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MinimalPrimes :: minprimes

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

Synopsis

Description

Given an ideal in a polynomial ring, or a quotient of a polynomial ring whose base ring is either QQ or ZZ/p, return a list of minimal primes of the ideal.

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 .00347902)  #primes = 0 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00011666)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00568224)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00966026)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .0146838)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00692882)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00544266)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00545372)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .0009614)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00068846)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00071674)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00467406)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .0052403)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .0069989)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00715112)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00465166)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00639378)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00536072)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .0058131)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00613882)  #primes = 0 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00003318)  #primes = 1 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00009256)  #primes = 1 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00002328)  #primes = 2 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .0000328)  #primes = 3 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00009168)  #primes = 3 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00002594)  #primes = 4 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .0033863)  #primes = 6 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00008894)  #primes = 6 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00007108)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .00059282)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .0005251)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .00208174)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .00243002)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .00040576)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .00033446)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .0006736)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .00064318)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .00272334)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .00302742)  #primes = 6 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .0000306)  #primes = 7 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00003508)  #primes = 8 #prunedViaCodim = 0
  Strategy: IndependentSet    (time .00004324)  #primes = 9 #prunedViaCodim = 0
  Strategy: IndependentSet    (time .00004424)  #primes = 10 #prunedViaCodim = 0
Converting annotated ideals to ideals and selecting minimal primes... Time taken : .0162383
#minprimes=6 #computed=10

                                  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 .00353718)  #primes = 0 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00011654)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00573724)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .0686746)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .0149857)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00696914)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00549086)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00552242)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00095168)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00075364)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00069672)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00480796)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00537324)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .0071163)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00730836)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00473056)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .0066518)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00543632)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00592866)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00626798)  #primes = 0 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00003044)  #primes = 1 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00009352)  #primes = 1 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .0000267)  #primes = 2 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00003604)  #primes = 3 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00009654)  #primes = 3 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00002422)  #primes = 4 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00340852)  #primes = 6 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00009864)  #primes = 6 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00007376)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .00058774)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .00053112)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .0020993)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .0024387)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .00041298)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .00033524)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .00069682)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .00065644)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .00274154)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .00312458)  #primes = 6 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00003342)  #primes = 7 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00003564)  #primes = 8 #prunedViaCodim = 0
  Strategy: Birational        (time .0137636)  #primes = 8 #prunedViaCodim = 0
  Strategy: Birational        (time .0126991)  #primes = 8 #prunedViaCodim = 0
  Strategy: Birational        (time .00058818)  #primes = 8 #prunedViaCodim = 0
  Strategy: Birational        (time .00056306)  #primes = 8 #prunedViaCodim = 0
  Strategy: Linear            (time .00014452)  #primes = 8 #prunedViaCodim = 0
  Strategy: Linear            (time .00014016)  #primes = 8 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00003212)  #primes = 9 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00003766)  #primes = 10 #prunedViaCodim = 0
Converting annotated ideals to ideals and selecting minimal primes... Time taken : .0168757
#minprimes=6 #computed=10

                                  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

Caveat

This will eventually be made to work over GF(q), and over other fields too.

Ways to use minprimes :