c c - 2a a a b b b c c + a a a b b b c c + a a b b b c c - a a b b b c c + a a b b b c c - 0 1 0 2 5 4 5 0 1 1 2 5 4 5 0 1 2 0 3 5 4 5 0 1 2 1 3 5 4 5 0 1 2 3 5 4 5 0 1 0 4 5 4 5 0 1 1 4 5 4 5 ---------------------------------------------------------------------------------------------------------------------------- 4 2 4 2 2 2 3 2 2 2 2 2 2 2 2 2 2 2 2 3 2 2 2 4 2 3 2 5 2 2 a a b b b c c + a a b b c - 2a a a b b b c + a a a b b c + 2a a a b b c - 2a a a b b c + a a b c - a a b b c + 0 1 3 4 5 4 5 1 2 0 3 5 0 1 2 0 1 3 5 0 1 2 1 3 5 0 1 2 0 3 5 0 1 2 1 3 5 0 2 3 5 1 2 0 4 5 ---------------------------------------------------------------------------------------------------------------------------- 4 2 2 3 2 2 2 3 2 3 2 2 4 2 2 6 2 2 5 2 2 4 2 2 2a a a b b b c - a a a b b c - 2a a a b b b c + 2a a a b b b c - a a a b b c + a b b c - 2a a b b b c + a a b b c + 0 1 2 0 1 4 5 0 1 2 1 4 5 0 1 2 0 3 4 5 0 1 2 1 3 4 5 0 1 2 3 4 5 1 0 5 5 0 1 0 1 5 5 0 1 1 5 5 ---------------------------------------------------------------------------------------------------------------------------- 2 4 2 3 3 2 4 2 2 2 6 3 5 2 2 4 2 3 3 3 6 2a a b b b c - 2a a b b b c + a a b b c - a b c c + 3a a b b c c - 3a a b b c c + a a b c c + 3a b b b c c - 0 1 0 3 5 5 0 1 1 3 5 5 0 1 3 5 5 2 1 0 6 1 2 1 2 0 6 1 2 1 2 0 6 1 2 2 0 6 2 0 1 3 0 6 ---------------------------------------------------------------------------------------------------------------------------- 5 5 4 2 5 5 2 2 4 4 3a a b b b c c - 3a a b b b c c + 3a a a b b c c - 3a a b b b c c + 3a a b b c c + 3a a b b b c c - 3a a a b b b c c 1 2 0 2 3 0 6 0 2 1 2 3 0 6 0 1 2 2 3 0 6 1 2 0 1 4 0 6 0 2 1 4 0 6 1 2 0 2 4 0 6 0 1 2 1 2 4 0 6 ---------------------------------------------------------------------------------------------------------------------------- 5 2 4 4 2 2 4 2 3 3 3 2 4 4 2 - 3a a b b b c c + 3a a b b b c c + 3a a a b b c c - 3a a b b c c + a a b c c + 3a a b b b c c - 6a a a b b c c - 0 2 0 3 4 0 6 0 2 2 3 4 0 6 0 1 2 0 4 0 6 0 2 1 4 0 6 0 2 4 0 6 1 2 0 1 5 0 6 0 1 2 1 5 0 6 ---------------------------------------------------------------------------------------------------------------------------- 3 3 2 3 3 2 2 4 2 4 2 3 3a a b b b c c + 9a a a b b b c c - 3a a a b b c c + 6a a a b b b c c + 3a a b b b c c - 9a a a b b b c c - 1 2 0 2 5 0 6 0 1 2 1 2 5 0 6 0 1 2 2 5 0 6 0 1 2 0 3 5 0 6 0 2 1 3 5 0 6 0 1 2 2 3 5 0 6 ---------------------------------------------------------------------------------------------------------------------------- 2 3 2 3 3 3 3 2 2 3 2 2 2 2 2 2 9a a a b b b c c + 9a a a b b b c c - 3a a b b b c c - 3a a a b b c c + 6a a a b b c c - 9a a a b b c c + 0 1 2 0 4 5 0 6 0 1 2 1 4 5 0 6 0 2 3 4 5 0 6 0 1 2 4 5 0 6 0 1 2 0 5 0 6 0 1 2 1 5 0 6 ---------------------------------------------------------------------------------------------------------------------------- 2 3 2 3 2 2 3 2 2 3 3 3 6 2 5 5 2 3a a a b b c c + 6a a a b b c c + 3a a a b b c c - 2a a b c c + a b b c c - 2a a b b b c c - a a b b c c + 0 1 2 2 5 0 6 0 1 2 3 5 0 6 0 1 2 4 5 0 6 0 1 5 0 6 2 0 1 1 6 1 2 0 1 2 1 6 0 2 1 2 1 6 ---------------------------------------------------------------------------------------------------------------------------- 2 4 2 4 2 2 3 3 6 2 5 2 4 2 5 2 a a b b c c + 2a a a b b c c - a a a b c c - 2a b b c c + 4a a b b b c c - 2a a b b c c + 2a a b b c c - 1 2 0 2 1 6 0 1 2 1 2 1 6 0 1 2 2 1 6 2 0 3 1 6 0 2 0 2 3 1 6 0 2 2 3 1 6 1 2 0 4 1 6 ---------------------------------------------------------------------------------------------------------------------------- 5 4 2 4 2 4 2 3 3 2 2 4 2 4 2a a b b b c c - 2a a a b b b c c + 2a a b b b c c + a a b b c c - a a b b c c - 2a a b b c c + 4a a a b b b c c + 0 2 0 1 4 1 6 0 1 2 0 2 4 1 6 0 2 1 2 4 1 6 0 2 0 4 1 6 0 2 2 4 1 6 1 2 0 5 1 6 0 1 2 0 1 5 1 6 ---------------------------------------------------------------------------------------------------------------------------- 2 4 2 2 3 2 2 2 2 2 4 3 3 3 3 3 2 a a b b c c - 6a a a b b b c c + 3a a a b b c c - 4a a b b b c c + 4a a b b b c c - 2a a b b b c c + 2a a a b b b c c 0 2 1 5 1 6 0 1 2 1 2 5 1 6 0 1 2 2 5 1 6 0 2 0 3 5 1 6 0 2 2 3 5 1 6 0 2 1 4 5 1 6 0 1 2 2 4 5 1 6 ---------------------------------------------------------------------------------------------------------------------------- 4 2 2 3 2 2 3 2 2 4 2 2 4 2 4 2 3 5 2 + a a b b c c + 4a a a b b c c - 4a a a b b c c - 2a a b b c c - 2a a a b b c c + 2a a b c c - a a b b c c + 0 2 4 5 1 6 0 1 2 1 5 1 6 0 1 2 2 5 1 6 0 2 3 5 1 6 0 1 2 4 5 1 6 0 1 5 1 6 1 2 0 1 2 6 ---------------------------------------------------------------------------------------------------------------------------- 5 3 2 4 4 2 3 3 2 2 3 2 5 2 5 a a b c c + 2a a b b b c c - 2a a a b b c c - a a b b c c + a a a b b c c + 2a a b b c c - 3a a b b b c c - 0 2 1 2 6 1 2 0 1 2 2 6 0 1 2 1 2 2 6 1 2 0 2 2 6 0 1 2 1 2 2 6 1 2 0 3 2 6 0 2 0 1 3 2 6 ---------------------------------------------------------------------------------------------------------------------------- 4 2 4 2 3 2 2 4 2 4 2 4 2 2 3 a a a b b b c c + 3a a b b b c c - a a a b b c c - 2a a b b c c + 5a a a b b b c c - 3a a b b c c - a a a b b b c c + 0 1 2 0 2 3 2 6 0 2 1 2 3 2 6 0 1 2 2 3 2 6 1 2 0 4 2 6 0 1 2 0 1 4 2 6 0 2 1 4 2 6 0 1 2 0 2 4 2 6 ---------------------------------------------------------------------------------------------------------------------------- 2 3 2 4 3 3 2 3 2 3 3 2 3 2 2 4 2 3 a a a b b b c c + 3a a b b b c c - 3a a b b b c c - 4a a a b b c c + 3a a b b c c + a a a b b c c - a a b c c + 0 1 2 1 2 4 2 6 0 2 0 3 4 2 6 0 2 2 3 4 2 6 0 1 2 0 4 2 6 0 2 1 4 2 6 0 1 2 2 4 2 6 0 2 4 2 6 ---------------------------------------------------------------------------------------------------------------------------- 3 3 2 2 3 2 3 2 3 2 2 2 2 2 3 2a a b b c c - 7a a a b b b c c + 5a a a b b c c + 3a a a b b b c c - 3a a a b b b c c - 2a a a b b b c c - 1 2 0 5 2 6 0 1 2 0 1 5 2 6 0 1 2 1 5 2 6 0 1 2 0 2 5 2 6 0 1 2 1 2 5 2 6 0 1 2 0 3 5 2 6 ---------------------------------------------------------------------------------------------------------------------------- 3 3 3 2 2 2 2 3 2 3 2 4 2 3a a b b b c c + 5a a a b b b c c + 9a a a b b b c c - 7a a a b b b c c - 2a a a b b b c c + 3a a b b b c c + 0 2 1 3 5 2 6 0 1 2 2 3 5 2 6 0 1 2 0 4 5 2 6 0 1 2 1 4 5 2 6 0 1 2 2 4 5 2 6 0 2 3 4 5 2 6 ---------------------------------------------------------------------------------------------------------------------------- 4 2 2 3 2 3 2 2 3 3 2 4 2 4 2 2 6 2 2a a a b b c c - 6a a a b b c c + 5a a a b b c c + a a b b c c - 4a a a b b c c - a a b b c c - a b b c c + 0 1 2 4 5 2 6 0 1 2 0 5 2 6 0 1 2 1 5 2 6 0 1 2 5 2 6 0 1 2 3 5 2 6 0 1 4 5 2 6 2 0 1 3 6 ---------------------------------------------------------------------------------------------------------------------------- 5 2 5 4 2 2 4 2 2 3 3 5 2 2 4 a a b b c c + 2a a b b b c c - 2a a a b b c c - a a b b c c + a a a b c c + a a b b c c - 2a a b b b c c + 1 2 0 2 3 6 0 2 0 1 2 3 6 0 1 2 0 