Uses Grenet's combinatorial construction (see B. Grenet: An Upper Bound for the Permanent versus Determinant Problem (2012)).
Here is the 7x7 matrix constructed from the 3x3 generic matrix of variables.
i1 : R = QQ[vars(0..8)] o1 = R o1 : PolynomialRing |
i2 : M = genericMatrix(R,a,3,3)
o2 = | a d g |
| b e h |
| c f i |
3 3
o2 : Matrix R <--- R
|
i3 : N = grenet M
o3 = | 1 0 e 0 h 0 0 |
| 0 1 b 0 0 h 0 |
| 0 0 1 0 0 0 i |
| 0 0 0 1 b e 0 |
| 0 0 0 0 1 0 f |
| 0 0 0 0 0 1 c |
| a d 0 g 0 0 0 |
7 7
o3 : Matrix R <--- R
|
i4 : det N o4 = c*e*g + b*f*g + c*d*h + a*f*h + b*d*i + a*e*i o4 : R |
Here is the 15x15 matrix constructed from a 4x4 generic matrix of variable (note that the even case has -1 on the diagonal).
i5 : R=QQ[a..p] o5 = R o5 : PolynomialRing |
i6 : M=genericMatrix(R,4,4)
o6 = | a e i m |
| b f j n |
| c g k o |
| d h l p |
4 4
o6 : Matrix R <--- R
|
i7 : N = grenet M
o7 = | -1 0 f 0 j 0 0 0 n 0 0 0 0 0 0 |
| 0 -1 b 0 0 j 0 0 0 n 0 0 0 0 0 |
| 0 0 -1 0 0 0 k 0 0 0 o 0 0 0 0 |
| 0 0 0 -1 b f 0 0 0 0 0 n 0 0 0 |
| 0 0 0 0 -1 0 g 0 0 0 0 0 o 0 0 |
| 0 0 0 0 0 -1 c 0 0 0 0 0 0 o 0 |
| 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 p |
| 0 0 0 0 0 0 0 -1 b f 0 j 0 0 0 |
| 0 0 0 0 0 0 0 0 -1 0 g 0 k 0 0 |
| 0 0 0 0 0 0 0 0 0 -1 c 0 0 k 0 |
| 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 l |
| 0 0 0 0 0 0 0 0 0 0 0 -1 c g 0 |
| 0 0 0 0 0 0 0 0 0 0 0 0 -1 0 h |
| 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 d |
| a e 0 i 0 0 0 m 0 0 0 0 0 0 0 |
15 15
o7 : Matrix R <--- R
|
i8 : det N
o8 = d*g*j*m + c*h*j*m + d*f*k*m + b*h*k*m + c*f*l*m + b*g*l*m + d*g*i*n +
------------------------------------------------------------------------
c*h*i*n + d*e*k*n + a*h*k*n + c*e*l*n + a*g*l*n + d*f*i*o + b*h*i*o +
------------------------------------------------------------------------
d*e*j*o + a*h*j*o + b*e*l*o + a*f*l*o + c*f*i*p + b*g*i*p + c*e*j*p +
------------------------------------------------------------------------
a*g*j*p + b*e*k*p + a*f*k*p
o8 : R
|
Here is the construction for a matrix of integers.
i9 : M = matrix{{1,2,3,4},{5,6,7,8},{9,10,11,12},{13,14,15,16}}
o9 = | 1 2 3 4 |
| 5 6 7 8 |
| 9 10 11 12 |
| 13 14 15 16 |
4 4
o9 : Matrix ZZ <--- ZZ
|
i10 : permanents(4,M) o10 = ideal 55456 o10 : Ideal of ZZ |
i11 : N = grenet M
o11 = | -1 0 6 0 7 0 0 0 8 0 0 0 0 0 0 |
| 0 -1 5 0 0 7 0 0 0 8 0 0 0 0 0 |
| 0 0 -1 0 0 0 11 0 0 0 12 0 0 0 0 |
| 0 0 0 -1 5 6 0 0 0 0 0 8 0 0 0 |
| 0 0 0 0 -1 0 10 0 0 0 0 0 12 0 0 |
| 0 0 0 0 0 -1 9 0 0 0 0 0 0 12 0 |
| 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 16 |
| 0 0 0 0 0 0 0 -1 5 6 0 7 0 0 0 |
| 0 0 0 0 0 0 0 0 -1 0 10 0 11 0 0 |
| 0 0 0 0 0 0 0 0 0 -1 9 0 0 11 0 |
| 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 15 |
| 0 0 0 0 0 0 0 0 0 0 0 -1 9 10 0 |
| 0 0 0 0 0 0 0 0 0 0 0 0 -1 0 14 |
| 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 13 |
| 1 2 0 3 0 0 0 4 0 0 0 0 0 0 0 |
15 15
o11 : Matrix ZZ <--- ZZ
|
i12 : det N o12 = 55456 |
The object grenet is a method function.