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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 .00132364)  #primes = 0 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000039194)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00238708)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00376091)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00587589)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00250082)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00199868)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00209827)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .000425712)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .000276718)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .000273652)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00167704)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00201131)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00263599)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00268941)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00168849)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00231327)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00194293)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00212706)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00227293)  #primes = 0 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00000761)  #primes = 1 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000025994)  #primes = 1 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000006666)  #primes = 2 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000007022)  #primes = 3 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00002476)  #primes = 3 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000006744)  #primes = 4 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00117667)  #primes = 6 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000025182)  #primes = 6 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000022166)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .000267794)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .000249756)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .000775604)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .000917622)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .000155182)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .000122752)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .00024711)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .00024338)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .00098293)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .00113749)  #primes = 6 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000006892)  #primes = 7 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000006878)  #primes = 8 #prunedViaCodim = 0
  Strategy: IndependentSet    (time .00001107)  #primes = 9 #prunedViaCodim = 0
  Strategy: IndependentSet    (time .000010696)  #primes = 10 #prunedViaCodim = 0
Converting annotated ideals to ideals and selecting minimal primes... Time taken : .0049138
#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 .0013254)  #primes = 0 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000038084)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00239289)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00371984)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00591313)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00249489)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00200862)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00214129)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .000429122)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .000301806)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .000276742)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00172406)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00204009)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00269437)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .0157168)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00170687)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00238886)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00194704)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .0021378)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00227849)  #primes = 0 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000008012)  #primes = 1 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000025086)  #primes = 1 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000007762)  #primes = 2 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000007918)  #primes = 3 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000023576)  #primes = 3 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000007664)  #primes = 4 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00119408)  #primes = 6 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000023888)  #primes = 6 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000021748)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .000268284)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .000250998)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .000792254)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .000917848)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .000155908)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .00011817)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .0002537)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .000246026)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .00100162)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .00111831)  #primes = 6 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000006838)  #primes = 7 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000008148)  #primes = 8 #prunedViaCodim = 0
  Strategy: Birational        (time .00487324)  #primes = 8 #prunedViaCodim = 0
  Strategy: Birational        (time .00443477)  #primes = 8 #prunedViaCodim = 0
  Strategy: Birational        (time .000224734)  #primes = 8 #prunedViaCodim = 0
  Strategy: Birational        (time .000219932)  #primes = 8 #prunedViaCodim = 0
  Strategy: Linear            (time .000055146)  #primes = 8 #prunedViaCodim = 0
  Strategy: Linear            (time .000050856)  #primes = 8 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000008132)  #primes = 9 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000007536)  #primes = 10 #prunedViaCodim = 0
Converting annotated ideals to ideals and selecting minimal primes... Time taken : .00499712
#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 :