2 3 6 0 2 1 2 3 6 0 1 2 2 3 6 0 2 0 4 3 6 0 2 0 2 4 3 6 ---------------------------------------------------------------------------------------------------------------------------- 3 3 2 4 2 2 4 2 3 3 3 3 2 2 3 3 a a b b c c - 2a a a b b c c - 2a a b b b c c + 6a a a b b b c c + 2a a b b b c c - 4a a a b b c c + 2a a b b b c c - 0 2 2 4 3 6 0 1 2 0 5 3 6 0 2 0 1 5 3 6 0 1 2 0 2 5 3 6 0 2 1 2 5 3 6 0 1 2 2 5 3 6 0 2 0 4 5 3 6 ---------------------------------------------------------------------------------------------------------------------------- 4 2 3 2 2 4 2 2 4 2 5 2 5 3 5 2 2a a b b b c c - 4a a a b b c c - a a b b c c + 5a a a b b c c + a a b b c c - 2a a b c c + a a b b c c - 0 2 2 4 5 3 6 0 1 2 0 5 3 6 0 2 1 5 3 6 0 1 2 2 5 3 6 0 2 4 5 3 6 0 1 5 3 6 1 2 0 1 4 6 ---------------------------------------------------------------------------------------------------------------------------- 5 2 2 4 2 2 4 2 2 3 2 2 3 2 5 2 2 4 a a b b c c - a a b b c c + a a b b c c + a a a b b c c - a a a b b c c + 2a a b b c c - 4a a b b b c c + 0 2 0 1 4 6 1 2 0 2 4 6 0 2 1 2 4 6 0 1 2 0 2 4 6 0 1 2 1 2 4 6 0 2 0 3 4 6 0 2 0 2 3 4 6 ---------------------------------------------------------------------------------------------------------------------------- 3 3 2 4 2 2 4 2 3 3 3 3 2 2 3 3 2 2a a b b c c - 3a a a b b c c + 2a a b b b c c + 4a a a b b b c c - 2a a b b b c c - a a a b b c c - a a b b c c + 0 2 2 3 4 6 0 1 2 0 4 4 6 0 2 0 1 4 4 6 0 1 2 0 2 4 4 6 0 2 1 2 4 4 6 0 1 2 2 4 4 6 0 2 0 4 4 6 ---------------------------------------------------------------------------------------------------------------------------- 4 2 2 2 3 2 2 3 3 3 2 2 2 2 3 2 3 2 2 a a b b c c + 4a a a b b c c - 2a a a b b b c c - a a b b c c - 6a a a b b b c c + 4a a a b b b c c + a a a b b c c + 0 2 2 4 4 6 0 1 2 0 5 4 6 0 1 2 0 1 5 4 6 0 2 1 5 4 6 0 1 2 0 2 5 4 6 0 1 2 1 2 5 4 6 0 1 2 2 5 4 6 ---------------------------------------------------------------------------------------------------------------------------- 3 3 4 2 3 2 4 2 5 2 3 2 2 4 2 4a a b b b c c - 4a a b b b c c - 2a a a b b b c c + 2a a b b b c c - a a b b c c + 4a a a b b c c - 3a a a b b c c - 0 2 0 3 5 4 6 0 2 2 3 5 4 6 0 1 2 0 4 5 4 6 0 2 1 4 5 4 6 0 2 4 5 4 6 0 1 2 0 5 4 6 0 1 2 1 5 4 6 ---------------------------------------------------------------------------------------------------------------------------- 4 2 2 5 2 5 2 2 4 2 4 2 2 4 3 3 3 2 2 3 a a b b c c + 2a a b b c c + a a b b c c - a a b b c c + 2a a a b b c c - a a b c c + a a b b c c - 2a a a b b b c c 0 1 2 5 4 6 0 2 3 5 4 6 0 1 4 5 4 6 1 2 0 1 5 6 0 1 2 0 1 5 6 0 2 1 5 6 1 2 0 2 5 6 0 1 2 0 1 2 5 6 ---------------------------------------------------------------------------------------------------------------------------- 2 3 2 4 2 2 4 2 3 3 3 3 2 2 + a a a b b c c - 4a a a b b c c + 3a a b b b c c + 5a a a b b b c c - 3a a b b b c c - a a a b b c c + 0 1 2 1 2 5 6 0 1 2 0 3 5 6 0 2 0 1 3 5 6 0 1 2 0 2 3 5 6 0 2 1 2 3 5 6 0 1 2 2 3 5 6 ---------------------------------------------------------------------------------------------------------------------------- 2 3 2 2 3 3 3 2 2 2 2 3 2 3 2 2 5a a a b b c c - 7a a a b b b c c + 3a a b b c c - 3a a a b b b c c + a a a b b b c c + a a a b b c c - 0 1 2 0 4 5 6 0 1 2 0 1 4 5 6 0 2 1 4 5 6 0 1 2 0 2 4 5 6 0 1 2 1 2 4 5 6 0 1 2 2 4 5 6 ---------------------------------------------------------------------------------------------------------------------------- 3 3 4 2 3 2 2 4 2 2 4 2 5 3 3 2 2 3a a b b b c c + 3a a b b b c c + 5a a a b b c c - 3a a b b c c - 2a a a b b c c + a a b c c - 6a a a b b c c + 0 2 0 3 4 5 6 0 2 2 3 4 5 6 0 1 2 0 4 5 6 0 2 1 4 5 6 0 1 2 2 4 5 6 0 2 4 5 6 0 1 2 0 5 5 6 ---------------------------------------------------------------------------------------------------------------------------- 2 2 2 3 2 2 2 3 3 2 3 3 2 3 2 9a a a b b b c c - 4a a a b b c c + 3a a a b b b c c - a a a b b b c c - a a b b c c - 2a a a b b b c c + 0 1 2 0 1 5 5 6 0 1 2 1 5 5 6 0 1 2 0 2 5 5 6 0 1 2 1 2 5 5 6 0 1 2 5 5 6 0 1 2 0 3 5 5 6 ---------------------------------------------------------------------------------------------------------------------------- 4 2 4 3 2 4 4 2 5 3a a b b b c c - a a a b b b c c - 7a a a b b b c c + 5a a a b b b c c + 2a a b b b c c - 3a a b b b c c - 0 2 1 3 5 5 6 0 1 2 2 3 5 5 6 0 1 2 0 4 5 5 6 0 1 2 1 4 5 5 6 0 1 2 4 5 5 6 0 2 3 4 5 5 6 ---------------------------------------------------------------------------------------------------------------------------- 5 2 3 3 2 4 2 2 5 2 6 3 2 5 2 2 2 4 2 2 3 3 3 2 2 4 2 2 a a b b c c + 2a a b b c c - 2a a b b c c + 2a a b b c c + a b c - 3a a b b c + 3a a b b c - a a b c + 3a a b b c - 0 1 4 5 5 6 0 1 0 5 5 6 0 1 1 5 5 6 0 1 3 5 5 6 2 0 6 0 2 0 2 6 0 2 0 2 6 0 2 2 6 0 2 0 5 6 ---------------------------------------------------------------------------------------------------------------------------- 3 3 2 4 2 2 2 4 2 2 2 5 2 2 6 3 2 5 3 2 4 2 3 3 2 4 2 3 6a a b b b c + 3a a b b c + 3a a b b c - 3a a b b c + a b c + a a b c c - 3a a b b c c + 3a a b b c c - a a b c c - 0 2 0 2 5 6 0 2 2 5 6 0 2 0 5 6 0 2 2 5 6 0 5 6 1 2 1 0 7 1 2 1 2 0 7 1 2 1 2 0 7 1 2 2 0 7 ---------------------------------------------------------------------------------------------------------------------------- 5 5 2 2 4 4 2 3 2 5 2 2 4 2 3a a b b b c c - a a b b c c + 3a a b b b c c + 5a a a b b b c c - 4a a a b b c c + 2a a b b c c - 2a a b b c c + 1 2 0 1 3 0 7 0 2 1 3 0 7 1 2 0 2 3 0 7 0 1 2 1 2 3 0 7 0 1 2 2 3 0 7 0 2 0 3 0 7 0 2 2 3 0 7 ---------------------------------------------------------------------------------------------------------------------------- 2 4 4 2 3 3 2 3 3 2 2 4 2 4 3a a b b b c c - 2a a a b b c c - 3a a b b b c c + a a a b b b c c + a a a b b c c - a a a b b b c c + 2a a b b b c c 1 2 0 1 4 0 7 0 1 2 1 4 0 7 1 2 0 2 4 0 7 0 1 2 1 2 4 0 7 0 1 2 2 4 0 7 0 1 2 0 3 4 0 7 0 2 1 3 4 0 7 ---------------------------------------------------------------------------------------------------------------------------- 2 3 2 3 2 2 3 2 3 3 2 3 3 2 3 2 4 2 - a a a b b b c c - a a a b b c c + a a a b b c c - a a b b c c - 3a a b b b c c + 5a a a b b c c + 3a a b b b c c - 0 1 2 2 3 4 0 7 0 1 2 0 4 0 7 0 1 2 1 4 0 7 0 2 3 4 0 7 1 2 0 1 5 0 7 0 1 2 1 5 0 7 1 2 0 2 5 0 7 ---------------------------------------------------------------------------------------------------------------------------- 3 2 4 2 2 3 2 3 2 2 2 3 3 2 7a a a b b b c c + 2a a a b b c c - 2a a a b b b c c - 7a a a b b b c c + 9a a a b b b c c + 2a a b b c c + 0 1 2 1 2 5 0 7 0 1 2 2 5 0 7 0 1 2 0 3 5 0 7 0 1 2 1 3 5 0 7 0 1 2 2 3 5 0 7 0 2 3 5 0 7 ---------------------------------------------------------------------------------------------------------------------------- 3 2 2 2 2 2 3 3 2 4 2 2 3 2 5a a a b b b c c - 3a a a b b b c c - 2a a a b b b c c + 3a a a b b b c c - 4a a a b b c c + 5a a a b b c c - 0 1 2 0 4 5 0 7 0 1 2 1 4 5 0 7 0 1 2 2 4 5 0 7 0 1 2 3 4 5 0 7 0 1 2 0 5 0 7 0 1 2 1 5 0 7 ---------------------------------------------------------------------------------------------------------------------------- 2 4 2 3 2 2 3 3 2 5 2 2 4 4 2 3 3 2 a a b b c c - 6a a a b b c c + a a b b c c - a a b b c c + 2a a b b b c c + a a a b b c c - a a b b c c - 0 1 2 5 0 7 0 1 2 3 5 0 7 0 1 4 5 0 7 1 2 0 1 1 7 1 2 0 1 2 1 7 0 1 2 1 2 1 7 1 2 0 2 1 7 ---------------------------------------------------------------------------------------------------------------------------- 2 3 2 3 2 3 5 2 5 4 2 4 2 3 2 2a a a b b c c + a a a b c c + 2a a b b c c + a a b b b c c - 5a a a b b b c c - a a b b b c c + 3a a a b b c c - 0 1 2 1 2 1 7 0 1 2 2 1 7 1 2 0 3 1 7 0 2 0 1 3 1 7 0 1 2 0 2 3 1 7 0 2 1 2 3 1 7 0 1 2 2 3 1 7 ---------------------------------------------------------------------------------------------------------------------------- 2 4 2 4 2 3 2 3 2 2 2 2 2 4 3 3 2a a b b c c + a a a b b b c c + 3a a a b b b c c - a a a b b b c c - a a a b b c c - a a b b b c c + a a b b b c c + 1 2 0 4 1 7 0 1 2 0 1 4 1 7 0 1 2 0 2 4 1 7 0 1 2 1 2 4 1 7 0 1 2 2 4 1 7 0 2 0 3 4 1 7 0 2 2 3 4 1 7 ---------------------------------------------------------------------------------------------------------------------------- 3 3 2 2 3 2 3 2 3 2 2 2 2 2 3 2 2a a b b c c - 3a a a b b b c c - a a a b b c c - a a a b b b c c + 5a a a b b b c c - 2a a a b b c c + 1 2 0 5 1 7 0 1 2 0 1 5 1 7 0 1 2 1 5 1 7 0 1 2 0 2 5 1 7 0 1 2 1 2 5 1 7 0 1 2 2 5 1 7 ---------------------------------------------------------------------------------------------------------------------------- 2 3 3 3 3 2 2 2 2 3 2 3 2 6a a a b b b c c + a a b b b c c - 7a a a b b b c c - 3a a a b b b c c + a a a b b b c c + 2a a a b b b c c - 0 1 2 0 3 5 1 7 0 2 1 3 5 1 7 0 1 2 2 3 5 1 7 0 1 2 0 4 5 1 7 0 1 2 1 4 5 1 7 0 1 2 2 4 5 1 7 ---------------------------------------------------------------------------------------------------------------------------- 4 2 2 3 2 3 2 2 3 3 2 4 2 4 2 2 2 4 2 a a b b b c c + 2a a a b b c c - 3a a a b b c c + a a b b c c + 4a a a b b c c - a a b b c c + a a b b c c - 0 2 3 4 5 1 7 0 1 2 0 5 1 7 0 1 2 1 5 1 7 0 1 2 5 1 7 0 1 2 3 5 1 7 0 1 4 5 1 7 1 2 0 1 2 7 ---------------------------------------------------------------------------------------------------------------------------- 4 3 3 3 2 3 2 4 2 2 3 2 2 2 4 2 4 a a a b c c - 2a a b b b c c + 2a a a b b c c + a a b b c c - a a a b b c c - 2a a b b c c + 2a a a b b b c c + 0 1 2 1 2 7 1 2 0 1 2 2 7 0 1 2 1 2 2 7 1 2 0 2 2 7 0 1 2 1 2 2 7 1 2 0 3 2 7 0 1 2 0 1 3 2 7 ---------------------------------------------------------------------------------------------------------------------------- 2 4 2 2 3 2 3 2 2 2 2 2 4 2 3 3 2 3 3 2 a a b b c c + 2a a a b b b c c - 4a a a b b b c c + a a a b b c c - 2a a b b c c + 2a a b b c c + 2a a b b c c - 0 2 1 3 2 7 0 1 2 0 2 3 2 7 0 1 2 1 2 3 2 7 0 1 2 2 3 2 7 0 2 0 3 2 7 0 2 2 3 2 7 1 2 0 4 2 7 ---------------------------------------------------------------------------------------------------------------------------- 2 3 2 3 2 2 3 3 3 2 2 2 2 3 2 2 4 2 2 4a a a b b b c c + 2a a a b b c c + 2a a a b b b c c - 2a a b b b c c + a a a b b c c - a a a b b c c + a a b b c c - 0 1 2 0 1 4 2 7 0 1 2 1 4 2 7 0 1 2 0 3 4 2 7 0 2 1 3 4 2 7 0 1 2 0 4 2 7 0 1 2 1 4 2 7 0 2 3 4 2 7 ---------------------------------------------------------------------------------------------------------------------------- 4 2 2 3 2 2 2 2 2 4 2 3 2 2 2 2a a b b c c + 6a a a b b b c c - 4a a a b b c c - 2a a a b b b c c + 2a a a b b b c c - 4a a a b b b c c + 1 2 0 5 2 7 0 1 2 0 1 5 2 7 0 1 2 1 5 2 7 0 1 2 0 2 5 2 7 0 1 2 1 2 5 2 7 0 1 2 0 3 5 2 7 ---------------------------------------------------------------------------------------------------------------------------- 3 2 3 2 4 2 2 2 3 3 2 4 6a a a b b b c c - 2a a a b b b c c - 2a a b b c c - 2a a a b b b c c + 2a a a b b b c c - 2a a a b b b c c + 0 1 2 1 3 5 2 7 0 1 2 2 3 5 2 7 0 2 3 5 2 7 0 1 2 0 4 5 2 7 0 1 2 1 4 5 2 7 0 1 2 3 4 5 2 7 ---------------------------------------------------------------------------------------------------------------------------- 2 4 2 3 3 2 4 2 2 5 2 2 4 2 4 2 3 2 2a a b b c c - 2a a b b c c + 2a a b b c c + a a b b c c - a a b b c c - 2a a a b b b c c + 2a a a b b c c + 0 1 0 5 2 7 0 1 1 5 2 7 0 1 3 5 2 7 1 2 0 1 3 7 1 2 0 2 3 7 0 1 2 0 1 2 3 7 0 1 2 0 2 3 7 ---------------------------------------------------------------------------------------------------------------------------- 2 3 2 2 2 2 3 5 2 2 4 3 3 2 4 2 2 3 a a a b b c c - a a a b c c - 2a a b b c c + 4a a b b b c c - 2a a b b c c + a a a b b c c - 2a a a b b b c c + 0 1 2 1 2 3 7 0 1 2 2 3 7 0 2 0 3 3 7 0 2 0 2 3 3 7 0 2 2 3 3 7 0 1 2 0 4 3 7 0 1 2 0 2 4 3 7 ---------------------------------------------------------------------------------------------------------------------------- 3 2 2 2 3 2 2 2 3 2 3 2 2 3 3 a a a b b c c + 2a a a b b b c c - 2a a a b b b c c - 2a a a b b b c c + 2a a a b b c c - 4a a b b b c c + 0 1 2 2 4 3 7 0 1 2 0 1 5 3 7 0 1 2 0 2 5 3 7 0 1 2 1 2 5 3 7 0 1 2 2 5 3 7 0 2 0 3 5 3 7 ---------------------------------------------------------------------------------------------------------------------------- 4 2 3 2 4 4 2 4 2 2 5 2 5 2 4a a b b b c c + 2a a a b b b c c - 2a a a b b b c c + a a a b b c c - a a b b c c - 2a a b b c c + a a b b c c - 0 2 2 3 5 3 7 0 1 2 0 4 5 3 7 0 1 2 2 4 5 3 7 0 1 2 1 5 3 7 0 1 2 5 3 7 0 2 3 5 3 7 0 1 4 5 3 7 ---------------------------------------------------------------------------------------------------------------------------- 2 4 2 4 2 3 3 2 2 3 2 3 2 2 2 2 2 2 2 4 a a b b c c + a a a b b c c + a a b b c c - a a a b b c c - a a a b b c c + a a a b b c c - a a b b b c c + 1 2 0 1 4 7 0 1 2 0 1 4 7 1 2 0 2 4 7 0 1 2 1 2 4 7 0 1 2 0 2 4 7 0 1 2 1 2 4 7 0 2 0 1 3 4 7 ---------------------------------------------------------------------------------------------------------------------------- 2 3 3 3 3 2 2 2 3 2 2 3 2 2 2 3 2 a a a b b b c c + a a b b b c c - a a a b b c c + a a a b b c c - a a a b b b c c - a a a b b b c c + a a a b b b c c 0 1 2 0 2 3 4 7 0 2 1 2 3 4 7 0 1 2 2 3 4 7 0 1 2 0 4 4 7 0 1 2 0 1 4 4 7 0 1 2 0 2 4 4 7 0 1 2 1 2 4 4 7 ---------------------------------------------------------------------------------------------------------------------------- 3 3 4 2 3 2 2 2 2 2 3 2 2 2 3 + a a b b b c c - a a b b b c c - 2a a a b b c c + a a a b b b c c + a a a b b c c + 3a a a b b b c c - 0 2 0 3 4 4 7 0 2 2 3 4 4 7 0 1 2 0 5 4 7 0 1 2 0 1 5 4 7 0 1 2 1 5 4 7 0 1 2 0 2 5 4 7 ---------------------------------------------------------------------------------------------------------------------------- 3 2 3 2 4 2 4 3 2 4 3a a a b b b c c - 2a a a b b b c c - a a b b b c c + 3a a a b b b c c + a a a b b b c c - a a a b b b c c + 0 1 2 1 2 5 4 7 0 1 2 0 3 5 4 7 0 2 1 3 5 4 7 0 1 2 2 3 5 4 7 0 1 2 0 4 5 4 7 0 1 2 1 4 5 4 7 ---------------------------------------------------------------------------------------------------------------------------- 5 3 3 2 4 2 2 5 2 3 3 2 2 3 2 2 