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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 .000884474)  #primes = 0 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000027066)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00133711)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00274417)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00205692)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00116555)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00127285)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .0002238)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .000151261)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .000150113)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00103929)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00116189)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .0016517)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00136511)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00113482)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00121353)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00143796)  #primes = 0 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000027906)  #primes = 1 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000022325)  #primes = 1 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000004963)  #primes = 2 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00000721)  #primes = 3 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000723017)  #primes = 5 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000032873)  #primes = 5 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000023825)  #primes = 5 #prunedViaCodim = 0
  Strategy: Factorization     (time .000221942)  #primes = 5 #prunedViaCodim = 0
  Strategy: Factorization     (time .000597723)  #primes = 5 #prunedViaCodim = 0
  Strategy: Factorization     (time .000597165)  #primes = 5 #prunedViaCodim = 0
  Strategy: Factorization     (time .000088307)  #primes = 5 #prunedViaCodim = 0
  Strategy: Factorization     (time .000069162)  #primes = 5 #prunedViaCodim = 0
  Strategy: Linear            (time .000151333)  #primes = 5 #prunedViaCodim = 0
  Strategy: Linear            (time .000569943)  #primes = 5 #prunedViaCodim = 0
  Strategy: Linear            (time .00081886)  #primes = 5 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000007086)  #primes = 6 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00000827)  #primes = 7 #prunedViaCodim = 0
  Strategy: IndependentSet    (time .000010646)  #primes = 8 #prunedViaCodim = 0
Converting annotated ideals to ideals and selecting minimal primes... Time taken : .00315414
#minprimes=6 #computed=8

                                  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 .000881482)  #primes = 0 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000030892)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00130172)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00316015)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00217826)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .0014255)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .00155575)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .000229429)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .000154772)  #primes = 0 #prunedViaCodim = 0
  Strategy: Factorization     (time .000152145)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00114842)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00112358)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00153328)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00149942)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00117071)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00132924)  #primes = 0 #prunedViaCodim = 0
  Strategy: Linear            (time .00133008)  #primes = 0 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000007737)  #primes = 1 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000018783)  #primes = 1 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000007149)  #primes = 2 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000006609)  #primes = 3 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .00075581)  #primes = 5 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000019712)  #primes = 5 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000016333)  #primes = 5 #prunedViaCodim = 0
  Strategy: Factorization     (time .000150247)  #primes = 5 #prunedViaCodim = 0
  Strategy: Factorization     (time .000531122)  #primes = 5 #prunedViaCodim = 0
  Strategy: Factorization     (time .000571788)  #primes = 5 #prunedViaCodim = 0
  Strategy: Factorization     (time .000092344)  #primes = 5 #prunedViaCodim = 0
  Strategy: Factorization     (time .000072319)  #primes = 5 #prunedViaCodim = 0
  Strategy: Linear            (time .000151374)  #primes = 5 #prunedViaCodim = 0
  Strategy: Linear            (time .000594127)  #primes = 5 #prunedViaCodim = 0
  Strategy: Linear            (time .000674868)  #primes = 5 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000005994)  #primes = 6 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000006899)  #primes = 7 #prunedViaCodim = 0
  Strategy: Birational        (time .00279784)  #primes = 7 #prunedViaCodim = 0
  Strategy: Birational        (time .00012377)  #primes = 7 #prunedViaCodim = 0
  Strategy: Linear            (time .000027779)  #primes = 7 #prunedViaCodim = 0
  Strategy: DecomposeMonomials(time .000007192)  #primes = 8 #prunedViaCodim = 0
Converting annotated ideals to ideals and selecting minimal primes... Time taken : .00274722
#minprimes=6 #computed=8

                                  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 :