Description
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 0.0172975 seconds
12 11 2 10 3 9 4 8 5 7 6
o3 = - 81t u - 1701t u - 15309t u - 76545t u - 229635t u - 413343t u
------------------------------------------------------------------------
6 7 5 8 11 10 2 9 3
- 413343t u - 177147t u + 567t u + 10206t u + 76545t u +
------------------------------------------------------------------------
8 4 7 5 6 6 5 7 4 8 3 9
306181t u + 688923t u + 826821t u + 413883t u + 1215t u + 1458t u
------------------------------------------------------------------------
2 10 10 9 2 8 3 7 4 6 5
+ 729t u - 1701t u - 25515t u - 153093t u - 459321t u - 689265t u
------------------------------------------------------------------------
5 6 4 7 3 8 2 9 9 8 2
- 414693t u - 2835t u - 3159t u - 1458t u + 2835t u + 33984t u +
------------------------------------------------------------------------
7 3 6 4 5 5 4 6 3 7 2 8
152565t u + 303135t u + 220905t u - 12150t u - 5913t u + 1215t u -
------------------------------------------------------------------------
8 7 2 6 3 5 4 4 5 3 6
2862t u - 25659t u - 75873t u - 70031t u + 16263t u + 13230t u -
------------------------------------------------------------------------
2 7 7 6 2 5 3 4 4 3 5
540t u + 1809t u + 10818t u + 15015t u - 6609t u - 9360t u +
------------------------------------------------------------------------
2 6 6 5 2 4 3 3 4 2 5 6
135t u - 729t u - 2313t u + 684t u + 3165t u - 81t u - 54t*u +
------------------------------------------------------------------------
5 4 2 3 3 2 4 5 4 3 2 2 3
189t u + 135t u - 583t u - 38t u + 54t*u - 27t u + 54t u + 47t u -
------------------------------------------------------------------------
4 2 2 3 2
18t*u - 9t u + 2t*u + 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
o4 = {a x + a x + a x , b x + b x x + b x + b x x + b x x + b x , c x
0 0 1 1 2 2 0 0 1 0 1 3 1 2 0 2 4 1 2 5 2 0 0
------------------------------------------------------------------------
2 2 3 2 2 2 2
+ c x x + c x x + c x + c x x + c x x x + c x x + c x x + c x x
1 0 1 3 0 1 6 1 2 0 2 4 0 1 2 7 1 2 5 0 2 8 1 2
------------------------------------------------------------------------
3
+ c x }
9 2
o4 : List
|
i5 : time resultant F
-- used 2.88785 seconds
6 3 2 5 2 2 2 4 2 2 3 3 3 2 2 4 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 -
2 3 0 1 2 3 4 0 1 2 3 4 0 1 2 4 0 1 2 3 5 0
------------------------------------------------------------------------
3 3 2 4 2 2 2 4 2 2 2 5 2 2 6 3 2
6a a b b b c + 3a a b b c + 3a a b b c - 3a a b b c + a b c -
1 2 3 4 5 0 1 2 4 5 0 1 2 3 5 0 1 2 4 5 0 1 5 0
------------------------------------------------------------------------
6 2 5 2 5 2 4
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 +
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
------------------------------------------------------------------------
5 2 2 4 2 3 3 2 4 2
a a b b c c - a a b b c c + a a b b c c - 2a a a b b c c +
0 2 3 4 0 1 1 2 1 4 0 1 1 2 2 4 0 1 0 1 2 3 4 0 1
------------------------------------------------------------------------
2 3 3 2 4 3 3 4 2
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 +
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
------------------------------------------------------------------------
3 3 4 2 2 3 3 2 2
2a a b b b c c - 2a 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 5 0 1 1 2 2 4 5 0 1 0 1 2 3 4 5 0 1 0 1 2 4 5 0 1
------------------------------------------------------------------------
4 2 2 5 2 3 2 2 4 2
- 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 -
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
------------------------------------------------------------------------
5 3 6 2 2 5 2 2 5 2 4 2
2a a b c c + a b b c - a a b b c - 2a a b b b c + 2a a a b b b c +
0 1 5 0 1 2 0 3 1 0 2 2 3 1 1 2 0 3 4 1 0 1 2 2 3 4 1
------------------------------------------------------------------------
2 4 2 2 2 3 2 2 2 4 2 2 3 2 2 4 2 2
a a b b c - a a a b b c + 2a a b b b c - 2a a a b b b c + a a b b c
1 2 0 4 1 0 1 2 2 4 1 1 2 0 3 5 1 0 1 2 2 3 5 1 0 2 3 5 1
------------------------------------------------------------------------
3 3 2 3 2 2 2 3 2 2 2 2 2 2
- 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 +
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
------------------------------------------------------------------------
4 2 2 2 4 2 2 2 2 2 2 2 2 3 2 2 2 4 3 2
a a 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 +
1 2 0 5 1 0 1 2 2 5 1 0 1 2 3 5 1 0 1 2 4 5 1 0 1 5 1
------------------------------------------------------------------------
5 2 2 4 2 5 3 2 4
a a b b c c - a a b b c c - 2a a b c c - 2a a b b b c c +
1 2 1 3 0 2 1 2 2 3 0 2 0 2 3 0 2 1 2 1 3 4 0 2
------------------------------------------------------------------------
3 3 4 2 3 3 2 4 2 2
2a a b b b c c + 5a a a b b c c + a a b b c c - a a b b c c -
1 2 2 3 4 0 2 0 1 2 3 4 0 2 1 2 1 4 0 2 1 2 2 4 0 2
------------------------------------------------------------------------
2 3 2 3 2 3 3 3 4 2
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 -
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
------------------------------------------------------------------------
2 3 2 4 2 5 3 2
4a a a b b c c - 2a a b b b c c + 2a a b b b c c + 6a a a b b b c c
0 1 2 3 5 0 2 1 2 1 4 5 0 2 1 2 2 4 5 0 2 0 1 2 3 4 5 0 2
------------------------------------------------------------------------
4 2 5 2 6 2 4 2
- 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 +
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
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 +
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
------------------------------------------------------------------------
2 4 4 2 3 2 4 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 b b c c -
1 2 0 3 4 1 2 0 1 2 1 3 4 1 2 0 1 2 2 3 4 1 2 0 2 3 4 1 2
------------------------------------------------------------------------
3 3 2 2 3 2 3 2 2 2 3 2
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 -
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
------------------------------------------------------------------------
2 2 2 3 3 3 2 3 3 2
a a a 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 2 4 1 2 1 2 0 3 5 1 2 0 1 2 1 3 5 1 2 0 1 2 2 3 5 1 2
------------------------------------------------------------------------
4 2 3 2 4
4a a b b b c c - 2a a a b b b c c - 2a a a b b b c c -
1 2 0 4 5 1 2 0 1 2 1 4 5 1 2 0 1 2 2 4 5 1 2
------------------------------------------------------------------------
2 2 2 2 3 2 5 2 4 2
2a a a b b b c c + 2a a a b b c c - 2a a b b c c + a a a b b c c +
0 1 2 3 4 5 1 2 0 1 2 4 5 1 2 1 2 0 5 1 2 0 1 2 1 5 1 2
------------------------------------------------------------------------
5 2 2 4 2 2 4 2 2 4 2 2 2 4 3 2
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 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
2a a b b b c + 2a a a b b b c - 2a a a b b c + a a 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
------------------------------------------------------------------------
3 2 2 2 2 2 2 2 2 4 2 2 3 2 2
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 +
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
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 +
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
------------------------------------------------------------------------
6 2 2 5 2 2 2 4 2 2 6 2 5
a b b c - a a b b c + a a b b c + a b b c c - 2a a b b b c c +
1 0 5 2 0 1 1 5 2 0 1 3 5 2 2 1 3 0 3 1 2 1 2 3 0 3
------------------------------------------------------------------------
2 4 2 6 2 5 2 5 2
a a b b c c - 2a b b c c + 2a a b b c c - a a b b c c +
1 2 2 3 0 3 2 0 3 0 3 0 2 2 3 0 3 1 2 1 4 0 3
------------------------------------------------------------------------
2 4 3 3 2 5 5
2a a b b b c c - a a b b c c + 4a a b b b c c - 2a a b b b c c -
1 2 1 2 4 0 3 1 2 2 4 0 3 1 2 0 3 4 0 3 0 2 1 3 4 0 3
------------------------------------------------------------------------
4 2 4 2 4 2 2 4 2
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 -
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
------------------------------------------------------------------------
2 3 3 2 4 2 3 3 4 2 2
a a a b c c + a a b b c c - 2a a b b b c c + a a b b c c -
0 1 2 4 0 3 1 2 1 5 0 3 1 2 1 2 5 0 3 1 2 2 5 0 3
------------------------------------------------------------------------
2 4 4 2 4 2 3 3
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 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
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
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
------------------------------------------------------------------------
3 2 2 4 2 2 3 2 2 4 3
+ 4a a 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 2 1 5 0 3 0 1 2 2 5 0 3 0 1 2 4 5 0 3 0 1 5 0 3
------------------------------------------------------------------------
6 5 5 4 2
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 +
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
------------------------------------------------------------------------
5 2 4 4 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 +
1 2 0 1 4 1 3 1 2 0 2 4 1 3 0 1 2 1 2 4 1 3 0 1 2 2 4 1 3
------------------------------------------------------------------------
5 2 4 4 2 2 3 2
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 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
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 -
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
------------------------------------------------------------------------
4 2 4 2 3
2a a a b b b c c - a a b b b c c + 3a a a b b b c c +
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
3a a a b b b c c + a a a b b b c c - 4a a a b 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