3 3 a a b b b c c - 2a a b b c c + 2a a b b c c - 2a a b b c c + a a b b c c - 2a a a b b c c + a a a b c c - 0 2 3 4 5 4 7 0 1 0 5 4 7 0 1 1 5 4 7 0 1 3 5 4 7 1 2 0 1 5 7 0 1 2 0 1 5 7 0 1 2 1 5 7 ---------------------------------------------------------------------------------------------------------------------------- 4 2 2 3 2 2 2 2 2 2 3 2 2 3 3 3 2 2 2 2 a a b b c c + 2a a a b b b c c - a a a b b c c + 2a a a b b c c - a a a b b b c c - a a b b c c - 3a a a b b b c c + 1 2 0 2 5 7 0 1 2 0 1 2 5 7 0 1 2 1 2 5 7 0 1 2 0 3 5 7 0 1 2 0 1 3 5 7 0 2 1 3 5 7 0 1 2 0 2 3 5 7 ---------------------------------------------------------------------------------------------------------------------------- 3 2 3 3 2 4 2 2 3 2 2 2 2 2 3 2 2 2 3 3a a a b b b c c + 2a a b b c c - 2a a b b c c - 3a a a b b c c + 5a a a b b b c c - 2a a a b b c c + a a a b b b c c 0 1 2 1 2 3 5 7 0 2 0 3 5 7 0 2 2 3 5 7 0 1 2 0 4 5 7 0 1 2 0 1 4 5 7 0 1 2 1 4 5 7 0 1 2 0 2 4 5 7 ---------------------------------------------------------------------------------------------------------------------------- 3 2 3 2 4 2 4 3 2 2 4 2 5 2 - a a a b b b c c - 3a a a b b b c c + 2a a b b b c c + a a a b b b c c - a a a b b c c + a a a b b c c - a a b b c c 0 1 2 1 2 4 5 7 0 1 2 0 3 4 5 7 0 2 1 3 4 5 7 0 1 2 2 3 4 5 7 0 1 2 0 4 5 7 0 1 2 1 4 5 7 0 2 3 4 5 7 ---------------------------------------------------------------------------------------------------------------------------- 4 2 2 3 3 2 2 2 4 3 3 3 2 + 4a a a b b c c - 7a a a b b b c c + 3a a a b b c c - a a b b b c c + a a b b b c c + 6a a a b b b c c - 0 1 2 0 5 5 7 0 1 2 0 1 5 5 7 0 1 2 1 5 5 7 0 1 0 2 5 5 7 0 1 1 2 5 5 7 0 1 2 0 3 5 5 7 ---------------------------------------------------------------------------------------------------------------------------- 4 4 2 5 2 3 3 4 2 5 5 3 5a a a b b b c c - a a b b b c c + 2a a b b c c + a a b b b c c - a a b b b c c + a a b b b c c - 2a a b c c + 0 1 2 1 3 5 5 7 0 1 2 3 5 5 7 0 2 3 5 5 7 0 1 0 4 5 5 7 0 1 1 4 5 5 7 0 1 3 4 5 5 7 1 2 0 6 7 ---------------------------------------------------------------------------------------------------------------------------- 5 2 4 2 2 4 2 3 2 3 3 2 3 2 3 2 4 2 a a b b c c + 5a a a b b c c - 2a a b b b c c - 4a a a b b c c + a a b b c c + a a a b c c - a a b b c c + 0 2 0 1 6 7 0 1 2 0 2 6 7 0 2 0 1 2 6 7 0 1 2 0 2 6 7 0 2 1 2 6 7 0 1 2 2 6 7 0 2 0 4 6 7 ---------------------------------------------------------------------------------------------------------------------------- 3 3 4 2 2 2 3 2 3 3 3 2 4 2 4 2 2a a b b b c c - a a b b c c - 4a a a b b c c + 2a a b b b c c + 6a a a b b b c c - 2a a b b b c c - 2a a a b b c c - 0 2 0 2 4 6 7 0 2 2 4 6 7 0 1 2 0 5 6 7 0 2 0 1 5 6 7 0 1 2 0 2 5 6 7 0 2 1 2 5 6 7 0 1 2 2 5 6 7 ---------------------------------------------------------------------------------------------------------------------------- 4 2 5 4 2 5 2 5 2 6 2 2 4 3 2 4 2 2 2a a b b b c c + 2a a b b b c c - 2a a a b b c c + a a b b c c + a a b b c c - a b b c c + a a b c - a a a b b c - 0 2 0 4 5 6 7 0 2 2 4 5 6 7 0 1 2 0 5 6 7 0 2 1 5 6 7 0 1 2 5 6 7 0 4 5 6 7 1 2 0 7 0 1 2 0 1 7 ---------------------------------------------------------------------------------------------------------------------------- 2 3 2 2 2 3 2 2 2 2 2 2 3 2 2 2 2 4 2 2 3 3 2 4 2 2 2 2 2 2 2 2 2a a a b b c + 2a a a b b b c + a a a b b c - a a a b b c + a a b b c - 2a a b b b c + a a b b c + 2a a a b b c - 0 1 2 0 2 7 0 1 2 0 1 2 7 0 1 2 0 2 7 0 1 2 1 2 7 0 2 0 3 7 0 2 0 2 3 7 0 2 2 3 7 0 1 2 0 5 7 ---------------------------------------------------------------------------------------------------------------------------- 3 2 2 3 2 2 4 2 4 2 2 5 2 4 2 2 2 5 2 2 6 2 2 2a a a b b b c - 2a a a b b b c + 2a a a b b b c + 2a a b b b c - 2a a b b b c + a a b b c - a a b b c + a b b c - 0 1 2 0 1 5 7 0 1 2 0 2 5 7 0 1 2 1 2 5 7 0 2 0 3 5 7 0 2 2 3 5 7 0 1 0 5 7 0 1 1 5 7 0 3 5 7 ---------------------------------------------------------------------------------------------------------------------------- 2 4 3 3 3 2 4 2 2 5 3 2 4 4 2 3 3 a a b c c + 3a a b b c c - 3a a b b c c + a a b c c + 3a a b b b c c + 2a a a b b c c - 3a a b b b c c - 1 2 1 0 8 1 2 1 2 0 8 1 2 1 2 0 8 1 2 2 0 8 1 2 0 1 3 0 8 0 1 2 1 3 0 8 1 2 0 2 3 0 8 ---------------------------------------------------------------------------------------------------------------------------- 2 3 3 2 2 4 2 2 4 2 2 3 2 3 3 2 3 2 7a a a b b b c c + 5a a a b b c c - 4a a a b b c c - a a b b c c + 5a a a b b c c - 3a a b b b c c + a a a b b c c + 0 1 2 1 2 3 0 8 0 1 2 2 3 0 8 0 1 2 0 3 0 8 0 2 1 3 0 8 0 1 2 2 3 0 8 1 2 0 1 4 0 8 0 1 2 1 4 0 8 ---------------------------------------------------------------------------------------------------------------------------- 4 2 3 2 4 2 2 3 2 3 2 2 2 3a a b b b c c + a a a b b b c c - 2a a a b b c c + 5a a a b b b c c - 2a a a b b b c c - 3a a a b b b c c + 1 2 0 2 4 0 8 0 1 2 1 2 4 0 8 0 1 2 2 4 0 8 0 1 2 0 3 4 0 8 0 1 2 1 3 4 0 8 0 1 2 2 3 4 0 8 ---------------------------------------------------------------------------------------------------------------------------- 3 3 2 3 2 2 2 3 2 4 2 3 2 2 5 4 a a b b c c - a a a b b c c + a a a b b c c + 3a a b b b c c - 4a a a b b c c - 3a a b b b c c + 5a a a b b b c c - 0 2 3 4 0 8 0 1 2 0 4 0 8 0 1 2 2 4 0 8 1 2 0 1 5 0 8 0 1 2 1 5 0 8 1 2 0 2 5 0 8 0 1 2 1 2 5 0 8 ---------------------------------------------------------------------------------------------------------------------------- 5 2 3 2 2 2 2 2 3 3 2 2 4 a a b b c c - 2a a a b b b c c + 9a a a b b b c c - 7a a a b b b c c - 6a a a b b c c - a a a b b b c c - 0 1 2 5 0 8 0 1 2 0 3 5 0 8 0 1 2 1 3 5 0 8 0 1 2 2 3 5 0 8 0 1 2 3 5 0 8 0 1 2 0 4 5 0 8 ---------------------------------------------------------------------------------------------------------------------------- 2 3 2 4 3 2 3 3 2 5 2 2 4 2 3 3 2 a a a b b b c c + 2a a b b b c c + 3a a a b b b c c - a a b b c c + 2a a b b c c - 2a a b b c c + 2a a b b c c + 0 1 2 1 4 5 0 8 0 1 2 4 5 0 8 0 1 2 3 4 5 0 8 0 1 4 5 0 8 0 1 0 5 0 8 0 1 1 5 0 8 0 1 3 5 0 8 ---------------------------------------------------------------------------------------------------------------------------- 2 4 2 3 3 2 3 2 4 2 2 3 2 2 4 3 2 4 2 a a b b c c - 2a a b b b c c - a a a b b c c + a a b b c c + 2a a a b b c c - a a a b c c - 2a a b b c c - 1 2 0 1 1 8 1 2 0 1 2 1 8 0 1 2 1 2 1 8 1 2 0 2 1 8 0 1 2 1 2 1 8 0 1 2 2 1 8 1 2 0 3 1 8 ---------------------------------------------------------------------------------------------------------------------------- 4 2 3 2 3 2 2 2 2 2 4 2 3 3 2 3 3 2 2a a a b b b c c + 6a a a b b b c c + 2a a a b b b c c - 4a a a b b c c + 2a a b b c c - 2a a b b c c + 2a a b b c c 0 1 2 0 1 3 1 8 0 1 2 0 2 3 1 8 0 1 2 1 2 3 1 8 0 1 2 2 3 1 8 0 2 0 3 1 8 0 2 2 3 1 8 1 2 0 4 1 8 ---------------------------------------------------------------------------------------------------------------------------- 3 2 2 3 2 2 3 3 2 2 2 2 2 3 2 2 - 4a a a b b b c c + 2a a a b b c c - 