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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 .0037532)  #primes = 0 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00012804)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00595068)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .010124)   #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .0155452)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .0764245)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00607804)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00595556)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00103454)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00075054)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .0007834)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .0050829)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00573356)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .0074222)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .0077545)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00510982)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00685838)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .0058753)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .0063278)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00678544)  #primes = 0 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00003648)  #primes = 1 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00010168)  #primes = 1 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .0000276)  #primes = 2 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00003386)  #primes = 3 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .0001058)  #primes = 3 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00002924)  #primes = 4 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00368568)  #primes = 6 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00010228)  #primes = 6 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00008606)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .00061856)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .00055568)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .00225812)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .00259214)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .00043324)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .00034966)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .0007397)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .00070848)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .00294814)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .0032151)  #primes = 6 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00003178)  #primes = 7 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00003534)  #primes = 8 #prunedViaCodim = 0
  Strategy: IndependentSet    (time .0000434)  #primes = 9 #prunedViaCodim = 0
  Strategy: IndependentSet    (time .0000478)  #primes = 10 #prunedViaCodim = 0
Converting annotated ideals to ideals and selecting minimal primes... Time taken : .017493
#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 .00390254)  #primes = 0 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00012976)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00605028)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .0105546)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .0159167)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00752598)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00591658)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00587246)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .0010464)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .0007742)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .0007798)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00514142)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00576476)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .0074747)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00770002)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00495322)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00674678)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00561526)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00623194)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .0065307)  #primes = 0 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00003258)  #primes = 1 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00010126)  #primes = 1 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00003406)  #primes = 2 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00003402)  #primes = 3 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00009948)  #primes = 3 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00003362)  #primes = 4 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00349634)  #primes = 6 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .0001028)  #primes = 6 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00009172)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .00062274)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .0005584)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .00216574)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .00259658)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .00042822)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .0003476)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .00071974)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .00069174)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .00279686)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .00319976)  #primes = 6 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00002848)  #primes = 7 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00003644)  #primes = 8 #prunedViaCodim = 0
  Strategy: Birational        (time .0137483)  #primes = 8 #prunedViaCodim = 0
  Strategy: Birational        (time .0126497)  #primes = 8 #prunedViaCodim = 0
  Strategy: Birational        (time .0005777)  #primes = 8 #prunedViaCodim = 0
  Strategy: Birational        (time .00057244)  #primes = 8 #prunedViaCodim = 0
  Strategy: Linear            (time .00013986)  #primes = 8 #prunedViaCodim = 0
  Strategy: Linear            (time .0001348)  #primes = 8 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00003966)  #primes = 9 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00003916)  #primes = 10 #prunedViaCodim = 0
Converting annotated ideals to ideals and selecting minimal primes... Time taken : .0167939
#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 :