------------------------------------------------------------------------
3 3 3 2 2 3 2 2 2 2 2 2
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 c c +
0 2 3 4 5 1 3 0 1 2 4 5 1 3 0 1 2 0 5 1 3 0 1 2 1 5 1 3
------------------------------------------------------------------------
2 3 2 3 2 2 3 2 2 3 3 3
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 +
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
------------------------------------------------------------------------
5 5 2 2 4 4
a a b b b c c - a a b b c c - a a b b b c c + a a a b b b c c +
1 2 0 1 3 2 3 0 2 1 3 2 3 1 2 0 2 3 2 3 0 1 2 1 2 3 2 3
------------------------------------------------------------------------
5 2 2 4 2 2 4 4 2
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 +
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
------------------------------------------------------------------------
3 3 2 3 4 2 4
a a b b b c c - a a a b b b c c - 5a a a b b b c c + 2a a b b b c c
1 2 0 2 4 2 3 0 1 2 1 2 4 2 3 0 1 2 0 3 4 2 3 0 2 1 3 4 2 3
------------------------------------------------------------------------
2 3 2 3 2 2 3 2 2 2 2 2
+ 3a a a b b b c c + 3a a a b b c c - 2a a a b b c c - a a a b b c c
0 1 2 2 3 4 2 3 0 1 2 0 4 2 3 0 1 2 1 4 2 3 0 1 2 2 4 2 3
------------------------------------------------------------------------
3 3 2 3 2 3 3 3 2 3 2
- 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 -
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
------------------------------------------------------------------------
4 2 3 2 2 3
a a b b b c c + a a a b b b c c + 6a a a b b b c c -
1 2 0 2 5 2 3 0 1 2 1 2 5 2 3 0 1 2 0 3 5 2 3
------------------------------------------------------------------------
2 3 2 2 2 3 3 2
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 1 3 5 2 3 0 1 2 2 3 5 2 3 0 2 3 5 2 3
------------------------------------------------------------------------
3 2 2 2 2 2 3
7a a a b b b c c + 5a a a b b b c c + 2a a a b b b c c -
0 1 2 0 4 5 2 3 0 1 2 1 4 5 2 3 0 1 2 2 4 5 2 3
------------------------------------------------------------------------
3 2 3 2 2 4 2 2 3 2
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 -
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
------------------------------------------------------------------------
2 4 2 3 2 2 3 3 2 6 2 2 5 2
a a b b c c + 2a a a b b c c + a a b b c c + a b b c - 2a a b b b c
0 1 2 5 2 3 0 1 2 3 5 2 3 0 1 4 5 2 3 2 0 3 3 0 2 0 2 3 3
------------------------------------------------------------------------
2 4 2 2 5 2 2 4 2 2 3 2 2 2 4 2 2
+ 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 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
- 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 -
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
------------------------------------------------------------------------
2 3 2 3 2 2 2 2 2 2 2 3 2 2 2
2a a a b b b c + 2a a a b b b c + 2a a a b b c - 2a a a b b c +
0 1 2 0 4 5 3 0 1 2 2 4 5 3 0 1 2 0 5 3 0 1 2 2 5 3
------------------------------------------------------------------------
4 2 2 2 4 2 2 4 2 3 2 5 2 2 4
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 -
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
------------------------------------------------------------------------
3 3 2 5 2 5 2 4 2
a a b b c c + 2a a b b c c + a a b b c c - 3a a a b b c c +
1 2 2 3 0 4 1 2 0 3 0 4 0 2 1 3 0 4 0 1 2 2 3 0 4
------------------------------------------------------------------------
2 4 2 3 3 4 2 2 2 4
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 +
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
------------------------------------------------------------------------
2 3 2 4 2 3 3 2 2 3 2
4a a a b b b c c - a a b b c c + 2a a b b c c - a a a b b c c -
0 1 2 2 3 4 0 4 0 2 3 4 0 4 1 2 0 4 0 4 0 1 2 1 4 0 4
------------------------------------------------------------------------
3 2 2 2 3 2 3 3 2 4 2
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 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
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 +
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
------------------------------------------------------------------------
2 3 2 4 2 3 2
4a a a b b c c - 4a a b b b c c + 4a a a b b b c c -
0 1 2 3 5 0 4 1 2 0 4 5 0 4 0 1 2 1 4 5 0 4
------------------------------------------------------------------------
2 2 2 2 3 2 5 2 4 2
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 +
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
------------------------------------------------------------------------
5 2 2 3 2 2 4 2 5
a a b b c c + 4a a a b b c c - a a b b c c + a a b b b c c -
0 1 2 5 0 4 0 1 2 3 5 0 4 0 1 4 5 0 4 1 2 0 1 3 1 4
------------------------------------------------------------------------
2 4 4 2 3 2 5 2
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 +
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
------------------------------------------------------------------------
2 4 2 2 4 3 3 2 3
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 -
0 2 2 3 1 4 1 2 0 1 4 1 4 1 2 0 2 4 1 4 0 1 2 1 2 4 1 4
------------------------------------------------------------------------
3 2 2 4 2 3 2 3 2
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
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
+ 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 +
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
------------------------------------------------------------------------
4 2 2 3 2 3 2 2 2
a a a b b c c - 2a 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 5 1 4 0 1 2 0 3 5 1 4 0 1 2 1 3 5 1 4 0 1 2 2 3 5 1 4
------------------------------------------------------------------------
3 3 2 3 2 2 2 2
- 2a a b b c c + a a a b b b c c - a a a b b b c c +
0 2 3 5 1 4 0 1 2 0 4 5 1 4 0 1 2 1 4 5 1 4
------------------------------------------------------------------------
3 2 3 2 2 2 3 2 2 4 2
3a 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 3 4 5 1 4 0 1 2 4 5 1 4 0 1 2 1 5 1 4 0 1 2 5 1 4
------------------------------------------------------------------------
3 2 2 3 3 2 2 4 4 2
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 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
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 +
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
------------------------------------------------------------------------
3 3 2 3 2 4 2 3 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 +
1 2 0 1 4 2 4 0 1 2 1 4 2 4 1 2 0 2 4 2 4 0 1 2 1 2 4 2 4
------------------------------------------------------------------------
2 3 2 2 2 3 3 2 3 2 2
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 +
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
------------------------------------------------------------------------
2 2 2 2 3 2 2 4 2 3 2 2
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 +
0 1 2 1 4 2 4 0 1 2 3 4 2 4 1 2 0 1 5 2 4 0 1 2 1 5 2 4
------------------------------------------------------------------------
5 4 3 2 2 2 2
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 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
+ a a a b b b c c - 2a a a b b c c + 3a a a b 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
------------------------------------------------------------------------
2 3 3 2 5 2 2 4 2
3a a a b b b c c + 3a a a b b b c c - 2a a b b c c + 2a a b b c c -
0 1 2 1 4 5 2 4 0 1 2 3 4 5 2 4 0 1 0 5 2 4 0 1 1 5 2 4
------------------------------------------------------------------------
3 3 2 5 2 5 4
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 -
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
------------------------------------------------------------------------
2 4 2 3 2 2 4 2 4
a a b 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 -
0 2 1 2 3 3 4 0 1 2 2 3 3 4 1 2 0 4 3 4 0 1 2 0 1 4 3 4
------------------------------------------------------------------------
2 3 2 3 2 2 2 2 2 4
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 +
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
------------------------------------------------------------------------
3 3 2 3 2 3 2 2 3 3 2
a a b b b c c + a a a b b c c - a a a b b c c - 2a a b b c c +
0 2 2 3 4 3 4 0 1 2 0 4 3 4 0 1 2 2 4 3 4 1 2 0 5 3 4
------------------------------------------------------------------------
2 3 3 2 2 2 2 2 3 2
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 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
- 2a a a b b b c c + a a b b b c c + a 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
------------------------------------------------------------------------
2 2 2 3 2 4 2 4 2
a a a b b b c c - a a a b b b c c - a a b b b c c + a a a b b c c -
0 1 2 0 4 5 3 4 0 1 2 1 4 5 3 4 0 2 3 4 5 3 4 0 1 2 4 5 3 4
------------------------------------------------------------------------
2 3 2 3 2 2 3 3 2 4 2 2
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 +
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
------------------------------------------------------------------------
2 4 2 2 4 2 2 3 2 2 3 2
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 +
1 2 0 3 4 0 1 2 0 1 3 4 0 1 2 0 2 3 4 0 1 2 1 2 3 4
------------------------------------------------------------------------
2 4 2 2 3 3 2 2 3 3 2 2 2 3 2 3 2 2
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 -
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
------------------------------------------------------------------------
2 2 2 2 2 3 2 3 2 2 4 2 2 2
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 -
0 1 2 1 2 4 4 0 1 2 0 3 4 4 0 1 2 2 3 4 4 1 2 0 5 4
------------------------------------------------------------------------
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 a b b b c + 2a a a b b b c -
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
