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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 .0013845)  #primes = 0 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000039474)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .0024144)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .0037916)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00596025)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00254814)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00202322)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .0021661)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00043232)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00027882)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .000278314)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00170949)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00206705)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00270396)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .0027793)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00173993)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00237983)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00196798)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00220786)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00233195)  #primes = 0 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000007718)  #primes = 1 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00002615)  #primes = 1 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00000979)  #primes = 2 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000007142)  #primes = 3 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000025964)  #primes = 3 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000006858)  #primes = 4 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00119545)  #primes = 6 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000025012)  #primes = 6 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000025282)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .000268544)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .0002513)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .00079342)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .000936774)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .000157528)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .000123264)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .000252544)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .00024788)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .00101563)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .00114721)  #primes = 6 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00000693)  #primes = 7 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000007166)  #primes = 8 #prunedViaCodim = 0
  Strategy: IndependentSet    (time .000011288)  #primes = 9 #prunedViaCodim = 0
  Strategy: IndependentSet    (time .0000125)  #primes = 10 #prunedViaCodim = 0
Converting annotated ideals to ideals and selecting minimal primes... Time taken : .00502669
#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 .00136544)  #primes = 0 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000037954)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00244306)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .0037997)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00602264)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00255617)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .002063)   #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00216554)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .000450204)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .000285634)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .000282592)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00173637)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00210894)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00272991)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00283471)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .0139522)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00244396)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00204522)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00219956)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00233392)  #primes = 0 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000008442)  #primes = 1 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000025766)  #primes = 1 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000006998)  #primes = 2 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000007148)  #primes = 3 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000025202)  #primes = 3 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000007014)  #primes = 4 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00122926)  #primes = 6 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000027298)  #primes = 6 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000025574)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .000274502)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .00025)    #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .0007858)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .00092994)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .00015841)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .00012332)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .000253542)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .00025718)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .00100453)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .00114002)  #primes = 6 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000007958)  #primes = 7 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000007264)  #primes = 8 #prunedViaCodim = 0
  Strategy: Birational        (time .0049815)  #primes = 8 #prunedViaCodim = 0
  Strategy: Birational        (time .00455381)  #primes = 8 #prunedViaCodim = 0
  Strategy: Birational        (time .00021902)  #primes = 8 #prunedViaCodim = 0
  Strategy: Birational        (time .000217222)  #primes = 8 #prunedViaCodim = 0
  Strategy: Linear            (time .000051586)  #primes = 8 #prunedViaCodim = 0
  Strategy: Linear            (time .00005026)  #primes = 8 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000008402)  #primes = 9 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000007804)  #primes = 10 #prunedViaCodim = 0
Converting annotated ideals to ideals and selecting minimal primes... Time taken : .00514336
#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 :