# listDetComplex -- 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.

## Synopsis

• Usage:
listOfTheDetOfTheComplex = listDetComplex(d,v,C)
• Inputs:
• d, an integer, integer corresponding to the degree of the strand of the chain complex.
• v, a list, list of variables of the polynomial ring R to take into account for elimination
• C, , a chain complex of free modules over a polynomial ring
• Optional inputs:
• Strategy => ..., default value null, choose between Exact and Numeric algorithms
• Outputs:
• a list, a list with the determinant polynomials of the maps computed by 'minorsComplex'

## Description

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 minorsComplex.

The input ChainComplex needs to be an exact complex of free modules over a polynomial ring. The polynomial ring must contain the list v 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 flattenRing for passing from $R=QQ[x,y][a,b,c]$ 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 : setRandomSeed 0 o15 = 0 i16 : dHPM = listDetComplex (2,l,koszul M, Strategy => Numeric) 3 2 2 2 2 3 2 2 o16 = {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} o16 : List