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 +
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
------------------------------------------------------------------------
3 2 2 4 2 2 4 2 2 3 3 2 2 4 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 +
0 1 2 1 4 5 4 0 1 2 3 4 5 4 0 1 0 5 4 0 1 1 5 4 0 1 3 5 4
------------------------------------------------------------------------
2 4 2 3 3 4 2 2 2 4 2
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 -
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
------------------------------------------------------------------------
4 2 2 3 2 2 4 3 3 3 2
2a a a b b c c + 4a a a b b c c + 2a a b c c - a a b b c c +
0 1 2 1 3 0 5 0 1 2 2 3 0 5 0 2 3 0 5 1 2 1 4 0 5
------------------------------------------------------------------------
4 2 5 2 3 3 2 3
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 -
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
------------------------------------------------------------------------
3 2 2 3 2 4 2 2 4 2
6a a a b b b c c - 4a a a b b c c - 2a a b b c c + 2a a a b b c c +
0 1 2 2 3 4 0 5 0 1 2 3 4 0 5 1 2 0 4 0 5 0 1 2 2 4 0 5
------------------------------------------------------------------------
2 2 2 2 2 3 3 4 2 2 5
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 +
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
------------------------------------------------------------------------
6 2 4 2 4 5
a b b c c - 4a a b b b c c + 4a a a b b b c c + 4a a b b b c c -
1 2 5 0 5 1 2 0 3 5 0 5 0 1 2 2 3 5 0 5 1 2 0 4 5 0 5
------------------------------------------------------------------------
4 5 2 4 2 6 2
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 +
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
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 +
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
------------------------------------------------------------------------
2 3 3 2 2 4 2 2 4 2
a a a b b b c c - a a a b b c c + 4a a a b b c c - a a b b c c -
0 1 2 1 2 3 1 5 0 1 2 2 3 1 5 0 1 2 0 3 1 5 0 2 1 3 1 5
------------------------------------------------------------------------
2 3 2 3 3 4 2 3 2
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 +
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
------------------------------------------------------------------------
4 2 2 3 2 3
a a a b b c c - 7a a a b b b c c + 2a a a b b b c c +
0 1 2 2 4 1 5 0 1 2 0 3 4 1 5 0 1 2 1 3 4 1 5
------------------------------------------------------------------------
2 2 2 3 3 2 3 2 2 2 2 2 2
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 -
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
------------------------------------------------------------------------
2 3 2 3 2 2 3 2 3 4 2
2a a a b b c c - 2a a a b b c c + a a a b c c - a a b b b c c +
0 1 2 2 4 1 5 0 1 2 3 4 1 5 0 1 2 4 1 5 1 2 0 1 5 1 5
------------------------------------------------------------------------
5 4 5 2 3 2
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 -
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
------------------------------------------------------------------------
2 2 2 2 3 3 2 2
3a a a b b b c c - 3a a a b b b c c + 2a a a b b c c -
0 1 2 1 3 5 1 5 0 1 2 2 3 5 1 5 0 1 2 3 5 1 5
------------------------------------------------------------------------
4 2 3 2 4
5a a a b b b c c + 3a a a b b b c c + 2a a b b b c c -
0 1 2 0 4 5 1 5 0 1 2 1 4 5 1 5 0 1 2 4 5 1 5
------------------------------------------------------------------------
3 2 3 3 2 5 2 2 4 2
a 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 +
0 1 2 3 4 5 1 5 0 1 4 5 1 5 0 1 0 5 1 5 0 1 1 5 1 5
------------------------------------------------------------------------
3 3 2 3 3 2 3 2 4 2
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 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
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 -
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
------------------------------------------------------------------------
3 3 3 4 2 3 2 2 5
2a 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 -
0 2 3 2 5 1 2 0 1 4 2 5 0 1 2 1 4 2 5 1 2 0 2 4 2 5
------------------------------------------------------------------------
4 3 2 2 2 2
a a a b b b c c + 3a a a b b b c c - 4a a a b b b c c +
0 1 2 1 2 4 2 5 0 1 2 0 3 4 2 5 0 1 2 1 3 4 2 5
------------------------------------------------------------------------
2 3 3 2 2 4 2 2 3 2
a a a b b b c c + 3a a a b b c c - a a a b b c c + a a a b b c c -
0 1 2 2 3 4 2 5 0 1 2 3 4 2 5 0 1 2 0 4 2 5 0 1 2 1 4 2 5
------------------------------------------------------------------------
3 2 2 5 4 2 6
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 +
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
------------------------------------------------------------------------
5 4 2 3 2 4
a a b b b c c - 2a a a b b b c c + 3a a a b b b c c - a a b b b c c
0 1 1 2 5 2 5 0 1 2 0 3 5 2 5 0 1 2 1 3 5 2 5 0 1 2 3 5 2 5
------------------------------------------------------------------------
3 2 2 5 2 4 3 3
- 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 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
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 +
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
------------------------------------------------------------------------
2 2 2 2 2 4 2 3 3 2 3 3 2
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 1 2 2 3 3 5 0 2 0 3 3 5 0 2 2 3 3 5 1 2 0 4 3 5
------------------------------------------------------------------------
2 3 2 3 2 3 2 2 3 2
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
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
------------------------------------------------------------------------
2 3 3 3 3 2
+ 6a a a b b b c c - 2a a b b b c c - 4a a a b b b c c -
0 1 2 0 3 4 3 5 0 2 1 3 4 3 5 0 1 2 2 3 4 3 5
------------------------------------------------------------------------
2 2 2 2 3 2 2 3 2 2 4 2 2
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 -
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
------------------------------------------------------------------------
4 3 4 2 2 3 2 2 2 2 2
a a a b c c + 2a a b b c c - 2a a a b b b c c + a a a b b c c -
0 1 2 4 3 5 1 2 0 5 3 5 0 1 2 0 1 5 3 5 0 1 2 1 5 3 5
------------------------------------------------------------------------
4 2 4 2 2 2 2 3 2
2a a a b b b c c + a a b b c c - 4a a a b b b c c + 2a a a b b b c c
0 1 2 0 2 5 3 5 0 1 2 5 3 5 0 1 2 0 3 5 3 5 0 1 2 1 3 5 3 5
------------------------------------------------------------------------
3 2 4 2 2 2 3
+ 2a a a b b b c c - 2a a b b c c + 6a a a b b b c c -
0 1 2 2 3 5 3 5 0 2 3 5 3 5 0 1 2 0 4 5 3 5
------------------------------------------------------------------------
3 2 3 3 4 4 2 2
4a a a b b b c c - 2a a b b b c c + 2a a a b b b c c + a a b b c c -
0 1 2 1 4 5 3 5 0 1 2 4 5 3 5 0 1 2 3 4 5 3 5 0 1 4 5 3 5
------------------------------------------------------------------------
2 4 2 3 3 2 4 2 2 3 3 2
2a 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 0 5 3 5 0 1 1 5 3 5 0 1 3 5 3 5 1 2 0 3 4 5
------------------------------------------------------------------------
2 3 2 3 2 3 2 2 2 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
0 1 2 0 1 3 4 5 0 1 2 1 3 4 5 0 1 2 0 2 3 4 5 0 1 2 1 2 3 4 5
------------------------------------------------------------------------
2 3 2 3 3 2 3 2 2 4 2 2
- 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 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
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
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
------------------------------------------------------------------------
2 2 2 3 2 4 2 2 2 3 2
+ 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 -
0 1 2 0 3 4 4 5 0 1 2 2 3 4 4 5 0 2 3 4 4 5 0 1 2 0 4 4 5
------------------------------------------------------------------------
3 2 2 4 2 5 2 4
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 -
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
------------------------------------------------------------------------
2 3 2 5 2 4 2 3
a a a b b c c + a a b b b c c - a a b b b c c - 2a a a b b b c c +
0 1 2 1 5 4 5 0 1 0 2 5 4 5 0 1 1 2 5 4 5 0 1 2 0 3 5 4 5
------------------------------------------------------------------------
3 2 3 3 2 4 3 3
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 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
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 +
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
------------------------------------------------------------------------
2 2 2 2 2 3 2 2 2 4 2 3 2 5 2 2 4 2
2a a a b b c - 2a a a b b c + a a b c - a a b b c + 2a a a b b b c
0 1 2 0 3 5 0 1 2 1 3 5 0 2 3 5 1 2 0 4 5 0 1 2 0 1 4 5
------------------------------------------------------------------------
2 3 2 2 2 3 2 3 2 2 4 2 2
- 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 +
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
------------------------------------------------------------------------
6 2 2 5 2 2 4 2 2 2 4 2 3 3 2
a b b c - 2a a b b b c + a a b b c + 2a a b b b c - 2a a b b b c +
1 0 5 5 0 1 0 1 5 5 0 1 1 5 5 0 1 0 3 5 5 0 1 1 3 5 5
------------------------------------------------------------------------
4 2 2 2 6 3 5 2 2 4 2 3 3 3
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 +
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
------------------------------------------------------------------------
6 5 5 4 2
3a b b b c c - 3a a b b b c c - 3a a b b b c c + 3a a a b b c c -
2 0 1 3 0 6 1 2 0 2 3 0 6 0 2 1 2 3 0 6 0 1 2 2 3 0 6
------------------------------------------------------------------------
5 5 2 2 4 4
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 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
