denseResultant -- dense resultant (classical resultant)

Synopsis

• Usage:

  denseResultant(d_0,d_1,...)

  denseResultant(f_0,f_1,...)

• Inputs:
  • (d_0,d_1,...), a sequence of integers.
  • (f_0,f_1,...), a sequence of $n+1$ Laurent polynomials in $n$ variables.
• Optional inputs:
  • Algorithm => ..., default value Poisson
  • CoefficientRing => ..., default value ZZ
• Outputs:
  • for (d_0,d_1,...), this is the same as sparseResultant exponentsMatrix genericLaurentPolynomials(d_0,d_1,...);
  • for (f_0,f_1,...), this is the same as affineResultant{f_0,f_1,...}.

Description

i1 : (f0,f1,f2) = genericLaurentPolynomials(1,2,2)

                           2               2                        2               2
o1 = (a x  + a x  + a , b x  + b x x  + b x  + b x  + b x  + b , c x  + c x x  + c x  + c x  + c x  + c )
       2 1    1 2    0   5 1    4 1 2    2 2    3 1    1 2    0   5 1    4 1 2    2 2    3 1    1 2    0

o1 : Sequence

i2 : time denseResultant(f0,f1,f2); -- using Poisson formula
     -- used 0.297241 seconds

i3 : time denseResultant(f0,f1,f2,Algorithm=>"Macaulay"); -- using Macaulay formula
     -- used 0.788567 seconds

i4 : time (denseResultant(1,2,2)) (f0,f1,f2); -- using sparseResultant
     -- used 0.594973 seconds

i5 : assert(o2 == o3 and o3 == o4)

See also

• sparseResultant -- sparse resultant (A-resultant)
• affineResultant -- affine resultant
• denseDiscriminant -- dense discriminant (classical discriminant)
• exponentsMatrix -- exponents in one or more polynomials
• genericLaurentPolynomials -- generic (Laurent) polynomials