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 .0011661)  #primes = 0 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .0000374)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00209769)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00409157)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00300625)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00196633)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00201608)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .000402576)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .000297719)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00029257)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00162247)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00153909)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00215352)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00221822)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00181057)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00198199)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .0139991)  #primes = 0 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00001046)  #primes = 1 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000027602)  #primes = 1 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000008055)  #primes = 2 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000008696)  #primes = 3 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00113706)  #primes = 5 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000027141)  #primes = 5 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000022863)  #primes = 5 #prunedViaCodim = 0
  Strategy: Factorization     (time .000251131)  #primes = 5 #prunedViaCodim = 0
  Strategy: Factorization     (time .000785225)  #primes = 5 #prunedViaCodim = 0
  Strategy: Factorization     (time .000888628)  #primes = 5 #prunedViaCodim = 0
  Strategy: Factorization     (time .000174548)  #primes = 5 #prunedViaCodim = 0
  Strategy: Factorization     (time .000152496)  #primes = 5 #prunedViaCodim = 0
  Strategy: Linear            (time .000232937)  #primes = 5 #prunedViaCodim = 0
  Strategy: Linear            (time .000947149)  #primes = 5 #prunedViaCodim = 0
  Strategy: Linear            (time .00109218)  #primes = 5 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000007163)  #primes = 6 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000008716)  #primes = 7 #prunedViaCodim = 0
  Strategy: IndependentSet    (time .000012513)  #primes = 8 #prunedViaCodim = 0
Converting annotated ideals to ideals and selecting minimal primes... Time taken : .00413803
#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 .00123417)  #primes = 0 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000040406)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00211125)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00418769)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00306291)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00207571)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00204073)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .000400211)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .000302097)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .000320402)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00162497)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00160119)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00217249)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00235961)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00183337)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00207996)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00218377)  #primes = 0 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000009688)  #primes = 1 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000027752)  #primes = 1 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000006392)  #primes = 2 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000009718)  #primes = 3 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00118284)  #primes = 5 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000029696)  #primes = 5 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000022923)  #primes = 5 #prunedViaCodim = 0
  Strategy: Factorization     (time .000255941)  #primes = 5 #prunedViaCodim = 0
  Strategy: Factorization     (time .000800573)  #primes = 5 #prunedViaCodim = 0
  Strategy: Factorization     (time .000924867)  #primes = 5 #prunedViaCodim = 0
  Strategy: Factorization     (time .000179246)  #primes = 5 #prunedViaCodim = 0
  Strategy: Factorization     (time .000151033)  #primes = 5 #prunedViaCodim = 0
  Strategy: Linear            (time .000246292)  #primes = 5 #prunedViaCodim = 0
  Strategy: Linear            (time .00102354)  #primes = 5 #prunedViaCodim = 0
  Strategy: Linear            (time .00114665)  #primes = 5 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000007875)  #primes = 6 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000009177)  #primes = 7 #prunedViaCodim = 0
  Strategy: Birational        (time .00456166)  #primes = 7 #prunedViaCodim = 0
  Strategy: Birational        (time .00018665)  #primes = 7 #prunedViaCodim = 0
  Strategy: Linear            (time .000048771)  #primes = 7 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000009408)  #primes = 8 #prunedViaCodim = 0
Converting annotated ideals to ideals and selecting minimal primes... Time taken : .00434218
#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