Description
Given a unimodular m \times \ n matrix over a polynomial ring with coefficients in
QQ,
ZZ, or
ZZ/p with
p a prime integer, this method uses the algorithm of Logar-Sturmfels to compute a solution of the unimodular matrix problem for
U. In other words, this method computes a square unimodular matrix
M such that if m \leq \ n then
U*M is of the form [I \ 0] where I is an m \times \ m identity matrix, and if m \geq \ n then
M is of the form [I \ 0]^T, where I is an n \times \ n identity matrix.
i1 : R = ZZ/101[x,y]
o1 = R
o1 : PolynomialRing
|
i2 : U = matrix{{x^2*y+1,x+y-2,2*x*y}}
o2 = | x2y+1 x+y-2 2xy |
1 3
o2 : Matrix R <-- R
|
i3 : isUnimodular U
o3 = true
|
i4 : M = qsAlgorithm U
o4 = | 1 2xy -x-y+2 |
| 0 0 1 |
| 50x -x2y-1 -50x2-50xy-x |
3 3
o4 : Matrix R <-- R
|
i5 : isUnimodular M
o5 = true
|
i6 : U*M
o6 = | 1 0 0 |
1 3
o6 : Matrix R <-- R
|
The inverse of the matrix obtained by qsAlgorithm gives a completion of the original unimodular matrix
U to a square invertible matrix over the polynomial ring. This completion can also be obtained directly by using the method
completeMatrix.
i7 : I = inverse M
o7 = {0} | x2y+1 x+y-2 2xy |
{0} | 50x 0 -1 |
{1} | 0 1 0 |
3 3
o7 : Matrix R <-- R
|
i8 : det I
o8 = 1
o8 : R
|
The method can also be used over a Laurent polynomial ring with coefficients in
QQ or
ZZ/p for
p a prime integer. The following example demonstrates how to construct a Laurent polynomial ring and also how to use the method on a unimodular matrix over the ring.
i9 : R = QQ[x,Inverses => true,MonomialOrder => Lex]
o9 = R
o9 : PolynomialRing
|
i10 : U = matrix{{3*x^-1-2-2*x+2*x^2, 3*x^-1-2*x,2*x},{6*x^-1+25-23*x-16*x^2+20*x^3, 6*x^-1+29-4*x-20*x^2,2+4*x+20*x^2}}
o10 = | 2x2-2x-2+3x-1 -2x+3x-1 2x |
| 20x3-16x2-23x+25+6x-1 -20x2-4x+29+6x-1 20x2+4x+2 |
2 3
o10 : Matrix R <-- R
|
i11 : M = qsAlgorithm U
o11 = | -2/3x+31+8x-1 -3x-1 -360x+1080x-1 |
| -2/3x2+65/3x-77/3-8x-1 -2+3x-1 -360x2+360x+720-1080x-1 |
| -10x-7/3 1 -180 |
3 3
o11 : Matrix R <-- R
|
i12 : det M
o12 = 180
o12 : R
|
i13 : U*M
o13 = | 1 0 0 |
| 0 1 0 |
2 3
o13 : Matrix R <-- R
|