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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 .00196884)  #primes = 0 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000056762)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00325132)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00534673)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00823544)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .0036095)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00282227)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00298118)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .000556686)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .000389524)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .000378604)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00246018)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00292771)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00385903)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00391611)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .0025365)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00343485)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00281803)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00315882)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00334542)  #primes = 0 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000013108)  #primes = 1 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000033058)  #primes = 1 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000010134)  #primes = 2 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000010306)  #primes = 3 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000035046)  #primes = 3 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000011666)  #primes = 4 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00165279)  #primes = 6 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000036088)  #primes = 6 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000033966)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .00035114)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .00032039)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .0010812)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .00129937)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .000210392)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .000163168)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .000358472)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .00035718)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .00143016)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .0016414)  #primes = 6 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00001035)  #primes = 7 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00001022)  #primes = 8 #prunedViaCodim = 0
  Strategy: IndependentSet    (time .000017554)  #primes = 9 #prunedViaCodim = 0
  Strategy: IndependentSet    (time .000016244)  #primes = 10 #prunedViaCodim = 0
Converting annotated ideals to ideals and selecting minimal primes... Time taken : .00842719
#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 .00197907)  #primes = 0 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000056028)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00329183)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .0158456)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .0082457)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00359731)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00282062)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00298301)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00056629)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .000382178)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .000382088)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00248824)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00294115)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00383882)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00392544)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00253829)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00338681)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00283338)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00315115)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00332796)  #primes = 0 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000012886)  #primes = 1 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00003418)  #primes = 1 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000011224)  #primes = 2 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000010158)  #primes = 3 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000033362)  #primes = 3 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000011152)  #primes = 4 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00166692)  #primes = 6 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00003599)  #primes = 6 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000034718)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .00034157)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .000318406)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .00108836)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .00134005)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .00021153)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .000161434)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .00036218)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .000357474)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .00144456)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .00169277)  #primes = 6 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000011348)  #primes = 7 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000010962)  #primes = 8 #prunedViaCodim = 0
  Strategy: Birational        (time .0069237)  #primes = 8 #prunedViaCodim = 0
  Strategy: Birational        (time .00641259)  #primes = 8 #prunedViaCodim = 0
  Strategy: Birational        (time .00027727)  #primes = 8 #prunedViaCodim = 0
  Strategy: Birational        (time .000272588)  #primes = 8 #prunedViaCodim = 0
  Strategy: Linear            (time .000084168)  #primes = 8 #prunedViaCodim = 0
  Strategy: Linear            (time .000082)   #primes = 8 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00001256)  #primes = 9 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000010792)  #primes = 10 #prunedViaCodim = 0
Converting annotated ideals to ideals and selecting minimal primes... Time taken : .00847548
#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 :