2a a a b b b c c + 2a a a b b b c c + a a a b b c c - a a a b b c c - 0 1 2 0 2 4 1 8 0 1 2 2 4 1 8 0 1 2 0 3 4 1 8 0 1 2 2 3 4 1 8 0 1 2 0 4 1 8 0 1 2 2 4 1 8 ---------------------------------------------------------------------------------------------------------------------------- 4 2 2 3 2 2 2 2 2 4 2 3 2 4 2 2a a b b c c + 2a a a b b b c c + a a a b b c c + 2a a a b b b c c - 4a a a b b b c c + a a b b c c - 1 2 0 5 1 8 0 1 2 0 1 5 1 8 0 1 2 1 5 1 8 0 1 2 0 2 5 1 8 0 1 2 1 2 5 1 8 0 1 2 5 1 8 ---------------------------------------------------------------------------------------------------------------------------- 2 2 2 3 2 3 2 4 2 2 2 3 3 3 4a a a b b b c c - 2a a a b b b c c + 6a a a b b b c c + 2a a b b c c + 2a a a b b b c c - 2a a b b b c c - 0 1 2 0 3 5 1 8 0 1 2 1 3 5 1 8 0 1 2 2 3 5 1 8 0 2 3 5 1 8 0 1 2 0 4 5 1 8 0 1 2 4 5 1 8 ---------------------------------------------------------------------------------------------------------------------------- 4 4 2 2 2 4 2 3 3 2 4 2 2 3 3 2 2 3 3 2a a a b b b c c + a a b b c c - 2a a b b c c + 2a a b b c c - 2a a b b c c - a a b b c c + a a a b c c + 0 1 2 3 4 5 1 8 0 1 4 5 1 8 0 1 0 5 1 8 0 1 1 5 1 8 0 1 3 5 1 8 1 2 0 1 2 8 0 1 2 1 2 8 ---------------------------------------------------------------------------------------------------------------------------- 4 2 3 2 2 5 2 4 2 3 3 2 2 3 2 3 2 2a a b b b c c - 2a a a b b c c - a a b b c c + a a a b b c c + 2a a b b c c - a a a b b b c c - 2a a a b b c c - 1 2 0 1 2 2 8 0 1 2 1 2 2 8 1 2 0 2 2 8 0 1 2 1 2 2 8 1 2 0 3 2 8 0 1 2 0 1 3 2 8 0 1 2 1 3 2 8 ---------------------------------------------------------------------------------------------------------------------------- 3 2 2 2 2 2 3 2 2 3 2 3 3 2 3 2 2 4 2 2 3a a a b b b c c + 5a a a b b b c c - a a a b b c c + 2a a a b b c c + a a b b c c - 3a a a b b c c - 2a a b b c c + 0 1 2 0 2 3 2 8 0 1 2 1 2 3 2 8 0 1 2 2 3 2 8 0 1 2 0 3 2 8 0 2 1 3 2 8 0 1 2 2 3 2 8 1 2 0 4 2 8 ---------------------------------------------------------------------------------------------------------------------------- 3 2 2 2 2 2 4 2 3 2 2 2 3 2 3a a a b b b c c - a a a b b c c + a a a b b b c c - a a a b b b c c - 3a a a b b b c c + 2a a a b b b c c + 0 1 2 0 1 4 2 8 0 1 2 1 4 2 8 0 1 2 0 2 4 2 8 0 1 2 1 2 4 2 8 0 1 2 0 3 4 2 8 0 1 2 1 3 4 2 8 ---------------------------------------------------------------------------------------------------------------------------- 3 2 4 2 2 5 2 4 2 3 2 5 2 4 a a a b b b c c - a a b b c c + 2a a b b c c - 5a a a b b b c c + 3a a a b b c c + a a b b b c c - a a b b b c c + 0 1 2 2 3 4 2 8 0 2 3 4 2 8 1 2 0 5 2 8 0 1 2 0 1 5 2 8 0 1 2 1 5 2 8 0 1 0 2 5 2 8 0 1 1 2 5 2 8 ---------------------------------------------------------------------------------------------------------------------------- 2 3 3 2 3 3 4 2 2 4 3 3 4 2 6a a a b b b c c - 7a a a b b b c c + a a b b b c c + 4a a a b b c c - a a b b b c c + a a b b b c c - a a b b b c c 0 1 2 0 3 5 2 8 0 1 2 1 3 5 2 8 0 1 2 3 5 2 8 0 1 2 3 5 2 8 0 1 0 4 5 2 8 0 1 1 4 5 2 8 0 1 3 4 5 2 8 ---------------------------------------------------------------------------------------------------------------------------- 2 4 2 3 3 2 2 3 3 2 2 2 2 2 2 2 3 3 4 2 - a a b b c c + a a b b c c + 2a a a b b b c c - 2a a a b b c c - a a a b b c c + a a a b c c + 4a a a b b c c - 1 2 0 1 3 8 1 2 0 2 3 8 0 1 2 0 1 2 3 8 0 1 2 0 2 3 8 0 1 2 1 2 3 8 0 1 2 2 3 8 0 1 2 0 3 3 8 ---------------------------------------------------------------------------------------------------------------------------- 2 4 2 3 3 3 3 2 2 2 3 2 2 3 a a b b b c c - 7a a a b b b c c + a a b b b c c + 3a a a b b c c - 3a a a b b c c + a a a b b b c c + 0 2 0 1 3 3 8 0 1 2 0 2 3 3 8 0 2 1 2 3 3 8 0 1 2 2 3 3 8 0 1 2 0 4 3 8 0 1 2 0 1 4 3 8 ---------------------------------------------------------------------------------------------------------------------------- 2 2 2 3 2 3 2 2 3 3 4 2 3 2 2 4 2 5a a a b b b c c - a a a b b b c c - 2a a a b b c c + a a b b b c c - a a b b b c c - a a a b b c c + a a a b b c c + 0 1 2 0 2 4 3 8 0 1 2 1 2 4 3 8 0 1 2 2 4 3 8 0 2 0 3 4 3 8 0 2 2 3 4 3 8 0 1 2 0 4 3 8 0 1 2 2 4 3 8 ---------------------------------------------------------------------------------------------------------------------------- 3 2 2 2 2 2 2 3 3 2 3 3 2 3 2 2a a a b b c c - 3a a a b b b c c - a a a b b b c c + 3a a a b b b c c - a a b b c c + 6a a a b b b c c - 0 1 2 0 5 3 8 0 1 2 0 1 5 3 8 0 1 2 0 2 5 3 8 0 1 2 1 2 5 3 8 0 1 2 5 3 8 0 1 2 0 3 5 3 8 ---------------------------------------------------------------------------------------------------------------------------- 4 2 4 3 2 4 4 2 5 5 2 a a b b b c c - 5a a a b b b c c - 3a a a b b b c c + a a a b b b c c + 2a a b b b c c + a a b b b c c - a a b b c c 0 2 1 3 5 3 8 0 1 2 2 3 5 3 8 0 1 2 0 4 5 3 8 0 1 2 1 4 5 3 8 0 1 2 4 5 3 8 0 2 3 4 5 3 8 0 1 4 5 3 8 ---------------------------------------------------------------------------------------------------------------------------- 3 3 2 4 2 2 5 2 3 3 2 2 3 2 4 2 2 2 2 2 2 + 2a a b b c c - 2a a b b c c + 2a a b b c c + a a b b c c - a a a b b c c - a a b b c c + a a a b b c c + 0 1 0 5 3 8 0 1 1 5 3 8 0 1 3 5 3 8 1 2 0 1 4 8 0 1 2 0 1 4 8 1 2 0 2 4 8 0 1 2 1 2 4 8 ---------------------------------------------------------------------------------------------------------------------------- 4 2 2 3 2 2 3 2 2 3 2 2 2 3 2 a a a b b c c - a a a b b c c - 2a a a b b c c + 3a a a b b b c c + a a a b b b c c - 3a a a b b b c c + 0 1 2 0 2 4 8 0 1 2 1 2 4 8 0 1 2 0 3 4 8 0 1 2 0 1 3 4 8 0 1 2 0 2 3 4 8 0 1 2 1 2 3 4 8 ---------------------------------------------------------------------------------------------------------------------------- 3 2 2 3 3 2 4 2 2 3 2 2 2 2 2 2 3 3 2 a a a b b c c - 2a a b b c c + 2a a b b c c + a a a b b c c - a a a b b b c c - a a a b b b c c + a a a b b b c c + 0 1 2 2 3 4 8 0 2 0 3 4 8 0 2 2 3 4 8 0 1 2 0 4 4 8 0 1 2 0 1 4 4 8 0 1 2 0 2 4 4 8 0 1 2 1 2 4 4 8 ---------------------------------------------------------------------------------------------------------------------------- 3 2 4 2 3 3 2 2 2 4 3 3 a a a b b b c c - a a a b b b c c + a a a b b b c c - a a a b b c c - a a b b b c c + a a b b b c c - 0 1 2 0 3 4 4 8 0 1 2 2 3 4 4 8 0 1 2 0 1 5 4 8 0 1 2 1 5 4 8 0 1 0 2 5 4 8 0 1 1 2 5 4 8 ---------------------------------------------------------------------------------------------------------------------------- 3 2 4 4 2 5 2 3 3 4 2 5 2a a a b b b c c + 3a a a b b b c c - a a b b b c c - 2a a b b c c + a a b b b c c - a a b b b c c + a a b b b c c - 0 1 2 0 3 5 4 8 0 1 2 1 3 5 4 8 0 1 2 3 5 4 8 0 2 3 5 4 8 0 1 0 4 5 4 8 0 1 1 4 5 4 8 0 1 3 4 5 4 8 ---------------------------------------------------------------------------------------------------------------------------- 4 2 2 3 2 2 2 2 2 3 5 2 4 2 3 2 2 2 2 a a b b c c + 2a a a b b c c - a a a b c c + a a b b c c - 2a a a b b b c c + a a a b b c c - 2a a a b b b c c + 1 2 0 1 5 8 0 1 2 0 1 5 8 0 1 2 1 5 8 1 2 0 2 5 8 0 1 2 0 1 2 5 8 0 