traceForm -- the trace quadratic form of a rational polynomial in an Artinian ring

Synopsis

• Usage:

  traceForm(f)

  traceForm(f,I),

• Inputs:
  • f, a ring element, a rational polynomial in an Artinian Ring
  • I, an ideal, the ideal generated by f
• Outputs:
  • a matrix, a symmetric matrix representing the trace quadratic form of f in the standard basis of the Artinian ring

Description

This computes the trace quadratic form of a polynomial f in an Artinian ring.

i1 : R = QQ[x,y]

o1 = R

o1 : PolynomialRing

i2 : F = {y^2 - x^2 - 1, x - y^2 + 4*y - 2}

         2    2         2
o2 = {- x  + y  - 1, - y  + x + 4y - 2}

o2 : List

i3 : I = ideal F

               2    2         2
o3 = ideal (- x  + y  - 1, - y  + x + 4y - 2)

o3 : Ideal of R

i4 : S = R/I

o4 = S

o4 : QuotientRing

i5 : f = y^2 - x^2 - x*y + 4

o5 = - x*y + 5

o5 : S

i6 : traceForm(f)

o6 = | 4    -86   -340  -42   |
     | -86  -266  -1262 -340  |
     | -340 -1262 -5884 -1454 |
     | -42  -340  -1454 -262  |

              4       4
o6 : Matrix QQ  <-- QQ

See also

• traceCount -- the number of real points of the spectrum of an Artinian ring (of characteristic 0)