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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 .00112244)  #primes = 0 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00004146)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00201471)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00348359)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .0141837)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00244394)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00204829)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00203869)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .000414361)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00032989)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .000311477)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00158378)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00171117)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00227838)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00236874)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00149742)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00202327)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00171759)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00191605)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00209325)  #primes = 0 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000010585)  #primes = 1 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000032679)  #primes = 1 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000008033)  #primes = 2 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000011261)  #primes = 3 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000030463)  #primes = 3 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000007896)  #primes = 4 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00120074)  #primes = 6 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000028871)  #primes = 6 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000025877)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .00022653)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .00021454)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .000799847)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .000923873)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .000164292)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .000137101)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .000219539)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .000210413)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .000939935)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .00106519)  #primes = 6 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00000944)  #primes = 7 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000010388)  #primes = 8 #prunedViaCodim = 0
  Strategy: IndependentSet    (time .000012243)  #primes = 9 #prunedViaCodim = 0
  Strategy: IndependentSet    (time .000013389)  #primes = 10 #prunedViaCodim = 0
Converting annotated ideals to ideals and selecting minimal primes... Time taken : .00520243
#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 .00115986)  #primes = 0 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000039926)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .0020237)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00356176)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .0056542)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00258604)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00211576)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00203728)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .000448473)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .000328727)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .000339982)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00157847)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00179549)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .0024629)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00239755)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .001488)   #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00205439)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00172317)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00193436)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00206846)  #primes = 0 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000010145)  #primes = 1 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000029746)  #primes = 1 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00000818)  #primes = 2 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000012574)  #primes = 3 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000028568)  #primes = 3 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .0000081)  #primes = 4 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00117982)  #primes = 6 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000030736)  #primes = 6 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000026047)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .000228585)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .000217107)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .000810742)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .000930569)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .000167155)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .000138296)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .000223791)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .00021265)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .000913206)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .0011037)  #primes = 6 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00000893)  #primes = 7 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .0000101)  #primes = 8 #prunedViaCodim = 0
  Strategy: Birational        (time .0044319)  #primes = 8 #prunedViaCodim = 0
  Strategy: Birational        (time .00399433)  #primes = 8 #prunedViaCodim = 0
  Strategy: Birational        (time .000180595)  #primes = 8 #prunedViaCodim = 0
  Strategy: Birational        (time .000206335)  #primes = 8 #prunedViaCodim = 0
  Strategy: Linear            (time .000043625)  #primes = 8 #prunedViaCodim = 0
  Strategy: Linear            (time .000042624)  #primes = 8 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000010086)  #primes = 9 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000011585)  #primes = 10 #prunedViaCodim = 0
Converting annotated ideals to ideals and selecting minimal primes... Time taken : .00513637
#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 :