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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 .00113963)  #primes = 0 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000038343)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00204452)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00337144)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00533287)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00233296)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00187435)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00192847)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .000382899)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .000253043)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .000248385)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00152885)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .0017747)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00234508)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00241933)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00151507)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00207241)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00173575)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00192133)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .0020411)  #primes = 0 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000007582)  #primes = 1 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000027658)  #primes = 1 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000007158)  #primes = 2 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000008371)  #primes = 3 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000027386)  #primes = 3 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000007329)  #primes = 4 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00106833)  #primes = 6 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000026827)  #primes = 6 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000022897)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .00022959)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .000217966)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .000725366)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .000842205)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .000141466)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .000117088)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .000227585)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .000214224)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .000896147)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .00103169)  #primes = 6 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00000766)  #primes = 7 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000008474)  #primes = 8 #prunedViaCodim = 0
  Strategy: IndependentSet    (time .000010831)  #primes = 9 #prunedViaCodim = 0
  Strategy: IndependentSet    (time .000011287)  #primes = 10 #prunedViaCodim = 0
Converting annotated ideals to ideals and selecting minimal primes... Time taken : .00442862
#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 .00115338)  #primes = 0 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000036357)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00204569)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00340192)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00536595)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00236765)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .0019195)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00197376)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .000384614)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .000268657)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .000258226)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00156208)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .0018341)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .0095883)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00264934)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00170898)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00227442)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .001887)   #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00210969)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00224518)  #primes = 0 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000009246)  #primes = 1 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000029734)  #primes = 1 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000007377)  #primes = 2 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000009143)  #primes = 3 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000026416)  #primes = 3 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000007907)  #primes = 4 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00116281)  #primes = 6 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000028991)  #primes = 6 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00002397)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .000242965)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .000229498)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .000775653)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .000909515)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .000153748)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .00012457)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .000243942)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .000241392)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .000968241)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .00112629)  #primes = 6 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000007546)  #primes = 7 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000008819)  #primes = 8 #prunedViaCodim = 0
  Strategy: Birational        (time .0047505)  #primes = 8 #prunedViaCodim = 0
  Strategy: Birational        (time .00439698)  #primes = 8 #prunedViaCodim = 0
  Strategy: Birational        (time .000227807)  #primes = 8 #prunedViaCodim = 0
  Strategy: Birational        (time .00022433)  #primes = 8 #prunedViaCodim = 0
  Strategy: Linear            (time .000048627)  #primes = 8 #prunedViaCodim = 0
  Strategy: Linear            (time .000045241)  #primes = 8 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00000868)  #primes = 9 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000009221)  #primes = 10 #prunedViaCodim = 0
Converting annotated ideals to ideals and selecting minimal primes... Time taken : .00488495
#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 :