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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 .00103037)  #primes = 0 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00003475)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00179352)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00299127)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00470524)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00209914)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00168114)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00172525)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .0252534)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .000264908)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .000251832)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00166654)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .0018215)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00210589)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00211771)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00131737)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00182218)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00151736)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00166736)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .0018048)  #primes = 0 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000009076)  #primes = 1 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000025221)  #primes = 1 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000006533)  #primes = 2 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000008669)  #primes = 3 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00002428)  #primes = 3 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000007066)  #primes = 4 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000990266)  #primes = 6 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00002405)  #primes = 6 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000021078)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .000205506)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .00018914)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .000639567)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .000744034)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .000131266)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .000104197)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .000208742)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .000200061)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .000811193)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .000920909)  #primes = 6 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000007447)  #primes = 7 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000008108)  #primes = 8 #prunedViaCodim = 0
  Strategy: IndependentSet    (time .000010177)  #primes = 9 #prunedViaCodim = 0
  Strategy: IndependentSet    (time .000011897)  #primes = 10 #prunedViaCodim = 0
Converting annotated ideals to ideals and selecting minimal primes... Time taken : .00464124
#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 .00104038)  #primes = 0 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000033899)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00181441)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .0030468)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00477048)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00209952)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00168615)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00172172)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .000332319)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .000223861)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .000217702)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00139763)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00158913)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00211995)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00218778)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00138543)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00207555)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00169577)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00186869)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00182088)  #primes = 0 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00000708)  #primes = 1 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000025006)  #primes = 1 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000006571)  #primes = 2 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000008001)  #primes = 3 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000023527)  #primes = 3 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000006794)  #primes = 4 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000952485)  #primes = 6 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000024715)  #primes = 6 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000021329)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .000201665)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .000188434)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .000626561)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .000743882)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .000128928)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .000101197)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .000201848)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .000189608)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .000790291)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .000899363)  #primes = 6 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000007222)  #primes = 7 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000008254)  #primes = 8 #prunedViaCodim = 0
  Strategy: Birational        (time .00385682)  #primes = 8 #prunedViaCodim = 0
  Strategy: Birational        (time .00351529)  #primes = 8 #prunedViaCodim = 0
  Strategy: Birational        (time .000162715)  #primes = 8 #prunedViaCodim = 0
  Strategy: Birational        (time .000158533)  #primes = 8 #prunedViaCodim = 0
  Strategy: Linear            (time .000035717)  #primes = 8 #prunedViaCodim = 0
  Strategy: Linear            (time .000037155)  #primes = 8 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000008644)  #primes = 9 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000008707)  #primes = 10 #prunedViaCodim = 0
Converting annotated ideals to ideals and selecting minimal primes... Time taken : .00466061
#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 :