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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 .0047814)  #primes = 0 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00013454)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00759104)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .0130243)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .0200128)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .0092561)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00729532)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00769304)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00118298)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00091418)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00115692)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00652078)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00697176)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00947316)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00972038)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00636946)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .0085451)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00729578)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00788042)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00856414)  #primes = 0 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00002958)  #primes = 1 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .0000932)  #primes = 1 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00004382)  #primes = 2 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .0000323)  #primes = 3 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00011984)  #primes = 3 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00002416)  #primes = 4 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00441986)  #primes = 6 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00011342)  #primes = 6 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00007188)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .00088218)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .00076638)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .00286872)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .00306572)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .00051454)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .00036656)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .00083174)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .00091594)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .00369002)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .00406904)  #primes = 6 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00004298)  #primes = 7 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00004636)  #primes = 8 #prunedViaCodim = 0
  Strategy: IndependentSet    (time .00003758)  #primes = 9 #prunedViaCodim = 0
  Strategy: IndependentSet    (time .00004204)  #primes = 10 #prunedViaCodim = 0
Converting annotated ideals to ideals and selecting minimal primes... Time taken : .0184588
#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 .0043943)  #primes = 0 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00014124)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .0075853)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .0124258)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .0205327)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00896254)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .0078055)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .0072401)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00117756)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00085724)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00084036)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00635384)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00703218)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00936942)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00894126)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .0067019)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .0087098)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00737256)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00739732)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .0077731)  #primes = 0 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .0000298)  #primes = 1 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00023756)  #primes = 1 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .0000424)  #primes = 2 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00004586)  #primes = 3 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .0000907)  #primes = 3 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00002256)  #primes = 4 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00454332)  #primes = 6 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00011468)  #primes = 6 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00008588)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .0007391)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .00094336)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .0026614)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .00327394)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .00052974)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .00035882)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .00084644)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .00082466)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .00371808)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .0040139)  #primes = 6 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00002666)  #primes = 7 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .0000305)  #primes = 8 #prunedViaCodim = 0
  Strategy: Birational        (time .0170456)  #primes = 8 #prunedViaCodim = 0
  Strategy: Birational        (time .0156581)  #primes = 8 #prunedViaCodim = 0
  Strategy: Birational        (time .00086726)  #primes = 8 #prunedViaCodim = 0
  Strategy: Birational        (time .00077226)  #primes = 8 #prunedViaCodim = 0
  Strategy: Linear            (time .00019936)  #primes = 8 #prunedViaCodim = 0
  Strategy: Linear            (time .00015108)  #primes = 8 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00003242)  #primes = 9 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00003996)  #primes = 10 #prunedViaCodim = 0
Converting annotated ideals to ideals and selecting minimal primes... Time taken : .0169346
#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 :