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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 .00378324)  #primes = 0 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00015446)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00611026)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .0102754)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .0156271)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .0073277)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00580412)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00582468)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00103152)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .0007432)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00074146)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00494304)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00563318)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00739982)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00761682)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00499718)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .0067757)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00564666)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .006217)   #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00655534)  #primes = 0 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00002838)  #primes = 1 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00009862)  #primes = 1 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .0000271)  #primes = 2 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00003022)  #primes = 3 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00009382)  #primes = 3 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00002332)  #primes = 4 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00356926)  #primes = 6 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000094)   #primes = 6 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00007424)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .0006258)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .0005856)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .00222024)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .00255874)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .00045282)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .00033862)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .00073968)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .00070502)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .00284076)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .00319836)  #primes = 6 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00002488)  #primes = 7 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00003098)  #primes = 8 #prunedViaCodim = 0
  Strategy: IndependentSet    (time .00003788)  #primes = 9 #prunedViaCodim = 0
  Strategy: IndependentSet    (time .00004518)  #primes = 10 #prunedViaCodim = 0
Converting annotated ideals to ideals and selecting minimal primes... Time taken : .0147092
#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 .00373932)  #primes = 0 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00014748)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00603736)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .0102981)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .0157183)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00741392)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .0058329)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00591928)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00103558)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .000814)   #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00070618)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .0050851)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .0057588)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .067557)   #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00790436)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .0050344)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .0069706)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00572268)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00631998)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00669126)  #primes = 0 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00002924)  #primes = 1 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00009616)  #primes = 1 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00002478)  #primes = 2 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00003018)  #primes = 3 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00008994)  #primes = 3 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00003072)  #primes = 4 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00353756)  #primes = 6 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00010278)  #primes = 6 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000077)   #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .00062082)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .00057334)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .00223124)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .0025814)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .00044644)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .0003802)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .00075362)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .00067732)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .00288204)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .0032536)  #primes = 6 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00002702)  #primes = 7 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00003132)  #primes = 8 #prunedViaCodim = 0
  Strategy: Birational        (time .0144311)  #primes = 8 #prunedViaCodim = 0
  Strategy: Birational        (time .0132566)  #primes = 8 #prunedViaCodim = 0
  Strategy: Birational        (time .00070688)  #primes = 8 #prunedViaCodim = 0
  Strategy: Birational        (time .0006936)  #primes = 8 #prunedViaCodim = 0
  Strategy: Linear            (time .00014846)  #primes = 8 #prunedViaCodim = 0
  Strategy: Linear            (time .00013964)  #primes = 8 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00003538)  #primes = 9 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00006276)  #primes = 10 #prunedViaCodim = 0
Converting annotated ideals to ideals and selecting minimal primes... Time taken : .0153268
#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 :