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

Description

This computes the symmetric matrix of the trace symmetric form of 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 : S = R/I

o3 = S

o3 : QuotientRing
 
i4 : f = y^2 - x^2 - x*y + 4

o4 = - x*y + 5

o4 : S
 
i5 : traceForm(f)

o5 = | 4    -86   -340  -42   |
     | -86  -266  -1262 -340  |
     | -340 -1262 -5884 -1454 |
     | -42  -340  -1454 -262  |

              4       4
o5 : Matrix QQ  <-- QQ

We can also compute the signature of the trace form of g, which is the number of real points in $V(J)$ where g is positive minus the number of real points in $V(J)$ where g is negative.

i6 : R = QQ[x,y]

o6 = R

o6 : PolynomialRing
 
i7 : J = ideal(y^2 - x^2 - 1,x - y^2 + 4*y - 2)

               2    2         2
o7 = ideal (- x  + y  - 1, - y  + x + 4y - 2)

o7 : Ideal of R
 
i8 : g = x + y

o8 = x + y

o8 : R
 
i9 : signature(traceForm(g,J))	 

o9 = 2

Additionally, we show an example computing the number of points in $V(I)$ using the rank of the trace form.

i10 : rank traceForm(1_R,J)	 

o10 = 4

See also

Ways to use traceForm:

  • traceForm(RingElement)
  • traceForm(RingElement,Ideal)

For the programmer

The object traceForm is a method function.

The source of this document is in RealRoots.m2:967:0.