regularRepresentation -- the regular representation of a rational polynomial

Synopsis

• Usage:

  regularRepresentation(f)

  regularRepresentation(g,I)

• Inputs:
  • f, a ring element, an element of an Artinian ring
  • g, a ring element, a rational polynomial
  • I, an ideal, a zero-dimensional ideal in the same ring as g
• Outputs:
  • a matrix, the standard basis of ring f (resp. (ring g)/I)
  • a matrix, the matrix of the linear map defined by multiplication by f (resp. g) in ring f (resp. (ring g)/I)

Description

This command gives the matrix of the linear map defined by multiplication by f (resp. g) in terms of the standard basis of the finite-dimensional vector space ring f (resp. (ring g)/I).

i1 : R = QQ[x,y]

o1 = R

o1 : PolynomialRing

i2 : I = ideal(y^2 - x^2 - 1,x - y^2 + 4*y - 2)

               2    2         2
o2 = ideal (- x  + y  - 1, - y  + x + 4y - 2)

o2 : Ideal of R

i3 : regularRepresentation(y,I)

o3 = (| 1 x xy y |, | 0 0 -3 -2 |)
                    | 0 0 -1 1  |
                    | 0 1 4  0  |
                    | 1 0 4  4  |

o3 : Sequence

i4 : S = R/I

o4 = S

o4 : QuotientRing

i5 : regularRepresentation(y)

o5 = (| 1 x xy y |, | 0 0 -3 -2 |)
                    | 0 0 -1 1  |
                    | 0 1 4  0  |
                    | 1 0 4  4  |

o5 : Sequence