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 +
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
------------------------------------------------------------------------
3 3 3 2 4 4 2 3 3
a a b c c + 3a a b b b c c - 6a a a b b c c - 3a a b b b c c +
0 2 4 0 6 1 2 0 1 5 0 6 0 1 2 1 5 0 6 1 2 0 2 5 0 6
------------------------------------------------------------------------
2 3 3 2 2 4
9a a a b b b c c - 3a a a b b c c + 6a a a b b b c c +
0 1 2 1 2 5 0 6 0 1 2 2 5 0 6 0 1 2 0 3 5 0 6
------------------------------------------------------------------------
2 4 2 3 2 3
3a a b b b c c - 9a a a b b b c c - 9a a a b b b c c +
0 2 1 3 5 0 6 0 1 2 2 3 5 0 6 0 1 2 0 4 5 0 6
------------------------------------------------------------------------
2 3 3 3 3 2 2 3 2 2
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
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
------------------------------------------------------------------------
2 2 2 2 2 3 2 3 2 2 3 2 2
- 9a a a b b c c + 3a a a b b c c + 6a a a b b c c + 3a a a b b c c
0 1 2 1 5 0 6 0 1 2 2 5 0 6 0 1 2 3 5 0 6 0 1 2 4 5 0 6
------------------------------------------------------------------------
3 3 3 6 2 5 5 2
- 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 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
a a b b c c + 2a a a b b c c - a a a b c c - 2a 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
------------------------------------------------------------------------
5 2 4 2 5 2 5
4a a b b b c c - 2a a b b c c + 2a a b b c c - 2a a b b b c c -
0 2 0 2 3 1 6 0 2 2 3 1 6 1 2 0 4 1 6 0 2 0 1 4 1 6
------------------------------------------------------------------------
4 2 4 2 4 2 3 3 2
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 -
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
------------------------------------------------------------------------
2 4 2 4 2 4 2 2 3
2a a b b c c + 4a a a b b b c c + a a b b c c - 6a a a b b b c c +
1 2 0 5 1 6 0 1 2 0 1 5 1 6 0 2 1 5 1 6 0 1 2 1 2 5 1 6
------------------------------------------------------------------------
2 2 2 2 2 4 3 3 3 3
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 +
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
------------------------------------------------------------------------
3 2 4 2 2 3 2 2 3 2 2
2a a a b b b c c + a a b b c c + 4a a a b b c c - 4a a a b b c c -
0 1 2 2 4 5 1 6 0 2 4 5 1 6 0 1 2 1 5 1 6 0 1 2 2 5 1 6
------------------------------------------------------------------------
4 2 2 4 2 4 2 3 5 2
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 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
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 +
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
------------------------------------------------------------------------
2 3 2 5 2 5 4
a a a b b c c + 2a a b b c c - 3a a b b b c c - a a a b b b c c +
0 1 2 1 2 2 6 1 2 0 3 2 6 0 2 0 1 3 2 6 0 1 2 0 2 3 2 6
------------------------------------------------------------------------
2 4 2 3 2 2 4 2 4
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 -
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
------------------------------------------------------------------------
2 4 2 2 3 2 3 2 4
3a a b b c c - a a a b b b c c + a a a b b b c c + 3a a b b b c c -
0 2 1 4 2 6 0 1 2 0 2 4 2 6 0 1 2 1 2 4 2 6 0 2 0 3 4 2 6
------------------------------------------------------------------------
3 3 2 3 2 3 3 2 3 2 2
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 -
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
------------------------------------------------------------------------
4 2 3 3 3 2 2 3 2 3 2
a a b c c + 2a a b b c c - 7a a a b b b c c + 5a a a b b c c +
0 2 4 2 6 1 2 0 5 2 6 0 1 2 0 1 5 2 6 0 1 2 1 5 2 6
------------------------------------------------------------------------
3 2 2 2 2 2 3
3a 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 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
3a a b b b c c + 5a a a b b b c c + 9a a 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
------------------------------------------------------------------------
3 2 3 2 4 2
7a a a b b b c c - 2a a a b b b c c + 3a a b b b c c +
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
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 -
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
------------------------------------------------------------------------
4 2 4 2 2 6 2 5 2
4a a a b b c c - a a b b c c - a b b c c + a a b b c c +
0 1 2 3 5 2 6 0 1 4 5 2 6 2 0 1 3 6 1 2 0 2 3 6
------------------------------------------------------------------------
5 4 2 2 4 2 2 3 3
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 +
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
------------------------------------------------------------------------
5 2 2 4 3 3 2 4 2
a 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 -
0 2 0 4 3 6 0 2 0 2 4 3 6 0 2 2 4 3 6 0 1 2 0 5 3 6
------------------------------------------------------------------------
2 4 2 3 3 3 3 2 2
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
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
------------------------------------------------------------------------
3 3 4 2 3 2 2 4 2 2
+ 2a a b b b c c - 2a a b b b c c - 4a a a b b c c - a a b b c c +
0 2 0 4 5 3 6 0 2 2 4 5 3 6 0 1 2 0 5 3 6 0 2 1 5 3 6
------------------------------------------------------------------------
4 2 5 2 5 3 5 2
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 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
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 -
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
------------------------------------------------------------------------
2 3 2 5 2 2 4 3 3 2
a 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 2 4 6 0 2 0 3 4 6 0 2 0 2 3 4 6 0 2 2 3 4 6
------------------------------------------------------------------------
4 2 2 4 2 3 3 3
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
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
------------------------------------------------------------------------
3 2 2 3 3 2 4 2 2 2 3 2
- a 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 -
0 1 2 2 4 4 6 0 2 0 4 4 6 0 2 2 4 4 6 0 1 2 0 5 4 6
------------------------------------------------------------------------
2 3 3 3 2 2 2 2 3 2
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
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
------------------------------------------------------------------------
3 2 2 3 3 4 2 3 2
+ a a a b b c c + 4a a b b b c c - 4a a b b b c c - 2a a a b b b c c
0 1 2 2 5 4 6 0 2 0 3 5 4 6 0 2 2 3 5 4 6 0 1 2 0 4 5 4 6
------------------------------------------------------------------------
4 2 5 2 3 2 2 4 2
+ 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 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
a a b b c c + 2a a b b c c + a a b b c c - a a 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
------------------------------------------------------------------------
4 2 2 4 3 3 3 2 2 3
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 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
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 -
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
------------------------------------------------------------------------
3 3 3 2 2 2 3 2 2 3
3a a b b b c c - a a a b b c c + 5a a a b b c c - 7a a a b b b c c +
0 2 1 2 3 5 6 0 1 2 2 3 5 6 0 1 2 0 4 5 6 0 1 2 0 1 4 5 6
------------------------------------------------------------------------
3 3 2 2 2 2 3 2 3 2 2
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 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
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 -
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
------------------------------------------------------------------------
4 2 5 3 3 2 2 2 2 2
2a a a b b c c + a a b c c - 6a a a b b c c + 9a a a b b b c c -
0 1 2 2 4 5 6 0 2 4 5 6 0 1 2 0 5 5 6 0 1 2 0 1 5 5 6
------------------------------------------------------------------------
3 2 2 2 3 3 2 3 3 2
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 -
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
------------------------------------------------------------------------
3 2 4 2 4
2a a a b b b c c + 3a a b b b c c - a a a b b b c c -
0 1 2 0 3 5 5 6 0 2 1 3 5 5 6 0 1 2 2 3 5 5 6
------------------------------------------------------------------------
3 2 4 4 2
7a a a b b b c c + 5a a a b b b c c + 2a a b b b c c -
0 1 2 0 4 5 5 6 0 1 2 1 4 5 5 6 0 1 2 4 5 5 6
------------------------------------------------------------------------
5 5 2 3 3 2 4 2 2
3a a b b b c c - a a b b c c + 2a a b b c c - 2a a b b c c +
0 2 3 4 5 5 6 0 1 4 5 5 6 0 1 0 5 5 6 0 1 1 5 5 6
------------------------------------------------------------------------
5 2 6 3 2 5 2 2 2 4 2 2 3 3 3 2
2a a b b c c + a b c - 3a a b b c + 3a a b b c - a a b c +
0 1 3 5 5 6 2 0 6 0 2 0 2 6 0 2 0 2 6 0 2 2 6
------------------------------------------------------------------------
2 4 2 2 3 3 2 4 2 2 2 4 2 2 2 5 2 2
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 +
0 2 0 5 6 0 2 0 2 5 6 0 2 2 5 6 0 2 0 5 6 0 2 2 5 6
------------------------------------------------------------------------
6 3 2 5 3 2 4 2 3 3 2 4 2 3
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 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
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 -
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
------------------------------------------------------------------------
2 3 2 5 2 2 4 2 2 4
4a a a b b c c + 2a a b b c c - 2a a b b c c + 3a a b b b c c -
