analyticSpread -- Compute the analytic spread of a module or ideal

Synopsis

• Usage:

  analyticSpread M

  analyticSpread(M,f)

• Inputs:
  • M, a module, or an ideal
  • f, a ring element, an optional element, which is a non-zerodivisor such that $M[f^{-1}]$ is a free module when $M$ is a module, an element in $M$ when $M$ is an ideal
• Optional inputs:
  • BasisElementLimit => ..., default value infinity, Bound the number of Groebner basis elements to compute in the saturation step
  • DegreeLimit => ..., default value {}, Bound the degrees considered in the saturation step. Defaults to infinity
  • MinimalGenerators => ..., default value true, Whether the saturation step returns minimal generators
  • PairLimit => ..., default value infinity, Bound the number of s-pairs considered in the saturation step
  • Strategy => ..., default value null, Choose a strategy for the saturation step
• Outputs:
  • an integer, the analytic spread of a module or an ideal $M$

Description

i1 : R=QQ[a..h]

o1 = R

o1 : PolynomialRing

i2 : M=matrix{{a,b,c,d},{e,f,g,h}}

o2 = | a b c d |
     | e f g h |

             2      4
o2 : Matrix R  <-- R

i3 : analyticSpread minors(2,M)

o3 = 5

i4 : specialFiberIdeal minors(2,M)

o4 = ideal(Z Z  - Z Z  + Z Z )
            2 3    1 4    0 5

o4 : Ideal of QQ[Z ..Z ]
                  0   5

i5 : R=QQ[a,b,c,d]

o5 = R

o5 : PolynomialRing

i6 : M=matrix{{a,b,c,d},{b,c,d,a}}

o6 = | a b c d |
     | b c d a |

             2      4
o6 : Matrix R  <-- R

i7 : analyticSpread minors(2,M)

o7 = 4

i8 : specialFiberIdeal minors(2,M)

                                 2                  2
o8 = ideal (Z Z  - Z Z  + Z Z , Z  - Z Z  - Z Z  - Z  + Z Z  + Z Z )
             2 3    1 4    0 5   1    0 2    0 3    4    2 5    3 5

o8 : Ideal of QQ[Z ..Z ]
                  0   5

See also

• specialFiberIdeal -- Special fiber of a blowup
• reesIdeal -- Compute the defining ideal of the Rees Algebra