Description
Given a unimodular row
f over a polynomial ring R = S[x_1,\ldots,x_{n-1}][x_n] (where S is either
QQ,
ZZ, or
ZZ/p) and a maximal ideal
M in S[x_1,\ldots,x_{n-1}] , horrocks computes a unimodular matrix
U over (S[x_1,\ldots,x_{n-1}]_M)[x_n] which solves the unimodular row problem for
f. That is,
f*U is of the form
|1 0 ... 0|.
i1 : S = ZZ[x]
o1 = S
o1 : PolynomialRing
|
i2 : M = ideal(2,x)
o2 = ideal (2, x)
o2 : Ideal of S
|
i3 : R = ZZ[x,y]
o3 = R
o3 : PolynomialRing
|
i4 : f = matrix{{y^2+x^2*y+x,3*y+1,x^2}}
o4 = | x2y+y2+x 3y+1 x2 |
1 3
o4 : Matrix R <-- R
|
i5 : U = horrocks(f,y,M)
o5 = | (-9)/(3x2-9x-1) (9y+3)/(3x2-9x-1)
| (3x2+3y-1)/(3x2-9x-1) (-3x2y-3y2-3x)/(3x2-9x-1)
| 0 0
------------------------------------------------------------------------
9x2/(3x2-9x-1) |
(-3x4-3x2y+x2)/(3x2-9x-1) |
1 |
3 3
o5 : Matrix (frac R) <-- (frac R)
|
It is worth noting that
M can be chosen to be an extension of a maximal ideal from S[x_1,\ldots,x_{n-1}] to the ring R. We repeat the above example with this convention.
i6 : R = ZZ[x,y]
o6 = R
o6 : PolynomialRing
|
i7 : f = matrix{{y^2+x^2*y+x,3*y+1,x^2}}
o7 = | x2y+y2+x 3y+1 x2 |
1 3
o7 : Matrix R <-- R
|
i8 : M = ideal(2,x)
o8 = ideal (2, x)
o8 : Ideal of R
|
i9 : U = horrocks(f,y,M)
o9 = | (-9)/(3x2-9x-1) (9y+3)/(3x2-9x-1)
| (3x2+3y-1)/(3x2-9x-1) (-3x2y-3y2-3x)/(3x2-9x-1)
| 0 0
------------------------------------------------------------------------
9x2/(3x2-9x-1) |
(-3x4-3x2y+x2)/(3x2-9x-1) |
1 |
3 3
o9 : Matrix (frac R) <-- (frac R)
|
One can check that
U is unimodular over (\mathbb{Z}[x]_{(2,x)})[y] (i.e. the determinant is a unit in this ring), and that
U solves the unimodular row problem for
f (i.e.
f*U = |1 0 0|).
i10 : det U
-3
o10 = ------------
2
3x - 9x - 1
o10 : frac R
|
i11 : f*U
o11 = | 1 0 0 |
1 3
o11 : Matrix (frac R) <-- (frac R)
|
Before applying horrocks one must guarantee that the first entry in the unimodular row is monic in the given variable. This can be accomplished, for example, by using the method
changeVar. One can also use the method
getMaxIdeal to construct a maximal ideal to localize at. We demonstrate the use of these methods in the next example.
i12 : R = ZZ/11[x,y]
o12 = R
o12 : PolynomialRing
|
i13 : f = matrix{{4*x^2-4*x*y+2*y^2+3*x-2*y+3,-2*x^2+2*x*y-4*y^2-2*x-2,-5*x^2-4*x*y-5*y^2+4*x+3*y+5}}
o13 = | 4x2-4xy+2y2+3x-2y+3 -2x2+2xy-4y2-2x-2 -5x2-4xy-5y2+4x+3y+5 |
1 3
o13 : Matrix R <-- R
|
i14 : isUnimodular f
o14 = true
|
i15 : M = getMaxIdeal(ideal(0_R),{x})
o15 = ideal x
o15 : Ideal of R
|
i16 : (N,subs,invSubs) = changeVar(f,{x,y})
o16 = (| -5 0 0 |, | x y |, | x y |)
| 0 1 0 |
| 0 0 1 |
o16 : Sequence
|
i17 : g = sub(f*N,subs)
o17 = | 2x2-2xy+y2-4x-y-4 -2x2+2xy-4y2-2x-2 -5x2-4xy-5y2+4x+3y+5 |
1 3
o17 : Matrix R <-- R
|
i18 : U = horrocks(g,y,M)
o18 = | (4x2-3xy-x-2y-2)/(x4+2x2+4x-3)
| (-2x2+2xy+4x+5y)/(x4+2x2+4x-3)
| 0
-----------------------------------------------------------------------
(-4x3+4x2y+3xy2-3x2-xy+2y2-3x+1)/(x4+2x2+4x-3)
(-4x3+4x2y-2xy2-2x2+xy-5y2-5x+5y-2)/(x4+2x2+4x-3)
0
-----------------------------------------------------------------------
(-2x4+x3y-3x2y2-4xy3+x3-3x2y-4xy2+y3-4x2-4xy-4y2+2x+5y-1)/(x4+2x2+4x-3)
(x4+2x3y-2x2y2-xy3-5x3-5x2y+xy2+3y3+5x2+2xy-4y2+2x-3y)/(x4+2x2+4x-3)
1
-----------------------------------------------------------------------
|
|
|
3 3
o18 : Matrix (frac R) <-- (frac R)
|
Again we can check that
U is unimodular over ((\mathbb{Z}/11\mathbb{Z})[x]_{(x)})[y] and that
U solves the unimodular row problem for
g.
i19 : det U
- 2x - 5
o19 = -----------------
4 2
x + 2x + 4x - 3
o19 : frac R
|
i20 : g*U
o20 = | 1 0 0 |
1 3
o20 : Matrix (frac R) <-- (frac R)
|