0 1 2 2 3 0 7 0 2 0 3 0 7 0 2 2 3 0 7 1 2 0 1 4 0 7
------------------------------------------------------------------------
4 2 3 3 2 3 3 2 2
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 -
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
------------------------------------------------------------------------
4 2 4 2 3 2 3 2
a 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 +
0 1 2 0 3 4 0 7 0 2 1 3 4 0 7 0 1 2 2 3 4 0 7 0 1 2 0 4 0 7
------------------------------------------------------------------------
2 3 2 3 3 2 3 3 2 3 2
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 +
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
------------------------------------------------------------------------
4 2 3 2 4 2
3a a b b b c c - 7a a a b b b c c + 2a a a b b c c -
1 2 0 2 5 0 7 0 1 2 1 2 5 0 7 0 1 2 2 5 0 7
------------------------------------------------------------------------
2 3 2 3 2 2 2
2a a a b b b c c - 7a a a b b b c c + 9a a a b b b c c +
0 1 2 0 3 5 0 7 0 1 2 1 3 5 0 7 0 1 2 2 3 5 0 7
------------------------------------------------------------------------
3 3 2 3 2 2 2 2
2a a b b c c + 5a a a b b b c c - 3a a a b b b c c -
0 2 3 5 0 7 0 1 2 0 4 5 0 7 0 1 2 1 4 5 0 7
------------------------------------------------------------------------
2 3 3 2 4 2
2a a a b b b c c + 3a a a b b b c c - 4a a a b b c c +
0 1 2 2 4 5 0 7 0 1 2 3 4 5 0 7 0 1 2 0 5 0 7
------------------------------------------------------------------------
2 3 2 2 4 2 3 2 2 3 3 2
5a a a b b c c - a a b b c c - 6a a a b b c c + a a b b c c -
0 1 2 1 5 0 7 0 1 2 5 0 7 0 1 2 3 5 0 7 0 1 4 5 0 7
------------------------------------------------------------------------
5 2 2 4 4 2 3 3 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 -
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
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 -
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
------------------------------------------------------------------------
4 2 4 2 3 2 2 4 2
5a a a b b b c c - a a b b b c c + 3a a a b b c c - 2a a b b c c +
0 1 2 0 2 3 1 7 0 2 1 2 3 1 7 0 1 2 2 3 1 7 1 2 0 4 1 7
------------------------------------------------------------------------
4 2 3 2 3 2 2 2 2
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 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
------------------------------------------------------------------------
2 4 3 3 3 3 2 2 3
- a a b b b c c + a a b b b c c + 2a a b b c c - 3a a a b b b c c -
0 2 0 3 4 1 7 0 2 2 3 4 1 7 1 2 0 5 1 7 0 1 2 0 1 5 1 7
------------------------------------------------------------------------
2 3 2 3 2 2 2 2 2 3 2
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
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
+ 6a a a b b b c c + a a b b b c c - 7a 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
------------------------------------------------------------------------
2 2 2 3 2 3 2
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 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
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 +
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
------------------------------------------------------------------------
4 2 4 2 2 2 4 2 4 3
4a a a b b c c - a a b b c c + a a b b c c - a a a b c c -
0 1 2 3 5 1 7 0 1 4 5 1 7 1 2 0 1 2 7 0 1 2 1 2 7
------------------------------------------------------------------------
3 3 2 3 2 4 2 2 3 2 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 -
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
------------------------------------------------------------------------
2 4 2 4 2 4 2 2 3
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 -
1 2 0 3 2 7 0 1 2 0 1 3 2 7 0 2 1 3 2 7 0 1 2 0 2 3 2 7
------------------------------------------------------------------------
2 3 2 2 2 2 2 4 2 3 3 2
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 +
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
------------------------------------------------------------------------
3 3 2 2 3 2 3 2 2 3
2a a b b c c - 4a a a b b b c c + 2a a a b b c c + 2a a a b b b c c
1 2 0 4 2 7 0 1 2 0 1 4 2 7 0 1 2 1 4 2 7 0 1 2 0 3 4 2 7
------------------------------------------------------------------------
3 3 2 2 2 2 3 2 2 4 2 2
- 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 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
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
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
------------------------------------------------------------------------
2 3 2 2 2 3 2
+ 2a a a b b b c c - 4a a a b b b c c + 6a a a b b b c c -
0 1 2 1 2 5 2 7 0 1 2 0 3 5 2 7 0 1 2 1 3 5 2 7
------------------------------------------------------------------------
3 2 4 2 2 2 3
2a a a b b b c c - 2a a b b c c - 2a a a b b b c c +
0 1 2 2 3 5 2 7 0 2 3 5 2 7 0 1 2 0 4 5 2 7
------------------------------------------------------------------------
3 2 4 2 4 2 3 3 2
2a a a b b b c c - 2a a a b b b c c + 2a a b b c c - 2a a b b c c +
0 1 2 1 4 5 2 7 0 1 2 3 4 5 2 7 0 1 0 5 2 7 0 1 1 5 2 7
------------------------------------------------------------------------
4 2 2 5 2 2 4 2 4
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 3 5 2 7 1 2 0 1 3 7 1 2 0 2 3 7 0 1 2 0 1 2 3 7
------------------------------------------------------------------------
2 3 2 2 3 2 2 2 2 3 5 2
2a a a b b c c + a a a b b c c - a a a b c c - 2a a b b c c +
0 1 2 0 2 3 7 0 1 2 1 2 3 7 0 1 2 2 3 7 0 2 0 3 3 7
------------------------------------------------------------------------
2 4 3 3 2 4 2 2 3
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 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
a a a b b c c + 2a a a b b b c c - 2a a 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
------------------------------------------------------------------------
3 2 3 2 2 3 3 4 2
2a a a b b b c c + 2a a a b b c c - 4a a b b b c c + 4a a b b b c c
0 1 2 1 2 5 3 7 0 1 2 2 5 3 7 0 2 0 3 5 3 7 0 2 2 3 5 3 7
------------------------------------------------------------------------
3 2 4 4 2 4 2 2
+ 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
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
------------------------------------------------------------------------
5 2 5 2 2 4 2 4 2
- 2a a b b c c + a a b b c c - a a b b c c + a a a b b c c +
0 2 3 5 3 7 0 1 4 5 3 7 1 2 0 1 4 7 0 1 2 0 1 4 7
------------------------------------------------------------------------
3 3 2 2 3 2 3 2 2 2 2 2 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 -
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
------------------------------------------------------------------------
2 4 2 3 3 3 3 2 2
a a b b b c c + a a a b b b c c + a a b b b c c - a a a b b c c +
0 2 0 1 3 4 7 0 1 2 0 2 3 4 7 0 2 1 2 3 4 7 0 1 2 2 3 4 7
------------------------------------------------------------------------
2 3 2 2 3 2 2 2 3 2
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 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
+ 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 +
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
------------------------------------------------------------------------
3 2 2 2 3 3 2
a a a b b c c + 3a a a b b b c c - 3a a a b b b c c -
0 1 2 1 5 4 7 0 1 2 0 2 5 4 7 0 1 2 1 2 5 4 7
------------------------------------------------------------------------
3 2 4 2 4
2a a a b b b c c - a a b b b c c + 3a a a b b b c c +
0 1 2 0 3 5 4 7 0 2 1 3 5 4 7 0 1 2 2 3 5 4 7
------------------------------------------------------------------------
3 2 4 5 3 3 2
a a a b b b c c - a a a b b b c c + a a b b b c c - 2a a b b c c +
0 1 2 0 4 5 4 7 0 1 2 1 4 5 4 7 0 2 3 4 5 4 7 0 1 0 5 4 7
------------------------------------------------------------------------
4 2 2 5 2 3 3 2 2 3 2
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 +
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
------------------------------------------------------------------------
2 3 3 4 2 2 3 2 2 2 2 2
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 +
0 1 2 1 5 7 1 2 0 2 5 7 0 1 2 0 1 2 5 7 0 1 2 1 2 5 7
------------------------------------------------------------------------
2 3 2 2 3 3 3 2 2 2 2
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 +
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
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 +
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
------------------------------------------------------------------------
2 2 2 3 2 2 2 3
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 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
a a a b b b c c - 3a a a b b b c c + 2a a b 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
------------------------------------------------------------------------
4 3 2 2 4 2 5 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 +
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
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 +
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
------------------------------------------------------------------------
3 3 3 2 4 4 2
a a b b b c c + 6a a a b b b c c - 5a a a b b b c c - a a b b b c c
0 1 1 2 5 5 7 0 1 2 0 3 5 5 7 0 1 2 1 3 5 5 7 0 1 2 3 5 5 7
------------------------------------------------------------------------
5 2 3 3 4 2 5
+ 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 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
------------------------------------------------------------------------
5 3 5 2 4 2 2 4
2a a b c c + a a b b c c + 5a a a b b c c - 2a a b b b c c -
1 2 0 6 7 0 2 0 1 6 7 0 1 2 0 2 6 7 0 2 0 1 2 6 7
------------------------------------------------------------------------
2 3 2 3 3 2 3 2 3 2 4 2
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 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
