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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 .00116119)  #primes = 0 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000034919)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00184689)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00344793)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .0206287)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00208867)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .0016563)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00167191)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .000313294)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .000214351)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .000215094)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00147181)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00207225)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00221112)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00227323)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00147667)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00203675)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00214258)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00191164)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00204967)  #primes = 0 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000008597)  #primes = 1 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000024348)  #primes = 1 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000005172)  #primes = 2 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000006264)  #primes = 3 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000020089)  #primes = 3 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000006619)  #primes = 4 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00103112)  #primes = 6 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000022625)  #primes = 6 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000020245)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .000199852)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .000279661)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .000844852)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .000880711)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .000122615)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .000096467)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .000289041)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .000422394)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .00107331)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .00122017)  #primes = 6 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000010247)  #primes = 7 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000006039)  #primes = 8 #prunedViaCodim = 0
  Strategy: IndependentSet    (time .000011314)  #primes = 9 #prunedViaCodim = 0
  Strategy: IndependentSet    (time .000008949)  #primes = 10 #prunedViaCodim = 0
Converting annotated ideals to ideals and selecting minimal primes... Time taken : .00597219
#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 .00121042)  #primes = 0 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000036256)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .0019435)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00373043)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00535895)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00236037)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00283513)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00189946)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .000333594)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .000241441)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .000242816)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00156372)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .0018029)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00237468)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00264114)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00146915)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00200383)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00164256)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00183944)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00195371)  #primes = 0 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000007948)  #primes = 1 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000021402)  #primes = 1 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000005066)  #primes = 2 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000005232)  #primes = 3 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00003306)  #primes = 3 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000008663)  #primes = 4 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00098717)  #primes = 6 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000023222)  #primes = 6 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000024486)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .000188396)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .000166801)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .00065151)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .000747657)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .00011629)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .000090102)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .00021614)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .000206781)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .000819113)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .0010044)  #primes = 6 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000005273)  #primes = 7 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000007072)  #primes = 8 #prunedViaCodim = 0
  Strategy: Birational        (time .00419133)  #primes = 8 #prunedViaCodim = 0
  Strategy: Birational        (time .00361872)  #primes = 8 #prunedViaCodim = 0
  Strategy: Birational        (time .000157432)  #primes = 8 #prunedViaCodim = 0
  Strategy: Birational        (time .000155577)  #primes = 8 #prunedViaCodim = 0
  Strategy: Linear            (time .000043068)  #primes = 8 #prunedViaCodim = 0
  Strategy: Linear            (time .000043284)  #primes = 8 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000005814)  #primes = 9 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000006569)  #primes = 10 #prunedViaCodim = 0
Converting annotated ideals to ideals and selecting minimal primes... Time taken : .00522117
#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 :