next | previous | forward | backward | up | top | index | toc | Macaulay2 web site
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 .00113907)  #primes = 0 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000034693)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00208639)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00348613)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00549057)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00247172)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00195033)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .0020057)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .000376843)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .000277354)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00024306)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .0015933)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .0018532)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00243706)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00252183)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00156624)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00216854)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00179359)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00198215)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00209928)  #primes = 0 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000008112)  #primes = 1 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000033303)  #primes = 1 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000006912)  #primes = 2 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000009986)  #primes = 3 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000028232)  #primes = 3 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000006887)  #primes = 4 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00113207)  #primes = 6 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000033404)  #primes = 6 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00002543)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .000222463)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .000212494)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .000761802)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .000866061)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .00016092)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .000127755)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .00022747)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .00023364)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .000934375)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .0010531)  #primes = 6 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000012615)  #primes = 7 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .0000101)  #primes = 8 #prunedViaCodim = 0
  Strategy: IndependentSet    (time .000020154)  #primes = 9 #prunedViaCodim = 0
  Strategy: IndependentSet    (time .000013886)  #primes = 10 #prunedViaCodim = 0
Converting annotated ideals to ideals and selecting minimal primes... Time taken : .00575039
#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 .00113188)  #primes = 0 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000046868)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .0020748)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00348223)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00552691)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00243645)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00195796)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00200959)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .000363047)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00028999)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .000248156)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00158578)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00180857)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00242345)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00250957)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00157647)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00217565)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00175388)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .0019675)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00213659)  #primes = 0 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000013174)  #primes = 1 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000028059)  #primes = 1 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000007073)  #primes = 2 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000009956)  #primes = 3 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000044497)  #primes = 3 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000007715)  #primes = 4 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00113913)  #primes = 6 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000028001)  #primes = 6 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000024171)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .000240191)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .00021571)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .000762071)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .000870674)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .00014773)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .000126218)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .000240171)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .000223278)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .000917781)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .00104194)  #primes = 6 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000008077)  #primes = 7 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00001056)  #primes = 8 #prunedViaCodim = 0
  Strategy: Birational        (time .00420351)  #primes = 8 #prunedViaCodim = 0
  Strategy: Birational        (time .00374701)  #primes = 8 #prunedViaCodim = 0
  Strategy: Birational        (time .000178882)  #primes = 8 #prunedViaCodim = 0
  Strategy: Birational        (time .00018353)  #primes = 8 #prunedViaCodim = 0
  Strategy: Linear            (time .000046257)  #primes = 8 #prunedViaCodim = 0
  Strategy: Linear            (time .000038602)  #primes = 8 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000012601)  #primes = 9 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000011487)  #primes = 10 #prunedViaCodim = 0
Converting annotated ideals to ideals and selecting minimal primes... Time taken : .00489889
#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 :