denseDiscriminant -- dense discriminant (classical discriminant)

Synopsis

• Usage:

  denseDiscriminant(d,n)

  denseDiscriminant f

• Inputs:
  • (d,n), a pair of two integers representing, respectively, the degree and the number of variables of a Laurent polynomial f.
  • f, a ring element, a Laurent polynomial of degree d in n variables.
• Optional inputs:
  • Algorithm => ...,
  • CoefficientRing => ...,
• Outputs:
  • for (d,n), this is the same of sparseDiscriminant exponentsMatrix(generic polynomial of degree d in n variables);
  • for f, this is the same of affineDiscriminant(f).

Description

i1 : (d,n) := (2,3);

i2 : time Disc = denseDiscriminant(d,n)
     -- used 0.209196 seconds

o2 = sparse discriminant associated to | 0 0 0 0 0 0 1 1 1 2 | over ZZ
                                       | 0 0 0 1 1 2 0 0 1 0 |
                                       | 0 1 2 0 1 0 0 1 0 0 |

o2 : SparseDiscriminant

i3 : f = first genericLaurentPolynomials prepend(d,n:0)

        2               2                        2
o3 = a x  + a x x  + a x  + a x x  + a x x  + a x  + a x  + a x  + a x  + a
      9 1    8 1 2    5 2    7 1 3    4 2 3    2 3    6 1    3 2    1 3    0

o3 : ZZ[a , a , a , a , a , a , a , a , a , a , b , c , d ][x , x , x ]
         0   1   2   3   4   5   6   7   8   9   0   0   0   1   2   3

i4 : assert(Disc(f) == denseDiscriminant(f))

See also

• sparseDiscriminant -- sparse discriminant (A-discriminant)
• affineDiscriminant -- affine discriminant
• denseResultant -- dense resultant (classical resultant)
• exponentsMatrix -- exponents in one or more polynomials
• genericLaurentPolynomials -- generic Laurent polynomials