minprimes -- minimal primes in a polynomial ring over a field

i1 : R = ZZ/32003[a..e]

o1 = R

o1 : PolynomialRing

i2 : I = ideal"a2b-c3,abd-c2e,ade-ce2"

             2     3           2              2
o2 = ideal (a b - c , a*b*d - c e, a*d*e - c*e )

o2 : Ideal of R

i3 : C = minprimes I;

i4 : netList C

     +---------------------------+
o4 = |ideal (c, a)               |
     +---------------------------+
     |              2     3      |
     |ideal (e, d, a b - c )     |
     +---------------------------+
     |ideal (e, c, b)            |
     +---------------------------+
     |ideal (d, c, b)            |
     +---------------------------+
     |ideal (d - e, b - c, a - c)|
     +---------------------------+
     |ideal (d + e, b - c, a + c)|
     +---------------------------+

i5 : C2 = minprimes(I, Strategy=>"NoBirational", Verbosity=>2)
  Strategy: Linear            (time .00163319)  #primes = 0 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000052108)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00300229)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00587743)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00431723)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00281743)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00287557)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .000611527)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .000421481)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .000430788)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .0023175)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00220435)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00303413)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00313453)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00248311)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .002794)   #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .0233028)  #primes = 0 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000015388)  #primes = 1 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000038051)  #primes = 1 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00001078)  #primes = 2 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000011331)  #primes = 3 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00170384)  #primes = 5 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000037921)  #primes = 5 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000033954)  #primes = 5 #prunedViaCodim = 0
  Strategy: Factorization     (time .000360486)  #primes = 5 #prunedViaCodim = 0
  Strategy: Factorization     (time .00108667)  #primes = 5 #prunedViaCodim = 0
  Strategy: Factorization     (time .0013042)  #primes = 5 #prunedViaCodim = 0
  Strategy: Factorization     (time .000224791)  #primes = 5 #prunedViaCodim = 0
  Strategy: Factorization     (time .000188383)  #primes = 5 #prunedViaCodim = 0
  Strategy: Linear            (time .000330229)  #primes = 5 #prunedViaCodim = 0
  Strategy: Linear            (time .00135113)  #primes = 5 #prunedViaCodim = 0
  Strategy: Linear            (time .0015712)  #primes = 5 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000013085)  #primes = 6 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000014107)  #primes = 7 #prunedViaCodim = 0
  Strategy: IndependentSet    (time .000017944)  #primes = 8 #prunedViaCodim = 0
Converting annotated ideals to ideals and selecting minimal primes... Time taken : .00580561
#minprimes=6 #computed=8

                                  2     3
o5 = {ideal (c, a), ideal (e, d, a b - c ), ideal (e, c, b), ideal (d, c, b), ideal (d - e, b - c, a - c), ideal (d + e, b - c, a
     ----------------------------------------------------------------------------------------------------------------------------
     + c)}

o5 : List

i6 : C1 = minprimes(I, Strategy=>"Birational", Verbosity=>2)
  Strategy: Linear            (time .00173269)  #primes = 0 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00005829)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00308776)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00610275)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00450595)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .0028678)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00300882)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00060262)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .000443201)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .000434625)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00235789)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00237121)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00317116)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .0032641)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00255947)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00296508)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00311412)  #primes = 0 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000038673)  #primes = 1 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00003754)  #primes = 1 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .0000108)  #primes = 2 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000013254)  #primes = 3 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00174789)  #primes = 5 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000039595)  #primes = 5 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000033733)  #primes = 5 #prunedViaCodim = 0
  Strategy: Factorization     (time .000359224)  #primes = 5 #prunedViaCodim = 0
  Strategy: Factorization     (time .00111801)  #primes = 5 #prunedViaCodim = 0
  Strategy: Factorization     (time .0013806)  #primes = 5 #prunedViaCodim = 0
  Strategy: Factorization     (time .000234099)  #primes = 5 #prunedViaCodim = 0
  Strategy: Factorization     (time .000195166)  #primes = 5 #prunedViaCodim = 0
  Strategy: Linear            (time .000332083)  #primes = 5 #prunedViaCodim = 0
  Strategy: Linear            (time .00136135)  #primes = 5 #prunedViaCodim = 0
  Strategy: Linear            (time .00169223)  #primes = 5 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000011402)  #primes = 6 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000011381)  #primes = 7 #prunedViaCodim = 0
  Strategy: Birational        (time .00641994)  #primes = 7 #prunedViaCodim = 0
  Strategy: Birational        (time .000257603)  #primes = 7 #prunedViaCodim = 0
  Strategy: Linear            (time .000075151)  #primes = 7 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000013565)  #primes = 8 #prunedViaCodim = 0
Converting annotated ideals to ideals and selecting minimal primes... Time taken : .00606067
#minprimes=6 #computed=8

                                  2     3
o6 = {ideal (c, a), ideal (e, d, a b - c ), ideal (e, c, b), ideal (d, c, b), ideal (d - e, b - c, a - c), ideal (d + e, b - c, a
     ----------------------------------------------------------------------------------------------------------------------------
     + c)}

o6 : List