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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 .00150343)  #primes = 0 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000055353)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00269141)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00447258)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .0373439)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00310652)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00285318)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .0025311)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00052488)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00029779)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .000369538)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00232067)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .0022979)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00301264)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00313692)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00205791)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00279679)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00235294)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00244365)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .0028005)  #primes = 0 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000017385)  #primes = 1 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000039561)  #primes = 1 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000009692)  #primes = 2 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000014908)  #primes = 3 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000037383)  #primes = 3 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00001037)  #primes = 4 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00150191)  #primes = 6 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000041294)  #primes = 6 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000032606)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .000374614)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .000259383)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .000936978)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .00107761)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .000229781)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .000152429)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .000384295)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .000277518)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .00110601)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .00132997)  #primes = 6 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000011511)  #primes = 7 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000016449)  #primes = 8 #prunedViaCodim = 0
  Strategy: IndependentSet    (time .000019892)  #primes = 9 #prunedViaCodim = 0
  Strategy: IndependentSet    (time .000017084)  #primes = 10 #prunedViaCodim = 0
Converting annotated ideals to ideals and selecting minimal primes... Time taken : .00720945
#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 .00155374)  #primes = 0 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000055073)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00265549)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00455941)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00683999)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00305849)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00236263)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00240328)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .000438874)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .000308926)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .000302424)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00205012)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00220992)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00312361)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00311114)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00187216)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00252436)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00209884)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00347934)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00270814)  #primes = 0 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00001634)  #primes = 1 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000040683)  #primes = 1 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000011988)  #primes = 2 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000012808)  #primes = 3 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000034534)  #primes = 3 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000010403)  #primes = 4 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00145025)  #primes = 6 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000042298)  #primes = 6 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000032816)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .000286625)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .000258382)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .000922623)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .00118843)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .00017505)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .000142369)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .000288355)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .000266716)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .00109737)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .0012435)  #primes = 6 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000012009)  #primes = 7 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000013445)  #primes = 8 #prunedViaCodim = 0
  Strategy: Birational        (time .0059511)  #primes = 8 #prunedViaCodim = 0
  Strategy: Birational        (time .00507981)  #primes = 8 #prunedViaCodim = 0
  Strategy: Birational        (time .000246578)  #primes = 8 #prunedViaCodim = 0
  Strategy: Birational        (time .000221099)  #primes = 8 #prunedViaCodim = 0
  Strategy: Linear            (time .000059734)  #primes = 8 #prunedViaCodim = 0
  Strategy: Linear            (time .000049001)  #primes = 8 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000015475)  #primes = 9 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000016282)  #primes = 10 #prunedViaCodim = 0
Converting annotated ideals to ideals and selecting minimal primes... Time taken : .00684577
#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 :