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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 .00169864) #primes = 0 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000059613 #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00704017) #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .0120872)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .0178719)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00732456) #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00764168) #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00680958) #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00195335) #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00125111) #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .0017193)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00297134) #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .0033884)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .0044683)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00492285) #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00166905) #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00387414) #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00281478) #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00317825) #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00449165) #primes = 0 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time 0)         #primes = 1 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00006656) #primes = 1 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000031983 #primes = 2 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000020428 #primes = 3 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000041346 #primes = 3 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000010703 #primes = 4 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .0323691)  #primes = 6 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time 2.4e-8)    #primes = 6 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time 2.4e-8)    #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .00116861) #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .00114344) #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .00379726) #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .00401333) #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .00109139) #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .000655174 #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .000338618 #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .000314824 #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .00210717) #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .00297402) #primes = 6 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000013665 #primes = 7 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000016327 #primes = 8 #prunedViaCodim = 0
  Strategy: IndependentSet    (time .00001804) #primes = 9 #prunedViaCodim = 0
  Strategy: IndependentSet    (time .000019304 #primes = 10 #prunedViaCodim = 0
Converting annotated ideals to ideals and selecting minimal primes... Time taken : .00612497
#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 .00179416) #primes = 0 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000099907 #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00794116) #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .0118708)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .0192099)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00681014) #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .0080126)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00747265) #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00240509) #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00156867) #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00172201) #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .003123)   #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00309537) #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00526969) #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00494806) #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00216294) #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00414057) #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00261965) #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00358125) #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00478106) #primes = 0 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000020145 #primes = 1 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000042257 #primes = 1 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000010484 #primes = 2 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000011999 #primes = 3 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000065521 #primes = 3 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000031631 #primes = 4 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00244534) #primes = 6 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000022686 #primes = 6 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time 0)         #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .000969864 #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .00108794) #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .0038008)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .00364981) #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .00105859) #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .00115455) #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .000428017 #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .000354392 #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .00203926) #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .00302919) #primes = 6 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000014711 #primes = 7 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000012967 #primes = 8 #prunedViaCodim = 0
  Strategy: Birational        (time .00914878) #primes = 8 #prunedViaCodim = 0
  Strategy: Birational        (time .00726322) #primes = 8 #prunedViaCodim = 0
  Strategy: Birational        (time .000289005 #primes = 8 #prunedViaCodim = 0
  Strategy: Birational        (time .000265788 #primes = 8 #prunedViaCodim = 0
  Strategy: Linear            (time .000094631 #primes = 8 #prunedViaCodim = 0
  Strategy: Linear            (time .000086244 #primes = 8 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000036036 #primes = 9 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000013931 #primes = 10 #prunedViaCodim = 0
Converting annotated ideals to ideals and selecting minimal primes... Time taken : .00691563
#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 :