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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 .0013899)  #primes = 0 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000051292)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .0025327)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00440367)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00637425)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00277566)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00234801)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00242029)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .000470412)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .000310194)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .000273581)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00188373)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .0022268)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00289449)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00303762)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00190769)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00377254)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00215023)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00229977)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00252808)  #primes = 0 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000010188)  #primes = 1 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000048199)  #primes = 1 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000008423)  #primes = 2 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000010722)  #primes = 3 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000033152)  #primes = 3 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000008578)  #primes = 4 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .0014621)  #primes = 6 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000037539)  #primes = 6 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000029598)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .000251741)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .000230215)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .000924009)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .00102199)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .00020856)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .000180928)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .00030035)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .000259245)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .00109554)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .00131104)  #primes = 6 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000014034)  #primes = 7 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000010778)  #primes = 8 #prunedViaCodim = 0
  Strategy: IndependentSet    (time .000020188)  #primes = 9 #prunedViaCodim = 0
  Strategy: IndependentSet    (time .000015243)  #primes = 10 #prunedViaCodim = 0
Converting annotated ideals to ideals and selecting minimal primes... Time taken : .00645597
#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 .00131595)  #primes = 0 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000044711)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00237669)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00403012)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00665679)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00294098)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00268403)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00246302)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00044486)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .000321386)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .000297291)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00227924)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00252624)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00286097)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00293432)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .0017853)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00268266)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00215941)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00239249)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00262014)  #primes = 0 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000014324)  #primes = 1 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000043606)  #primes = 1 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000007751)  #primes = 2 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000010559)  #primes = 3 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000038673)  #primes = 3 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000007859)  #primes = 4 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00143275)  #primes = 6 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00003258)  #primes = 6 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000027788)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .000291163)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .000273676)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .000879798)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .00106706)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .000184187)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .000147834)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .000280357)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .000282605)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .00109926)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .00122402)  #primes = 6 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000009816)  #primes = 7 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000011479)  #primes = 8 #prunedViaCodim = 0
  Strategy: Birational        (time .00483444)  #primes = 8 #prunedViaCodim = 0
  Strategy: Birational        (time .00432122)  #primes = 8 #prunedViaCodim = 0
  Strategy: Birational        (time .000205294)  #primes = 8 #prunedViaCodim = 0
  Strategy: Birational        (time .000192227)  #primes = 8 #prunedViaCodim = 0
  Strategy: Linear            (time .000044758)  #primes = 8 #prunedViaCodim = 0
  Strategy: Linear            (time .000043847)  #primes = 8 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000010082)  #primes = 9 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000013196)  #primes = 10 #prunedViaCodim = 0
Converting annotated ideals to ideals and selecting minimal primes... Time taken : .00593404
#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 :