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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 .00114893)  #primes = 0 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000036832)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00205849)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00336282)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00535594)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00234489)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00187286)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00192906)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .000379524)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00025693)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .000250045)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00152554)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00178359)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00235143)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00242513)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00153103)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00207908)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00173431)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00192226)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .0020537)  #primes = 0 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000007461)  #primes = 1 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000026572)  #primes = 1 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000008512)  #primes = 2 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000007717)  #primes = 3 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000027689)  #primes = 3 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000007444)  #primes = 4 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00109259)  #primes = 6 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000026758)  #primes = 6 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000024163)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .000235205)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .000220037)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .00073316)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .000843772)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .000143358)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .000113307)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .000233139)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .000216895)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .000891843)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .00102006)  #primes = 6 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000007112)  #primes = 7 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00000844)  #primes = 8 #prunedViaCodim = 0
  Strategy: IndependentSet    (time .000012062)  #primes = 9 #prunedViaCodim = 0
  Strategy: IndependentSet    (time .000011024)  #primes = 10 #prunedViaCodim = 0
Converting annotated ideals to ideals and selecting minimal primes... Time taken : .00443453
#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 .00115855)  #primes = 0 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00003788)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00205809)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00339857)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00539493)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00236336)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00189355)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00197246)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .000393238)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .000256991)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .000256554)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00156944)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00183025)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00929417)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00217941)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00141277)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00188061)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .0015456)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00171866)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00181286)  #primes = 0 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000007321)  #primes = 1 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000025403)  #primes = 1 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000007142)  #primes = 2 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000007646)  #primes = 3 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000024165)  #primes = 3 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000007301)  #primes = 4 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000999243)  #primes = 6 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000024984)  #primes = 6 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000020952)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .00021187)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .000196396)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .00064943)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .000754291)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .000126767)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .000104507)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .000208649)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .000197186)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .000798109)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .00091345)  #primes = 6 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00000649)  #primes = 7 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000007736)  #primes = 8 #prunedViaCodim = 0
  Strategy: Birational        (time .00399181)  #primes = 8 #prunedViaCodim = 0
  Strategy: Birational        (time .00364325)  #primes = 8 #prunedViaCodim = 0
  Strategy: Birational        (time .000190698)  #primes = 8 #prunedViaCodim = 0
  Strategy: Birational        (time .000187961)  #primes = 8 #prunedViaCodim = 0
  Strategy: Linear            (time .000036801)  #primes = 8 #prunedViaCodim = 0
  Strategy: Linear            (time .000035638)  #primes = 8 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000007503)  #primes = 9 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000009221)  #primes = 10 #prunedViaCodim = 0
Converting annotated ideals to ideals and selecting minimal primes... Time taken : .00426576
#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 :