This function calculates the list with the determinants of some minors of the maps of a graded ChainComplex with respect to a subset of the variables of the polynomial ring in a fixed degree. Precisely, this list corresponds to the list with the determinant polynomials of the maps computed by
.
The input ChainComplex needs to be an exact complex of free modules over a polynomial ring. The polynomial ring must contain the list
as variables.
It is recommended not to defines rings as R=QQ[x,y][a,b,c] when the variables to eliminate are '{x,y}'. In this case, see
to QQ[x,y,a,b,c].
i1 : R=QQ[a,b,c,x,y]
o1 = R
o1 : PolynomialRing
|
i2 : f1 = a*x^2+b*x*y+c*y^2
2 2
o2 = a*x + b*x*y + c*y
o2 : R
|
i3 : f2 = 2*a*x+b*y
o3 = 2a*x + b*y
o3 : R
|
i4 : M = matrix{{f1,f2}}
o4 = | ax2+bxy+cy2 2ax+by |
1 2
o4 : Matrix R <--- R
|
i5 : l = {x,y}
o5 = {x, y}
o5 : List
|
i6 : dHPM = listDetComplex (2,l,koszul M)
2 2
o6 = {- a*b + 4a c, 1}
o6 : List
|
i7 : dHPM = listDetComplex (3,l,koszul M)
2 2 2
o7 = {- a*b c + 4a c , c}
o7 : List
|
i8 : R=QQ[a,b,c,d,e,f,g,h,i,x,y,z]
o8 = R
o8 : PolynomialRing
|
i9 : f1 = a*x+b*y+c*z
o9 = a*x + b*y + c*z
o9 : R
|
i10 : f2 = d*x+e*y+f*z
o10 = d*x + e*y + f*z
o10 : R
|
i11 : f3 = g*x+h*y+i*z
o11 = g*x + h*y + i*z
o11 : R
|
i12 : M = matrix{{f1,f2,f3}}
o12 = | ax+by+cz dx+ey+fz gx+hy+iz |
1 3
o12 : Matrix R <--- R
|
i13 : l = {x,y,z}
o13 = {x, y, z}
o13 : List
|
i14 : dHPM = listDetComplex (1,l,koszul M, Strategy => Exact)
o14 = {- c*e*g + b*f*g + c*d*h - a*f*h - b*d*i + a*e*i, 1, 1}
o14 : List
|
i15 : dHPM = listDetComplex (2,l,koszul M, Strategy => Numeric)
3 2 2 2 2 3 2 2
o15 = {c e g - 2b*c e*f*g + b c*f g - c d*e*h + b*c d*f*h + a*c e*f*h -
-----------------------------------------------------------------------
2 2 2 2 2 2
a*b*c*f h + b*c d*e*i - a*c e i - b c*d*f*i + a*b*c*e*f*i, - c e +
-----------------------------------------------------------------------
b*c*f, 1}
o15 : List
|