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 +
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
------------------------------------------------------------------------
3 2 4 2 4 2 4 2
6a a 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
0 1 2 0 2 5 6 7 0 2 1 2 5 6 7 0 1 2 2 5 6 7 0 2 0 4 5 6 7
------------------------------------------------------------------------
5 4 2 5 2 5 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 -
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
------------------------------------------------------------------------
6 2 2 4 3 2 4 2 2 2 3 2 2 2 3 2
a b b c c + a a b c - a a a b b c - 2a a a b b c + 2a a a b b b c +
0 4 5 6 7 1 2 0 7 0 1 2 0 1 7 0 1 2 0 2 7 0 1 2 0 1 2 7
------------------------------------------------------------------------
2 2 2 2 2 3 2 2 2 2 4 2 2 3 3 2 4 2 2 2
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 +
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
------------------------------------------------------------------------
2 2 2 2 2 3 2 2 3 2 2 4 2
2a a a b b c - 2a a a b b b c - 2a a a b b b c + 2a a a b b b c +
0 1 2 0 5 7 0 1 2 0 1 5 7 0 1 2 0 2 5 7 0 1 2 1 2 5 7
------------------------------------------------------------------------
4 2 2 5 2 4 2 2 2 5 2 2 6 2 2
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 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
a a b c c + 3a a b b c c - 3a a b b c c + a a 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
------------------------------------------------------------------------
2 4 4 2 3 3 2 3
3a a b b b c c + 2a a a b b c c - 3a a b b b c c - 7a a a b b b c c
1 2 0 1 3 0 8 0 1 2 1 3 0 8 1 2 0 2 3 0 8 0 1 2 1 2 3 0 8
------------------------------------------------------------------------
3 2 2 4 2 2 4 2 2 3 2
+ 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 -
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
------------------------------------------------------------------------
3 3 2 3 2 4 2 3 2
3a a b b b c c + a a a b b c c + 3a a b b b c c + a a a b b b c c -
1 2 0 1 4 0 8 0 1 2 1 4 0 8 1 2 0 2 4 0 8 0 1 2 1 2 4 0 8
------------------------------------------------------------------------
4 2 2 3 2 3
2a a a b b c c + 5a a a b b b c c - 2a a a b b b c c -
0 1 2 2 4 0 8 0 1 2 0 3 4 0 8 0 1 2 1 3 4 0 8
------------------------------------------------------------------------
2 2 2 3 3 2 3 2 2 2 3 2
3a 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 +
0 1 2 2 3 4 0 8 0 2 3 4 0 8 0 1 2 0 4 0 8 0 1 2 2 4 0 8
------------------------------------------------------------------------
4 2 3 2 2 5 4
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
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
- a a b b c c - 2a a a b b b c c + 9a 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
------------------------------------------------------------------------
2 3 3 2 2 4
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 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
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 +
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
------------------------------------------------------------------------
5 2 2 4 2 3 3 2 2 4 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 -
0 1 0 5 0 8 0 1 1 5 0 8 0 1 3 5 0 8 1 2 0 1 1 8
------------------------------------------------------------------------
3 3 2 3 2 4 2 2 3 2 2
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 -
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
------------------------------------------------------------------------
4 3 2 4 2 4 2 3
a a a b c c - 2a a b b c c - 2a a a b b b c c + 6a a a b b b c c +
0 1 2 2 1 8 1 2 0 3 1 8 0 1 2 0 1 3 1 8 0 1 2 0 2 3 1 8
------------------------------------------------------------------------
2 3 2 2 2 2 2 4 2 3 3 2
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 +
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
------------------------------------------------------------------------
3 3 2 3 2 2 3 2 2 3
2a a b b c c - 4a a a b b b c c + 2a a a b b c c - 2a a a b b b c c
1 2 0 4 1 8 0 1 2 0 2 4 1 8 0 1 2 2 4 1 8 0 1 2 0 3 4 1 8
------------------------------------------------------------------------
3 2 2 2 2 2 3 2 2 4 2 2
+ 2a a a b b b c c + a a a b b c c - a a a b b c c - 2a a b b c c +
0 1 2 2 3 4 1 8 0 1 2 0 4 1 8 0 1 2 2 4 1 8 1 2 0 5 1 8
------------------------------------------------------------------------
3 2 2 2 2 2 4
2a 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 0 1 5 1 8 0 1 2 1 5 1 8 0 1 2 0 2 5 1 8
------------------------------------------------------------------------
2 3 2 4 2 2 2 2 3 2
4a a a b b b c c + a a b b c c - 4a a a b b b c c - 2a a a b b b c c
0 1 2 1 2 5 1 8 0 1 2 5 1 8 0 1 2 0 3 5 1 8 0 1 2 1 3 5 1 8
------------------------------------------------------------------------
3 2 4 2 2 2 3
+ 6a a a b b b c c + 2a a b b c c + 2a a a b b b c c -
0 1 2 2 3 5 1 8 0 2 3 5 1 8 0 1 2 0 4 5 1 8
------------------------------------------------------------------------
3 3 4 4 2 2 2 4 2
2a 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 +
0 1 2 4 5 1 8 0 1 2 3 4 5 1 8 0 1 4 5 1 8 0 1 0 5 1 8
------------------------------------------------------------------------
3 3 2 4 2 2 3 3 2 2 3 3
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 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
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 +
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
------------------------------------------------------------------------
3 3 2 2 3 2 3 2 3 2
2a a b b c c - a a a b b b c c - 2a a a b b c c - 3a a a b b b c c +
1 2 0 3 2 8 0 1 2 0 1 3 2 8 0 1 2 1 3 2 8 0 1 2 0 2 3 2 8
------------------------------------------------------------------------
2 2 2 2 3 2 2 3 2 3 3 2
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 -
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
------------------------------------------------------------------------
3 2 2 4 2 2 3 2 2 2 2 2
3a 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 1 2 2 3 2 8 1 2 0 4 2 8 0 1 2 0 1 4 2 8 0 1 2 1 4 2 8
------------------------------------------------------------------------
4 2 3 2 2 2
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 0 2 4 2 8 0 1 2 1 2 4 2 8 0 1 2 0 3 4 2 8
------------------------------------------------------------------------
3 2 3 2 4 2 2 5 2
2a a a b b b c c + a a a b b b c c - a a b b c c + 2a a b b c c -
0 1 2 1 3 4 2 8 0 1 2 2 3 4 2 8 0 2 3 4 2 8 1 2 0 5 2 8
------------------------------------------------------------------------
4 2 3 2 5 2 4
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 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
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
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
------------------------------------------------------------------------
2 4 3 3 4 2 2 4 2
- 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 c c +
0 1 0 4 5 2 8 0 1 1 4 5 2 8 0 1 3 4 5 2 8 1 2 0 1 3 8
------------------------------------------------------------------------
3 3 2 2 3 3 2 2 2 2 2 2
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 +
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
------------------------------------------------------------------------
2 3 3 4 2 2 4 2 3
a a a b c c + 4a a a b b c c - a a b b b c c - 7a a a b b b c c +
0 1 2 2 3 8 0 1 2 0 3 3 8 0 2 0 1 3 3 8 0 1 2 0 2 3 3 8
------------------------------------------------------------------------
3 3 3 2 2 2 3 2 2 3
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 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
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
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
------------------------------------------------------------------------
4 2 3 2 2 4 2 3 2 2
- a a b 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 -
0 2 2 3 4 3 8 0 1 2 0 4 3 8 0 1 2 2 4 3 8 0 1 2 0 5 3 8
------------------------------------------------------------------------
2 2 2 2 3 3 2 3 3 2
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
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
------------------------------------------------------------------------
3 2 4 2 4
+ 6a a a b b b c c - a a b b b c c - 5a a a b b b c c -
0 1 2 0 3 5 3 8 0 2 1 3 5 3 8 0 1 2 2 3 5 3 8
------------------------------------------------------------------------
3 2 4 4 2 5
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
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
------------------------------------------------------------------------
5 2 3 3 2 4 2 2 5 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 +
0 1 4 5 3 8 0 1 0 5 3 8 0 1 1 5 3 8 0 1 3 5 3 8
------------------------------------------------------------------------
3 3 2 2 3 2 4 2 2 2 2 2 2
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 +
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
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 +
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
------------------------------------------------------------------------
2 2 2 3 2 3 2 2 3 3 2
a a a b b b c c - 3a a a b b b c c + a a a b b c c - 2a a b b c c +
0 1 2 0 2 3 4 8 0 1 2 1 2 3 4 8 0 1 2 2 3 4 8 0 2 0 3 4 8
------------------------------------------------------------------------
4 2 2 3 2 2 2 2 2 2 3
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 +
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
------------------------------------------------------------------------
3 2 3 2 4
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 1 2 4 4 8 0 1 2 0 3 4 4 8 0 1 2 2 3 4 4 8
------------------------------------------------------------------------
2 3 3 2 2 2 4 3 3
