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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 .00113316)  #primes = 0 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000037168)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00194341)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00377279)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00559896)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00250195)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00198295)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00201429)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .000369494)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .000271176)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .000251129)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00161623)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00183032)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00238921)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00255786)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00160527)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00230951)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .0019324)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00210993)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00215121)  #primes = 0 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000012034)  #primes = 1 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000040274)  #primes = 1 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000008331)  #primes = 2 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000024355)  #primes = 3 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000033231)  #primes = 3 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000008334)  #primes = 4 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00128423)  #primes = 6 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000032764)  #primes = 6 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000027122)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .000293527)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .000242434)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .000853533)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .000936349)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .000153343)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .000132536)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .000243708)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .000251984)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .000988454)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .00108885)  #primes = 6 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000009023)  #primes = 7 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000011242)  #primes = 8 #prunedViaCodim = 0
  Strategy: IndependentSet    (time .0000177)  #primes = 9 #prunedViaCodim = 0
  Strategy: IndependentSet    (time .000014553)  #primes = 10 #prunedViaCodim = 0
Converting annotated ideals to ideals and selecting minimal primes... Time taken : .0059663
#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 .00115003)  #primes = 0 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000057238)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .0100164)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00397765)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00589601)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00269868)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00226939)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00228739)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .000468481)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .000285426)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .000274328)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00188639)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00211956)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00285769)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00291016)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00178875)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .002408)   #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00207907)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00235583)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00250063)  #primes = 0 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000016817)  #primes = 1 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000068357)  #primes = 1 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000010812)  #primes = 2 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000013692)  #primes = 3 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000036054)  #primes = 3 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000022674)  #primes = 4 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00135079)  #primes = 6 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00003603)  #primes = 6 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000027982)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .000263023)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .000224541)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .00084863)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .000996456)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .000185641)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .000146339)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .000299773)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .000269931)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .00101389)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .00116392)  #primes = 6 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000012)   #primes = 7 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00001322)  #primes = 8 #prunedViaCodim = 0
  Strategy: Birational        (time .0057183)  #primes = 8 #prunedViaCodim = 0
  Strategy: Birational        (time .00493721)  #primes = 8 #prunedViaCodim = 0
  Strategy: Birational        (time .000224906)  #primes = 8 #prunedViaCodim = 0
  Strategy: Birational        (time .000249685)  #primes = 8 #prunedViaCodim = 0
  Strategy: Linear            (time .00005115)  #primes = 8 #prunedViaCodim = 0
  Strategy: Linear            (time .0000446)  #primes = 8 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000011759)  #primes = 9 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000013273)  #primes = 10 #prunedViaCodim = 0
Converting annotated ideals to ideals and selecting minimal primes... Time taken : .00643017
#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 :