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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 .00221745)  #primes = 0 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .0000782)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00355953)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .0062854)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00954073)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .0045876)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .0729112)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00366628)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .000652475)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .000523525)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .0005405)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00302068)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .0035258)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00466815)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .0046373)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00313317)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00414545)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00347818)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00390755)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00407065)  #primes = 0 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00002205)  #primes = 1 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .0000675)  #primes = 1 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000019075)  #primes = 2 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000022875)  #primes = 3 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000065075)  #primes = 3 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000019475)  #primes = 4 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .0022342)  #primes = 6 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .0000683)  #primes = 6 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000059025)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .00037895)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .0003626)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .0014054)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .00163313)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .0002814)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .000235325)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .000443375)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .0004401)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .00186117)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .00203357)  #primes = 6 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00002)    #primes = 7 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00002525)  #primes = 8 #prunedViaCodim = 0
  Strategy: IndependentSet    (time .000028025)  #primes = 9 #prunedViaCodim = 0
  Strategy: IndependentSet    (time .000028525)  #primes = 10 #prunedViaCodim = 0
Converting annotated ideals to ideals and selecting minimal primes... Time taken : .0107779
#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 .0023909)  #primes = 0 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000080675)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00377058)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00647168)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00989383)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00472765)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00371283)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00370637)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .0006537)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .000496425)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .000493825)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00306688)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .0036077)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .0046591)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00481555)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00302485)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00414643)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00349958)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00376962)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00399613)  #primes = 0 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000022)   #primes = 1 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000066675)  #primes = 1 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000018)   #primes = 2 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00002285)  #primes = 3 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .0000608)  #primes = 3 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00001905)  #primes = 4 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00217523)  #primes = 6 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000065275)  #primes = 6 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .0000541)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .000364375)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .00035165)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .00133482)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .0015702)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .000273775)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .000229825)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .000434575)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .00043385)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .0018972)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .00192155)  #primes = 6 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00002075)  #primes = 7 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000023375)  #primes = 8 #prunedViaCodim = 0
  Strategy: Birational        (time .00839215)  #primes = 8 #prunedViaCodim = 0
  Strategy: Birational        (time .00771125)  #primes = 8 #prunedViaCodim = 0
  Strategy: Birational        (time .00037345)  #primes = 8 #prunedViaCodim = 0
  Strategy: Birational        (time .0003591)  #primes = 8 #prunedViaCodim = 0
  Strategy: Linear            (time .0000878)  #primes = 8 #prunedViaCodim = 0
  Strategy: Linear            (time .0000854)  #primes = 8 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00002245)  #primes = 9 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000026675)  #primes = 10 #prunedViaCodim = 0
Converting annotated ideals to ideals and selecting minimal primes... Time taken : .0105054
#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 :