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 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
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 +
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
------------------------------------------------------------------------
3 3 4 2 5 4 2 2
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 c c +
0 1 0 4 5 4 8 0 1 1 4 5 4 8 0 1 3 4 5 4 8 1 2 0 1 5 8
------------------------------------------------------------------------
3 2 2 2 2 2 3 5 2 4
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 +
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
------------------------------------------------------------------------
2 3 2 2 2 2 3 2 2 2 3
a a a b b c c - 2a a a b b b c c + 2a a a b b c c + 2a a a b b b c c
0 1 2 1 2 5 8 0 1 2 0 1 3 5 8 0 1 2 1 3 5 8 0 1 2 0 2 3 5 8
------------------------------------------------------------------------
3 2 4 2 2 4 2 4 2
- 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 -
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
------------------------------------------------------------------------
2 3 3 2 2 3 2
2a 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 0 1 4 5 8 0 1 2 1 4 5 8 0 1 2 0 3 4 5 8
------------------------------------------------------------------------
4 5 2 5 2 2 4
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 -
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
------------------------------------------------------------------------
3 3 2 3 3 4 2 5 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 +
0 1 1 5 5 8 0 1 0 3 5 5 8 0 1 1 3 5 5 8 0 1 3 5 5 8
------------------------------------------------------------------------
2 4 3 4 2 2 4 2 2 3 2
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 +
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
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 -
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
------------------------------------------------------------------------
2 4 2 3 3 4 2 2 2 3 2
2a a b b c c + 4a a b b b c c - 2a a b b c c + 4a a a b b c c -
0 2 0 3 6 8 0 2 0 2 3 6 8 0 2 2 3 6 8 0 1 2 0 4 6 8
------------------------------------------------------------------------
3 3 3 2 4 2 4 2
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 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
------------------------------------------------------------------------
4 2 2 5 2 4 2 2 4
+ a 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 2 0 4 6 8 0 2 2 4 6 8 0 2 1 5 6 8 0 1 2 1 2 5 6 8
------------------------------------------------------------------------
4 2 2 4 2 5 4
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 -
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
------------------------------------------------------------------------
5 5 6 2 4 2 2
2a a b b b c c - 2a a b b b c c + a b b c c - 2a a b b c c +
0 2 1 4 5 6 8 0 1 2 4 5 6 8 0 4 5 6 8 0 1 0 5 6 8
------------------------------------------------------------------------
5 2 6 2 3 3 3 2 3 2
2a a b b c c - 2a b b c c - 2a a b c c + 3a a a b b c c -
0 1 1 5 6 8 0 3 5 6 8 1 2 0 7 8 0 1 2 0 1 7 8
------------------------------------------------------------------------
2 3 2 3 2 2 2 2 2 3 2 2
a 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 0 1 7 8 0 1 2 0 2 7 8 0 1 2 0 1 2 7 8 0 1 2 1 2 7 8
------------------------------------------------------------------------
2 3 2 3 2 2 2 3 2 3 3
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 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
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 +
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
------------------------------------------------------------------------
3 2 3 2 4 4 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 b c c
0 1 2 0 1 4 7 8 0 1 2 0 2 4 7 8 0 1 2 1 2 4 7 8 0 2 0 3 4 7 8
------------------------------------------------------------------------
5 2 3 2 3 2 4 2
+ 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
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
------------------------------------------------------------------------
3 3 4 2 4 5
+ a a b b b c c - a a b b b c c - 2a a a b b b c c + a a b b b c c +
0 1 0 2 5 7 8 0 1 1 2 5 7 8 0 1 2 0 3 5 7 8 0 2 1 3 5 7 8
------------------------------------------------------------------------
5 4 2 5 6
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 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
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
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
------------------------------------------------------------------------
3 2 2 2 2 2 2 2 2 3 2 2 3 2 2
- a a a b b c + 2a a a b b c - 2a a a b b b c - 2a a a b b b c +
0 1 2 1 2 8 0 1 2 0 3 8 0 1 2 0 1 3 8 0 1 2 0 2 3 8
------------------------------------------------------------------------
4 2 4 2 2 2 5 2 2 2 4 2 2 3 3 2
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 +
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
------------------------------------------------------------------------
4 2 2 2 4 2 2 5 2 6 2 2 3 3 3
a a b b c + 2a a b b b c - 2a a b b b c + a b b c + a a b c c -
0 1 1 5 8 0 1 0 3 5 8 0 1 1 3 5 8 0 3 5 8 1 2 1 0 9
------------------------------------------------------------------------
4 2 2 5 2 6 3 3 3
3a a b b c c + 3a a b b c c - a b c c - 3a a b b b c c -
1 2 1 2 0 9 1 2 1 2 0 9 1 2 0 9 1 2 0 1 3 0 9
------------------------------------------------------------------------
2 3 2 4 2 3 2 4 2
3a a a b b c c + 3a a b b b c c + 9a a a b b b c c - 6a a a b b c c
0 1 2 1 3 0 9 1 2 0 2 3 0 9 0 1 2 1 2 3 0 9 0 1 2 2 3 0 9
------------------------------------------------------------------------
2 3 2 2 3 2 2 2 2 2 3 3 3
+ 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 +
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
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 -
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
------------------------------------------------------------------------
3 2 2 3 3 2 2
9a a a b b b c c + 9a a a b b b c c + 3a a a b b c c +
0 1 2 0 3 4 0 9 0 1 2 2 3 4 0 9 0 1 2 3 4 0 9
------------------------------------------------------------------------
4 2 2 4 2 3 2 2 3 3 3
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 -
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
------------------------------------------------------------------------
5 4 2 6 5
3a a b b b c c + 3a a a b b c c + 3a b b b c c - 3a a b b b c c +
1 2 0 1 5 0 9 0 1 2 1 5 0 9 1 0 2 5 0 9 0 1 1 2 5 0 9
------------------------------------------------------------------------
4 2 3 2 4
6a a a b b b c c - 9a a a b b b c c + 3a a b b b c c +
0 1 2 0 3 5 0 9 0 1 2 1 3 5 0 9 0 1 2 3 5 0 9
------------------------------------------------------------------------
3 2 2 5 2 4 3 3
6a a a 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 -
0 1 2 3 5 0 9 0 1 0 4 5 0 9 0 1 1 4 5 0 9 0 1 3 4 5 0 9
------------------------------------------------------------------------
3 3 2 4 2 3 2 2 5 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 -
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
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 -
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
------------------------------------------------------------------------
3 2 2 2 2 2 3 2
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 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
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 -
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
------------------------------------------------------------------------
3 2 4 2 3 2 4 2
a a a b b b c c + 5a a a b b b c c + a a a b b b c c - 3a a b b c c
0 1 2 0 1 4 1 9 0 1 2 0 2 4 1 9 0 1 2 1 2 4 1 9 0 1 2 4 1 9
------------------------------------------------------------------------
2 2 2 3 2 3 2
+ 9a a a b b b c c - 2a a a b b b c c - 7a a a b b b c c -
0 1 2 0 3 4 1 9 0 1 2 1 3 4 1 9 0 1 2 2 3 4 1 9
------------------------------------------------------------------------
4 2 2 2 3 2 3 2 2 3 3 2
a a b b c c - 4a a a b b c c + a a a b b c c + 3a a b b c c +
0 2 3 4 1 9 0 1 2 0 4 1 9 0 1 2 1 4 1 9 0 1 2 4 1 9
------------------------------------------------------------------------
4 2 4 2 3 5 2 4
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 -
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
------------------------------------------------------------------------
2 3 2 5 2 4 2 3
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 +
0 1 2 1 5 1 9 0 1 0 2 5 1 9 0 1 1 2 5 1 9 0 1 2 0 3 5 1 9
------------------------------------------------------------------------
3 2 3 3 4 2 2 4
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 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
- 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 -
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
------------------------------------------------------------------------
5 4 2 6 2 5 2
2a a b b b c c + 2a a a b b c c + a b b c c - a a b b c c -
1 2 0 1 2 2 9 0 1 2 1 2 2 9 1 0 2 2 9 0 1 1 2 2 9
------------------------------------------------------------------------
4 2 2 2 2 2 2 4 2 3
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
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
------------------------------------------------------------------------
2 4 2 3 2 2 3 2 2 4 2 3
+ a a b b c c - 4a a a b b c c + 4a a a b b c c + 2a a b c c +
0 1 2 3 2 9 0 1 2 1 3 2 9 0 1 2 2 3 2 9 0 2 3 2 9
------------------------------------------------------------------------
5 2 4 5 2 4
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 +
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
------------------------------------------------------------------------
3 2 3 3 4 2 2 4 2
2a 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 -
