minorsComplex -- calculate some minors of the maps of a graded ChainComplex in a subset of variables and fixed degree

Synopsis

• Usage:

  minorsOfTheComplex = minorsComplex(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, 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 of square (full-rank) matrices

Description

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 = minorsComplex (2,l,koszul M)

o6 = {{2} | a 2a 0  |, 0}
      {2} | b b  2a |
      {2} | c 0  b  |

o6 : List

i7 : dHPM = minorsComplex (3,l,koszul M)

o7 = {{3} | a 0 2a 0  |, {2} | c |}
      {3} | b a b  2a |
      {3} | c b 0  b  |
      {3} | 0 c 0  0  |

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 = minorsComplex (1,l,koszul M, Strategy => Exact)

o14 = {{1} | a d g |, 0, 0}
       {1} | b e h |
       {1} | c f i |

o14 : List

i15 : setRandomSeed 0

o15 = 0

i16 : dHPM = minorsComplex (2,l,koszul M, Strategy => Numeric)

o16 = {{2} | a 0 0 d 0 g |, {1} | c 0 -i |, 0}
       {2} | b a 0 e d h |  {1} | 0 b e  |
       {2} | c 0 a f 0 i |  {1} | 0 c f  |
       {2} | 0 b 0 0 e 0 |
       {2} | 0 c b 0 f 0 |
       {2} | 0 0 c 0 0 0 |

o16 : List

See also

• degHomPolMap -- return the base of monomials in a subset of variables, and the matrix of coefficients of a morphism of free modules f:R(d1)+...+R(dn)->R_d with respect to these variables
• 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.
• detComplex -- 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.
• mapsComplex -- This function calculates the maps of a graded ChainComplex with respect to a subset of the variables of the polynomial ring in a fixed degree.