This function calculates the determinant of a graded ChainComplex with respect to a subset of the variables of the polynomial ring in a fixed degree. It corresponds to the alternate product of the elements in the list
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
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i2 : f1 = a*x^2+b*x*y+c*y^2
2 2
o2 = a*x + b*x*y + c*y
o2 : R
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i3 : f2 = 2*a*x+b*y
o3 = 2a*x + b*y
o3 : R
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i4 : M = matrix{{f1,f2}}
o4 = | ax2+bxy+cy2 2ax+by |
1 2
o4 : Matrix R <--- R
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i5 : l = {x,y}
o5 = {x, y}
o5 : List
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i6 : dHPM = detComplex (2,l,koszul M)
2 2
o6 = - a*b + 4a c
o6 : frac(R)
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i7 : dHPM = detComplex (3,l,koszul M)
2 2
o7 = - a*b + 4a c
o7 : frac(R)
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i8 : R=QQ[a,b,c,d,e,f,g,h,i,x,y,z]
o8 = R
o8 : PolynomialRing
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i9 : f1 = a*x+b*y+c*z
o9 = a*x + b*y + c*z
o9 : R
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i10 : f2 = d*x+e*y+f*z
o10 = d*x + e*y + f*z
o10 : R
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i11 : f3 = g*x+h*y+i*z
o11 = g*x + h*y + i*z
o11 : R
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i12 : M = matrix{{f1,f2,f3}}
o12 = | ax+by+cz dx+ey+fz gx+hy+iz |
1 3
o12 : Matrix R <--- R
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i13 : l = {x,y,z}
o13 = {x, y, z}
o13 : List
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i14 : dHPM = detComplex (1,l,koszul M)
o14 = - c*e*g + b*f*g + c*d*h - a*f*h - b*d*i + a*e*i
o14 : frac(R)
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i15 : dHPM = detComplex (2,l,koszul M)
o15 = - c*e*g + b*f*g + c*d*h - a*f*h - b*d*i + a*e*i
o15 : frac(R)
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i16 : dHPM = detComplex (1,l,koszul M, Strategy => Exact)
o16 = - c*e*g + b*f*g + c*d*h - a*f*h - b*d*i + a*e*i
o16 : frac(R)
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i17 : dHPM = detComplex (1,l,koszul M, Strategy => Numeric)
o17 = - c*e*g + b*f*g + c*d*h - a*f*h - b*d*i + a*e*i
o17 : frac(R)
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