0 1 2 1 3 4 2 9 0 1 2 3 4 2 9 0 1 2 3 4 2 9 0 1 0 4 2 9
------------------------------------------------------------------------
3 3 2 4 2 2 6 2 5
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 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
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 1 5 2 9 0 1 0 3 5 2 9 0 1 1 3 5 2 9 0 1 3 5 2 9
------------------------------------------------------------------------
3 3 2 4 2 2 3 2 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 +
1 2 0 1 3 9 1 2 0 2 3 9 0 1 2 0 1 2 3 9 0 1 2 0 2 3 9
------------------------------------------------------------------------
2 3 2 2 4 3 2 3 2 2 3
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 -
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
------------------------------------------------------------------------
3 3 2 2 2 2 3 2 3 2 2
a a b b c c + 9a a a b b b c c - a a a b b b c c - 4a a a b b c c +
0 2 1 3 3 9 0 1 2 0 2 3 3 9 0 1 2 1 2 3 3 9 0 1 2 2 3 3 9
------------------------------------------------------------------------
3 3 2 4 2 2 3 2 2 2 2 2
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 +
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
------------------------------------------------------------------------
3 2 2 2 3 3 2 3 3 2
a a a b b c c - 7a a a b b b c c + a a a b b b c c + 3a a b b c c -
0 1 2 1 4 3 9 0 1 2 0 2 4 3 9 0 1 2 1 2 4 3 9 0 1 2 4 3 9
------------------------------------------------------------------------
3 2 4 2 4
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 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
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 +
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
------------------------------------------------------------------------
5 3 4 2 2 3 3 2 2
a a b c c - 4a a a b b c c + 5a a a b b b c c - a a a b b c c +
0 1 4 3 9 0 1 2 0 5 3 9 0 1 2 0 1 5 3 9 0 1 2 1 5 3 9
------------------------------------------------------------------------
2 4 3 3 3 2 4
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
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
------------------------------------------------------------------------
4 2 5 2 3 3 4 2
+ 3a a b b b c c + 2a a b b c c - 3a a b b b c c + 3a a b b b c c -
0 1 2 3 5 3 9 0 2 3 5 3 9 0 1 0 4 5 3 9 0 1 1 4 5 3 9
------------------------------------------------------------------------
5 4 2 2 3 2 2 5 2
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 -
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
------------------------------------------------------------------------
2 3 2 5 2 2 4 2 3 2 2
a 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 -
0 1 2 1 2 4 9 0 1 0 2 4 9 0 1 1 2 4 9 0 1 2 0 3 4 9
------------------------------------------------------------------------
2 2 2 3 2 2 2 3
6a 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 0 1 3 4 9 0 1 2 1 3 4 9 0 1 2 0 2 3 4 9
------------------------------------------------------------------------
3 2 3 3 2 3 2 2 4 2 2
4a a a b b b c c - a a b b c c + 4a a a b b c c - a a b b c c -
0 1 2 1 2 3 4 9 0 1 2 3 4 9 0 1 2 0 3 4 9 0 2 1 3 4 9
------------------------------------------------------------------------
4 2 4 2 2 3 3 2 2
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 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
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
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
------------------------------------------------------------------------
5 2 3 3 2 4 2 2 5 2
+ a a b b c c - a a b b c c + a a b b c c - a a b b c c +
0 2 3 4 4 9 0 1 0 4 4 9 0 1 1 4 4 9 0 1 3 4 4 9
------------------------------------------------------------------------
5 2 2 4 3 3 2 3 3
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 -
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
------------------------------------------------------------------------
4 2 5 2 5 2 4 2
4a a b b b c c + 2a a b b c c + a a b b c c - 2a a a b b c c +
0 1 1 3 5 4 9 0 1 3 5 4 9 1 2 0 1 5 9 0 1 2 0 1 5 9
------------------------------------------------------------------------
2 3 3 6 2 5 2 4 2
a a 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 1 5 9 1 0 2 5 9 0 1 0 1 2 5 9 0 1 1 2 5 9
------------------------------------------------------------------------
4 2 2 3 3 2 2 2 4
2a a a b b c c + 6a a a b b b c c - 4a a a b b c c - 2a a b b b c c
0 1 2 0 3 5 9 0 1 2 0 1 3 5 9 0 1 2 1 3 5 9 0 1 0 2 3 5 9
------------------------------------------------------------------------
3 3 3 2 2 4 2 4 2 2
+ 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 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
2a a b c c + a a b b c c - 2a a b b b c c + a a b 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
------------------------------------------------------------------------
3 3 4 2 5 2 3 3 3
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 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
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 -
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
------------------------------------------------------------------------
2 3 2 3 3 3 2 3 2 3 3
3a a a b b c c + a a b c c + 6a a a b b c c - 3a a b b b c c -
0 1 2 0 2 6 9 0 1 2 6 9 0 1 2 0 3 6 9 0 2 0 1 3 6 9
------------------------------------------------------------------------
3 2 4 2 4 2 2 2 2 2
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
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
------------------------------------------------------------------------
3 2 4 2 2 3 2
+ 9a a a b b b c c - 3a a b b c c + 9a a a b b b c c -
0 1 2 0 1 4 6 9 0 2 1 4 6 9 0 1 2 0 2 4 6 9
------------------------------------------------------------------------
4 4 2 2 4 2 5
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 -
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
------------------------------------------------------------------------
4 2 5 2 5 2 6 3
6a a a b b c c + 3a a b b c c + 3a a b b c c - a b c c +
0 1 2 0 4 6 9 0 2 1 4 6 9 0 1 2 4 6 9 0 4 6 9
------------------------------------------------------------------------
2 3 2 3 2 4 2 3 3
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 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
+ 3a a b b b c c + 6a a a b b b c c - 3a a 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
------------------------------------------------------------------------
5 4 2 5 6
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 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
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 -
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
------------------------------------------------------------------------
4 2 2 3 2 4 2 3 3 2
2a a a b b c c + 2a a a b b b c c + a a b b c c - a a b b c c +
0 1 2 0 2 7 9 0 1 2 0 1 2 7 9 0 1 0 2 7 9 0 1 1 2 7 9
------------------------------------------------------------------------
4 2 2 4 4 2 2 4 2 2
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 +
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
------------------------------------------------------------------------
5 2 2 3 2 3 2 4 2
2a 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 -
0 2 2 3 7 9 0 1 2 0 4 7 9 0 1 2 0 1 4 7 9 0 1 2 1 4 7 9
------------------------------------------------------------------------
3 3 4 2 4 5
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 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
- 2a a b b b c c + a a b b c c - a a b b c c + 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
------------------------------------------------------------------------
2 4 2 3 3 4 2 2 4 2
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 +
0 1 0 5 7 9 0 1 0 1 5 7 9 0 1 1 5 7 9 0 1 0 3 5 7 9
------------------------------------------------------------------------
5 6 2 5 3 4 2
4a a b b b c c - 2a b b c c - 2a a b c c + 5a a a b b c c -
0 1 1 3 5 7 9 0 3 5 7 9 1 2 0 8 9 0 1 2 0 1 8 9
------------------------------------------------------------------------
2 3 2 3 2 3 5 2 2 4
4a 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 +
0 1 2 0 1 8 9 0 1 2 1 8 9 0 1 0 2 8 9 0 1 0 1 2 8 9
------------------------------------------------------------------------
3 3 2 2 3 2 3 2 4 2
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 +
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
------------------------------------------------------------------------
3 3 4 2 4 2 5 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 +
0 1 0 2 3 8 9 0 1 1 2 3 8 9 0 1 2 0 3 8 9 0 2 1 3 8 9
------------------------------------------------------------------------
5 2 2 4 2 3 3 4 2 2
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 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
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 +
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
------------------------------------------------------------------------
2 4 2 2 3 3 3 2 2 4 2 2 3 3 2 4 2 2 2
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 +
0 1 0 1 9 0 1 1 9 0 1 0 3 9 0 1 0 1 3 9 0 1 1 3 9
------------------------------------------------------------------------
4 2 2 2 5 2 2 6 3 2
3a a b b c - 3a a b b c + a b c
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
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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
o6 = {a x + a x x + a x + a x x + a x x + a x , b x + b x x + b x +
0 0 1 0 1 3 1 2 0 2 4 1 2 5 2 0 0 1 0 1 3 1
------------------------------------------------------------------------
2 2 2 2
b x x + b x x + b x , c x + c x x + c x + c x x + c x x + c x }
2 0 2 4 1 2 5 2 0 0 1 0 1 3 1 2 0 2 4 1 2 5 2
o6 : List
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i7 : time # terms resultant F
-- used 0.627472 seconds
o7 = 21894
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