1 2 1 2 5 8 0 1 2 0 1 3 5 8 ---------------------------------------------------------------------------------------------------------------------------- 3 2 2 2 3 3 2 4 2 2 4 2 4 2 2 3 2a a a b b c c + 2a a a b b b c c - 2a a a b b b c c - a a b b c c + a a a b b c c + a a a b b c c - 2a a a b b b c c 0 1 2 1 3 5 8 0 1 2 0 2 3 5 8 0 1 2 1 2 3 5 8 0 2 1 3 5 8 0 1 2 2 3 5 8 0 1 2 0 4 5 8 0 1 2 0 1 4 5 8 ---------------------------------------------------------------------------------------------------------------------------- 3 2 2 3 2 4 5 2 5 2 2 4 3 3 2 + a a a b b c c + 2a a a b b b c c - 2a a a b b b c c + a a b b c c - 2a a b b c c + 4a a b b b c c - 2a a b b c c - 0 1 2 1 4 5 8 0 1 2 0 3 4 5 8 0 1 2 1 3 4 5 8 0 2 3 4 5 8 0 1 0 5 5 8 0 1 0 1 5 5 8 0 1 1 5 5 8 ---------------------------------------------------------------------------------------------------------------------------- 3 3 4 2 5 2 2 4 3 4 2 2 4 2 2 3 2 4a a b b b c c + 4a a b b b c c - 2a a b b c c + 2a a b c c - 2a a a b b c c + a a b b c c - 4a a a b b c c + 0 1 0 3 5 5 8 0 1 1 3 5 5 8 0 1 3 5 5 8 1 2 0 6 8 0 1 2 0 1 6 8 0 2 0 1 6 8 0 1 2 0 2 6 8 ---------------------------------------------------------------------------------------------------------------------------- 2 3 3 3 2 2 2 2 2 3 2 3 2 4 2 3 3 4 2 2 2a a a b b b c c - a a b b c c + 3a a a b b c c - a a a b c c - 2a a b b c c + 4a a b b b c c - 2a a b b c c + 0 1 2 0 1 2 6 8 0 2 1 2 6 8 0 1 2 0 2 6 8 0 1 2 2 6 8 0 2 0 3 6 8 0 2 0 2 3 6 8 0 2 2 3 6 8 ---------------------------------------------------------------------------------------------------------------------------- 2 3 2 3 3 3 2 4 2 4 2 4 2 2 5 2 4a a a b b c c - 2a a b b b c c - 6a a a b b b c c + 2a a b b b c c + 2a a a b b c c + a a b b c c - a a b b c c + 0 1 2 0 4 6 8 0 2 0 1 4 6 8 0 1 2 0 2 4 6 8 0 2 1 2 4 6 8 0 1 2 2 4 6 8 0 2 0 4 6 8 0 2 2 4 6 8 ---------------------------------------------------------------------------------------------------------------------------- 4 2 2 4 4 2 2 4 2 5 4 5 a a b b c c - 2a a a b b b c c + a a b b c c - 4a a b b b c c + 4a a b b b c c + 4a a a b b b c c - 2a a b b b c c - 0 2 1 5 6 8 0 1 2 1 2 5 6 8 0 1 2 5 6 8 0 2 0 3 5 6 8 0 2 2 3 5 6 8 0 1 2 0 4 5 6 8 0 2 1 4 5 6 8 ---------------------------------------------------------------------------------------------------------------------------- 5 6 2 4 2 2 5 2 6 2 3 3 3 2 3 2 2 3 2 2a a b b b c c + a b b c c - 2a a b b c c + 2a a b b c c - 2a b b c c - 2a a b c c + 3a a a b b c c - a a a b b c c 0 1 2 4 5 6 8 0 4 5 6 8 0 1 0 5 6 8 0 1 1 5 6 8 0 3 5 6 8 1 2 0 7 8 0 1 2 0 1 7 8 0 1 2 0 1 7 8 ---------------------------------------------------------------------------------------------------------------------------- 3 2 2 2 2 2 3 2 2 2 3 2 3 2 2 2 3 2 3 3 + 3a a a b b c c - 4a a a b b b c c + a a a b b c c - a a a b b c c + a a a b b c c - 2a a a b b c c + a a b b b c c 0 1 2 0 2 7 8 0 1 2 0 1 2 7 8 0 1 2 1 2 7 8 0 1 2 0 2 7 8 0 1 2 1 2 7 8 0 1 2 0 3 7 8 0 2 0 1 3 7 8 ---------------------------------------------------------------------------------------------------------------------------- 3 2 4 2 4 2 2 2 2 2 3 2 3 2 + 3a a a b b b c c - a a b b b c c - a a a b b c c - a a a b b c c + a a a b b b c c + a a a b b b c c - 0 1 2 0 2 3 7 8 0 2 1 2 3 7 8 0 1 2 2 3 7 8 0 1 2 0 4 7 8 0 1 2 0 1 4 7 8 0 1 2 0 2 4 7 8 ---------------------------------------------------------------------------------------------------------------------------- 4 4 2 5 2 3 2 3 2 4 2 3 3 a a a b b b c c - a a b b b c c + a a b b b c c - 2a a a b b c c + 3a a a b b b c c - a a a b b c c + a a b b b c c - 0 1 2 1 2 4 7 8 0 2 0 3 4 7 8 0 2 2 3 4 7 8 0 1 2 0 5 7 8 0 1 2 0 1 5 7 8 0 1 2 1 5 7 8 0 1 0 2 5 7 8 ---------------------------------------------------------------------------------------------------------------------------- 4 2 4 5 5 4 2 5 6 a a b b b c c - 2a a a b b b c c + a a b b b c c + a a b b b c c - a a b b b c c + a a b b b c c - a b b b c c + 0 1 1 2 5 7 8 0 1 2 0 3 5 7 8 0 2 1 3 5 7 8 0 1 2 3 5 7 8 0 1 0 4 5 7 8 0 1 1 4 5 7 8 0 3 4 5 7 8 ---------------------------------------------------------------------------------------------------------------------------- 4 2 3 2 3 2 2 2 2 2 2 2 2 4 2 2 2 3 2 3 2 2 2 2 2 2 2 2 3 2 2 a a b c - 2a a a b b c + a a a b b c - a a a b b c + 2a a a b b b c - a a a b b c + 2a a a b b c - 2a a a b b b c - 1 2 0 8 0 1 2 0 1 8 0 1 2 0 1 8 0 1 2 0 2 8 0 1 2 0 1 2 8 0 1 2 1 2 8 0 1 2 0 3 8 0 1 2 0 1 3 8 ---------------------------------------------------------------------------------------------------------------------------- 3 2 2 4 2 4 2 2 2 5 2 2 2 4 2 2 3 3 2 4 2 2 2 4 2 2 2a a a b b b c + 2a a a b b b c + a a b b c - a a b b c + a a b b c - 2a a b b b c + a a b b c + 2a a b b b c - 0 1 2 0 2 3 8 0 1 2 1 2 3 8 0 2 0 3 8 0 2 2 3 8 0 1 0 5 8 0 1 0 1 5 8 0 1 1 5 8 0 1 0 3 5 8 ---------------------------------------------------------------------------------------------------------------------------- 5 2 6 2 2 3 3 3 4 2 2 5 2 6 3 3 3 2 3 2 2a a b b b c + a b b c + a a b c c - 3a a b b c c + 3a a b b c c - a b c c - 3a a b b b c c - 3a a a b b c c + 0 1 1 3 5 8 0 3 5 8 1 2 1 0 9 1 2 1 2 0 9 1 2 1 2 0 9 1 2 0 9 1 2 0 1 3 0 9 0 1 2 1 3 0 9 ---------------------------------------------------------------------------------------------------------------------------- 4 2 3 2 4 2 2 3 2 2 3 2 2 2 2 2 3 3 3 3a a b b b c c + 9a a a b b b c c - 6a a a b b c c + 6a a a b b c c + 3a a a b b c c - 9a a a b b c c - 2a a b c c + 1 2 0 2 3 0 9 0 1 2 1 2 3 0 9 0 1 2 2 3 0 9 0 1 2 0 3 0 9 0 1 2 1 3 0 9 0 1 2 2 3 0 9 0 2 3 0 9 ---------------------------------------------------------------------------------------------------------------------------- 4 2 5 4 5 2 3 2 2 3 3a a b b b c c - 3a a b b b c c - 3a a a b b b c c + 3a a b b c c - 9a a a b b b c c + 9a a a b b b c c + 1 2 0 1 4 0 9 1 2 0 2 4 0 9 0 1 2 1 2 4 0 9 0 1 2 4 0 9 0 1 2 0 3 4 0 9 0 1 2 2 3 4 0 9 ---------------------------------------------------------------------------------------------------------------------------- 3 2 2 4 2 2 4 2 3 2 2 3 3 3 5 4 2 3a a a b b c c + 3a a a b b c c - 3a a b b c c - 3a a a b b c c + a a b c c - 3a a b b b c c + 3a a a b b c c + 0 1 2 3 4 0 9 0 1 2 0 4 0 9 0 1 2 4 0 9 0 1 2 3 4 0 9 0 1 4 0 9 1 2 0 1 5 0 9 0 1 2 1 5 0 9 ---------------------------------------------------------------------------------------------------------------------------- 6 5 4 2 3 2 4 3 2 2 3a b b b c c - 3a a b b b c c + 6a a a b b b c c - 9a a a b b b c c + 3a a b b b c c + 6a a a b b c c - 1 0 2 5 0 9 0 1 1 2 5 0 9 0 1 2 0 3 5 0 9 0 1 2 1 3 5 0 9 0 1 2 3 5 0 9 0 1 2 3 5 0 9 ---------------------------------------------------------------------------------------------------------------------------- 5 2 4 3 3 3 3 2 4 2 3 2 2 5 2 3a a b b b c c + 3a a b b b c c - 3a a b b b c c - a a b b c c + 2a a b b b c c + a a a b b c c - a a b b c c - 0 1 0 4 5 0 9 0 1 1 4 5 0 9 0 1 3 4 5 0 9 1 2 0 1 1 9 1 2 0 1 2 1 9 0 1 2 1 2 1 9 1 2 0 2 1 9 ---------------------------------------------------------------------------------------------------------------------------- 4 2 5 3 3 3 2 2 3 3 2 2 2 2 2 3 2 2a a a b b c c + a a b c c + 2a a b b c c + 3a a a b b b c c - 7a a a b b b c c - 3a a a b b b c c + 5a a a b b c c - 0 1 2 1 2 1 9 0 1 2 1 9 1 2 0 3 1 9 0 1 2 0 1 3 1 9 0 1 2 0 2 3 1 9 0 1 2 1 2 3 1 9 0 1 2 2 3 1 9 ---------------------------------------------------------------------------------------------------------------------------- 2 3 2 3 3 2 3 2 2 4 2 2 3 2 4 2 3 6a a a b b c c + a a b b c c + 5a a a b b c c - 2a a b b c c - a a a b b b c c + 5a a a b b b c c + a a a b b b c c - 0 1 2 0 3 1 9 0 2 1 3 1 9 0 1 2 2 3 1 9 1 2 0 4 1 9 0 1 2 0 1 4 1 9 0 1 2 0 2 4 1 9 0 1 2 1 2 4 1 9 ---------------------------------------------------------------------------------------------------------------------------- 2 4 2 2 2 2 3 2 3 2 4 2 2 2 3 2 3 2 2 3a a b b c c + 9a a a b b b c c - 2a a a b b b c c - 7a a a b b b c c - a a b b c c - 4a a a b b c c + a a a b b c c 0 1 2 4 1 9 0 1 2 0 3 4 1 9 0 1 2 1 3 4 1 9 0 1 2 2 3 4 1 9 0 2 3 4 1 9 0 1 2 0 4 1 9 0 1 2 1 4 1 9 ---------------------------------------------------------------------------------------------------------------------------- 3 3 2 4 2 4 2 3 5 2 4 2 3 2 5 + 3a a b b c c + 2a a a b b c c - a a b c c + 2a a b b c c - a a a b b b c c - a a a b b c c - 3a a b b b c c + 0 1 2 4 1 9 0 1 2 3 4 1 9 0 1 4 1 9 1 2 0 5 1 9 0 1 2 0 1 5 1 9 0 1 2 1 5 1 9 0 1 0 2 5 1 9 ---------------------------------------------------------------------------------------------------------------------------- 2 4 2 3 3 2 3 3 4 2 2 4 3a a b b b c c - 2a a a b b b c c + 5a a a b b b c c - 3a a b b b c c - 4a a a b b c c + 3a a b b b c c - 0 1 1 2 5 1 9 0 1 2 0 3 5 1 9 0 1 2 1 3 5 1 9 0 1 2 3 5 1 9 0 1 2 3 5 1 9 0 1 0 4 5 1 9 ---------------------------------------------------------------------------------------------------------------------------- 3 3 4 2 4 2 2 3 2 3 5 4 2 6 2 3a a b b b c c + 3a a b b b c c + a a b b c c - a a a b c c - 2a a b b b c c + 2a a a b b c c + a b b c c - 0 1 1 4 5 1 9 0 1 3 4 5 1 9 1 2 0 1 2 9 0 1 2 1 2 9 1 2 0 1 2 2 9 0 1 2 1 2 2 9 1 0 2 2 9 ---------------------------------------------------------------------------------------------------------------------------- 5 2 4 2 2 2 2 2 2 4 2 3 2 4 2 3 2 2 a a b b c c - 2a a b b c c + 3a a a b b c c + 4a a a b b b c c - 6a a a b b b c c + a a b b c c - 4a a a b b c c + 0 1 1 2 2 9 1 2 0 3 2 9 0 1 2 1 3 2 9 0 1 2 0 2 3 2 9 0 1 2 1 2 3 2 9 0 1 2 3 2 9 0 1 2 1 3 2 9 ---------------------------------------------------------------------------------------------------------------------------- 3 2 2 4 2 3 5 2 4 5 2 4 3 2 4a a a b b c c + 2a a b c c + 2a a b b c c - 2a a a b b b c c - 2a a b b b c c + 2a a b b b c c + 2a a a b b b c c - 0 1 2 2 3 2 9 0 2 3 2 9 1 2 0 4 2 9 0 1 2 0 1 4 2 9 0 1 0 2 4 2 9 0 1 1 2 4 2 9 0 1 2 1 3 4 2 9 ---------------------------------------------------------------------------------------------------------------------------- 3 3 4 2 2 4 2 3 3 2 4 2 2 6 2 5 2a a b b b c c - 2a a a b b c c + a a b b c c - a a b b c c + a a b b c c - 2a b b c c + 4a a b b b c c - 0 1 2 3 4 2 9 0 1 2 3 4 2 9 0 1 0 4 2 9 0 1 1 4 2 9 0 1 3 4 2 9 1 0 5 2 9 0 1 0 1 5 2 9 ---------------------------------------------------------------------------------------------------------------------------- 2 4 2 2 4 3 3 4 2 2 3 3 2 4 2 2 3 2 2a a b b c c - 4a a b b b c c + 4a a b b b c c - 2a a b b c c + a a b b c c - a a b b c c - 2a a a b b b c c + 0 1 1 5 2 9 0 1 0 3 5 2 9 0 1 1 3 5 2 9 0 1 3 5 2 9 1 2 0 1 3 9 1 2 0 2 3 9 0 1 2 0 1 2 3 9 ---------------------------------------------------------------------------------------------------------------------------- 4 2 2 3 2 2 4 3 2 3 2 2 3 3 3 2 2 2 2 2a a a b b c c + a a a b b c c - a a b c c - 6a a a b b c c + 3a a a b b b c c - a a b b c c + 9a a a b b b c c - 0 1 2 0 2 3 9 0 1 2 1 2 3 9 0 1 2 3 9 0 1 2 0 3 3 9 0 1 2 0 1 3 3 9 0 2 1 3 3 9 0 1 2 0 2 3 3 9 ---------------------------------------------------------------------------------------------------------------------------- 3 2 3 2 2 3 3 2 4 2 2 3 2 2 2 2 2 3 2 2 a a a b b b c c - 4a a a b b c c + 2a a b b c c - 2a a b b c c + 5a a a b b c c - 3a a a b b b c c + a a a b b c c - 0 1 2 1 2 3 3 9 0 1 2 2 3 3 9 0 2 0 3 3 9 0 2 2 3 3 9 0 1 2 0 4 3 9 0 1 2 0 1 4 3 9 0 1 2 1 4 3 9 ---------------------------------------------------------------------------------------------------------------------------- 2 3 3 2 3 3 2 3 2 4 2 4 7a a a b b b c c + a a a b b b c c + 3a a b b c c - 7a a a b b b c c + 2a a b b b c c + 5a a a b b b c c + 0 1 2 0 2 4 3 9 0 1 2 1 2 4 3 9 0 1 2 4 3 9 0 1 2 0 3 4 3 9 0 2 1 3 4 3 9 0 1 2 2 3 4 3 9 ---------------------------------------------------------------------------------------------------------------------------- 3 2 2 4 2 4 2 2 5 2 5 3 4 2 2 3 5a a a b b c c - 2a a a b b c c - 3a a b b c c - a a b b c c + a a b c c - 4a a a b b c c + 5a a a b b b c c - 0 1 2 0 4 3 9 0 1 2 1 4 3 9 0 1 2 4 3 9 0 2 3 4 3 9 0 1 4 3 9 0 1 2 0 5 3 9 0 1 2 0 1 5 3 9 ---------------------------------------------------------------------------------------------------------------------------- 3 2 2 2 4 3 3 3 2 4 4 2 5 2 a a a b b c c + 3a a b b b c c - 3a a b b b c c - 2a a a b b b c c - a a a b b b c c + 3a a b b b c c + 2a a b b c c 0 1 2 1 5 3 9 0 1 0 2 5 3 9 0 1 1 2 5 3 9 0 1 2 0 3 5 3 9 0 1 2 1 3 5 3 9 0 1 2 3 5 3 9 0 2 3 5 3 9 ---------------------------------------------------------------------------------------------------------------------------- 3 3 4 2 5 4 2 2 3 2 2 5 2 2 3 2 - 3a a b b b c c + 3a a b b b c c - 3a a b b b c c - a a b b c c + a a a b b c c + a a b b c c - a a a b b c c - 0 1 0 4 5 3 9 0 1 1 4 5 3 9 0 1 3 4 5 3 9 1 2 0 1 4 9 0 1 2 0 1 4 9 1 2 0 2 4 9 0 1 2 1 2 4 9 ---------------------------------------------------------------------------------------------------------------------------- 5 2 2 4 2 3 2 2 2 2 2 3 2 2 2 3 3 2 a a b b c c + a a b b c c + 4a a a b b c c - 6a a a b b b c c + a a a b b c c - 2a a a b b b c c + 4a a a b b b c c - 0 1 0 2 4 9 0 1 1 2 4 9 0 1 2 0 3 4 9 0 1 2 0 1 3 4 9 0 1 2 1 3 4 9 0 1 2 0 2 3 4 9 0 1 2 1 2 3 4 9 ---------------------------------------------------------------------------------------------------------------------------- 3 3 2 3 2 2 4 2 2 4 2 4 2 2 3 3 2 2 a a b b c c + 4a a a b b c c - a a b b c c - 3a a a b b c c - 3a a a b b c c + 4a a a b b b c c - a a a b b c c + 0 1 2 3 4 9 0 1 2 0 3 4 9 0 2 1 3 4 9 0 1 2 2 3 4 9 0 1 2 0 4 4 9 0 1 2 0 1 4 4 9 0 1 2 1 4 4 9 ---------------------------------------------------------------------------------------------------------------------------- 2 4 3 3 3 2 4 2 5 2 3 3 2 4 2 2 2a a b b b c c - 2a a b b b c c - 2a a a b b b c c + 2a a b b b c c + a a b b c c - a a b b c c + a a b b c c - 0 1 0 2 4 4 9 0 1 1 2 4 4 9 0 1 2 0 3 4 4 9 0 1 2 3 4 4 9 0 2 3 4 4 9 0 1 0 4 4 9 0 1 1 4 4 9 ---------------------------------------------------------------------------------------------------------------------------- 