macaulayFormula -- returns two matrices such that the ratio of their determinants is the Macaulay resultant

Synopsis

• Usage:

  macaulayFormula(v,m)

• Inputs:
  • v, a list, a list of n variables such that the polynomials $f_1,...,f_n$ are homogeneous with respect to these variables
  • m, a matrix, a single row matrix with polynomials $f_1,...,f_n$
• Outputs:
  • a list, a list of two matrices such that the ratio of their determinants is the Macaulay resultant of f_1,...,f_n with respect to the variables v

Description

Let $f_1,...,f_n$ be a polynomials two groups of variables $X_1,...,X_n$ and $a_1,...,a_s$ and such that $f_1,...,f_n$ are homogeneous polynomials with respect to the variables $X_1,...,X_n$. This function returns two matrices M1 and M2 such that $det(D_1)/det(D_2)$ is the Macaulay resultant of $f_1,...,f_n$ providing det(D_2) is nonzero.

Remark: if D2 is the empty matrix, its determinant has to be understood as 1 (and not zero, which is the case in Macaulay2 since the empty matrix is identified to the zero.

i1 : R=QQ[a..i,x,y,z]

o1 = R

o1 : PolynomialRing

i2 : f1 = a*x+b*y+c*z

o2 = a*x + b*y + c*z

o2 : R

i3 : f2 = d*x+e*y+f*z

o3 = d*x + e*y + f*z

o3 : R

i4 : f3 = g*x+h*y+i*z

o4 = g*x + h*y + i*z

o4 : R

i5 : M = matrix{{f1,f2,f3}}

o5 = | ax+by+cz dx+ey+fz gx+hy+iz |

             1       3
o5 : Matrix R  <--- R

i6 : l = {x,y,z}

o6 = {x, y, z}

o6 : List

i7 : MR = macaulayFormula (l,M)

o7 = ({1} | a d g |, 0)
      {1} | b e h |
      {1} | c f i |

o7 : Sequence

See also

• eliminationMatrix -- returns a matrix that represents the image of the map
• 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.
• 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.
• minorsComplex -- calculate some minors of the maps of a graded ChainComplex in a subset of variables and 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.