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 .00334514)  #primes = 0 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .0001059)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00541938)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .009172)   #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .0139177)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00652136)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00516812)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00515068)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00090574)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .0006533)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00065984)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00442478)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00504198)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00664386)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00684326)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00446784)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00609116)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .0050572)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00557104)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .0058774)  #primes = 0 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00002602)  #primes = 1 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .0000829)  #primes = 1 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .0000228)  #primes = 2 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00002702)  #primes = 3 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00007978)  #primes = 3 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00002378)  #primes = 4 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00314724)  #primes = 6 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00007842)  #primes = 6 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00006742)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .00054556)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .000501)   #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .00197284)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .00230444)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .0003838)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .00031116)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .00066194)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .00063674)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .00253892)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .00283288)  #primes = 6 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00002416)  #primes = 7 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00002636)  #primes = 8 #prunedViaCodim = 0
  Strategy: IndependentSet    (time .00003388)  #primes = 9 #prunedViaCodim = 0
  Strategy: IndependentSet    (time .00003962)  #primes = 10 #prunedViaCodim = 0
Converting annotated ideals to ideals and selecting minimal primes... Time taken : .0154103
#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 .0527946)  #primes = 0 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00013324)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00578158)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00937564)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .0142013)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00666706)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .0053196)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .0053592)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00094222)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00066616)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00066428)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .0046128)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00517982)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .0068879)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00707042)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00463508)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00633944)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .0051848)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00586172)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00621506)  #primes = 0 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00002882)  #primes = 1 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00008106)  #primes = 1 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00001968)  #primes = 2 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00002646)  #primes = 3 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00007868)  #primes = 3 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .0000211)  #primes = 4 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .0032637)  #primes = 6 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .0000852)  #primes = 6 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00006438)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .00055066)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .00050878)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .00201348)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .00234036)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .00040208)  #primes = 6 #prunedViaCodim = 0
  Strategy: Factorization     (time .00031956)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .0006693)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .00065912)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .00268356)  #primes = 6 #prunedViaCodim = 0
  Strategy: Linear            (time .00299166)  #primes = 6 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .0000258)  #primes = 7 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00002732)  #primes = 8 #prunedViaCodim = 0
  Strategy: Birational        (time .0131386)  #primes = 8 #prunedViaCodim = 0
  Strategy: Birational        (time .0120489)  #primes = 8 #prunedViaCodim = 0
  Strategy: Birational        (time .00055016)  #primes = 8 #prunedViaCodim = 0
  Strategy: Birational        (time .00053602)  #primes = 8 #prunedViaCodim = 0
  Strategy: Linear            (time .0001393)  #primes = 8 #prunedViaCodim = 0
  Strategy: Linear            (time .00013288)  #primes = 8 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .0000293)  #primes = 9 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00002866)  #primes = 10 #prunedViaCodim = 0
Converting annotated ideals to ideals and selecting minimal primes... Time taken : .0160744
#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 :