5 2 5 2 2 4 3 3 2 3 3 4 2 5 2 a a b b c c + 2a a b b c c - 4a a b b b c c + 2a a b b c c + 4a a b b b c c - 4a a b b b c c + 2a a b b c c + 0 1 3 4 4 9 0 1 0 5 4 9 0 1 0 1 5 4 9 0 1 1 5 4 9 0 1 0 3 5 4 9 0 1 1 3 5 4 9 0 1 3 5 4 9 ---------------------------------------------------------------------------------------------------------------------------- 5 2 4 2 2 3 3 6 2 5 2 4 2 4 2 a a b b c c - 2a a a b b c c + a a a b c c - a b b c c + 2a a b b b c c - a a b b c c - 2a a a b b c c + 1 2 0 1 5 9 0 1 2 0 1 5 9 0 1 2 1 5 9 1 0 2 5 9 0 1 0 1 2 5 9 0 1 1 2 5 9 0 1 2 0 3 5 9 ---------------------------------------------------------------------------------------------------------------------------- 2 3 3 2 2 2 4 3 3 3 2 2 4 2 4 2 2 6a a a b b b c c - 4a a a b b c c - 2a a b b b c c + 2a a b b b c c - 4a a a b b c c + 5a a a b b c c - a a b b c c - 0 1 2 0 1 3 5 9 0 1 2 1 3 5 9 0 1 0 2 3 5 9 0 1 1 2 3 5 9 0 1 2 0 3 5 9 0 1 2 1 3 5 9 0 1 2 3 5 9 ---------------------------------------------------------------------------------------------------------------------------- 5 3 5 2 2 4 3 3 2 3 3 4 2 5 2 3 3 3 2a a b c c + a a b b c c - 2a a b b b c c + a a b b c c + 2a a b b b c c - 2a a b b b c c + a a b b c c - 2a a b c c 0 2 3 5 9 0 1 0 4 5 9 0 1 0 1 4 5 9 0 1 1 4 5 9 0 1 0 3 4 5 9 0 1 1 3 4 5 9 0 1 3 4 5 9 1 2 0 6 9 ---------------------------------------------------------------------------------------------------------------------------- 2 3 2 2 3 2 3 3 3 3 2 2 2 3 2 3 3 3 2 3 2 + 3a a a b b c c - 3a a a b b c c + a a b c c + 3a a a b b c c - 3a a a b b c c + a a b c c + 6a a a b b c c - 0 1 2 0 1 6 9 0 1 2 0 1 6 9 0 2 1 6 9 0 1 2 0 2 6 9 0 1 2 0 2 6 9 0 1 2 6 9 0 1 2 0 3 6 9 ---------------------------------------------------------------------------------------------------------------------------- 3 3 3 2 4 2 4 2 2 2 2 2 3 2 3a a b b b c c - 9a a a b b b c c + 3a a b b b c c + 3a a a b b c c - 9a a a b b c c + 9a a a b b b c c - 0 2 0 1 3 6 9 0 1 2 0 2 3 6 9 0 2 1 2 3 6 9 0 1 2 2 3 6 9 0 1 2 0 4 6 9 0 1 2 0 1 4 6 9 ---------------------------------------------------------------------------------------------------------------------------- 4 2 2 3 2 4 4 2 2 4 2 5 4 2 3a a b b c c + 9a a a b b b c c - 3a a a b b b c c - 3a a b b c c + 3a a b b b c c - 3a a b b b c c - 6a a a b b c c 0 2 1 4 6 9 0 1 2 0 2 4 6 9 0 1 2 1 2 4 6 9 0 1 2 4 6 9 0 2 0 3 4 6 9 0 2 2 3 4 6 9 0 1 2 0 4 6 9 ---------------------------------------------------------------------------------------------------------------------------- 5 2 5 2 6 3 2 3 2 3 2 4 2 3 3 + 3a a b b c c + 3a a b b c c - a b c c + 6a a a b b c c - 9a a a b b b c c + 3a a a b b c c - 3a a b b b c c + 0 2 1 4 6 9 0 1 2 4 6 9 0 4 6 9 0 1 2 0 5 6 9 0 1 2 0 1 5 6 9 0 1 2 1 5 6 9 0 1 0 2 5 6 9 ---------------------------------------------------------------------------------------------------------------------------- 4 2 4 5 5 4 2 5 6 3a a b b b c c + 6a a a b b b c c - 3a a b b b c c - 3a a b b b c c + 3a a b b b c c - 3a a b b b c c + 3a b b b c c 0 1 1 2 5 6 9 0 1 2 0 3 5 6 9 0 2 1 3 5 6 9 0 1 2 3 5 6 9 0 1 0 4 5 6 9 0 1 1 4 5 6 9 0 3 4 5 6 9 ---------------------------------------------------------------------------------------------------------------------------- 4 2 3 3 2 2 2 2 2 2 3 2 3 4 2 2 3 2 4 2 + 2a a b c c - 4a a a b b c c + 3a a a b b c c - a a a b c c - 2a a a b b c c + 2a a a b b b c c + a a b b c c - 1 2 0 7 9 0 1 2 0 1 7 9 0 1 2 0 1 7 9 0 1 2 1 7 9 0 1 2 0 2 7 9 0 1 2 0 1 2 7 9 0 1 0 2 7 9 ---------------------------------------------------------------------------------------------------------------------------- 3 3 2 4 2 2 4 4 2 2 4 2 2 5 2 2 3 2 a a b b c c + a a b b c c - 2a a a b b b c c + a a b b c c - 2a a b b c c + 2a a b b c c + 4a a a b b c c - 0 1 1 2 7 9 0 2 1 3 7 9 0 1 2 1 2 3 7 9 0 1 2 3 7 9 0 2 0 3 7 9 0 2 2 3 7 9 0 1 2 0 4 7 9 ---------------------------------------------------------------------------------------------------------------------------- 3 2 4 2 3 3 4 2 4 5 6a a a b b b c c + 2a a a b b c c - 2a a b b b c c + 2a a b b b c c + 4a a a b b b c c - 2a a b b b c c - 0 1 2 0 1 4 7 9 0 1 2 1 4 7 9 0 1 0 2 4 7 9 0 1 1 2 4 7 9 0 1 2 0 3 4 7 9 0 2 1 3 4 7 9 ---------------------------------------------------------------------------------------------------------------------------- 5 4 2 2 5 2 6 2 2 4 2 3 3 4 2 2 2a a b b b c c + a a b b c c - a a b b c c + a b b c c - 2a a b b c c + 4a a b b b c c - 2a a b b c c - 0 1 2 3 4 7 9 0 1 0 4 7 9 0 1 1 4 7 9 0 3 4 7 9 0 1 0 5 7 9 0 1 0 1 5 7 9 0 1 1 5 7 9 ---------------------------------------------------------------------------------------------------------------------------- 4 2 5 6 2 5 3 4 2 2 3 2 3 2 3 4a a b b b c c + 4a a b b b c c - 2a b b c c - 2a a b c c + 5a a a b b c c - 4a a a b b c c + a a a b c c + 0 1 0 3 5 7 9 0 1 1 3 5 7 9 0 3 5 7 9 1 2 0 8 9 0 1 2 0 1 8 9 0 1 2 0 1 8 9 0 1 2 1 8 9 ---------------------------------------------------------------------------------------------------------------------------- 5 2 2 4 3 3 2 2 3 2 3 2 4 2 3 3 a a b b c c - 2a a b b b c c + a a b b c c - 4a a a b b c c + 6a a a b b b c c - 2a a a b b c c + 2a a b b b c c - 0 1 0 2 8 9 0 1 0 1 2 8 9 0 1 1 2 8 9 0 1 2 0 3 8 9 0 1 2 0 1 3 8 9 0 1 2 1 3 8 9 0 1 0 2 3 8 9 ---------------------------------------------------------------------------------------------------------------------------- 4 2 4 2 5 2 5 2 2 4 2 3 3 4 2 2 2a a b b b c c - 2a a a b b c c + a a b b c c + a a b b c c - a a b b c c + 2a a b b b c c - a a b b c c - 0 1 1 2 3 8 9 0 1 2 0 3 8 9 0 2 1 3 8 9 0 1 2 3 8 9 0 1 0 4 8 9 0 1 0 1 4 8 9 0 1 1 4 8 9 ---------------------------------------------------------------------------------------------------------------------------- 4 2 5 6 2 6 3 2 5 2 2 2 4 2 2 3 3 3 2 2 4 2 2 3 3 2 2a a b b b c c + 2a a b b b c c - a b b c c + a b c - 3a a b b c + 3a a b b c - a a b c + 3a a b b c - 6a a b b b c 0 1 0 3 4 8 9 0 1 1 3 4 8 9 0 3 4 8 9 1 0 9 0 1 0 1 9 0 1 0 1 9 0 1 1 9 0 1 0 3 9 0 1 0 1 3 9 ---------------------------------------------------------------------------------------------------------------------------- 4 2 2 2 4 2 2 2 5 2 2 6 3 2 + 3a a b b c + 3a a b b c - 3a a b b c + a b c 0 1 1 3 9 0 1 0 3 9 0 1 1 3 9 0 3 9 o5 : ZZ[a ..a , b ..b , c ..c ] 0 2 0 5 0 9 |
On page 88 of Using Algebraic Geometry is stated that the resultant of three ternary quadrics, written out in its full glory, has 21.894 terms.
i6 : F = genericPolynomials({2,2,2},ZZ) 2 2 2 2 2 2 2 2 o6 = {a x + a x x + a x + a x x + a x x + a x , b x + b x x + b x + b x x + b x x + b x , c x + c x x + c x + c x x 0 0 1 0 1 3 1 2 0 2 4 1 2 5 2 0 0 1 0 1 3 1 2 0 2 4 1 2 5 2 0 0 1 0 1 3 1 2 0 2 ---------------------------------------------------------------------------------------------------------------------------- 2 + c x x + c x } 4 1 2 5 2 o6 : List |
i7 : time # terms resultant F -- used 1.72545 seconds o7 = 21894 |
The object resultant is a method function with options.